R Analysis 7: Time-Series Data and Streaming Services

Hello everybody,

Michael here, and today’s post will be an R analysis on streaming services using time-series data (yes, it’s my first R post since July 27). For this analysis, I will be analyzing Google Trends data from the past year (12/7/18-12/6/19) for ten major streaming services-two of which haven’t launched yet.

  • Don’t worry guys, I wasn’t paid by any of these services for this post. I just thought this would be an interesting topic to analyze since there seems to be so many streaming services on the way.

As we should always do in R, let’s remember to read our csv of the data (Streaming Services) and understand our variables:

Screen Shot 2019-12-07 at 8.31.26 PM

The logic here is the same as that of the time-series analysis I did in R Lesson 9: Time Series Data, except you would replace the names of people with the names of streaming services. The numbers still represent search metrics, and 100 is still the highest while 0 is still the lowest. Also, just as with the aforementioned R post, I replaced any instances of <1 with 0. The Week variable is still the same, and we still have to convert it to a date as follows:

Screen Shot 2019-12-07 at 8.39.34 PM

  • The weeks also start on Sunday, so the first week listed is 12/9/2018 while the last week listed is 12/1/2019

Now that we’ve explained our variables, let’s get ready to graph by first installing the ggplot2 package.

  • Also keep in mind that just because a graph fluctuates a lot, that doesn’t mean all graphs have the same maximum and/or minimum.

Next, let’s start by looking at the graph for the first service listed, HBO Max (this is one of the services that hasn’t launched yet, but will debut in May 2020):

Screen Shot 2019-12-07 at 9.06.47 PMScreen Shot 2019-12-07 at 9.06.28 PM

  • Maximum-100
  • Mean-9.6
  • Minimum-0

The first streaming service that I will analyze is HBO Max, which is scheduled to launch in May 2020 (exact date TBA). HBO Max is expected to have a massive library of content not only consisting of all of HBO’s programming but also content owned by WarnerMedia, HBO’s parent company. That includes content from Cartoon Network, TBS, CNN, and other networks.

Three popular shows that will be on HBO Max are Friends, The Big Bang Theory, and South Park. In fact, HBO Max had a search metric of 100 on the week of October 27, 2019-the week the service had secured the rights to all 23 seasons of South Park, plus the rights to stream any new episodes of South Park 24 hours after they air on Comedy Central.

Screen Shot 2019-12-08 at 8.21.44 PMScreen Shot 2019-12-08 at 8.21.26 PM

  • Maximum-100
  • Mean-25.4
  • Minimum-20

The next service I will be analyzing is Disney+, which launched on November 12, 2019. Even though the average and minimum Disney+ search metric is higher than those of HBO Max, the overall search metric remains surprisingly low until the week of November 10, 2019 (the week Disney+ launched). I thought Disney+ would’ve trended much higher on the weeks of March 17, and April 7, 2019, as those were the weeks of Disney’s FOX acquisition and Disney+ securing the rights to stream all three decades of The Simpsons, respectively.

Screen Shot 2019-12-08 at 8.22.44 PMScreen Shot 2019-12-08 at 8.22.23 PM

  • Maximum-100
  • Mean-55.6
  • Minimum-47

The next service I will analyze is Netflix, which was one of the first services to stream original content. Granted, the service has been around since 1997, but Netflix didn’t start streaming their own content until 2013 (they simply carried other media companies’ content). The average and minimum search metric is higher than those of either Disney+ or Netflix, which implies that public interest in Netflix has not waned, despite the rise of multiple other streaming services.

Screen Shot 2019-12-08 at 8.23.31 PMScreen Shot 2019-12-08 at 8.23.17 PM

  • Maximum-100
  • Mean-58.2
  • Min-44

The next service I will analyze is Hulu, which has a higher average search metric than Netflix but a lower minimum metric. Interestingly, Hulu’s search metric hit 100 on the week of November 10, 2019-the same week Disney+’s metric hit 100. The only possible theory I can give as to why this is the case is that Disney+ offered Disney+, Hulu, and ESPN+ as a $12.99/month bundle for all three services upon the launch of Disney+ (remember that all three services are all owned by Disney).

Another tidbit to note is that Hulu surprisingly didn’t peak on the week of May 12, 2019, when it was announced Disney would be taking full control of Hulu (Disney previously shared a stake in Hulu with other media companies such as Comcast and FOX).

Screen Shot 2019-12-08 at 8.24.48 PMScreen Shot 2019-12-08 at 8.24.35 PM

  • Maximum-100
  • Mean-40.9
  • Minimum-20

The next service I will be analyzing is CBS All Access. This service isn’t as broad as the others I previously analyzed with regards to content, as CBS All Access consists of mostly CBS programming such as Criminal Minds and The Late Show with Stephen Colbert. To their credit, they do have original programming, such as Star Trek Discovery, but they are about to lose streaming rights to The Big Bang Theory to HBO Max. Another thing to note is that the search metric for CBS All Access hit 100 on the week of February 3, 2019, which was the week of Super Bowl LIII (CBS had the rights to broadcast the Super Bowl in 2019, and CBS All Access broadcast the game live).

Screen Shot 2019-12-08 at 8.25.42 PMScreen Shot 2019-12-08 at 8.25.23 PM

  • Maximum-100
  • Mean-38.9
  • Minimum-26

The next service I will analyze is Apple TV+, which is Apple’s streaming platform that launched on November 1, 2019-just 11 days before Disney+. Apple TV+ also happens to be the narrowest streaming service with regards to content, as there were only eight original series plus a documentary at launch; unlike the other streaming services analyzed here, Apple TV+ doesn’t carry content from other networks.

An interesting tidbit about Apple TV+ is that its search metric hit 100 on the week of November 10, 2019 (the week of the Disney+ launch)-not the week of October 27, 2019, when Apple TV+ launched.

Screen Shot 2019-12-08 at 8.26.29 PMScreen Shot 2019-12-08 at 8.26.16 PM

  • Maximum-100
  • Mean-79
  • Minimum-66

The next service I will analyze is Amazon Prime Video, which is the Amazon counterpart of Netflix. So far, Amazon Prime Video has the highest mean and minimum search metric of all the streaming services I analyzed. I find this interesting considering it’s not a new service (it’s been around since 2006) and that, unlike services such as HBO Max, they haven’t acquired the streaming rights to any major shows such as the Big Bang Theory or South Park.

Screen Shot 2019-12-08 at 8.27.10 PMScreen Shot 2019-12-08 at 8.26.55 PM

  • Maximum-100
  • Mean-8.4
  • Minimum-0

The next service I will analyze is Peacock, which, like HBO Max, has yet to launch (Peacock is scheduled for an April 2020 debut-exact date TBA). Peacock is Universal’s streaming service, thus it will contain Universal movies (e.g. Fast and Furious series) and NBC programming such as Parks and Recreation and The Office (remember that NBC is owned by Universal).

The mean and minimum search metrics for Peacock are the lowest of all the streaming services I’ve analyzed so far, which isn’t surprising given the name and launch date weren’t even announced until the week of September 15, 2019 (when the search metric hit 100). A possible reason for the high search metric that week could be that, aside from the launch date and name announcement, the service will be free to pay-TV subscribers (although the free version will have ads)

Screen Shot 2019-12-08 at 8.28.01 PMScreen Shot 2019-12-08 at 8.27.44 PM

  • Maximum-100
  • Mean-38.5
  • Minimum-15

The next service I will analyze is DC Universe, which is the most niche service I’ve analyzed so far, as DC Universe carries nothing but DC Comics content-both original programming starring DC Comics characters and DC Comics movies. Given this service’s highly narrow scope of content, I’m surprised its mean and minimum search metrics are higher than those of Peacock.

Screen Shot 2019-12-08 at 8.28.43 PMScreen Shot 2019-12-08 at 8.28.28 PM

  • Maximum-100
  • Mean-65.5
  • Minimum-45

The last service I will analyze is ESPN+, which, like DC Universe, has a narrow scope of content (albeit their focus is sports content rather than comic book content). However, the mean and minimum search metric are higher than those for DC Universe.

The only major event for ESPN+ occurred on the week of August 4, 2019, when it was announced that Disney would offer a three-service bundle for ESPN+, Disney+, and Hulu for the price of $12.99/month for the trio.

As always, thanks for reading,

Michael

R Analysis 6: ANOVA & American Cartoons

Hello everybody,

It’s Michael, and today’s post will be the 6th R analysis; the focus of this analysis, as you might have guessed from the title, will be ANOVA. The topic of this ANOVA analysis will be American cartoons.

Here’s the dataset-Cartoons.

Now, as always, let’s load our dataset into R and learn about each variable:

Screen Shot 2019-07-13 at 4.19.13 PM

As you can see, we’ve got 78 cartoons (the observations) listed along with 9 other variables that describe the cartoons (such as the name, rating, network, etc.). Here’s a variable-by-variable breakdown:

  • Name-The name of the cartoon
  • Debut.Year-The year the cartoon premiered
  • Seasons-The number of seasons the cartoon lasted or, if the show is still on the air, the amount of seasons that have aired as of July 1, 2019.
  • Episodes-The number of episodes the cartoon lasted, or, of the show is still on the air, the amount of episodes that have aired as of July 1, 2019.
  • Creator-The creator(s) of the cartoon.
  • Network-The network that currently airs new episodes of the cartoon. Some of the cartoons on this list, such as American Dad, switched networks (in that case, from FOX to TBS), so I list the network that currently airs new episodes.
  • Ended-Whether or not the show ended; this can either be a “Yes” or “No”.
  • Contract.Length-If Ended is a no, then this denotes the length of a show’s current contract. If Ended is a yes, then this field is blank.
  • Rating-The shows TV rating, which can be one of these five:
    • TV-Y
    • TV-Y7
    • TV-PG
    • TV-14
    • TV-MA

Now, let’s check for missing values (remember to install the Amelia package). Also remember to use the line missmap(name of file variable) to see the missmap:

Screen Shot 2019-07-13 at 4.42.38 PM

Interestingly, the graph doesn’t list any of our variables as missing, even though the Contract.Length variable has several blank spots. However, those blank spots might’ve counted as a value.

Now, let’s do some one-way ANOVA:

Screen Shot 2019-07-16 at 9.38.28 PM

In this model, I used Episodes as the dependent variable and Rating as the independent variable. I am trying to analyze the relationship (if there is any) between how many episodes a cartoon has aired and the cartoon’s TV rating (in other words, whether or not the cartoon is for kids). As you can see, Rating has 3 asterisks beside it, meaning that it is a very significant variable.

Now, let’s do a Tukey’s HSD Test to analyze pair-wise differences:

Screen Shot 2019-07-16 at 9.46.57 PM

This Tukey’s HSD Test gives us the pair-wise differences among pairs of TV content ratings. Remember that a pair-wise difference with a p adj that is less than 0.01 is statistically significant. There are three statistically significant pair-wise differences, which include:

  • TV-PG; TV-14-this one is interesting because cartoons with TV-PG and TV-14 ratings often air on network TV. For example, FOX airs the TV-PG Simpsons and Bob’s Burgers along with the TV-14 Family Guy.
  • TV-PG; TV-MA-this one isn’t as interesting. Well, cartoons with both of these ratings air on cartoon network, though you’ll find plenty more cartoons with a TV-MA rating on Cartoon Network’s Adult Swim block (e.g. Robot Chicken, Aqua Teen Hunger Force)
  • TV-Y7; TV-PG-this one is quite interesting, since Cartoon Network has aired several shows with both of these ratings, such as the TV-Y7 Ed, Edd n Eddy and Camp Lazlo along with the TV-PG Regular Show and Adventure Time.

Now let’s create a two-way ANOVA model:

Screen Shot 2019-07-20 at 5.56.26 PM

In this model, I once again used Episodes as a dependent variable and Rating as an independent variable. However, I also included Network (the network a show airs on) as another independent variable.

Just as with the previous model, Rating is a statistically significant variable, except this time it has slightly less significance, as there are two asterisks instead of three. Network, on the other hand, isn’t significant at all, as its p-value is much greater than 0.01.

Now let’s do a Tukey’s HSD test to analyze pair-wise differences. Since this is a two-way ANOVA, we will see pair-wise differences for :

Tukey multiple comparisons of means
99% family-wise confidence level

Fit: aov(formula = file$Episodes ~ file$Network + file$Rating)

$`file$Network`
diff lwr upr p adj
Comedy Central-Cartoon Network -18.6166667 -217.13053 179.89720 1.0000000
Crackle-Cartoon Network -79.3666667 -458.47404 299.74071 0.9997177
Disney Channel-Cartoon Network -39.8666667 -312.22545 232.49211 0.9999939
FOX-Cartoon Network 84.7583333 -63.63972 233.15638 0.5350384
FX-Cartoon Network -0.3666667 -379.47404 378.74071 1.0000000
G4-Cartoon Network 25.6333333 -353.47404 404.74071 1.0000000
Hulu-Cartoon Network -77.3666667 -456.47404 301.74071 0.9997826
Netflix-Cartoon Network -80.3666667 -222.10667 61.37334 0.5465350
Nickelodeon-Cartoon Network 0.6333333 -141.10667 142.37334 1.0000000
PBS-Cartoon Network 0.7444444 -140.99556 142.48445 1.0000000
Showtime-Cartoon Network -85.3666667 -464.47404 293.74071 0.9994113
TBS-Cartoon Network 33.6333333 -238.72545 305.99211 0.9999991
Crackle-Comedy Central -60.7500000 -477.71252 356.21252 0.9999942
Disney Channel-Comedy Central -21.2500000 -344.22778 301.72778 1.0000000
FOX-Comedy Central 103.3750000 -125.00478 331.75478 0.8359614
FX-Comedy Central 18.2500000 -398.71252 435.21252 1.0000000
G4-Comedy Central 44.2500000 -372.71252 461.21252 0.9999998
Hulu-Comedy Central -58.7500000 -475.71252 358.21252 0.9999960
Netflix-Comedy Central -61.7500000 -285.86062 162.36062 0.9959708
Nickelodeon-Comedy Central 19.2500000 -204.86062 243.36062 1.0000000
PBS-Comedy Central 19.3611111 -204.74951 243.47173 1.0000000
Showtime-Comedy Central -66.7500000 -483.71252 350.21252 0.9999835
TBS-Comedy Central 52.2500000 -270.72778 375.22778 0.9999815
Disney Channel-Crackle 39.5000000 -417.25956 496.25956 1.0000000
FOX-Crackle 164.1250000 -231.44038 559.69038 0.9020872
FX-Crackle 79.0000000 -448.42051 606.42051 0.9999921
G4-Crackle 105.0000000 -422.42051 632.42051 0.9998317
Hulu-Crackle 2.0000000 -525.42051 529.42051 1.0000000
Netflix-Crackle -1.0000000 -394.11604 392.11604 1.0000000
Nickelodeon-Crackle 80.0000000 -313.11604 473.11604 0.9997889
PBS-Crackle 80.1111111 -313.00492 473.22715 0.9997858
Showtime-Crackle -6.0000000 -533.42051 521.42051 1.0000000
TBS-Crackle 113.0000000 -343.75956 569.75956 0.9985243
FOX-Disney Channel 124.6250000 -170.21203 419.46203 0.8900907
FX-Disney Channel 39.5000000 -417.25956 496.25956 1.0000000
G4-Disney Channel 65.5000000 -391.25956 522.25956 0.9999951
Hulu-Disney Channel -37.5000000 -494.25956 419.25956 1.0000000
Netflix-Disney Channel -40.5000000 -332.04266 251.04266 0.9999966
Nickelodeon-Disney Channel 40.5000000 -251.04266 332.04266 0.9999966
PBS-Disney Channel 40.6111111 -250.93154 332.15377 0.9999965
Showtime-Disney Channel -45.5000000 -502.25956 411.25956 0.9999999
TBS-Disney Channel 73.5000000 -299.44262 446.44262 0.9998485
FX-FOX -85.1250000 -480.69038 310.44038 0.9996265
G4-FOX -59.1250000 -454.69038 336.44038 0.9999922
Hulu-FOX -162.1250000 -557.69038 233.44038 0.9094227
Netflix-FOX -165.1250000 -346.34254 16.09254 0.0287424
Nickelodeon-FOX -84.1250000 -265.34254 97.09254 0.8118260
PBS-FOX -84.0138889 -265.23143 97.20365 0.8131464
Showtime-FOX -170.1250000 -565.69038 225.44038 0.8778977
TBS-FOX -51.1250000 -345.96203 243.71203 0.9999609
G4-FX 26.0000000 -501.42051 553.42051 1.0000000
Hulu-FX -77.0000000 -604.42051 450.42051 0.9999940
Netflix-FX -80.0000000 -473.11604 313.11604 0.9997889
Nickelodeon-FX 1.0000000 -392.11604 394.11604 1.0000000
PBS-FX 1.1111111 -392.00492 394.22715 1.0000000
Showtime-FX -85.0000000 -612.42051 442.42051 0.9999823
TBS-FX 34.0000000 -422.75956 490.75956 1.0000000
Hulu-G4 -103.0000000 -630.42051 424.42051 0.9998622
Netflix-G4 -106.0000000 -499.11604 287.11604 0.9966917
Nickelodeon-G4 -25.0000000 -418.11604 368.11604 1.0000000
PBS-G4 -24.8888889 -418.00492 368.22715 1.0000000
Showtime-G4 -111.0000000 -638.42051 416.42051 0.9997021
TBS-G4 8.0000000 -448.75956 464.75956 1.0000000
Netflix-Hulu -3.0000000 -396.11604 390.11604 1.0000000
Nickelodeon-Hulu 78.0000000 -315.11604 471.11604 0.9998375
PBS-Hulu 78.1111111 -315.00492 471.22715 0.9998351
Showtime-Hulu -8.0000000 -535.42051 519.42051 1.0000000
TBS-Hulu 111.0000000 -345.75956 567.75956 0.9987571
Nickelodeon-Netflix 81.0000000 -94.80684 256.80684 0.8192765
PBS-Netflix 81.1111111 -94.69572 256.91795 0.8179377
Showtime-Netflix -5.0000000 -398.11604 388.11604 1.0000000
TBS-Netflix 114.0000000 -177.54266 405.54266 0.9335787
PBS-Nickelodeon 0.1111111 -175.69572 175.91795 1.0000000
Showtime-Nickelodeon -86.0000000 -479.11604 307.11604 0.9995591
TBS-Nickelodeon 33.0000000 -258.54266 324.54266 0.9999997
Showtime-PBS -86.1111111 -479.22715 307.00492 0.9995533
TBS-PBS 32.8888889 -258.65377 324.43154 0.9999997
TBS-Showtime 119.0000000 -337.75956 575.75956 0.9975931

$`file$Rating`
diff lwr upr p adj
TV-MA-TV-14 46.329035 -63.199178 155.85725 0.6047355
TV-PG-TV-14 147.497292 8.667629 286.32695 0.0052996
TV-Y-TV-14 56.355123 -77.349896 190.06014 0.6079772
TV-Y7-TV-14 51.949774 -50.169122 154.06867 0.4226375
TV-PG-TV-MA 101.168257 -32.481774 234.81829 0.0874937
TV-Y-TV-MA 10.026088 -118.292607 138.34478 0.9988693
TV-Y7-TV-MA 5.620739 -89.336747 100.57822 0.9996220
TV-Y-TV-PG -91.142168 -245.229589 62.94525 0.2719733
TV-Y7-TV-PG -95.547518 -223.196137 32.10110 0.0934011
TV-Y7-TV-Y -4.405350 -126.460775 117.65008 0.9999471

Just like the Tukey’s Test for the model I created in R Lesson 18: ANOVA part 2, each variable’s pair-wise differences are analyzed separately. In other words,  the pair-wise differences for Network are analyzed separately from the pair-wise differences for Rating.

Another thing I want to note is that since all of the p-values for the Network pair-wise differences are well above 0.01, there aren’t any statistically significant pair-wise differences for Network. On the other hand, there is a statistically significant pair-wise difference for Rating, which would be the TV-PG - TV-14 pair. Unlike the one-way ANOVA, Rating has a lower statistical significance (since there are two asterisks by Rating instead of three) and thus has fewer statistically significant pair-wise differences.

From the data, I can conclude that a cartoon’s rating (if you don’t factor in any other variables) is a good indicator as to how many episodes that cartoon will air. However, the combination of a cartoon’s rating and the network it airs on makes it less certain to predict how many episodes might air.

Thanks for reading,

Michael

 

 

 

 

R Lesson 18: ANOVA part 2

Hello everybody,

It’s Michael, and today I will discuss two-way ANOVA. This post will act as a continuation for my previous lesson on one-way ANOVA, although I will not use the same dataset from the previous post. As you may recall from the previous post, two-way ANOVA is just ANOVA with two independent variables instead of one.

Here’s the dataset-Celebrities.

Now, let’s load our dataset into R and learn more about our variables:

Screen Shot 2019-07-05 at 4.49.23 PM

This dataset contains the names of 104 celebrities who died in the 2010s (the observations) along with 6 variables (the main focus is on the celebrities’ net worth at the time of their deaths):

  • Name-the celebrity’s name
  • Date.of.Death-the date of the celebrity’s death
  • Age.of.Death-how old the celebrity was when he/she died
  • Occupation-the celebrity’s profession
  • Net.Worth-the celebrity’s net worth at the time of their death; this doesn’t factor in the possibility of the celebrity’s estate continuing to earn money long after their death
    • Net.Worth is given in millions, so for instance Nipsey Hussle’s net worth is listed as 8, which means he had $8 million net worth at the time of his death.
    • There are some special cases, such as Mark Salling, whose net worth is listed as 0.1, meaning that he only had a net worth of $100 thousand when he died. Steve Jobs is the richest person on this list, as his net worth is listed at 10200, or $10.2 billion. On the other hand, Whitney Houston is the poorest person on this list, as her net worth is listed at -20, which means she died with $20 million in debt.
  • Cause-the celebrity’s cause of death

Now, I would usually check for missing data (which I didn’t do for R Lesson 17: ANOVA Part 1), but since I wrote the datasets for both ANOVA posts, I know there’s no missing data.

Now let’s set up our model:

Screen Shot 2019-07-07 at 12.51.15 PM

In this model, I used Net.Worth as the dependent variable and Occupation and Age.at.Death as independent variables. Why did I make Age.at.Death a factor? In ANOVA, the independent variables need to be categorical (non-numerical) rather than continuous (numerical). Since Age.at.Death is a numerical variable, using the as.factor function would convert it into a non-numerical/categorical variable.

Both of the variables have high f-values and in turn, high significance levels (due to the p-values being much less than 0.01). This means that we can reject the null hypothesis; in other words, a celebrity’s occupation and age at death are related to their net worth at the time of their death.

Now let’s do a Tukey’s HSD test on our model, using the line TukeyHSD(model2,conf.level=0.99). Since we have two variables, pairwise comparisons for each variable are displayed separately.

First, here are the pairwise comparisons for the Occupation variable:

diff lwr upr p adj
activist-41st president -2.237500e+01 -353.196211 308.44621 1.0000000
actor-41st president -5.898696e+00 -281.822660 270.02527 1.0000000
actress-41st president 9.444444e+01 -190.281121 379.17001 0.9912907
animator-41st president 9.500000e+01 -286.999431 476.99943 0.9998272
artist-41st president 1.533333e+01 -296.567896 327.23456 1.0000000
astronaut-41st president -1.850000e+01 -349.321211 312.32121 1.0000000
athlete-41st president -1.440714e+01 -303.171570 274.35728 1.0000000
author-41st president 5.200000e+01 -243.895487 347.89549 0.9999996
businessman-41st president 1.711750e+03 1419.993115 2003.50688 0.0000000
chef-41st president -9.000000e+00 -390.999431 372.99943 1.0000000
comedian-41st president 1.250000e+02 -256.999431 506.99943 0.9925625
director-41st president 1.666667e+01 -295.234563 328.56790 1.0000000
DJ-41st president 4.000000e+01 -290.821211 370.82121 1.0000000
fashion designer-41st president 1.516667e+02 -160.234563 463.56790 0.7685610
film critic-41st president -1.600000e+01 -397.999431 365.99943 1.0000000
first lady-41st president -1.666667e+00 -313.567896 310.23456 1.0000000
football coach-41st president -1.500000e+01 -396.999431 366.99943 1.0000000
journalist-41st president -2.100000e+01 -402.999431 360.99943 1.0000000
pastor-41st president -3.268497e-13 -381.999431 381.99943 1.0000000
physicist-41st president -5.000000e+00 -386.999431 376.99943 1.0000000
producer-41st president 8.250000e+01 -248.321211 413.32121 0.9998196
rapper-41st president -2.019167e+01 -311.948551 271.56522 1.0000000
singer-41st president 4.140000e+01 -236.545412 319.34541 1.0000000
sportscaster-41st president -1.550000e+01 -397.499431 366.49943 1.0000000
US Senator-41st president -9.000000e+00 -390.999431 372.99943 1.0000000
actor-activist 1.647630e+01 -182.654664 215.60727 1.0000000
actress-activist 1.168194e+02 -94.338686 327.97757 0.5702341
animator-activist 1.173750e+02 -213.446211 448.19621 0.9817419
artist-activist 3.770833e+01 -208.871239 284.28791 1.0000000
astronaut-activist 3.875000e+00 -266.239388 273.98939 1.0000000
athlete-activist 7.967857e+00 -208.605463 224.54118 1.0000000
author-activist 7.437500e+01 -151.618911 300.36891 0.9920491
businessman-activist 1.734125e+03 1513.577526 1954.67247 0.0000000
chef-activist 1.337500e+01 -317.446211 344.19621 1.0000000
comedian-activist 1.473750e+02 -183.446211 478.19621 0.8670545
director-activist 3.904167e+01 -207.537906 285.62124 1.0000000
DJ-activist 6.237500e+01 -207.739388 332.48939 0.9999475
fashion designer-activist 1.740417e+02 -72.537906 420.62124 0.1927916
film critic-activist 6.375000e+00 -324.446211 337.19621 1.0000000
first lady-activist 2.070833e+01 -225.871239 267.28791 1.0000000
football coach-activist 7.375000e+00 -323.446211 338.19621 1.0000000
journalist-activist 1.375000e+00 -329.446211 332.19621 1.0000000
pastor-activist 2.237500e+01 -308.446211 353.19621 1.0000000
physicist-activist 1.737500e+01 -313.446211 348.19621 1.0000000
producer-activist 1.048750e+02 -165.239388 374.98939 0.9559022
rapper-activist 2.183333e+00 -218.364141 222.73081 1.0000000
singer-activist 6.377500e+01 -138.147661 265.69766 0.9951348
sportscaster-activist 6.875000e+00 -323.946211 337.69621 1.0000000
US Senator-activist 1.337500e+01 -317.446211 344.19621 1.0000000
actress-actor 1.003431e+02 -5.860043 206.54632 0.0181925
animator-actor 1.008987e+02 -175.025269 376.82266 0.9751205
artist-actor 2.123203e+01 -144.577638 187.04170 1.0000000
astronaut-actor -1.260130e+01 -211.732273 186.52966 1.0000000
athlete-actor -8.508447e+00 -125.107603 108.09071 1.0000000
author-actor 5.789870e+01 -75.385242 191.18263 0.8890175
businessman-actor 1.717649e+03 1593.824027 1841.47336 0.0000000
chef-actor -3.101304e+00 -279.025269 272.82266 1.0000000
comedian-actor 1.308987e+02 -145.025269 406.82266 0.7997856
director-actor 2.256536e+01 -143.244305 188.37503 1.0000000
DJ-actor 4.589870e+01 -153.232273 245.02966 0.9999491
fashion designer-actor 1.575654e+02 -8.244305 323.37503 0.0171557
film critic-actor -1.010130e+01 -286.025269 265.82266 1.0000000
first lady-actor 4.232029e+00 -161.577638 170.04170 1.0000000
football coach-actor -9.101304e+00 -285.025269 266.82266 1.0000000
journalist-actor -1.510130e+01 -291.025269 260.82266 1.0000000
pastor-actor 5.898696e+00 -270.025269 281.82266 1.0000000
physicist-actor 8.986957e-01 -275.025269 276.82266 1.0000000
producer-actor 8.839870e+01 -110.732273 287.52966 0.8702814
rapper-actor -1.429297e+01 -138.117640 109.53170 1.0000000
singer-actor 4.729870e+01 -39.096449 133.69384 0.5878943
sportscaster-actor -9.601304e+00 -285.525269 266.32266 1.0000000
US Senator-actor -3.101304e+00 -279.025269 272.82266 1.0000000
animator-actress 5.555556e-01 -284.170009 285.28112 1.0000000
artist-actress -7.911111e+01 -259.187370 100.96515 0.8795454
astronaut-actress -1.129444e+02 -324.102575 98.21369 0.6264662
athlete-actress -1.088516e+02 -244.976444 27.27327 0.0813248
author-actress -4.244444e+01 -193.107052 108.21816 0.9989104
businessman-actress 1.617306e+03 1474.942773 1759.66834 0.0000000
chef-actress -1.034444e+02 -388.170009 181.28112 0.9766920
comedian-actress 3.055556e+01 -254.170009 315.28112 1.0000000
director-actress -7.777778e+01 -257.854036 102.29848 0.8936327
DJ-actress -5.444444e+01 -265.602575 156.71369 0.9996991
fashion designer-actress 5.722222e+01 -122.854036 237.29848 0.9947597
film critic-actress -1.104444e+02 -395.170009 174.28112 0.9562763
first lady-actress -9.611111e+01 -276.187370 83.96515 0.6299987
football coach-actress -1.094444e+02 -394.170009 175.28112 0.9597755
journalist-actress -1.154444e+02 -400.170009 169.28112 0.9355653
pastor-actress -9.444444e+01 -379.170009 190.28112 0.9912907
physicist-actress -9.944444e+01 -384.170009 185.28112 0.9845452
producer-actress -1.194444e+01 -223.102575 199.21369 1.0000000
rapper-actress -1.146361e+02 -256.998894 27.72667 0.0769968
singer-actress -5.304444e+01 -164.394048 58.30516 0.7947154
sportscaster-actress -1.099444e+02 -394.670009 174.78112 0.9580517
US Senator-actress -1.034444e+02 -388.170009 181.28112 0.9766920
artist-animator -7.966667e+01 -391.567896 232.23456 0.9997391
astronaut-animator -1.135000e+02 -444.321211 217.32121 0.9872684
athlete-animator -1.094071e+02 -398.171570 179.35728 0.9648528
author-animator -4.300000e+01 -338.895487 252.89549 1.0000000
businessman-animator 1.616750e+03 1324.993115 1908.50688 0.0000000
chef-animator -1.040000e+02 -485.999431 277.99943 0.9993283
comedian-animator 3.000000e+01 -351.999431 411.99943 1.0000000
director-animator -7.833333e+01 -390.234563 233.56790 0.9997987
DJ-animator -5.500000e+01 -385.821211 275.82121 0.9999999
fashion designer-animator 5.666667e+01 -255.234563 368.56790 0.9999993
film critic-animator -1.110000e+02 -492.999431 270.99943 0.9983352
first lady-animator -9.666667e+01 -408.567896 215.23456 0.9961502
football coach-animator -1.100000e+02 -491.999431 271.99943 0.9985272
journalist-animator -1.160000e+02 -497.999431 265.99943 0.9970282
pastor-animator -9.500000e+01 -476.999431 286.99943 0.9998272
physicist-animator -1.000000e+02 -481.999431 281.99943 0.9996220
producer-animator -1.250000e+01 -343.321211 318.32121 1.0000000
rapper-animator -1.151917e+02 -406.948551 176.56522 0.9487637
singer-animator -5.360000e+01 -331.545412 224.34541 0.9999978
sportscaster-animator -1.105000e+02 -492.499431 271.49943 0.9984337
US Senator-animator -1.040000e+02 -485.999431 277.99943 0.9993283
astronaut-artist -3.383333e+01 -280.412906 212.74624 1.0000000
athlete-artist -2.974048e+01 -216.137112 156.65616 0.9999999
author-artist 3.666667e+01 -160.596991 233.93032 0.9999989
businessman-artist 1.696417e+03 1505.416951 1887.41638 0.0000000
chef-artist -2.433333e+01 -336.234563 287.56790 1.0000000
comedian-artist 1.096667e+02 -202.234563 421.56790 0.9834023
director-artist 1.333333e+00 -219.214141 221.88081 1.0000000
DJ-artist 2.466667e+01 -221.912906 271.24624 1.0000000
fashion designer-artist 1.363333e+02 -84.214141 356.88081 0.3818703
film critic-artist -3.133333e+01 -343.234563 280.56790 1.0000000
first lady-artist -1.700000e+01 -237.547474 203.54747 1.0000000
football coach-artist -3.033333e+01 -342.234563 281.56790 1.0000000
journalist-artist -3.633333e+01 -348.234563 275.56790 1.0000000
pastor-artist -1.533333e+01 -327.234563 296.56790 1.0000000
physicist-artist -2.033333e+01 -332.234563 291.56790 1.0000000
producer-artist 6.716667e+01 -179.412906 313.74624 0.9993232
rapper-artist -3.552500e+01 -226.524716 155.47472 0.9999989
singer-artist 2.606667e+01 -143.085525 195.21886 1.0000000
sportscaster-artist -3.083333e+01 -342.734563 281.06790 1.0000000
US Senator-artist -2.433333e+01 -336.234563 287.56790 1.0000000
athlete-astronaut 4.092857e+00 -212.480463 220.66618 1.0000000
author-astronaut 7.050000e+01 -155.493911 296.49391 0.9958237
businessman-astronaut 1.730250e+03 1509.702526 1950.79747 0.0000000
chef-astronaut 9.500000e+00 -321.321211 340.32121 1.0000000
comedian-astronaut 1.435000e+02 -187.321211 474.32121 0.8902022
director-astronaut 3.516667e+01 -211.412906 281.74624 1.0000000
DJ-astronaut 5.850000e+01 -211.614388 328.61439 0.9999823
fashion designer-astronaut 1.701667e+02 -76.412906 416.74624 0.2201389
film critic-astronaut 2.500000e+00 -328.321211 333.32121 1.0000000
first lady-astronaut 1.683333e+01 -229.746239 263.41291 1.0000000
football coach-astronaut 3.500000e+00 -327.321211 334.32121 1.0000000
journalist-astronaut -2.500000e+00 -333.321211 328.32121 1.0000000
pastor-astronaut 1.850000e+01 -312.321211 349.32121 1.0000000
physicist-astronaut 1.350000e+01 -317.321211 344.32121 1.0000000
producer-astronaut 1.010000e+02 -169.114388 371.11439 0.9690196
rapper-astronaut -1.691667e+00 -222.239141 218.85581 1.0000000
singer-astronaut 5.990000e+01 -142.022661 261.82266 0.9978071
sportscaster-astronaut 3.000000e+00 -327.821211 333.82121 1.0000000
US Senator-astronaut 9.500000e+00 -321.321211 340.32121 1.0000000
author-athlete 6.640714e+01 -91.755648 224.56993 0.9143646
businessman-athlete 1.726157e+03 1575.879370 1876.43492 0.0000000
chef-athlete 5.407143e+00 -283.357284 294.17157 1.0000000
comedian-athlete 1.394071e+02 -149.357284 428.17157 0.7779470
director-athlete 3.107381e+01 -155.322827 217.47045 0.9999999
DJ-athlete 5.440714e+01 -162.166178 270.98046 0.9997979
fashion designer-athlete 1.660738e+02 -20.322827 352.47045 0.0322196
film critic-athlete -1.592857e+00 -290.357284 287.17157 1.0000000
first lady-athlete 1.274048e+01 -173.656160 199.13711 1.0000000
football coach-athlete -5.928571e-01 -289.357284 288.17157 1.0000000
journalist-athlete -6.592857e+00 -295.357284 282.17157 1.0000000
pastor-athlete 1.440714e+01 -274.357284 303.17157 1.0000000
physicist-athlete 9.407143e+00 -279.357284 298.17157 1.0000000
producer-athlete 9.690714e+01 -119.666178 313.48046 0.8629084
rapper-athlete -5.784524e+00 -156.062296 144.49325 1.0000000
singer-athlete 5.580714e+01 -65.498180 177.11247 0.8348572
sportscaster-athlete -1.092857e+00 -289.857284 287.67157 1.0000000
US Senator-athlete 5.407143e+00 -283.357284 294.17157 1.0000000
businessman-author 1.659750e+03 1496.187615 1823.31238 0.0000000
chef-author -6.100000e+01 -356.895487 234.89549 0.9999925
comedian-author 7.300000e+01 -222.895487 368.89549 0.9998476
director-author -3.533333e+01 -232.596991 161.93032 0.9999995
DJ-author -1.200000e+01 -237.993911 213.99391 1.0000000
fashion designer-author 9.966667e+01 -97.596991 296.93032 0.7152060
film critic-author -6.800000e+01 -363.895487 227.89549 0.9999515
first lady-author -5.366667e+01 -250.930325 143.59699 0.9993352
football coach-author -6.700000e+01 -362.895487 228.89549 0.9999622
journalist-author -7.300000e+01 -368.895487 222.89549 0.9998476
pastor-author -5.200000e+01 -347.895487 243.89549 0.9999996
physicist-author -5.700000e+01 -352.895487 238.89549 0.9999978
producer-author 3.050000e+01 -195.493911 256.49391 1.0000000
rapper-author -7.219167e+01 -235.754051 91.37072 0.8754649
singer-author -1.060000e+01 -148.019890 126.81989 1.0000000
sportscaster-author -6.750000e+01 -363.395487 228.39549 0.9999571
US Senator-author -6.100000e+01 -356.895487 234.89549 0.9999925
chef-businessman -1.720750e+03 -2012.506885 -1428.99312 0.0000000
comedian-businessman -1.586750e+03 -1878.506885 -1294.99312 0.0000000
director-businessman -1.695083e+03 -1886.083049 -1504.08362 0.0000000
DJ-businessman -1.671750e+03 -1892.297474 -1451.20253 0.0000000
fashion designer-businessman -1.560083e+03 -1751.083049 -1369.08362 0.0000000
film critic-businessman -1.727750e+03 -2019.506885 -1435.99312 0.0000000
first lady-businessman -1.713417e+03 -1904.416382 -1522.41695 0.0000000
football coach-businessman -1.726750e+03 -2018.506885 -1434.99312 0.0000000
journalist-businessman -1.732750e+03 -2024.506885 -1440.99312 0.0000000
pastor-businessman -1.711750e+03 -2003.506885 -1419.99312 0.0000000
physicist-businessman -1.716750e+03 -2008.506885 -1424.99312 0.0000000
producer-businessman -1.629250e+03 -1849.797474 -1408.70253 0.0000000
rapper-businessman -1.731942e+03 -1887.892281 -1575.99105 0.0000000
singer-businessman -1.670350e+03 -1798.616000 -1542.08400 0.0000000
sportscaster-businessman -1.727250e+03 -2019.006885 -1435.49312 0.0000000
US Senator-businessman -1.720750e+03 -2012.506885 -1428.99312 0.0000000
comedian-chef 1.340000e+02 -247.999431 515.99943 0.9838100
director-chef 2.566667e+01 -286.234563 337.56790 1.0000000
DJ-chef 4.900000e+01 -281.821211 379.82121 1.0000000
fashion designer-chef 1.606667e+02 -151.234563 472.56790 0.6861785
film critic-chef -7.000000e+00 -388.999431 374.99943 1.0000000
first lady-chef 7.333333e+00 -304.567896 319.23456 1.0000000
football coach-chef -6.000000e+00 -387.999431 375.99943 1.0000000
journalist-chef -1.200000e+01 -393.999431 369.99943 1.0000000
pastor-chef 9.000000e+00 -372.999431 390.99943 1.0000000
physicist-chef 4.000000e+00 -377.999431 385.99943 1.0000000
producer-chef 9.150000e+01 -239.321211 422.32121 0.9991584
rapper-chef -1.119167e+01 -302.948551 280.56522 1.0000000
singer-chef 5.040000e+01 -227.545412 328.34541 0.9999993
sportscaster-chef -6.500000e+00 -388.499431 375.49943 1.0000000
US Senator-chef -2.842171e-13 -381.999431 381.99943 1.0000000
director-comedian -1.083333e+02 -420.234563 203.56790 0.9854411
DJ-comedian -8.500000e+01 -415.821211 245.82121 0.9997146
fashion designer-comedian 2.666667e+01 -285.234563 338.56790 1.0000000
film critic-comedian -1.410000e+02 -522.999431 240.99943 0.9726971
first lady-comedian -1.266667e+02 -438.567896 185.23456 0.9346884
football coach-comedian -1.400000e+02 -521.999431 241.99943 0.9745592
journalist-comedian -1.460000e+02 -527.999431 235.99943 0.9618505
pastor-comedian -1.250000e+02 -506.999431 256.99943 0.9925625
physicist-comedian -1.300000e+02 -511.999431 251.99943 0.9883604
producer-comedian -4.250000e+01 -373.321211 288.32121 1.0000000
rapper-comedian -1.451917e+02 -436.948551 146.56522 0.7370064
singer-comedian -8.360000e+01 -361.545412 194.34541 0.9973727
sportscaster-comedian -1.405000e+02 -522.499431 241.49943 0.9736403
US Senator-comedian -1.340000e+02 -515.999431 247.99943 0.9838100
DJ-director 2.333333e+01 -223.246239 269.91291 1.0000000
fashion designer-director 1.350000e+02 -85.547474 355.54747 0.3980601
film critic-director -3.266667e+01 -344.567896 279.23456 1.0000000
first lady-director -1.833333e+01 -238.880808 202.21414 1.0000000
football coach-director -3.166667e+01 -343.567896 280.23456 1.0000000
journalist-director -3.766667e+01 -349.567896 274.23456 1.0000000
pastor-director -1.666667e+01 -328.567896 295.23456 1.0000000
physicist-director -2.166667e+01 -333.567896 290.23456 1.0000000
producer-director 6.583333e+01 -180.746239 312.41291 0.9994942
rapper-director -3.685833e+01 -227.858049 154.14138 0.9999978
singer-director 2.473333e+01 -144.418858 193.88552 1.0000000
sportscaster-director -3.216667e+01 -344.067896 279.73456 1.0000000
US Senator-director -2.566667e+01 -337.567896 286.23456 1.0000000
fashion designer-DJ 1.116667e+02 -134.912906 358.24624 0.8512039
film critic-DJ -5.600000e+01 -386.821211 274.82121 0.9999998
first lady-DJ -4.166667e+01 -288.246239 204.91291 0.9999998
football coach-DJ -5.500000e+01 -385.821211 275.82121 0.9999999
journalist-DJ -6.100000e+01 -391.821211 269.82121 0.9999990
pastor-DJ -4.000000e+01 -370.821211 290.82121 1.0000000
physicist-DJ -4.500000e+01 -375.821211 285.82121 1.0000000
producer-DJ 4.250000e+01 -227.614388 312.61439 1.0000000
rapper-DJ -6.019167e+01 -280.739141 160.35581 0.9993042
singer-DJ 1.400000e+00 -200.522661 203.32266 1.0000000
sportscaster-DJ -5.550000e+01 -386.321211 275.32121 0.9999998
US Senator-DJ -4.900000e+01 -379.821211 281.82121 1.0000000
film critic-fashion designer -1.676667e+02 -479.567896 144.23456 0.6182614
first lady-fashion designer -1.533333e+02 -373.880808 67.21414 0.2109002
football coach-fashion designer -1.666667e+02 -478.567896 145.23456 0.6280663
journalist-fashion designer -1.726667e+02 -484.567896 139.23456 0.5691260
pastor-fashion designer -1.516667e+02 -463.567896 160.23456 0.7685610
physicist-fashion designer -1.566667e+02 -468.567896 155.23456 0.7237307
producer-fashion designer -6.916667e+01 -315.746239 177.41291 0.9989741
rapper-fashion designer -1.718583e+02 -362.858049 19.14138 0.0293733
singer-fashion designer -1.102667e+02 -279.418858 58.88552 0.2986636
sportscaster-fashion designer -1.671667e+02 -479.067896 144.73456 0.6231663
US Senator-fashion designer -1.606667e+02 -472.567896 151.23456 0.6861785
first lady-film critic 1.433333e+01 -297.567896 326.23456 1.0000000
football coach-film critic 1.000000e+00 -380.999431 382.99943 1.0000000
journalist-film critic -5.000000e+00 -386.999431 376.99943 1.0000000
pastor-film critic 1.600000e+01 -365.999431 397.99943 1.0000000
physicist-film critic 1.100000e+01 -370.999431 392.99943 1.0000000
producer-film critic 9.850000e+01 -232.321211 429.32121 0.9976983
rapper-film critic -4.191667e+00 -295.948551 287.56522 1.0000000
singer-film critic 5.740000e+01 -220.545412 335.34541 0.9999923
sportscaster-film critic 5.000000e-01 -381.499431 382.49943 1.0000000
US Senator-film critic 7.000000e+00 -374.999431 388.99943 1.0000000
football coach-first lady -1.333333e+01 -325.234563 298.56790 1.0000000
journalist-first lady -1.933333e+01 -331.234563 292.56790 1.0000000
pastor-first lady 1.666667e+00 -310.234563 313.56790 1.0000000
physicist-first lady -3.333333e+00 -315.234563 308.56790 1.0000000
producer-first lady 8.416667e+01 -162.412906 330.74624 0.9879665
rapper-first lady -1.852500e+01 -209.524716 172.47472 1.0000000
singer-first lady 4.306667e+01 -126.085525 212.21886 0.9997516
sportscaster-first lady -1.383333e+01 -325.734563 298.06790 1.0000000
US Senator-first lady -7.333333e+00 -319.234563 304.56790 1.0000000
journalist-football coach -6.000000e+00 -387.999431 375.99943 1.0000000
pastor-football coach 1.500000e+01 -366.999431 396.99943 1.0000000
physicist-football coach 1.000000e+01 -371.999431 391.99943 1.0000000
producer-football coach 9.750000e+01 -233.321211 428.32121 0.9979882
rapper-football coach -5.191667e+00 -296.948551 286.56522 1.0000000
singer-football coach 5.640000e+01 -221.545412 334.34541 0.9999944
sportscaster-football coach -5.000000e-01 -382.499431 381.49943 1.0000000
US Senator-football coach 6.000000e+00 -375.999431 387.99943 1.0000000
pastor-journalist 2.100000e+01 -360.999431 402.99943 1.0000000
physicist-journalist 1.600000e+01 -365.999431 397.99943 1.0000000
producer-journalist 1.035000e+02 -227.321211 434.32121 0.9956713
rapper-journalist 8.083333e-01 -290.948551 292.56522 1.0000000
singer-journalist 6.240000e+01 -215.545412 340.34541 0.9999672
sportscaster-journalist 5.500000e+00 -376.499431 387.49943 1.0000000
US Senator-journalist 1.200000e+01 -369.999431 393.99943 1.0000000
physicist-pastor -5.000000e+00 -386.999431 376.99943 1.0000000
producer-pastor 8.250000e+01 -248.321211 413.32121 0.9998196
rapper-pastor -2.019167e+01 -311.948551 271.56522 1.0000000
singer-pastor 4.140000e+01 -236.545412 319.34541 1.0000000
sportscaster-pastor -1.550000e+01 -397.499431 366.49943 1.0000000
US Senator-pastor -9.000000e+00 -390.999431 372.99943 1.0000000
producer-physicist 8.750000e+01 -243.321211 418.32121 0.9995594
rapper-physicist -1.519167e+01 -306.948551 276.56522 1.0000000
singer-physicist 4.640000e+01 -231.545412 324.34541 0.9999999
sportscaster-physicist -1.050000e+01 -392.499431 371.49943 1.0000000
US Senator-physicist -4.000000e+00 -385.999431 377.99943 1.0000000
rapper-producer -1.026917e+02 -323.239141 117.85581 0.8216250
singer-producer -4.110000e+01 -243.022661 160.82266 0.9999941
sportscaster-producer -9.800000e+01 -428.821211 232.82121 0.9978473
US Senator-producer -9.150000e+01 -422.321211 239.32121 0.9991584
singer-rapper 6.159167e+01 -66.674333 189.85767 0.7847897
sportscaster-rapper 4.691667e+00 -287.065218 296.44855 1.0000000
US Senator-rapper 1.119167e+01 -280.565218 302.94855 1.0000000
sportscaster-singer -5.690000e+01 -334.845412 221.04541 0.9999934
US Senator-singer -5.040000e+01 -328.345412 227.54541 0.9999993
US Senator-sportscaster 6.500000e+00 -375.499431 388.49943 1.0000000

Recall that diff is the pairwise difference between the two observations, lwr and upr are the lower and upper limits for the pair at a 99% confidence level, respectively, and p adj is the p-value for the pair. Remember that a pairwise difference is statistically significant if the p-value is 0.01 or less.

Some interesting pairwise differences include (rounded to two decimal places):

  • artist-activist-(-208.87)
    • Makes sense. After all, artists (e.g. Thomas Kinkade) and activists (e.g. Elie Wiesel) have very different lines of work.
  • fashion designer-artist-136.33
    • This is an interesting difference. After all, if you’ve got a very broad definition of the word “artist”, fashion designers such as Kate Spade and Alexander McQueen could be considered artists.
  • producer-DJ-42.5
    • Another interesting difference since DJs such as AVICII can be viewed as producers of music.
  • singer-rapper-61.59
    • Yet another interesting difference since rappers are singers (I just decided to put them in separate categories when I wrote this dataset).

Now here are the pairwise differences for the Age.at.Death variable:

diff lwr upr p adj
22-20 -6.579167e+01 -490.449623 358.866290 1.0000000
26-20 4.000000e+00 -420.657956 428.657956 1.0000000
27-20 -2.553972e+01 -372.271490 321.192049 1.0000000
28-20 -1.519167e+01 -439.849623 409.466290 1.0000000
31-20 -1.729297e+01 -441.950927 407.364985 1.0000000
33-20 3.607738e+00 -364.156840 371.372316 1.0000000
34-20 -1.329297e+01 -437.950927 411.364985 1.0000000
35-20 -1.919297e+01 -443.850927 405.464985 1.0000000
38-20 -1.928797e+01 -443.945927 405.369985 1.0000000
40-20 -7.057565e+01 -438.340230 297.188926 1.0000000
41-20 -2.004583e+01 -387.810412 347.718745 1.0000000
42-20 -1.028452e+01 -434.942480 414.373433 1.0000000
44-20 -4.650000e+00 -429.307956 420.007956 1.0000000
45-20 -1.421085e-12 -424.657956 424.657956 1.0000000
46-20 1.570703e+01 -408.950927 440.364985 1.0000000
47-20 -3.175595e+00 -370.940173 364.588983 1.0000000
48-20 -5.613377e+01 -402.865538 290.598001 1.0000000
49-20 -1.914297e+01 -443.800927 405.514985 1.0000000
51-20 2.192435e+01 -345.840230 389.688926 1.0000000
52-20 -7.942319e+00 -375.706897 359.822259 1.0000000
53-20 1.334083e+02 -291.249623 558.066290 0.9997168
54-20 6.299168e+00 -340.432601 353.030938 1.0000000
55-20 -8.319000e+02 -1199.664578 -464.135422 0.0000000
56-20 8.463058e+03 8038.400377 8887.716290 0.0000000
57-20 1.166083e+02 -251.156245 484.372912 0.9996587
60-20 -9.463611e+01 -519.294068 330.021845 1.0000000
61-20 6.774565e+00 -339.957204 353.506335 1.0000000
63-20 3.070703e+01 -393.950927 455.364985 1.0000000
65-20 -8.585667e+02 -1226.331245 -490.802088 0.0000000
66-20 1.256083e+02 -242.156245 493.372912 0.9986051
67-20 -1.068452e+01 -435.342480 413.973433 1.0000000
69-20 5.608568e+01 -290.646093 402.817446 1.0000000
70-20 -3.547315e+01 -382.204918 311.258621 1.0000000
71-20 -1.629297e+01 -440.950927 408.364985 1.0000000
73-20 -1.129297e+01 -435.950927 413.364985 1.0000000
74-20 -2.432262e+01 -371.054389 322.409150 1.0000000
75-20 -9.269167e+01 -517.349623 331.966290 1.0000000
76-20 -2.581944e+01 -361.541037 309.902148 1.0000000
77-20 -1.636942e+03 -2061.599623 -1212.283710 0.0000000
79-20 4.803639e+02 55.705932 905.021845 0.0021269
80-20 -5.963611e+01 -484.294068 365.021845 1.0000000
81-20 -1.916667e-01 -367.956245 367.572912 1.0000000
82-20 1.310807e+01 -315.830566 342.046711 1.0000000
83-20 -1.025920e+01 -356.990972 336.472567 1.0000000
84-20 -1.696454e+01 -384.729119 350.800037 1.0000000
85-20 1.110463e+01 -335.627140 357.836399 1.0000000
86-20 -6.719167e+01 -491.849623 357.466290 1.0000000
87-20 2.183333e+00 -422.474623 426.841290 1.0000000
88-20 -1.129297e+01 -435.950927 413.364985 1.0000000
89-20 -3.926906e+02 -721.629194 -63.751917 0.0010138
90-20 -3.952500e+01 -464.182956 385.132956 1.0000000
91-20 -5.711421e+02 -917.873871 -224.410332 0.0000060
92-20 1.475000e+00 -423.182956 426.132956 1.0000000
93-20 -3.525000e+00 -428.182956 421.132956 1.0000000
94-20 6.416667e-01 -367.122912 368.406245 1.0000000
95-20 3.891667e+00 -363.872912 371.656245 1.0000000
97-20 8.036389e+01 -344.294068 505.021845 1.0000000
99-20 -1.916667e-01 -424.849623 424.466290 1.0000000
26-22 6.979167e+01 -354.866290 494.449623 1.0000000
27-22 4.025195e+01 -306.479823 386.983716 1.0000000
28-22 5.060000e+01 -374.057956 475.257956 1.0000000
31-22 4.849870e+01 -376.159261 473.156652 1.0000000
33-22 6.939940e+01 -298.365173 437.163983 1.0000000
34-22 5.249870e+01 -372.159261 477.156652 1.0000000
35-22 4.659870e+01 -378.059261 471.256652 1.0000000
38-22 4.650370e+01 -378.154261 471.161652 1.0000000
40-22 -4.783986e+00 -372.548564 362.980593 1.0000000
41-22 4.574583e+01 -322.018745 413.510412 1.0000000
42-22 5.550714e+01 -369.150814 480.165099 1.0000000
44-22 6.114167e+01 -363.516290 485.799623 1.0000000
45-22 6.579167e+01 -358.866290 490.449623 1.0000000
46-22 8.149870e+01 -343.159261 506.156652 1.0000000
47-22 6.261607e+01 -305.148507 430.380650 1.0000000
48-22 9.657899e+00 -337.073871 356.389668 1.0000000
49-22 4.664870e+01 -378.009261 471.306652 1.0000000
51-22 8.771601e+01 -280.048564 455.480593 0.9999997
52-22 5.784935e+01 -309.915230 425.613926 1.0000000
53-22 1.992000e+02 -225.457956 623.857956 0.9034355
54-22 7.209084e+01 -274.640934 418.822605 1.0000000
55-22 -7.661083e+02 -1133.872912 -398.343755 0.0000001
56-22 8.528850e+03 8104.192044 8953.507956 0.0000000
57-22 1.824000e+02 -185.364578 550.164578 0.8447064
60-22 -2.884444e+01 -453.502401 395.813512 1.0000000
61-22 7.256623e+01 -274.165538 419.298001 1.0000000
63-22 9.649870e+01 -328.159261 521.156652 0.9999999
65-22 -7.927750e+02 -1160.539578 -425.010422 0.0000000
66-22 1.914000e+02 -176.364578 559.164578 0.7787857
67-22 5.510714e+01 -369.550814 479.765099 1.0000000
69-22 1.218773e+02 -224.854426 468.609112 0.9977002
70-22 3.031852e+01 -316.413251 377.050288 1.0000000
71-22 4.949870e+01 -375.159261 474.156652 1.0000000
73-22 5.449870e+01 -370.159261 479.156652 1.0000000
74-22 4.146905e+01 -305.262722 388.200817 1.0000000
75-22 -2.690000e+01 -451.557956 397.757956 1.0000000
76-22 3.997222e+01 -295.749370 375.693814 1.0000000
77-22 -1.571150e+03 -1995.807956 -1146.492044 0.0000000
79-22 5.461556e+02 121.497599 970.813512 0.0003430
80-22 6.155556e+00 -418.502401 430.813512 1.0000000
81-22 6.560000e+01 -302.164578 433.364578 1.0000000
82-22 7.889974e+01 -250.038899 407.838378 0.9999996
83-22 5.553246e+01 -291.199306 402.264233 1.0000000
84-22 4.882713e+01 -318.937453 416.591704 1.0000000
85-22 7.689630e+01 -269.835473 423.628066 1.0000000
86-22 -1.400000e+00 -426.057956 423.257956 1.0000000
87-22 6.797500e+01 -356.682956 492.632956 1.0000000
88-22 5.449870e+01 -370.159261 479.156652 1.0000000
89-22 -3.268989e+02 -655.837527 2.039750 0.0107514
90-22 2.626667e+01 -398.391290 450.924623 1.0000000
91-22 -5.053504e+02 -852.082204 -158.618665 0.0000481
92-22 6.726667e+01 -357.391290 491.924623 1.0000000
93-22 6.226667e+01 -362.391290 486.924623 1.0000000
94-22 6.643333e+01 -301.331245 434.197912 1.0000000
95-22 6.968333e+01 -298.081245 437.447912 1.0000000
97-22 1.461556e+02 -278.502401 570.813512 0.9984024
99-22 6.560000e+01 -359.057956 490.257956 1.0000000
27-26 -2.953972e+01 -376.271490 317.192049 1.0000000
28-26 -1.919167e+01 -443.849623 405.466290 1.0000000
31-26 -2.129297e+01 -445.950927 403.364985 1.0000000
33-26 -3.922619e-01 -368.156840 367.372316 1.0000000
34-26 -1.729297e+01 -441.950927 407.364985 1.0000000
35-26 -2.319297e+01 -447.850927 401.464985 1.0000000
38-26 -2.328797e+01 -447.945927 401.369985 1.0000000
40-26 -7.457565e+01 -442.340230 293.188926 1.0000000
41-26 -2.404583e+01 -391.810412 343.718745 1.0000000
42-26 -1.428452e+01 -438.942480 410.373433 1.0000000
44-26 -8.650000e+00 -433.307956 416.007956 1.0000000
45-26 -4.000000e+00 -428.657956 420.657956 1.0000000
46-26 1.170703e+01 -412.950927 436.364985 1.0000000
47-26 -7.175595e+00 -374.940173 360.588983 1.0000000
48-26 -6.013377e+01 -406.865538 286.598001 1.0000000
49-26 -2.314297e+01 -447.800927 401.514985 1.0000000
51-26 1.792435e+01 -349.840230 385.688926 1.0000000
52-26 -1.194232e+01 -379.706897 355.822259 1.0000000
53-26 1.294083e+02 -295.249623 554.066290 0.9998497
54-26 2.299168e+00 -344.432601 349.030938 1.0000000
55-26 -8.359000e+02 -1203.664578 -468.135422 0.0000000
56-26 8.459058e+03 8034.400377 8883.716290 0.0000000
57-26 1.126083e+02 -255.156245 480.372912 0.9998336
60-26 -9.863611e+01 -523.294068 326.021845 0.9999998
61-26 2.774565e+00 -343.957204 349.506335 1.0000000
63-26 2.670703e+01 -397.950927 451.364985 1.0000000
65-26 -8.625667e+02 -1230.331245 -494.802088 0.0000000
66-26 1.216083e+02 -246.156245 489.372912 0.9992286
67-26 -1.468452e+01 -439.342480 409.973433 1.0000000
69-26 5.208568e+01 -294.646093 398.817446 1.0000000
70-26 -3.947315e+01 -386.204918 307.258621 1.0000000
71-26 -2.029297e+01 -444.950927 404.364985 1.0000000
73-26 -1.529297e+01 -439.950927 409.364985 1.0000000
74-26 -2.832262e+01 -375.054389 318.409150 1.0000000
75-26 -9.669167e+01 -521.349623 327.966290 0.9999999
76-26 -2.981944e+01 -365.541037 305.902148 1.0000000
77-26 -1.640942e+03 -2065.599623 -1216.283710 0.0000000
79-26 4.763639e+02 51.705932 901.021845 0.0023779
80-26 -6.363611e+01 -488.294068 361.021845 1.0000000
81-26 -4.191667e+00 -371.956245 363.572912 1.0000000
82-26 9.108072e+00 -319.830566 338.046711 1.0000000
83-26 -1.425920e+01 -360.990972 332.472567 1.0000000
84-26 -2.096454e+01 -388.729119 346.800037 1.0000000
85-26 7.104630e+00 -339.627140 353.836399 1.0000000
86-26 -7.119167e+01 -495.849623 353.466290 1.0000000
87-26 -1.816667e+00 -426.474623 422.841290 1.0000000
88-26 -1.529297e+01 -439.950927 409.364985 1.0000000
89-26 -3.966906e+02 -725.629194 -67.751917 0.0008783
90-26 -4.352500e+01 -468.182956 381.132956 1.0000000
91-26 -5.751421e+02 -921.873871 -228.410332 0.0000053
92-26 -2.525000e+00 -427.182956 422.132956 1.0000000
93-26 -7.525000e+00 -432.182956 417.132956 1.0000000
94-26 -3.358333e+00 -371.122912 364.406245 1.0000000
95-26 -1.083333e-01 -367.872912 367.656245 1.0000000
97-26 7.636389e+01 -348.294068 501.021845 1.0000000
99-26 -4.191667e+00 -428.849623 420.466290 1.0000000
28-27 1.034805e+01 -336.383716 357.079823 1.0000000
31-27 8.246749e+00 -338.485020 354.978519 1.0000000
33-27 2.914746e+01 -244.968074 303.262991 1.0000000
34-27 1.224675e+01 -334.485020 358.978519 1.0000000
35-27 6.346749e+00 -340.385020 353.078519 1.0000000
38-27 6.251749e+00 -340.480020 352.983519 1.0000000
40-27 -4.503593e+01 -319.151464 229.079600 1.0000000
41-27 5.493887e+00 -268.621645 279.609419 1.0000000
42-27 1.525520e+01 -331.476573 361.986966 1.0000000
44-27 2.088972e+01 -325.842049 367.621490 1.0000000
45-27 2.553972e+01 -321.192049 372.271490 1.0000000
46-27 4.124675e+01 -305.485020 387.978519 1.0000000
47-27 2.236413e+01 -251.751407 296.479657 1.0000000
48-27 -3.059405e+01 -275.770433 214.582338 1.0000000
49-27 6.396749e+00 -340.335020 353.128519 1.0000000
51-27 4.746407e+01 -226.651464 321.579600 1.0000000
52-27 1.759740e+01 -256.518131 291.712934 1.0000000
53-27 1.589481e+02 -187.783716 505.679823 0.9223188
54-27 3.183889e+01 -213.337497 277.015274 1.0000000
55-27 -8.063603e+02 -1080.475812 -532.244747 0.0000000
56-27 8.488598e+03 8141.866284 8835.329823 0.0000000
57-27 1.421481e+02 -131.967478 416.263586 0.7841873
60-27 -6.909639e+01 -415.828160 277.635379 1.0000000
61-27 3.231429e+01 -212.862100 277.490671 1.0000000
63-27 5.624675e+01 -290.485020 402.978519 1.0000000
65-27 -8.330269e+02 -1107.142478 -558.911414 0.0000000
66-27 1.511481e+02 -122.967478 425.263586 0.6830173
67-27 1.485520e+01 -331.876573 361.586966 1.0000000
69-27 8.162540e+01 -163.550989 326.801782 0.9991241
70-27 -9.933428e+00 -255.109813 235.242958 1.0000000
71-27 9.246749e+00 -337.485020 355.978519 1.0000000
73-27 1.424675e+01 -332.485020 360.978519 1.0000000
74-27 1.217101e+00 -243.959284 246.393487 1.0000000
75-27 -6.715195e+01 -413.883716 279.579823 1.0000000
76-27 -2.797239e-01 -229.621232 229.061784 1.0000000
77-27 -1.611402e+03 -1958.133716 -1264.670177 0.0000000
79-27 5.059036e+02 159.171840 852.635379 0.0000473
80-27 -3.409639e+01 -380.828160 312.635379 1.0000000
81-27 2.534805e+01 -248.767478 299.463586 1.0000000
82-27 3.864779e+01 -180.644633 257.940219 1.0000000
83-27 1.528052e+01 -229.895868 260.456903 1.0000000
84-27 8.575179e+00 -265.540353 282.690712 1.0000000
85-27 3.664435e+01 -208.532035 281.820736 1.0000000
86-27 -4.165195e+01 -388.383716 305.079823 1.0000000
87-27 2.772305e+01 -319.008716 374.454823 1.0000000
88-27 1.424675e+01 -332.485020 360.978519 1.0000000
89-27 -3.671508e+02 -586.443261 -147.858409 0.0000045
90-27 -1.398528e+01 -360.717049 332.746490 1.0000000
91-27 -5.456024e+02 -790.778766 -300.425996 0.0000000
92-27 2.701472e+01 -319.717049 373.746490 1.0000000
93-27 2.201472e+01 -324.717049 368.746490 1.0000000
94-27 2.618139e+01 -247.934145 300.296919 1.0000000
95-27 2.943139e+01 -244.684145 303.546919 1.0000000
97-27 1.059036e+02 -240.828160 452.635379 0.9998422
99-27 2.534805e+01 -321.383716 372.079823 1.0000000
31-28 -2.101304e+00 -426.759261 422.556652 1.0000000
33-28 1.879940e+01 -348.965173 386.563983 1.0000000
34-28 1.898696e+00 -422.759261 426.556652 1.0000000
35-28 -4.001304e+00 -428.659261 420.656652 1.0000000
38-28 -4.096304e+00 -428.754261 420.561652 1.0000000
40-28 -5.538399e+01 -423.148564 312.380593 1.0000000
41-28 -4.854167e+00 -372.618745 362.910412 1.0000000
42-28 4.907143e+00 -419.750814 429.565099 1.0000000
44-28 1.054167e+01 -414.116290 435.199623 1.0000000
45-28 1.519167e+01 -409.466290 439.849623 1.0000000
46-28 3.089870e+01 -393.759261 455.556652 1.0000000
47-28 1.201607e+01 -355.748507 379.780650 1.0000000
48-28 -4.094210e+01 -387.673871 305.789668 1.0000000
49-28 -3.951304e+00 -428.609261 420.706652 1.0000000
51-28 3.711601e+01 -330.648564 404.880593 1.0000000
52-28 7.249348e+00 -360.515230 375.013926 1.0000000
53-28 1.486000e+02 -276.057956 573.257956 0.9978696
54-28 2.149084e+01 -325.240934 368.222605 1.0000000
55-28 -8.167083e+02 -1184.472912 -448.943755 0.0000000
56-28 8.478250e+03 8053.592044 8902.907956 0.0000000
57-28 1.318000e+02 -235.964578 499.564578 0.9968218
60-28 -7.944444e+01 -504.102401 345.213512 1.0000000
61-28 2.196623e+01 -324.765538 368.698001 1.0000000
63-28 4.589870e+01 -378.759261 470.556652 1.0000000
65-28 -8.433750e+02 -1211.139578 -475.610422 0.0000000
66-28 1.408000e+02 -226.964578 508.564578 0.9911771
67-28 4.507143e+00 -420.150814 429.165099 1.0000000
69-28 7.127734e+01 -275.454426 418.009112 1.0000000
70-28 -2.028148e+01 -367.013251 326.450288 1.0000000
71-28 -1.101304e+00 -425.759261 423.556652 1.0000000
73-28 3.898696e+00 -420.759261 428.556652 1.0000000
74-28 -9.130952e+00 -355.862722 337.600817 1.0000000
75-28 -7.750000e+01 -502.157956 347.157956 1.0000000
76-28 -1.062778e+01 -346.349370 325.093814 1.0000000
77-28 -1.621750e+03 -2046.407956 -1197.092044 0.0000000
79-28 4.955556e+02 70.897599 920.213512 0.0013926
80-28 -4.444444e+01 -469.102401 380.213512 1.0000000
81-28 1.500000e+01 -352.764578 382.764578 1.0000000
82-28 2.829974e+01 -300.638899 357.238378 1.0000000
83-28 4.932464e+00 -341.799306 351.664233 1.0000000
84-28 -1.772874e+00 -369.537453 365.991704 1.0000000
85-28 2.629630e+01 -320.435473 373.028066 1.0000000
86-28 -5.200000e+01 -476.657956 372.657956 1.0000000
87-28 1.737500e+01 -407.282956 442.032956 1.0000000
88-28 3.898696e+00 -420.759261 428.556652 1.0000000
89-28 -3.774989e+02 -706.437527 -48.560250 0.0017504
90-28 -2.433333e+01 -448.991290 400.324623 1.0000000
91-28 -5.559504e+02 -902.682204 -209.218665 0.0000096
92-28 1.666667e+01 -407.991290 441.324623 1.0000000
93-28 1.166667e+01 -412.991290 436.324623 1.0000000
94-28 1.583333e+01 -351.931245 383.597912 1.0000000
95-28 1.908333e+01 -348.681245 386.847912 1.0000000
97-28 9.555556e+01 -329.102401 520.213512 0.9999999
99-28 1.500000e+01 -409.657956 439.657956 1.0000000
33-31 2.090071e+01 -346.863869 388.665287 1.0000000
34-31 4.000000e+00 -420.657956 428.657956 1.0000000
35-31 -1.900000e+00 -426.557956 422.757956 1.0000000
38-31 -1.995000e+00 -426.652956 422.662956 1.0000000
40-31 -5.328268e+01 -421.047259 314.481897 1.0000000
41-31 -2.752862e+00 -370.517440 365.011716 1.0000000
42-31 7.008447e+00 -417.649509 431.666404 1.0000000
44-31 1.264297e+01 -412.014985 437.300927 1.0000000
45-31 1.729297e+01 -407.364985 441.950927 1.0000000
46-31 3.300000e+01 -391.657956 457.657956 1.0000000
47-31 1.411738e+01 -353.647202 381.881954 1.0000000
48-31 -3.884080e+01 -385.572567 307.890972 1.0000000
49-31 -1.850000e+00 -426.507956 422.807956 1.0000000
51-31 3.921732e+01 -328.547259 406.981897 1.0000000
52-31 9.350652e+00 -358.413926 377.115230 1.0000000
53-31 1.507013e+02 -273.956652 575.359261 0.9972989
54-31 2.359214e+01 -323.139630 370.323909 1.0000000
55-31 -8.146070e+02 -1182.371607 -446.842451 0.0000000
56-31 8.480351e+03 8055.693348 8905.009261 0.0000000
57-31 1.339013e+02 -233.863274 501.665883 0.9958949
60-31 -7.734314e+01 -502.001097 347.314816 1.0000000
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97-80 1.400000e+02 -284.657956 564.657956 0.9992710
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90-81 -3.933333e+01 -407.097912 328.431245 1.0000000
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85-82 -2.003443e+00 -221.295869 217.288983 1.0000000
86-82 -8.029974e+01 -409.238378 248.638899 0.9999994
87-82 -1.092474e+01 -339.863378 318.013899 1.0000000
88-82 -2.440104e+01 -353.339682 304.537595 1.0000000
89-82 -4.057986e+02 -595.711440 -215.885816 0.0000000
90-82 -5.263307e+01 -381.571711 276.305566 1.0000000
91-82 -5.842502e+02 -803.542600 -364.957748 0.0000000
92-82 -1.163307e+01 -340.571711 317.305566 1.0000000
93-82 -1.663307e+01 -345.571711 312.305566 1.0000000
94-82 -1.246641e+01 -263.697441 238.764629 1.0000000
95-82 -9.216406e+00 -260.447441 242.014629 1.0000000
97-82 6.725582e+01 -261.682822 396.194455 1.0000000
99-82 -1.329974e+01 -342.238378 315.638899 1.0000000
84-83 -6.705338e+00 -280.820870 267.410194 1.0000000
85-83 2.136383e+01 -223.812553 266.540218 1.0000000
86-83 -5.693246e+01 -403.664233 289.799306 1.0000000
87-83 1.244254e+01 -334.289233 359.174306 1.0000000
88-83 -1.033768e+00 -347.765538 345.698001 1.0000000
89-83 -3.824314e+02 -601.723778 -163.138927 0.0000022
90-83 -2.926580e+01 -375.997567 317.465972 1.0000000
91-83 -5.608829e+02 -806.059284 -315.706513 0.0000000
92-83 1.173420e+01 -334.997567 358.465972 1.0000000
93-83 6.734203e+00 -339.997567 353.465972 1.0000000
94-83 1.090087e+01 -263.214663 285.016402 1.0000000
95-83 1.415087e+01 -259.964663 288.266402 1.0000000
97-83 9.062309e+01 -256.108678 437.354861 0.9999960
99-83 1.006754e+01 -336.664233 356.799306 1.0000000
85-84 2.806917e+01 -246.046361 302.184703 1.0000000
86-84 -5.022713e+01 -417.991704 317.537453 1.0000000
87-84 1.914787e+01 -348.616704 386.912453 1.0000000
88-84 5.671570e+00 -362.093008 373.436148 1.0000000
89-84 -3.757260e+02 -626.957050 -124.494979 0.0000314
90-84 -2.256046e+01 -390.325037 345.204119 1.0000000
91-84 -5.541776e+02 -828.293093 -280.062028 0.0000001
92-84 1.843954e+01 -349.325037 386.204119 1.0000000
93-84 1.343954e+01 -354.325037 381.204119 1.0000000
94-84 1.760621e+01 -282.672313 317.884728 1.0000000
95-84 2.085621e+01 -279.422313 321.134728 1.0000000
97-84 9.732843e+01 -270.436148 465.093008 0.9999945
99-84 1.677287e+01 -350.991704 384.537453 1.0000000
86-85 -7.829630e+01 -425.028066 268.435473 0.9999999
87-85 -8.921296e+00 -355.653066 337.810473 1.0000000
88-85 -2.239760e+01 -369.129370 324.334169 1.0000000
89-85 -4.037952e+02 -623.087611 -184.502759 0.0000008
90-85 -5.062963e+01 -397.361399 296.102140 1.0000000
91-85 -5.822467e+02 -827.423117 -337.070346 0.0000000
92-85 -9.629630e+00 -356.361399 337.102140 1.0000000
93-85 -1.462963e+01 -361.361399 332.102140 1.0000000
94-85 -1.046296e+01 -284.578495 263.652569 1.0000000
95-85 -7.212963e+00 -281.328495 266.902569 1.0000000
97-85 6.925926e+01 -277.472510 415.991029 1.0000000
99-85 -1.129630e+01 -358.028066 335.435473 1.0000000
87-86 6.937500e+01 -355.282956 494.032956 1.0000000
88-86 5.589870e+01 -368.759261 480.556652 1.0000000
89-86 -3.254989e+02 -654.437527 3.439750 0.0112988
90-86 2.766667e+01 -396.991290 452.324623 1.0000000
91-86 -5.039504e+02 -850.682204 -157.218665 0.0000503
92-86 6.866667e+01 -355.991290 493.324623 1.0000000
93-86 6.366667e+01 -360.991290 488.324623 1.0000000
94-86 6.783333e+01 -299.931245 435.597912 1.0000000
95-86 7.108333e+01 -296.681245 438.847912 1.0000000
97-86 1.475556e+02 -277.102401 572.213512 0.9981131
99-86 6.700000e+01 -357.657956 491.657956 1.0000000
88-87 -1.347630e+01 -438.134261 411.181652 1.0000000
89-87 -3.948739e+02 -723.812527 -65.935250 0.0009374
90-87 -4.170833e+01 -466.366290 382.949623 1.0000000
91-87 -5.733254e+02 -920.057204 -226.593665 0.0000056
92-87 -7.083333e-01 -425.366290 423.949623 1.0000000
93-87 -5.708333e+00 -430.366290 418.949623 1.0000000
94-87 -1.541667e+00 -369.306245 366.222912 1.0000000
95-87 1.708333e+00 -366.056245 369.472912 1.0000000
97-87 7.818056e+01 -346.477401 502.838512 1.0000000
99-87 -2.375000e+00 -427.032956 422.282956 1.0000000
89-88 -3.813976e+02 -710.336223 -52.458946 0.0015213
90-88 -2.823203e+01 -452.889985 396.425927 1.0000000
91-88 -5.598491e+02 -906.580900 -213.117361 0.0000085
92-88 1.276797e+01 -411.889985 437.425927 1.0000000
93-88 7.767971e+00 -416.889985 432.425927 1.0000000
94-88 1.193464e+01 -355.829940 379.699216 1.0000000
95-88 1.518464e+01 -352.579940 382.949216 1.0000000
97-88 9.165686e+01 -333.001097 516.314816 1.0000000
99-88 1.110130e+01 -413.556652 435.759261 1.0000000
90-89 3.531656e+02 24.226917 682.104194 0.0042028
91-89 -1.784515e+02 -397.743972 40.840880 0.0808911
92-89 3.941656e+02 65.226917 723.104194 0.0009616
93-89 3.891656e+02 60.226917 718.104194 0.0011506
94-89 3.933322e+02 142.101187 644.563257 0.0000145
95-89 3.965822e+02 145.351187 647.813257 0.0000126
97-89 4.730544e+02 144.115806 801.993083 0.0000598
99-89 3.924989e+02 63.560250 721.437527 0.0010208
91-90 -5.316171e+02 -878.348871 -184.885332 0.0000207
92-90 4.100000e+01 -383.657956 465.657956 1.0000000
93-90 3.600000e+01 -388.657956 460.657956 1.0000000
94-90 4.016667e+01 -327.597912 407.931245 1.0000000
95-90 4.341667e+01 -324.347912 411.181245 1.0000000
97-90 1.198889e+02 -304.769068 544.546845 0.9999729
99-90 3.933333e+01 -385.324623 463.991290 1.0000000
92-91 5.726171e+02 225.885332 919.348871 0.0000058
93-91 5.676171e+02 220.885332 914.348871 0.0000067
94-91 5.717838e+02 297.668236 845.899300 0.0000001
95-91 5.750338e+02 300.918236 849.149300 0.0000001
97-91 6.515060e+02 304.774221 998.237760 0.0000006
99-91 5.709504e+02 224.218665 917.682204 0.0000061
93-92 -5.000000e+00 -429.657956 419.657956 1.0000000
94-92 -8.333333e-01 -368.597912 366.931245 1.0000000
95-92 2.416667e+00 -365.347912 370.181245 1.0000000
97-92 7.888889e+01 -345.769068 503.546845 1.0000000
99-92 -1.666667e+00 -426.324623 422.991290 1.0000000
94-93 4.166667e+00 -363.597912 371.931245 1.0000000
95-93 7.416667e+00 -360.347912 375.181245 1.0000000
97-93 8.388889e+01 -340.769068 508.546845 1.0000000
99-93 3.333333e+00 -421.324623 427.991290 1.0000000
95-94 3.250000e+00 -297.028521 303.528521 1.0000000
97-94 7.972222e+01 -288.042356 447.486800 1.0000000
99-94 -8.333333e-01 -368.597912 366.931245 1.0000000
97-95 7.647222e+01 -291.292356 444.236800 1.0000000
99-95 -4.083333e+00 -371.847912 363.681245 1.0000000
99-97 -8.055556e+01 -505.213512 344.102401 1.0000000

This Tukey’s HSD test displays the pairwise difference for different values of Age.at.Death. All ages come from the dataset and range from 20 (XXXTentacion) to 99 (Billy Graham).

Some interesting pairwise differences include (rounded to two decimal places):

  • 85-77-1648.05
    • Let’s take any person on this dataset who died at 85-Karl Lagerfeld or Shirley Temple-and anybody who died at 77-Roger Ailes-and analyze how much they have in common. Roger Ailes has more in common with Karl Lagerfeld than with Shirley Temple, as Ailes and Lagerfeld both ran successful businesses.
  • 69-65-914.65
    • Granted, either of the two people on this dataset who died at 69-David Bowie and Alan Rickman-wouldn’t have much in common with Craig Sager (the only person in this dataset who died at 65). Well, the only thing that connects all three of these people is that all died from cancer.
  • 22-20-(-65.79)
    • This isn’t a huge difference, but still interesting since the people in this dataset who died at 22 and 20 are Christina Grimmie and XXXTentacion, respectively. Both were young singers who were sadly victims of gun violence. Both also happened to die in Florida in the month of June.
  • 56-45-8463.06
    • This is a considerably large difference which is made even more interesting given that the two people in this dataset who died at these ages-Steve Jobs at 56 and Phife Dawg at 45-have almost nothing in common..

Thanks for reading,

Michael

 

R Lesson 17: ANOVA Part 1

Hello everybody,

It’s Michael, and today I will be discussing ANOVA in R. ANOVA is a statisical technique in R that stands for analysis of variance. ANOVA is used to analyze data in R by comparing the means of subsets in the data. There are two types of ANOVA-one way and two way-the first of which I’ll discuss in this post. I’ll discuss two-way ANOVA in my next post.

Here’s the dataset-US Cities.

This dataset contains information about 150 randomly selected US cities (three for each state) such as population and climate. As always let’s load our file into R and learn more about our data:

Screen Shot 2019-06-28 at 1.09.41 PM

There are 150 observations of 11 variables. Here’s a detailed breakdown of each variable:

  • City-the name of the US city
    • The reason R says City only displays 149 levels, despite there being 150 cities is because I used “Jackson” twice  (for Jackson, Mississippi and Jackson, Wyoming).
  • State-I used the state’s postal code here, rather than the state’s name. For instance, Alaska is AK, Alabama is AL, and so on.
  • Population-the city’s population; I tried to get the most recent data possible, but in some cases, I had to rely on data from the 2010 US Census.
  • January.High and January.Low-The city’s high & low temperatures for the month of January, respectively.
  • July.High and July.Low-The city’s high & low temperatures for the month of July, respectively.
    • I chose January and July to get a gauge of a city’s climate since they are the coldest and hottest months of the year, respectively.
  • Time.Zone-the time zone where the city is located. I used single letters to represent each time zone, which are as follows:
    • A-Alaska (the state has its own time zone)
    • H-Hawaii (the one other state with its own time zone)
    • E-Eastern
    • C-Central
    • M-Mountain
    • P-Pacific
  • Elevation-How many feet a city is above sea level.
    • There is one negative number in the dataset-(-6)-which corresponds to New Orleans. This means that New Orleans is the only city in this dataset that is below sea level.
  • Landlocked-Whether or not the city is located in a landlocked state-0 means the state isn’t landlocked while 1 means the state is landlocked.
  • Founded-the year a city was founded/settled. Sometimes I used the settlement year if I couldn’t find the year a city was founded.

Now let’s start the ANOVA. I will start with one-way ANOVA, which is just ANOVA using  one independent variable.

Here’s our model along with the model’s summary:

Screen Shot 2019-06-28 at 4.24.40 PM

In this model, I used January.Low as a dependent variable and State as an independent variable. If you’re wondering what I’m trying to analyze, I’m looking the relationship between states and the pleasantness/misery of their winters. I used January.Low rather than January.Highbecause I felt that the former would be a better measure of how good/bad a state’s winter is.

Now, before I discuss the output below summary, let me explain the concepts of null hypothesis and alternative hypothesis, as they are important to one-way ANOVA. A null hypothesis states that there is NO statistical significance between the dependent and independent variable. In this case, the null hypothesis is that there is no statistically significant relationship between a state and the pleasantness/misery of their winters.This is the hypothesis that we are trying to disprove.

On the other hand, the alternative hypothesis is simply the opposite of the null hypothesis; this hypothesis states that there is a statistically significant relationship between our dependent variable (January.Low) and our independent variable (State).

Now let’s explain the output below summary. First of all, the Df stands for degrees of freedom, which represents the amount of values that are free to vary in a dataset. The Df for State is 49, which means that there are 49 values in State that may vary.

We got 49 degrees of freedom because there are 50 possible values for State. Degrees of freedom for a variable/dataset will always be equal to N-1, with N being either the number of possible values in a variable or number of observations in a dataset. Why do we always subtract by 1? Here’s why:

Let’s say we’re looking for a set a four numbers that add up to 100. Some possibilities include (15, 20, 25, 40) or (15, 18, 30, 37) or (15, 16, 30, 39). However, notice that only three of the four numbers in each set vary, while the fourth is fixed, since it will always be 15. Therefore, only three of the four numbers are free to vary.

The f-value is 17.46 which is highly significant, given that the corresponding p-value is much less than the benchmark* level of significance-0.01 or 1%. This means that there is a statistically significant relationship between a state and how good/bad their winters are.

  • The “benchmark level of significance” refers to the threshold where you should reject the null hypothesis. Remember that if the Pr(>f)-which refers to the p-value of your independent variable-is less than 0.01 for your independent variable, reject the null hypothesis.

The Sum Sq and Mean Sq columns, which represent sum of squares and mean of sqaures, respectively (for both the independent variable and the residuals(, aren’t important for our ANOVA analysis.

Now it’s time to analyze pair-wise differences. In this case, we are analyzing the pair-wise differences between states to see which state pair has the greatest winter temperature difference. To analyze pair-wise differences, let’s use Tukey’s HSD test, which stands for Tukey’s Honest Significance Test. Here’s how it works (and I couldn’t post a screenshot since the output is too long). By the way, the line to set up the Tukey’s HSD test is TukeyHSD(model, conf.level=0.99)-remember that we’re analyzing our data with a 99% confidence level:

diff lwr upr p adj
AL-AK 34.33333333 15.0921361 53.574530535 0.0000000
AR-AK 26.90000000 7.6588028 46.141197201 0.0000048
AZ-AK 42.16666667 22.9254695 61.407863868 0.0000000
CA-AK 43.03333333 23.7921361 62.274530535 0.0000000
CO-AK 16.86666667 -2.3745305 36.107863868 0.0650303
CT-AK 18.80000000 -0.4411972 38.041197201 0.0145141
DE-AK 25.06666667 5.8254695 44.307863868 0.0000350
FL-AK 48.80000000 29.5588028 68.041197201 0.0000000
GA-AK 33.16666667 13.9254695 52.407863868 0.0000000
HI-AK 61.10000000 41.8588028 80.341197201 0.0000000
IA-AK 10.23333333 -9.0078639 29.474530535 0.9446692
ID-AK 13.10000000 -6.1411972 32.341197201 0.5202584
IL-AK 13.83333333 -5.4078639 33.074530535 0.3867198
IN-AK 16.80000000 -2.4411972 36.041197201 0.0681855
KS-AK 18.13333333 -1.1078639 37.374530535 0.0249778
KY-AK 22.23333333 2.9921361 41.474530535 0.0006339
LA-AK 39.13333333 19.8921361 58.374530535 0.0000000
MA-AK 18.06666667 -1.1745305 37.307863868 0.0263343
MD-AK 23.90000000 4.6588028 43.141197201 0.0001185
ME-AK 13.50000000 -5.7411972 32.741197201 0.4457807
MI-AK 15.26666667 -3.9745305 34.507863868 0.1844181
MN-AK 5.33333333 -13.9078639 24.574530535 1.0000000
MO-AK 20.20000000 0.9588028 39.441197201 0.0043056
MS-AK 36.26666667 17.0254695 55.507863868 0.0000000
MT-AK 12.33333333 -6.9078639 31.574530535 0.6652421
NC-AK 28.03333333 8.7921361 47.274530535 0.0000014
ND-AK -2.43333333 -21.6745305 16.807863868 1.0000000
NE-AK 11.63333333 -7.6078639 30.874530535 0.7858266
NH-AK 11.86666667 -7.3745305 31.107863868 0.7478742
NJ-AK 23.30000000 4.0588028 42.541197201 0.0002188
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WV-MT 9.26666667 -9.9745305 28.507863868 0.9866689
WY-MT -2.06666667 -21.3078639 17.174530535 1.0000000
ND-NC -30.46666667 -49.7078639 -11.225469465 0.0000001
NE-NC -16.40000000 -35.6411972 2.841197201 0.0899896
NH-NC -16.16666667 -35.4078639 3.074530535 0.1052106
NJ-NC -4.73333333 -23.9745305 14.507863868 1.0000000
NM-NC -6.96666667 -26.2078639 12.274530535 0.9999654
NV-NC -1.36666667 -20.6078639 17.874530535 1.0000000
NY-NC -8.26666667 -27.5078639 10.974530535 0.9982931
OH-NC -8.93333333 -28.1745305 10.307863868 0.9927615
OK-NC -4.03333333 -23.2745305 15.207863868 1.0000000
OR-NC 3.96666667 -15.2745305 23.207863868 1.0000000
PA-NC -7.70000000 -26.9411972 11.541197201 0.9996197
RI-NC -7.53333333 -26.7745305 11.707863868 0.9997685
SC-NC 4.86666667 -14.3745305 24.107863868 1.0000000
SD-NC -20.33333333 -39.5745305 -1.092136132 0.0038167
TN-NC -2.10000000 -21.3411972 17.141197201 1.0000000
TX-NC 7.96666667 -11.2745305 27.207863868 0.9992021
UT-NC -11.33333333 -30.5745305 7.907863868 0.8303487
VA-NC -0.86666667 -20.1078639 18.374530535 1.0000000
VT-NC -20.46666667 -39.7078639 -1.225469465 0.0033807
WA-NC 2.70000000 -16.5411972 21.941197201 1.0000000
WI-NC -20.20000000 -39.4411972 -0.958802799 0.0043056
WV-NC -6.43333333 -25.6745305 12.807863868 0.9999959
WY-NC -17.76666667 -37.0078639 1.474530535 0.0332992
NE-ND 14.06666667 -5.1745305 33.307863868 0.3476397
NH-ND 14.30000000 -4.9411972 33.541197201 0.3107477
NJ-ND 25.73333333 6.4921361 44.974530535 0.0000172
NM-ND 23.50000000 4.2588028 42.741197201 0.0001786
NV-ND 29.10000000 9.8588028 48.341197201 0.0000004
NY-ND 22.20000000 2.9588028 41.441197201 0.0006549
OH-ND 21.53333333 2.2921361 40.774530535 0.0012479
OK-ND 26.43333333 7.1921361 45.674530535 0.0000080
OR-ND 34.43333333 15.1921361 53.674530535 0.0000000
PA-ND 22.76666667 3.5254695 42.007863868 0.0003741
RI-ND 22.93333333 3.6921361 42.174530535 0.0003167
SC-ND 35.33333333 16.0921361 54.574530535 0.0000000
SD-ND 10.13333333 -9.1078639 29.374530535 0.9512161
TN-ND 28.36666667 9.1254695 47.607863868 0.0000009
TX-ND 38.43333333 19.1921361 57.674530535 0.0000000
UT-ND 19.13333333 -0.1078639 38.374530535 0.0109637
VA-ND 29.60000000 10.3588028 48.841197201 0.0000002
VT-ND 10.00000000 -9.2411972 29.241197201 0.9590483
WA-ND 33.16666667 13.9254695 52.407863868 0.0000000
WI-ND 10.26666667 -8.9745305 29.507863868 0.9423542
WV-ND 24.03333333 4.7921361 43.274530535 0.0001033
WY-ND 12.70000000 -6.5411972 31.941197201 0.5963928
NH-NE 0.23333333 -19.0078639 19.474530535 1.0000000
NJ-NE 11.66666667 -7.5745305 30.907863868 0.7805710
NM-NE 9.43333333 -9.8078639 28.674530535 0.9823527
NV-NE 15.03333333 -4.2078639 34.274530535 0.2108675
NY-NE 8.13333333 -11.1078639 27.374530535 0.9987715
OH-NE 7.46666667 -11.7745305 26.707863868 0.9998117
OK-NE 12.36666667 -6.8745305 31.607863868 0.6590833
OR-NE 20.36666667 1.1254695 39.607863868 0.0037030
PA-NE 8.70000000 -10.5411972 27.941197201 0.9954754
RI-NE 8.86666667 -10.3745305 28.107863868 0.9936474
SC-NE 21.26666667 2.0254695 40.507863868 0.0016077
SD-NE -3.93333333 -23.1745305 15.307863868 1.0000000
TN-NE 14.30000000 -4.9411972 33.541197201 0.3107477
TX-NE 24.36666667 5.1254695 43.607863868 0.0000730
UT-NE 5.06666667 -14.1745305 24.307863868 1.0000000
VA-NE 15.53333333 -3.7078639 34.774530535 0.1572874
VT-NE -4.06666667 -23.3078639 15.174530535 1.0000000
WA-NE 19.10000000 -0.1411972 38.341197201 0.0112786
WI-NE -3.80000000 -23.0411972 15.441197201 1.0000000
WV-NE 9.96666667 -9.2745305 29.207863868 0.9608524
WY-NE -1.36666667 -20.6078639 17.874530535 1.0000000
NJ-NH 11.43333333 -7.8078639 30.674530535 0.8160859
NM-NH 9.20000000 -10.0411972 28.441197201 0.9881374
NV-NH 14.80000000 -4.4411972 34.041197201 0.2399005
NY-NH 7.90000000 -11.3411972 27.141197201 0.9993331
OH-NH 7.23333333 -12.0078639 26.474530535 0.9999117
OK-NH 12.13333333 -7.1078639 31.374530535 0.7015694
OR-NH 20.13333333 0.8921361 39.374530535 0.0045717
PA-NH 8.46666667 -10.7745305 27.707863868 0.9972766
RI-NH 8.63333333 -10.6078639 27.874530535 0.9960707
SC-NH 21.03333333 1.7921361 40.274530535 0.0020022
SD-NH -4.16666667 -23.4078639 15.074530535 1.0000000
TN-NH 14.06666667 -5.1745305 33.307863868 0.3476397
TX-NH 24.13333333 4.8921361 43.374530535 0.0000931
UT-NH 4.83333333 -14.4078639 24.074530535 1.0000000
VA-NH 15.30000000 -3.9411972 34.541197201 0.1808481
VT-NH -4.30000000 -23.5411972 14.941197201 1.0000000
WA-NH 18.86666667 -0.3745305 38.107863868 0.0137286
WI-NH -4.03333333 -23.2745305 15.207863868 1.0000000
WV-NH 9.73333333 -9.5078639 28.974530535 0.9718711
WY-NH -1.60000000 -20.8411972 17.641197201 1.0000000
NM-NJ -2.23333333 -21.4745305 17.007863868 1.0000000
NV-NJ 3.36666667 -15.8745305 22.607863868 1.0000000
NY-NJ -3.53333333 -22.7745305 15.707863868 1.0000000
OH-NJ -4.20000000 -23.4411972 15.041197201 1.0000000
OK-NJ 0.70000000 -18.5411972 19.941197201 1.0000000
OR-NJ 8.70000000 -10.5411972 27.941197201 0.9954754
PA-NJ -2.96666667 -22.2078639 16.274530535 1.0000000
RI-NJ -2.80000000 -22.0411972 16.441197201 1.0000000
SC-NJ 9.60000000 -9.6411972 28.841197201 0.9769993
SD-NJ -15.60000000 -34.8411972 3.641197201 0.1510067
TN-NJ 2.63333333 -16.6078639 21.874530535 1.0000000
TX-NJ 12.70000000 -6.5411972 31.941197201 0.5963928
UT-NJ -6.60000000 -25.8411972 12.641197201 0.9999917
VA-NJ 3.86666667 -15.3745305 23.107863868 1.0000000
VT-NJ -15.73333333 -34.9745305 3.507863868 0.1390297
WA-NJ 7.43333333 -11.8078639 26.674530535 0.9998304
WI-NJ -15.46666667 -34.7078639 3.774530535 0.1637665
WV-NJ -1.70000000 -20.9411972 17.541197201 1.0000000
WY-NJ -13.03333333 -32.2745305 6.207863868 0.5329043
NV-NM 5.60000000 -13.6411972 24.841197201 0.9999999
NY-NM -1.30000000 -20.5411972 17.941197201 1.0000000
OH-NM -1.96666667 -21.2078639 17.274530535 1.0000000
OK-NM 2.93333333 -16.3078639 22.174530535 1.0000000
OR-NM 10.93333333 -8.3078639 30.174530535 0.8811182
PA-NM -0.73333333 -19.9745305 18.507863868 1.0000000
RI-NM -0.56666667 -19.8078639 18.674530535 1.0000000
SC-NM 11.83333333 -7.4078639 31.074530535 0.7534555
SD-NM -13.36666667 -32.6078639 5.874530535 0.4702642
TN-NM 4.86666667 -14.3745305 24.107863868 1.0000000
TX-NM 14.93333333 -4.3078639 34.174530535 0.2229925
UT-NM -4.36666667 -23.6078639 14.874530535 1.0000000
VA-NM 6.10000000 -13.1411972 25.341197201 0.9999991
VT-NM -13.50000000 -32.7411972 5.741197201 0.4457807
WA-NM 9.66666667 -9.5745305 28.907863868 0.9745346
WI-NM -13.23333333 -32.4745305 6.007863868 0.4951198
WV-NM 0.53333333 -18.7078639 19.774530535 1.0000000
WY-NM -10.80000000 -30.0411972 8.441197201 0.8957091
NY-NV -6.90000000 -26.1411972 12.341197201 0.9999730
OH-NV -7.56666667 -26.8078639 11.674530535 0.9997438
OK-NV -2.66666667 -21.9078639 16.574530535 1.0000000
OR-NV 5.33333333 -13.9078639 24.574530535 1.0000000
PA-NV -6.33333333 -25.5745305 12.907863868 0.9999974
RI-NV -6.16666667 -25.4078639 13.074530535 0.9999988
SC-NV 6.23333333 -13.0078639 25.474530535 0.9999983
SD-NV -18.96666667 -38.2078639 0.274530535 0.0126238
TN-NV -0.73333333 -19.9745305 18.507863868 1.0000000
TX-NV 9.33333333 -9.9078639 28.574530535 0.9850575
UT-NV -9.96666667 -29.2078639 9.274530535 0.9608524
VA-NV 0.50000000 -18.7411972 19.741197201 1.0000000
VT-NV -19.10000000 -38.3411972 0.141197201 0.0112786
WA-NV 4.06666667 -15.1745305 23.307863868 1.0000000
WI-NV -18.83333333 -38.0745305 0.407863868 0.0141163
WV-NV -5.06666667 -24.3078639 14.174530535 1.0000000
WY-NV -16.40000000 -35.6411972 2.841197201 0.0899896
OH-NY -0.66666667 -19.9078639 18.574530535 1.0000000
OK-NY 4.23333333 -15.0078639 23.474530535 1.0000000
OR-NY 12.23333333 -7.0078639 31.474530535 0.6835500
PA-NY 0.56666667 -18.6745305 19.807863868 1.0000000
RI-NY 0.73333333 -18.5078639 19.974530535 1.0000000
SC-NY 13.13333333 -6.1078639 32.374530535 0.5139523
SD-NY -12.06666667 -31.3078639 7.174530535 0.7133999
TN-NY 6.16666667 -13.0745305 25.407863868 0.9999988
TX-NY 16.23333333 -3.0078639 35.474530535 0.1006599
UT-NY -3.06666667 -22.3078639 16.174530535 1.0000000
VA-NY 7.40000000 -11.8411972 26.641197201 0.9998474
VT-NY -12.20000000 -31.4411972 7.041197201 0.6895908
WA-NY 10.96666667 -8.2745305 30.207863868 0.8772854
WI-NY -11.93333333 -31.1745305 7.307863868 0.7365647
WV-NY 1.83333333 -17.4078639 21.074530535 1.0000000
WY-NY -9.50000000 -28.7411972 9.741197201 0.9803441
OK-OH 4.90000000 -14.3411972 24.141197201 1.0000000
OR-OH 12.90000000 -6.3411972 32.141197201 0.5582869
PA-OH 1.23333333 -18.0078639 20.474530535 1.0000000
RI-OH 1.40000000 -17.8411972 20.641197201 1.0000000
SC-OH 13.80000000 -5.4411972 33.041197201 0.3924661
SD-OH -11.40000000 -30.6411972 7.841197201 0.8209066
TN-OH 6.83333333 -12.4078639 26.074530535 0.9999790
TX-OH 16.90000000 -2.3411972 36.141197201 0.0635003
UT-OH -2.40000000 -21.6411972 16.841197201 1.0000000
VA-OH 8.06666667 -11.1745305 27.307863868 0.9989634
VT-OH -11.53333333 -30.7745305 7.707863868 0.8012360
WA-OH 11.63333333 -7.6078639 30.874530535 0.7858266
WI-OH -11.26666667 -30.5078639 7.974530535 0.8395194
WV-OH 2.50000000 -16.7411972 21.741197201 1.0000000
WY-OH -8.83333333 -28.0745305 10.407863868 0.9940553
OR-OK 8.00000000 -11.2411972 27.241197201 0.9991286
PA-OK -3.66666667 -22.9078639 15.574530535 1.0000000
RI-OK -3.50000000 -22.7411972 15.741197201 1.0000000
SC-OK 8.90000000 -10.3411972 28.141197201 0.9932164
SD-OK -16.30000000 -35.5411972 2.941197201 0.0962725
TN-OK 1.93333333 -17.3078639 21.174530535 1.0000000
TX-OK 12.00000000 -7.2411972 31.241197201 0.7250692
UT-OK -7.30000000 -26.5411972 11.941197201 0.9998897
VA-OK 3.16666667 -16.0745305 22.407863868 1.0000000
VT-OK -16.43333333 -35.6745305 2.807863868 0.0879728
WA-OK 6.73333333 -12.5078639 25.974530535 0.9999858
WI-OK -16.16666667 -35.4078639 3.074530535 0.1052106
WV-OK -2.40000000 -21.6411972 16.841197201 1.0000000
WY-OK -13.73333333 -32.9745305 5.507863868 0.4040725
PA-OR -11.66666667 -30.9078639 7.574530535 0.7805710
RI-OR -11.50000000 -30.7411972 7.741197201 0.8062496
SC-OR 0.90000000 -18.3411972 20.141197201 1.0000000
SD-OR -24.30000000 -43.5411972 -5.058802799 0.0000783
TN-OR -6.06666667 -25.3078639 13.174530535 0.9999992
TX-OR 4.00000000 -15.2411972 23.241197201 1.0000000
UT-OR -15.30000000 -34.5411972 3.941197201 0.1808481
VA-OR -4.83333333 -24.0745305 14.407863868 1.0000000
VT-OR -24.43333333 -43.6745305 -5.192136132 0.0000681
WA-OR -1.26666667 -20.5078639 17.974530535 1.0000000
WI-OR -24.16666667 -43.4078639 -4.925469465 0.0000899
WV-OR -10.40000000 -29.6411972 8.841197201 0.9324101
WY-OR -21.73333333 -40.9745305 -2.492136132 0.0010302
RI-PA 0.16666667 -19.0745305 19.407863868 1.0000000
SC-PA 12.56666667 -6.6745305 31.807863868 0.6216609
SD-PA -12.63333333 -31.8745305 6.607863868 0.6090496
TN-PA 5.60000000 -13.6411972 24.841197201 0.9999999
TX-PA 15.66666667 -3.5745305 34.907863868 0.1449218
UT-PA -3.63333333 -22.8745305 15.607863868 1.0000000
VA-PA 6.83333333 -12.4078639 26.074530535 0.9999790
VT-PA -12.76666667 -32.0078639 6.474530535 0.5837044
WA-PA 10.40000000 -8.8411972 29.641197201 0.9324101
WI-PA -12.50000000 -31.7411972 6.741197201 0.6342129
WV-PA 1.26666667 -17.9745305 20.507863868 1.0000000
WY-PA -10.06666667 -29.3078639 9.174530535 0.9552573
SC-RI 12.40000000 -6.8411972 31.641197201 0.6528992
SD-RI -12.80000000 -32.0411972 6.441197201 0.5773525
TN-RI 5.43333333 -13.8078639 24.674530535 1.0000000
TX-RI 15.50000000 -3.7411972 34.741197201 0.1605020
UT-RI -3.80000000 -23.0411972 15.441197201 1.0000000
VA-RI 6.66666667 -12.5745305 25.907863868 0.9999891
VT-RI -12.93333333 -32.1745305 6.307863868 0.5519339
WA-RI 10.23333333 -9.0078639 29.474530535 0.9446692
WI-RI -12.66666667 -31.9078639 6.574530535 0.6027260
WV-RI 1.10000000 -18.1411972 20.341197201 1.0000000
WY-RI -10.23333333 -29.4745305 9.007863868 0.9446692
SD-SC -25.20000000 -44.4411972 -5.958802799 0.0000304
TN-SC -6.96666667 -26.2078639 12.274530535 0.9999654
TX-SC 3.10000000 -16.1411972 22.341197201 1.0000000
UT-SC -16.20000000 -35.4411972 3.041197201 0.1029146
VA-SC -5.73333333 -24.9745305 13.507863868 0.9999999
VT-SC -25.33333333 -44.5745305 -6.092136132 0.0000263
WA-SC -2.16666667 -21.4078639 17.074530535 1.0000000
WI-SC -25.06666667 -44.3078639 -5.825469465 0.0000350
WV-SC -11.30000000 -30.5411972 7.941197201 0.8349684
WY-SC -22.63333333 -41.8745305 -3.392136132 0.0004272
TN-SD 18.23333333 -1.0078639 37.474530535 0.0230616
TX-SD 28.30000000 9.0588028 47.541197201 0.0000010
UT-SD 9.00000000 -10.2411972 28.241197201 0.9917758
VA-SD 19.46666667 0.2254695 38.707863868 0.0082348
VT-SD -0.13333333 -19.3745305 19.107863868 1.0000000
WA-SD 23.03333333 3.7921361 42.274530535 0.0002864
WI-SD 0.13333333 -19.1078639 19.374530535 1.0000000
WV-SD 13.90000000 -5.3411972 33.141197201 0.3753454
WY-SD 2.56666667 -16.6745305 21.807863868 1.0000000
TX-TN 10.06666667 -9.1745305 29.307863868 0.9552573
UT-TN -9.23333333 -28.4745305 10.007863868 0.9874204
VA-TN 1.23333333 -18.0078639 20.474530535 1.0000000
VT-TN -18.36666667 -37.6078639 0.874530535 0.0207147
WA-TN 4.80000000 -14.4411972 24.041197201 1.0000000
WI-TN -18.10000000 -37.3411972 1.141197201 0.0256479
WV-TN -4.33333333 -23.5745305 14.907863868 1.0000000
WY-TN -15.66666667 -34.9078639 3.574530535 0.1449218
UT-TX -19.30000000 -38.5411972 -0.058802799 0.0095084
VA-TX -8.83333333 -28.0745305 10.407863868 0.9940553
VT-TX -28.43333333 -47.6745305 -9.192136132 0.0000009
WA-TX -5.26666667 -24.5078639 13.974530535 1.0000000
WI-TX -28.16666667 -47.4078639 -8.925469465 0.0000012
WV-TX -14.40000000 -33.6411972 4.841197201 0.2956591
WY-TX -25.73333333 -44.9745305 -6.492136132 0.0000172
VA-UT 10.46666667 -8.7745305 29.707863868 0.9270184
VT-UT -9.13333333 -28.3745305 10.107863868 0.9894722
WA-UT 14.03333333 -5.2078639 33.274530535 0.3530942
WI-UT -8.86666667 -28.1078639 10.374530535 0.9936474
WV-UT 4.90000000 -14.3411972 24.141197201 1.0000000
WY-UT -6.43333333 -25.6745305 12.807863868 0.9999959
VT-VA -19.60000000 -38.8411972 -0.358802799 0.0073328
WA-VA 3.56666667 -15.6745305 22.807863868 1.0000000
WI-VA -19.33333333 -38.5745305 -0.092136132 0.0092399
WV-VA -5.56666667 -24.8078639 13.674530535 0.9999999
WY-VA -16.90000000 -36.1411972 2.341197201 0.0635003
WA-VT 23.16666667 3.9254695 42.407863868 0.0002504
WI-VT 0.26666667 -18.9745305 19.507863868 1.0000000
WV-VT 14.03333333 -5.2078639 33.274530535 0.3530942
WY-VT 2.70000000 -16.5411972 21.941197201 1.0000000
WI-WA -22.90000000 -42.1411972 -3.658802799 0.0003275
WV-WA -9.13333333 -28.3745305 10.107863868 0.9894722
WY-WA -20.46666667 -39.7078639 -1.225469465 0.0033807
WV-WI 13.76666667 -5.4745305 33.007863868 0.3982507
WY-WI 2.43333333 -16.8078639 21.674530535 1.0000000
WY-WV -11.33333333 -30.5745305 7.907863868 0.8303487

In case you were wondering how many pairs are being analyzed, it’s 2450. How did I figure that out? Well, each state is paired up with every other state, so that makes for 49 possible pairings for each state, since states aren’t paired up with themselves. There are also 50 states with pairings, so multiply 49 by 50 to get 2450 pairings.

This table shows the pair-wise differences in average January low temperatures amongst pairs of states. The lwr and upr columns show the lower and upper 99% confidence bounds, respectively, for mean temperature differences between two states. The p adj column gives the p-values for each state pair adjusted for the number of comparisons made. Any p-value that is less than 0.01 means that the pair-wise difference is statistically significant. Some notably significant pair-wise differences include (rounded to two decimal places):

  • AZ-AK, Arizona-Alaska, 42.17
    • Makes sense. After all, Arizona is a dry desert, while Alaska is very close to the Arctic Circle
  • VT-HI, Vermont-Hawaii, -53.53
    • Vermont is in New England, which is known for cold winters, while Hawaii is made up of several islands that are situated on active volcanoes.
  • OR-MN, Oregon-Minnesota, 26.67
    • This one I found interesting, since both Oregon and Minnesota can get very cold winters, but if you live in Oregon, you’re likelier to experience rougher winters if you live west of the Cascade Mountains. Minnesota (and most of the Midwest for that matter) are known for their brutal winters.
  • LA-HI, Louisiana-Hawaii, -21.97
    • Another interesting observation, since neither state recieves plenty of snow (or gets very cold for that matter). Then again, Lousiana has lots of bayous, while Hawaii consists of several islands sitting on active volcanoes.

Thanks for reading, and stay tuned for my post on two-way ANOVA,

Michael

R Analysis 5: Random Forests, Decision Trees and Merit Badges

Hello everybody,

Its Michael, and today’s post will be an R analysis about Boy Scout merit badges earned in 2018 utilizing random forests and decision trees. I thought this would be a fun analysis to do since I am an Eagle Scout (Class of ’14).  Also, since I haven’t done so yet, I will show you a regression random forest and a regression decision tree in this analysis.

Here’s the dataset-2018 merit badges.

As always, let’s load our file into R and try to understand our data:

Screen Shot 2019-06-15 at 2.46.40 PM

So as you can see, we’ve got 137 observations of 7 variables. Here’s a variable-by-variable breakdown:

  • Rank-These badges are ranked based on how many Scouts earned them; the ranks go from 1 (most earned badge) to 137 (least earned badge)
    • The original dataset I found on ScoutingMagazine.org had 139 observations, but since the bottom 2 had discontinued or rebranded merit badges (Computers and Cinematography), I removed those two observations from the dataset
  • Merit.Badge-The name of the merit badge
  • X2018.earned-How many Scouts earned that particular badge
  • Year.Introduced-The year the badge was introduced into the BSA
  • Requirements.Revision-The year of the most recent requirements revision for the badge
  • Change.from.17-The percentage change of Scouts who earned a particular merit badge from 2017 to 2018
    • I mentioned that this variable was a percentage change, however I had to list these values as decimals so R would read them as num and not factor.
  • Eagle.Required-Whether or not a badge is required for the Eagle Scout rank; the two values are “Yes” and “No”
    • Just so you know, there are 21 merit badges required for the Eagle Scout rank.

Now let’s set up our model (remember to install the randomForest package). Since the model is fairly small-with only 137 observations-there’s no need to split the data into training and validation sets. So, here’s the model along with the corresponding summary:

Screen Shot 2019-06-15 at 2.56.14 PM

In this model, I used X2018.earned as my dependent variable and Year.Introduced and Rank as my independent variables. In other words, I am attempting to figure out whether the amount of Scouts that earned a particular merit badge has to do with when the badge was introduced into the BSA and/or the badge’s rank of popularity. Since my dependent variable is non-binary, I will be performing a linear regression.

Now, you will see two unfamiliar terms at the bottom of the output. Here’s an explanation of each term:

  • The mean of squared residuals is the average of the squares of the residuals.
    • In regression analysis (linear & logistic), residuals are the differences between the observed value and the predicted value of the dependent variable-X2018.earned.
  • The % var explained is a measure  of how well out-of-bag predictions explain the variance of the training set; a low % var explained could be due to either true random behavior or a lack of fit. Our % var explained-97.07%-is high, which indicates plenty of fit in the model.

Now, you might recall that when dealing with classification random forests, we create several classification matrices. However, when dealing with regression random forests, we create a graph of the random forest using the handy ggplot2 package (which you must remember to install):

Screen Shot 2019-06-15 at 3.43.07 PM

And here’s the actual graph:

Screen Shot 2019-06-15 at 3.42.50 PM

OK, so what’s the deal with the actual and predicted columns? Both of these columns demonstrate the concepts of residuals. Remember that earlier in this post, I mentioned that residuals are the differences between the actual and predicted values of the dependent variable-X2018.earned.

This graph shows all of the predicted and actualvalues of X2018.earned. In case you were wondering, here are all of the predicted values of X2018.earned in order of their appearance on the dataset. Since I sorted all the badges (and other information in turn) alphabetically, the first number-2073.128-corresponds to the American Business merit badge, while each subsequent number corresponds to the next merit badge alphabetically. The last number-5492.116-corresponds to the alphabetically last merit badge-Woodworking:

Screen Shot 2019-06-15 at 4.06.52 PM

  • In case you’re wondering how I got this output of numbers, I wrote the line file$predicted <- predict(model1) and then typed in file$predicted to get this output.

But that’s not all. See, we’re then going to take the predicted values and place them onto two separate graphs-one with Rank and the other with Year.Introduced. The predicted values will be the y-axis in both graphs, while the dependent variables will each serve as the x-axes. The whole point of making these graphs is to see how well the predicted values correlate with each of the independent variables.

First, here’s the code and the graph for Year.Introduced:

Screen Shot 2019-06-16 at 12.09.54 PMScreen Shot 2019-06-16 at 12.09.34 PM

From this graph, you can see there is a small negative correlation between the predicted values and Year.Introduced. This implies that, even with all of the new merit badges that have been introduced in the last decade (e.g. Robotics, Game Design, etc.), many Scouts still went for the traditional merit badges (e.g. Archery, First Aid, Cycling) that have been around since the BSA’s founding in 1910. Most of the traditional badges, such as Swimming and Cooking, are Eagle-required badges, which could explain why plenty of Scouts earned them in 2018.

  • Granted, many of the traditional badges had been around since 1910, but their Year.Introduced is 1911, since that’s the year the first Scout handbook came out.

Want to figure out the exact correlation between Year.Introduced and the predicted values? Here’s how:

Screen Shot 2019-06-16 at 11.55.59 AM

The number -0.1470237 indicates that there is some negative correlation between the predicted values and Year.Introduced. This means that the more recently a badge was introduced, the fewer scouts earned it-SOMETIMES. There are exceptions to this rule. For instance, twice as many Scouts earned the Chess badge-which was introduced in 2011-than the Aviation badge-which was introduced 59 years earlier. And here’s a more interesting example-four times as many Scouts earned the Exploration badge (which was just introduced in 2017) than the Stamp Collecting badge (which has been around since 1932).

Now here’s the code and the graph for Rank:

Screen Shot 2019-06-16 at 12.11.49 PM

Screen Shot 2019-06-16 at 12.11.34 PM

This graph shows that Rank has a much stronger correlation than Year.Introducedwith the predicted values. This makes perfect sense; after all, the higher a certain badge’s popularity ranking is, the more Scouts earned that badge in 2018, and vice versa.

Want to know the exact correlation? Check this out:

Screen Shot 2019-06-16 at 12.17.43 PM

Just as with Year.Introduced, there is also a negative correlation. However, the number -0.8173855 implies a much stronger inverse correlation (stronger than the correlation with Year.Introduced).

Now let’s make some regression decision trees (remember to install the rpart package):

Screen Shot 2019-06-16 at 12.29.41 PM

The formula used to create the tree is the same one you would use for classification decision trees, except use anova for the method.

  • ANOVA stands for analysis of variance, which is a type of regression that I will likely cover in a future R post.

You’ll recognize the pruning table at the bottom of the printcp(tree) output. Remember that the optimal place to prune the tree would be at the lowest level where  rel error-xstd < xerror.

Now let’s plot the pre-pruned tree (remember to keep the window with the tree open when you are writing the text line):

Screen Shot 2019-06-16 at 12.43.07 PMScreen Shot 2019-06-16 at 12.42.52 PM

In this tree, only Rank was used, even though Year.Introduced was likely factored into the tree’s construction (though I’m not certain that it was). By the way, I’ll round up on all of the ranks.

Now, here’s a breakdown of the tree:

  • The top row contains all 137 observations and any badge ranked 22 or lower.
  • The top row splits into two branches:
    • The left branch contains badges ranked 55 or lower and 116 observations.
    • The right branch contains badges ranked 12 or lower and 21 observations.
  • The branch with the badges ranked 55 or higher is further split into two branches (this branch has 116 observations):
    • The left branch contains badges ranked 95 or lower and 83 observations.
      • This branch further splits into two terminal nodes, each containing the number of Scouts and the observations in the group.
        • The left node contains 2,587 Scouts and 43 merit badges.
        • The right node contains 6,563 Scouts and 40 merit badges.
    • The right branch contains badges ranked 31 or lower and 33 observations.
      • The right branch also further splits into two terminal nodes, each containing the number of Scouts and observations in the group.
        • The left node contains 1,287 Scouts and 24 merit badges.
        • The right node contains 2,083 scouts and 9 merit badges.
  • The branch containing badges ranked 12 or lower (and with 21 observations) splits into two terminal nodes.
    • The left node contains 3,424 Scouts and 9 merit badges.
    • The right node contains 5,364 Scouts and 12 merit badges.

So here’s a summary of all the groups in the tree, going from left to right:

  • Group 1 (the leftmost) contains 2,587 Scouts and 43 merit badges.
  • Group 2 contains 6,563 Scouts and 40 merit badges.
  • Group 3 contains 1,287 Scouts and 24 merit badges.
  • Group 4 contains 2,083 Scouts and 9 merit badges.
  • Group 5 contains 3,424 Scouts and 9 merit badges.
  • Group 6 (the rightmost) contains 5,364 Scouts and 12 merit badges.

Now let’s see if our tree needs pruning:

Screen Shot 2019-06-16 at 2.56.43 PM

Screen Shot 2019-06-16 at 2.56.34 PM

And here’s the pruned tree:

Screen Shot 2019-06-16 at 2.56.13 PM

So as it turns out, this tree did need to be pruned. As you can see, there are only 3 terminal nodes now as opposed to the six that were in the pre-pruned tree. The top branch is still the same-all of the merit badges and ranking 22 or lower. The top branch’s splits are also the same-116 badges and ranking 55 or lower on the left, 21 badges on the right.

However, that’s where the similarities end. Unlike the pre-pruned tree, the right split from the top branch is a terminal node; this node contains 4,532 Scouts and 21 merit badges and doesn’t split any further. The left split, on the other hand, splits into two branches (just as it does in the pre-pruned tree), but unlike the pre-pruned tree, those branches are terminal nodes. Those two terminal nodes contain 83 badges/4,503 Scouts on the left split and 33 badges/1,504 Scouts on the right split.

So here’s a breakdown of each group in this tree:

  • Group 1 (left)-4,532 Scouts and 21 merit badges
  • Group 2 (center)-4,503 Scouts and 83 merit badges
  • Group 3 (right)-1,504 Scouts and 33 merit badges

How did I know where to prune the tree? Here’s how:

Screen Shot 2019-06-16 at 3.24.10 PM

When I typed in plotcp(tree), I got this handy little diagram, which is a visual summary of the output I got with printcp(tree). The area I circled, which corresponds to Node 3, is the optimal place to prune the tree, since it’s the one closest above the dashed line (which I’ll call the pruning line). You might think Node 4 would have been a better place to prune the tree, but since it touches the pruning line, that would not have a good place to prune the tree.

So after determining the best place to prune the tree, I look up the CP for Node 3-0.061173-and use that for the pfit <- prune(tree, cp=0.061173) line (this is the line that prunes the tree-the CP tells it where to prune).

Thanks for reading,

Michael

R Lesson 16: R Loops

Hello everybody,

It’s Michael, and today’s lesson will be on loops in R. I’ve posted about loops before, but those were for my Java lessons. However, while there may be some differences between Java loops and R loops, the basic ideas are the same. In fact, two of the three main types of Java loops-while and for-are also the two main types of R loops. The do-while loop doesn’t exist in R…..at least not yet. In its place, R has something called the repeatloop.

Let’s demonstrate the forloop first:

Screen Shot 2019-06-02 at 1.02.22 PM

In this for loop, the factorials for the numbers 1-10 are printed. The i is a common symbol in R loops-it represents the indexes (or elements) of a loop. The i will traverse through all 10 elements of the loop-in this case, the numbers 1 through 10-and print out the factorials of each number along with the corresponding text-The factorial for , i , is, factorial(i).

  • In case you weren’t aware, the factorial of a number is the product of that number and all numbers below it (stopping at 1). So 4 factorial would be 24, since 4*3*2*1 equals 24. Factorials are denoted with an exclamation point by the number, so 4 factorial would be written as 4!. You might have come across this concept in your algebra class.

Now let’s take a look at the differences in syntax between a Java for loop and an R for loop (using the same factorial output)

R for loop:

for (i in 1:10)

{print(paste(“The factorial for” , i , “is” , factorial(i)))}

Java for loop:

int fac = 1;

for (int i = 1; i <= 10; i++)

{

fac *= i;

System.out.println(“The factorial of  ” + i + ” is ” + fac );

}

So, where are all of the differences? Let me name 5:

  1. The commands to print the output are different. Java requires System.out.println (or System.out.print). R requires print(paste()).
  2. Java needs 6 lines of code to do what R can accomplish in two lines-which is to calculate the factorials of the numbers 1-10 and print the specified output.
  3. The way to concatenate strings and numbers is different-Java requires a plus sign while R uses a comma
  4. You will often need to declare a variable outside the loop in Java (like I did with fac = 1). That is usually not necessary in R.
  5. The forloop parameters in R are much simpler than those in Java. Here’s how:
    • The Java for loop parameters go (initializing condition; condition to stop the loop; action to perform after each iteration)
    • The R for loop parameters go (index variable in starting value:ending value). The starting value:ending value part tells the loop the startpoint and endpoint.

Next, let’s demonstrate a nested for loop. Just like in Java, you can make nested for loops:

Screen Shot 2019-06-02 at 3.45.40 PM

And here’s the output (it was too large to capture in a single screenshot):

[1] “I times J is 6”
[1] “I times J is 8”
[1] “I times J is 10”
[1] “I times J is 12”
[1] “I times J is 14”
[1] “I times J is 16”
[1] “I times J is 18”
[1] “I times J is 9”
[1] “I times J is 12”
[1] “I times J is 15”
[1] “I times J is 18”
[1] “I times J is 21”
[1] “I times J is 24”
[1] “I times J is 27”
[1] “I times J is 12”
[1] “I times J is 16”
[1] “I times J is 20”
[1] “I times J is 24”
[1] “I times J is 28”
[1] “I times J is 32”
[1] “I times J is 36”
[1] “I times J is 15”
[1] “I times J is 20”
[1] “I times J is 25”
[1] “I times J is 30”
[1] “I times J is 35”
[1] “I times J is 40”
[1] “I times J is 45”
[1] “I times J is 18”
[1] “I times J is 24”
[1] “I times J is 30”
[1] “I times J is 36”
[1] “I times J is 42”
[1] “I times J is 48”
[1] “I times J is 54”
[1] “I times J is 21”
[1] “I times J is 28”
[1] “I times J is 35”
[1] “I times J is 42”
[1] “I times J is 49”
[1] “I times J is 56”
[1] “I times J is 63”
[1] “I times J is 24”
[1] “I times J is 32”
[1] “I times J is 40”
[1] “I times J is 48”
[1] “I times J is 56”
[1] “I times J is 64”
[1] “I times J is 72”

In this nested forloop, I have two important segments. The I segment traverses through the numbers 2-8 while the J segment traverses through the numbers 3-9. However, the statements on their own are useless. The only line that really matters is the output line-"I times J is" , i*j. This line will print out the product of i and j.

OK, so that isn’t as simple as it looks. That is because the loop will traverse through every element of the I and J statements. In other words, the products of every element in the I statement and every element in the J statement will be displayed. For instance, the answer to 2*3 will be displayed, followed by 2*4, 2*5, all the way up to 2*9. Then all of the products from 3*3 to 3*9 will be displayed, and so forth, until you hit the 8 range (meaning from 8*3 to 8*9).

Now let’s create a while loop:

Screen Shot 2019-06-03 at 3.43.39 PM

And here is the output (it was too long to fit on the same screenshot as the loop):

Screen Shot 2019-06-03 at 3.43.55 PM

In this loop, I use 1902 as a starting point and 2018 as a stopping point. For each iteration, num will increase by 4 (starting from 1902) until 2018 is reached. In case you are wondering where I’m going with this, I’m printing out every even-numbered non-leap year between 1902 and 2018 inclusive.

  • Unfortunately, R doesn’t recognize increment/decrement symbols like ++, --, +=, -=, /=, *= the way Java does. So you’ll just have to increment/decrement in long form, as I did with the num <- num + 4 line.

Unlike for loops between Java and R, while loops between Java and R are very similar. Here’s the R loop for this program:

num <- 1902

while (num <= 2018)

{

print (num)

num <- num + 4

}

And here’s a Java whileloop that will do the exact same thing as the R loop:

num = 1902;

while (num <= 2018)

{

System.out.println(num)

num += 4;

}

One of the similarities between the two while loops are the parameters of the loop. In both loops, the lone parameter is what I’d like to call the running condition, which is the condition that tells the loop that while that condition is true, keep the loop running. In both cases, the running condition is num <= 2018.

The differences between the two loops are related to the difference in functions between Java and R. For instance, Java output requires System.out.printlnwhile R output simply requires print(or print(paste()) for outputs with strings). The other main difference is how to denote incrementing/decrementing in the loop, which I already discussed earlier in this post.

The third type of R loop is the repeat; this is the loop that is exclusive to R (just as the do-while loop is exclusive to Java). In R, a repeat loop is similar to a while loop in the sense that both loops have a stop condition (which is the condition that stops the loop from running). However, in while loops, the stop condition also functions as the running condition. For instance, the statement num <= 2018acts as both a running condition and a stop condition, since it tells the loop to keep running as long as num is less than or equal to 2018 but it also tells the loop to stop running as soon as 2018 (or the closest number to it) is reached.

Repeat loops don’t have a running condition per se, since they can be infinite. As a result, including a stop condition is totally optional. However, if you wish to include a stop condition, write an ifstatement (they’re formatted just like Java if statements) ending with the word break, which indicates that the loop will stop if the if statement condition is met.

Here’s an example of a repeat loop with a breakclause:

Screen Shot 2019-06-03 at 10.34.01 PM

In this loop, I start with num <- 5 just outside the loop. Once I get inside the loop, numkeeps getting multiplied by 2 until a number that is either greater than or equal to 1000 is reached. All of the products of num * 2 are printed, and the last output is 1280, which is 640*2.

  • Had I not included a breakstatement, this loop would have gone on and on. Inf would be displayed after a certain point to denote that the loop would be infinite.

Thanks for reading,

Michael

R Lesson 15: Decision Trees

Hello everybody,

It’s Michael, and today’s lesson will be about how to create decision trees in R. I briefly explained the concept of decision trees in my previous post. This post serves as the second part of a two-part lesson (with R Lesson 14 being the first part); I will be using the same dataset.

Now I know you’ve learned the basic concept of a decision tree from my previous post, so I’ll start off this post by going further in depth with more decision tree terminology.

Here’s an example of the decision tree (it’s not the one you saw in my previous post):

Decision Tree

The uppermost question-“Income range of applicant?”-is the root node of the tree, which represents the entire population/sample. Splitting the tree involves dividing a node (ovals represent the nodes here) into sub-nodes (represented by the other questions in this tree); the opposite of splitting is pruning, which is when sub-nodes are removed. When sub-nodes split into further sub-nodes, they are referred to as decision nodes (but you can use the terms sub-node and decision node interchangeably). Also, when dealing with sub-nodes, the main node is called the parent node while the sub-node is called the child node. Nodes can be both a parent and child node; for instance, the “Years in present job?” node is the child node of the “Income range of applicant?” node but it is the parent node of the “Makes credit card payments?” node. Nodes that don’t split are called terminal nodes, or leaves (the ovals with loan and no loan are terminal nodes in this tree).

There are two main types of decision trees-regression and classification (which are also the two types of random forest models). Regression trees are more appropriate for QUANTITATIVE problems (such as the percentage likelihood that something will or wont happen) while classification trees are more appropriate for QUALITATIVE problems (such as Yes/No questions). Since the models we made in the previous post are classification problems, then we will learn how to build classification trees.

  • An easy way to remember how decision trees work is to think of them as questions that need more questions to be answered.

To create our decision tree, let’s start by installing a package called rpart.  Then let’s use the printcp function to display a summary of the tree:

Screen Shot 2019-05-24 at 3.46.46 PM

To create the classification tree, use the `Convicting.Offense.Type` variable (and only this variable) along with `trainSet` as your data. Since we are creating a classification tree, use `class` as the method; if we were creating a regression tree, use `anova`for the method.

Now what do all the unfamiliar terms in the output actually mean? The first term, root node error, sounds like a misnomer since it refers to percentage of correctly sorted records at the first root or split. The higher this number is, the better. Here, the root node error is 69.3%, which is OK, but we should try to figure out how to increase this number to at least 75%.

The other part of the output is the pruning table that you can see at the bottom of the output (that six-by-five matrix is referred to as a pruning table). The numbers 1-5 refer to each of the five values of Convicting.Offense.TypeDrug, Property, Violent, Public Order, and Other. The CP refers to the complexity parameter, which is a metric that is used to control the size of the tree and select the optimal tree size. It decreases with each successive level (the numbers 1-5 are referred to as levels), and tree building stops when the CP reaches 0. nsplit refers to the amount of splits at that level of the tree; in this case, the number of splits equals the level minus 1 (e.g. the 1st level has no splits, while the 2nd level has 1 split). However, that won’t always be the case, since the 2nd level of a tree might have more than 1 split sometimes.

The `relerror` for each level is 1 minus the root mean squared error (or the mean squared error, which is the square of the RMSE); this is the error for predictions of the data that were used to estimate the model-`trainSet` in this case. The lower the `relerror`, the better.

Next is the xerror, which is the cross-validated error rate for each level. Cross-validation is the process of testing the results of the model on an independent, unseen dataset in order to avoid overfitting or underfitting the model, as well as to analyze how well the data will fit with an independent data set. The xerror is used to measure the error generated from the cross-validation process; the lower the `xerror` the better. In this case, all of the `xerror` values are equal to all of the `relerror` values, but that won’t always be the case.

Last is the xtsd, which represents the standard deviation, which is a measure of how much observations in a certain level are spread out from one another. The smaller the number, the less spread out the observations are from each other.

The graph below is simply a visual representation of the matrix I just described:

Screen Shot 2019-05-24 at 4.18.32 PM

Screen Shot 2019-05-24 at 3.46.26 PM

So, what do we do next? We try to find the best place to prune the tree. If you don’t know, pruning means to remove nodes from the tree that aren’t significant to the model. It’s similar to the concept of pruning your tree, since you would be removing unnecessary branches from the tree.

And how do we find the best place to prune the tree? A rule of thumb is to find the lowest level where rel error-xstd < xerror. But, since this is an unusual case where all of the rel error equals all of the xerror, I’ll let R decide where to prune the tree (if doing so is necessary)

But before I do that, let me plot a pre-pruned decision tree:

Screen Shot 2019-05-27 at 4.21.02 PMScreen Shot 2019-05-27 at 4.20.44 PM

The first thing you would do is to use the `plot` function (along with any parameters) to plot the tree then use the `text` function (along with any parameters) to place the text on the tree. Remember-KEEP the window with the decision tree open when calling your `text` function.

  • The `margin` parameter is completely optional; it’s just a good idea since the default tree tends to cut off some of the text. A margin of 0.25 or 0.3 is ideal.
  • Even though I didn’t use Convicting.Offense.Subtype in the model, it is included in the tree since the tree couldn’t be created with only one variable (plus `Convicting.Offense.Subtype` and `Convicting.Offense.Type` are closely related to one another).

Now, you might be wondering about all the letters on the tree right by Convicting.Offense.Subtype. Here’s the thing-the actual values of this variable are too long to show on the tree, and there are 26 possible values, so it wouldn’t have been practical to create 26 nodes (one for each value). Since there are 26 possible values, each value is represented by a letter of the alphabet. Each of the values are represented by a letter of the alphabet; the values and their corresponding letters are listed alphabetically. Here’s the list of values along with the letters they correspond to:

  • A-alcohol (I’m guessing public intoxication is a part of this)
  • B-animals (namely animal cruelty offenses)
  • C-arson
  • D-assault
  • E-burglary
  • F-drug possession
  • G-flight/escape (probably referring to prison escapes but could also mean offenders who flee town before trial/sentencing)
  • H-forgery/fraud
  • I-kidnap
  • J-murder/manslaughter (not sure why these are grouped together, since murder would suggest malicious intent while manslaughter is mostly negligence)
  • K-other criminal (meaning some other criminal offense that won’t fit into the other 25 types)
  • L-other drug (any other drug offense that isn’t possession, so sale of drug paraphernalia would count towards this)
  • M-other public order (so something like indecent exposure)
  • N-other violent (other violent crimes that won’t fall into any of the other mentioned subtypes)
  • O-OWI (operating while intoxicated, the legalese term for a DUI)
  • P-prostitution/pimping
  • Q-robbery
  • R-sex (possibly sexual assault, maybe human trafficking)
  • S-sex offender registry/residency (sex crimes against kids)
  • T-special sentence revocation (usually for those who violate probation/parole)
  • U-stolen property
  • V-theft (I guess this means both petty theft and grand theft)
  • W-traffic (any other traffic offense that is not a DUI/OWI)
  • X-trafficking (I think this means drug trafficking)
  • Y-vandalism (graffiti, destruction of property, etc.)
  • Z-weapons (probably illegal possession of weapons, i.e. convicted felon possessing a gun)

So, knowing what each of the letters refer to, how do we read this tree? First, let’s look at the uppermost node. It starts with offense sub-types F, L, and X (which are all drug-related offenses) and has the type of offense-Drug-on the bottom of the Convicting.Offense.Subtype line. The string of numbers you see after the word Drug5982/974/5490/2706/4363-refer to how many observations belong in each level. Since this string of numbers is on the root node, all 19,515 observations in trainSet are considered. As we move down the tree, you’ll start to see more 0s in each section and as a result, you’ll figure out exactly how many observations are in each level.

So, the root node splits into two branches. The left branch has the word Drug and the number string 5982/0/0/0/0 on it. Since this branch doesn’t split any further, we can conclude that 5,982 of the 19,515 offenses are drug crimes. The right branch contains offense sub-types C, E, H, U, V, and Y, which are all property-related offenses, along with the number string 0/974/5490/2706/4363, which refers to all 13,533 of the offenses that aren’t drug-related.

The Property branch (the right branch from the Drug branch split) also splits in two. The left branch contains the word Property and the number string 0/0/5490/0/0. Since this branch doesn’t split any further, we can conclude that 5,490 of the 19,515 offenses are property crimes.  The right branch contains offense sub-types A, B, G, K, M, O, P, S, T, W, and Z, which are all violent crimes, and the number string 0/974/0/2706/4363, which refers to the 8,043 crimes that aren’t drug or property related.

The Violent branch (the right branch from the Property branch split) also splits in two, just like the previous two branches. The right branch has the word Violent and the number string 0/37/0/0/4363, which indicates that 4,400 (4363+37) of the offenses are violent crimes. The left branch has the word Public Order and the offense sub-types B, K, and T, which are public order offenses, along with the number string 0/937/0/2706/0, which refers to the 3,643 observations that haven’t yet been classified. If you’re wondering how 937 was derived, the Violent branch had the number string 0/974/0/2706/4363. When that branch split in two, 37 of the 974 observations went into the right split while the other 937 went into the left split.

The Public Order branch (with the number string 0/937/0/2706/0) also splits into two. The left branch has the word Other and the number string 0/937/0/0/0, meaning that 937 of the offenses are miscellaneous crimes that don’t fit into the other 4 categories and the right branch has the number string 0/0/0/2706/0, meaning that 2,706 of the offenses are public order crimes (i.e. prostitution).

At this point, all observations have been classified. Here’s a breakdown of the results:

  • 5,982 Drug crimes
  • 5,490 Property crimes
  • 937 Other crimes
  • 2,706 Public Order crimes
  • 4,400 Violent crimes

Now, we will let R do the magic of pruning the tree (if R finds it necessary to do so). Here’s the code that you need to prune the tree (remember to not close the window with the tree when you are writing the text line):

Screen Shot 2019-05-29 at 3.07.18 PM

And here’s the tree. Turns out, R didn’t think it was necessary to prune the tree, so we got the exact same tree. Screen Shot 2019-05-29 at 3.06.48 PM

How come R didn’t prune the tree. Well, let’s analyze this graph from earlier:

Screen Shot 2019-05-24 at 3.46.26 PM

Notice the dashed horizontal line. See how it meets the curve at the 5th level. Well, that line gives us an idea of where the tree should be pruned. Since it’s at the 5th level, the tree won’t be pruned, as all 5 levels are significant. This could also be because all of the rel error values are equal to all of the xerror values.

Thanks for reading,

Michael

R Lesson 14: Random Forests

Hello everybody,

It’s Michael, and today’s post will be on random forests in R.

You’re probably wondering what a random forest does? Me too, since I’ve never actually done any random forest problems (the other R lessons covered concepts from my undergrad business analytics coursework, so I was familiar with them).

Random forests are a form of supervised learning (look at R Lesson 10: Intro to Machine Learning-Supervised and Unsupervised for an explanation on supervised and unsupervised learning). But to better explain the concept of random forests, let me provide an easy to understand example.

Let’s say you wanted to visit the new ice cream shop in town, but aren’t sure if its any good. Would you go in and visit the place anyway, regardless of any doubts you have? It’s unlikely, since you’d probably want to ask all of your buddies and/or relatives about their thoughts on the new ice cream place. You would ideally ask multiple people for their opinions, since the opinion of one person is usually biased based on his/her preferences. Therefore, by asking several people about their opinions, we can eliminate bias in our decision making process as much as possible. After all, one person might not like the ice cream shop in question because they weren’t satisfied with the service, or. they don’t like the shop’s options, or a myriad of other reasons.

In data analytics jargon, the above paragraph provides a great example of a technique called ensembling, which is a type of supervised learning technique where several models are trained on a training set with their individual outputs combined by a set rule to determine the final output.

Now, what does all of that mean? First of all, when I mentioned that several models are trained on a training set, this means that either the same model with different parameters or different models can utilize the training set.

When I mentioned that their [referring to the “several models”] individual outputs are combined by a set rule to determine the final output, this means that the final output is derived by using a certain rule to combine all other outputs. There are plenty of rules that can be utilized, but the two most common are averages (in terms of numerical output) and votes (in terms of non-numerical output). In the case of numerical output, we can take the average of all the outputs and use that average as the final output. In the case of categorial (or non–numerical) output, we can use vote (or the output that occurs the most times) as the final output.

Random forests are an example of an ensembling algorithm. Random forests work by creating several decision trees (which I’ll explain a little more) and combining their output to derive a final output. Decision trees are easy to understand models, however they don’t have much predictive power, which is why they have been referred to as weak learners.

Here’s a basic example of a decision tree:

Decision-Trees-modified-1

This tree isn’t filled with analytical jargon, but the idea is the same. See, this tree answers a question-“Is a person fit?”-just as decision trees do. It then starts with a basic question on the top branch-“Is the person younger than 30?” Depending on the answer to that question (Yes/No), another question is then posed, which can be either “Eats a lot of pizzas?” or “Exercises in the morning?”. And depending on the answer to either question (both are Yes/No questions), you get the answer to the question the tree is trying to answer-“Is a person fit?”. For instance, if the answer to “Eats a lot of pizzas?” is “Yes”, then the answer to the question “Is a person fit?” would be “No”.

This tree is a very simple example of decision trees, but the basic idea is the same with R decision trees (which I will show you more of in the next post).

Now let’s get into the dataset-Iowa Recidivism-which gives data on people serving a prison term in the state of Iowa between 2010 and 2015, with recidivism follow–up between 2013 and 2018. If you are wondering, I got this dataset from Kaggle.

Now, let’s start our analysis by first understanding our data:

Screen Shot 2019-05-15 at 3.00.29 PM

Here, we have 26020 observations of 12 variables, which is a lot, but lets break it down variable-by-variable:

  • Fiscal.Year.Released-the year the person was released from prison/jail
  • Recidivism.Reporting.Year-the year recidivism follow-up was conducted, which is 3 years after the person was released
  • Race..Ethnicity-the race/ethnicity of the person who was released
  • Age.At.Release-the age a person was when he/she was released. Exact ages aren’t given; instead, age brackets are mentioned, of which there are five. They include:
    • Under 25
    • 25-34
    • 35-44
    • 45-54
    • 55 and over
  •  Convicting.Offense.Classification-The level of seriousness of the crime, which often dictates the length of the sentence. They include:
    • [Class] A Felony-Up to life in prison
    • [Class] B Felony-25 to 50 years in prison
    • [Class] C Felony-Up to 10 years in prison
    • [Class] D Felony-Up to 5 years in prison
    • Aggravated Misdemeanor-Up to 2 years in prison
    • Serious Misdemeanor-Up to 1 year in prison
    • Simple Misdemeanor-Up to 30 days in prison
  • Convicting.Offense.Type-the general category of the crime the person was convicted of, which can include:
    • Drug-drug crimes (anything from narcotics trafficking to possession of less than an ounce of marijuana)
    • Violent-any violent crime (murder, sexual assault, etc.)
    • Property-any crimes against property (like robbery/burglary)
    • Public order-usually victimless and non-violent crimes that go against societal norms (prostitution, public drunkenness, etc.)
    • Other-any other crimes that don’t fit the four categories mentioned above (like any white collar crimes)
  • Convicting.Offense.Subtype-the more specific category of the crime the person was convicted of, which can include “animal” (for animal cruelty crimes), “trafficking” (which I’m guessing refers to human trafficking), among others.
  • Main.Supervising.District-the jurisdiction supervising the offender during the 3 year period (Iowa has 11 judicial districts)
  • Release.Type-why the person was released, which can include parole, among other things
  • Release.type..Paroled.to.detainder.united-this is pretty much the same as Release.Type; I won’t be using this variable in my analysis
  • Part.of.Target.Population-whether or not a prisoner is a parolee
  • Recidivism...Return.to.Prison.numeric-Whether or not a person returned to prison within 3 years of release; 1 means “no”, they didn’t return and 0 means “yes”, they did return

Now let’s see if we’ve got any missing observations (remember to install the Amelia package):

Screen Shot 2019-05-15 at 3.50.30 PMScreen Shot 2019-05-15 at 3.50.06 PM

Looks like there’s nothing missing in our dataset, which is always a good sign (though not all datasets will be perfectly tidy).

Next let’s split our data into training and validation sets. If you’re wondering what a validation set is, it’s essentially the dataset you would use to fine tune the parameters of a model to prevent overfitting. You can make a testing set for this model if you want, but I’ll just go with training and validation sets.

Here’s the training/validation split, following the 75-25 rule I used for splitting training and testing sets:

Screen Shot 2019-05-15 at 10.22.50 PM

  • This will be a different approach to splitting the data than what you’ve seen in previous R posts. The above line of code is used as a guideline to splitting the data into training (trainSet) and validation (validationSet) sets. It tells R to utilize 75% of the data (19651 observations) for the training set. However, the interesting thing is the nrow function, which tells R to utilize ANY 19651 observations SELECTED AT RANDOM, as opposed to simply using observations 1-19651.
  • On the lines of code seen below, train selects the 75% of the observations chosen with the sample function while -train selects the other 25% of observations that are NOT part of the training set. By the way, the minus sign means NOT.

Screen Shot 2019-05-15 at 10.23.03 PM

Now here’s our random forest model, using default parameters and the Convicting.Offense.Type. variable  (remember to install the randomForest package):

Screen Shot 2019-05-15 at 10.23.17 PM

In case you were wondering, the default number of decision trees is 500, the default number of variables tried at each split is 2 (it’s 3 in this case), and the default OOB estimate of error rate is 3.6% (it’s much lower here-0.22%).

The confusion matrix you see details how many observations from the Convicting.Offense.Type variable are correctly classified and how many are misclassified. A margin of error-class.error-in this matrix tells you the percentage of observations that were misclassified. Here’s a breakdown:

  • Drug-No misclassification
  • Other-3.5% misclassification
  • Property-0.02% misclassification
  • Public Order-0.04% misclassification
  • Violent-0.1% misclassification

So all the crimes that were classified as Drug from trainSet were classified correctly, while Other had the most misclassification (and even 3.5% is a relatively small misclassification rate).

  • One important thing to note is that random forests can be used for both classification and regression analyses. If I wanted to do a regression analysis, I would’ve put my binary variable-Recidivism...Return.to.Prison.numeric-to the left of the ~ sign and other variables to the right of the ~ sign.

Now let’s change the model by change ntree and mtry, which refer to the number of decision trees in the model and the number of variables sampled at each stage, respectively:

Screen Shot 2019-05-16 at 3.19.08 PM

By reducing ntree and increasing mtry, the OOB estimate of error rate decreases (albeit by only 0.02%) as does the amount of variables that have a misclassification error. By that I mean that the previous model had only one variable with no misclassification error-Drug-while this model has 3 variables with no misclassification error-Drug, Property, and Public Order. Also, the misclassification error for the other two variables-Other and Violent-is both smaller and larger than that of the previous model. Other had a 3.5% error in the previous model while it has a 2.2% error in this model, so the misclassification error decreased in this case. On the other hand, Violent had a 0.1% error in the previous model but it has a 0.44% error in this model, so the misclassification error increased in this case.

Now let’s do some predictions, first on our training set and then on our validation set (using the second model):

Screen Shot 2019-05-16 at 8.28.31 PM

And then let’s check the accuracy of each prediction:

Screen Shot 2019-05-16 at 8.30.51 PM

For both the training set and validation set, the classifications are very accruate. trainSet has a 99.9% accuracy rate (only 10 of the 19,651 observations were misclassified) while `validationSet` has a 99.8% accuracy rate (only 13 of the 19,651 observations were misclassified).

Now let’s analyze the importance of each variable using the `importance` function:

Screen Shot 2019-05-17 at 4.03.04 PM

So what exactly does this complex matrix mean? It shows the importance of each of the 12 variables when classifying a crime based on the five types of offenses derived from the variable Convicting.Offense.Type.  More specifically, this matrix shows the importance of the 12 variables when classifying a variable-Convicting.Offense.Type-that has five possible values. The two important metrics to focus on in this matrix are MeanDecreaseAccuracy and MeanDecreaseGini.

A variable’s MeanDecreaseAccuracy is a metric of how much the accuracy of the model would decrease if that variable was to be excluded. The higher the number, the more accuracy the model will lose if that variable will be excluded. For instance, if I was to exclude the Recidivism...Return.to.Prison.numeric variable, the accuracy of the model wouldn’t be impacted much since it has the lowest MeanDecreaseAccuracy-3.42. Another reason the exclusion of this variable wouldn’t impact the model much is because this is a binary variable, which would be better suited to a regression problem as opposed to a classification problem (as this model is). On the other hand, excluding the variable Convicting.Offense.Subtype would have a significant impact on the model, since it has the largest MeanDecreaseAccuracy at 13,358.53.

A variable’s MeanDecreaseGini is a metric of how much Gini impurity will decrease when a certain variable is chosen to split a node. What exactly is Gini impurity? Well, let’s take a random point in the dataset. Using the guidelines of trainSet, let’s classify that point as Drug 5982/19651 of the time and as Property 5490/19651 of the time. The odds that we misclassify the data point based on class distribution is the Gini impurity. With that said, the MeanDecreaseGini is a measure of how much the misclassification likelihood will decrease if we include a certain variable. The higher this number is, the less likely the model will make misclassifications if that variable is included. For instance, Convicting.Offense.Subtype has the highest MeanDecreaseGini, so misclassification likelihood will greatly decrease if we include this variable in our model. On the other hand, Part.of.Target.Population has the lowest MeanDecreaseGini, so misclassification likelihood won’t be significantly impacted if this variable is included. This could be because this variable is a binary variable, which is better suited for regression problems rather than classification problems.

But what about all of the numbers in the first five columns? What do they mean? The first five columns, along with all the numbers, show the importance of each of the 12 variables when classifying a crime as one of the five possible types of crimes listed in Convicting.Offense.Type, which is the variable I used to build both random forest models. The higher the number, the more important that variable is when classifying a crime as one of the five possible crimes. For instance, the highest number (rounded) in the Fiscal.Year.Released row is 11.16, which corresponds to the Violent column. This implies that many Iowa prisoners convicted of violent crimes were likely released around the same time. Another example would be the highest number (rounded) in the Main.Supervising.District row-14.45-which corresponds to the `Other` column. This implies that Iowa prisoners who were convicted of crimes that don’t fall under the `Drug`, `Violent`, `Public Order`, or `Property` umbrellas would be, for the most part, located in the same Iowa jurisdiction.

  • If you wish to see the numbers in this matrix as a percent of 1, simply write scale = FALSEafter model2 in the parentheses. Remember to separate these two things with a comma.

Thanks for reading,

Michael

R Analysis 4: Naive Bayes and Amazon Alexa

Hello everybody,

It’s Michael, and today’s post will be an R analysis (my fourth one overall). I’ll be going back to R lessons now, but for those who want more Java content, don’t worry, it will be back soon.

First of all, let’s upload and understand our data. Here is the file-amazon_alexa.

Screen Shot 2019-05-04 at 12.46.49 PM

This dataset contains reviews for various Amazon Alexa products (such as the Echo Sub and Echo Dot) along with information about those reviews (such as star rating). Here’s a variable-by-variable breakdown of the data:

  • rating-the star rating the user gave the product, with 1 being the worst and 5 being the best
  • date-the date the review was posted; all of the reviews are from June or July 2018
  • variation-the exact product that is being reviewed, of which there are 16
  • verified_reviews-what the actual review said
  • feedback-whether the review was positive or negative-0 denotes negative reviews and 1 denotes positive reviews

In total, there are 3,150 reviews along with five aspects of information about the reviews (such as star rating, date posted, etc.)

Next, let’s convert feedback into a factored create a missmap in order to see if we have any missing observations (remember to install the Amelia package):

Screen Shot 2019-05-04 at 1.07.54 PM

Screen Shot 2019-05-04 at 1.07.39 PM

Screen Shot 2019-05-04 at 1.07.21 PM

According to the diagram, there are no missing observations, which is always a good thing.

Now here’s a table using the feedback variable that shows us how many positive (1) and negative (0) reviews are in the dataset:

Screen Shot 2019-05-04 at 1.15.30 PM

According to the table, of the 3150 reviews, only 257 are negative, while 2893 are positive. I’m guessing this is because Amazon’s various Alexa products are widely praised (though there are always going to be people who weren’t impressed with the products).

Let’s create some word-clouds next in order to analyze which words are common amongst positive and negative reviews. We’ll create two word-clouds, one for the positive reviews (feedback == 1) and another for the negative reviews (feedback == 0). Remember to install the wordcloud package:

Screen Shot 2019-05-05 at 3.10.52 PM

  • Before creating the word-clouds, remember to subset the data. In this case, create subsets for positive (feedback == 1) and negative (feedback == 0) reviews.

Here are the wordclouds for both the positive and negative reviews:

Screen Shot 2019-05-05 at 3.10.33 PM

Screen Shot 2019-05-05 at 10.15.08 PM

Screen Shot 2019-05-05 at 3.09.39 PM

Screen Shot 2019-05-05 at 10.12.27 PM

Some of the words that are common amongst both the positive and negative reviews include echo, Alexa, and work. This is likely because customers usually mention the name of the product in the review (most of which contain either Alexa or echo) in both positive and negative reviews. Customers’ reviews, whether positive or negative, are also likely to use the word work because customers usually mention whether or not their product worked (and if it did, how well it worked).

Some of the words that are common amongst positive reviews include great, love, can, like, and easy, which are all words that are commonly found in positive reviews (e.g. “The Echo is super easy to use. Love it will buy again!”). On the other hand, some of the words that are common amongst negative reviews include doesn't, didn't, stopped, and never, which you are very likely to find in critical reviews (e.g.”The Alexa speaker doesn’t work AT ALL! Never gonna buy again!”). An interesting observation is that the word refurbished is commonly found in negative reviews; this could be because many Alexa users who are dissatisfied with their product usually buy refurbished Alexa devices.

Next up, let’s clean up the data and prepare a corpus, which will be the text document that is derived from the actual text of the reviews. Our corpus is then used to create a document term matrix (referred to in the program as dtm), which contains the text of the review in the rows and the words in the reviews in the columns (remember to install the tm package):

Screen Shot 2019-05-06 at 8.28.44 PM

You may recall from R Lesson 13: Naive Bayes Classification that there are four functions to include when creating the dtm. They are:

  • toLower-sets all words to lowercase
  • removeNumbers-remove any numbers present in the reviews (like years)
  • removePunctuation-remove any punctuation present in the reviews
  • stemming-simplify analysis by combining similar words (such as verbs in different tenses and pairs of plural/singular nouns). For instance, the words “trying”, “tries”, “tried” would be combined into the word “try”.

Remember to set each function to TRUE.

Next we’ll create training and testing labels for our dataset. I’ve probably said it before, but I always like to use the 75-25 split when it comes to splitting my data into training and testing sets. This means that I like to use 75% of my dataset to train and build the model before testing it on the other 25% of my dataset. Here’s how to split the data:

Screen Shot 2019-05-06 at 8.42.14 PM

Using the 75-25 split, observations 1-2363 will be part of my trainLabels while observations 2364-3150 will be part of my testLabels. I included the feedback variable since you should always include the binary variable when creating your trainLabels and testLabels.

Now let’s make some tables analyzing the proportions of positive-to-negative reviews for both our trainLabels and testLabels. Remember to use the prop.table function:

Screen Shot 2019-05-07 at 3.06.20 PM

According to the tables, the difference between the proportions of positive-to-negative reviews for our trainLabels and testLabels is rather small-1% to be exact.

Now let’s split our dtm into training and testing sets, using the same guidelines that were used to create trainLabels and testLabels:

Screen Shot 2019-05-07 at 3.14.44 PM

To further tidy up the model, let’s only include words that appear at least 3 times:

Screen Shot 2019-05-07 at 3.20.43 PM

Our DTM uses 1s and 0s depending on whether a certain word can be found in the review (1 means the word is present and 0 means the word isn’t present). Since Naive Bayes works with categorical features, 1 and 0 are converted into Yes and No. This conversion is applied to every column (hence MARGIN = 2):

Screen Shot 2019-05-07 at 3.33.22 PM

Last but not least, it’s time to create the Naive Bayes classifier (remember to install the package e1071):

Screen Shot 2019-05-07 at 3.38.59 PM

Now let’s test out the classifier on two words, great and disappoint:

Screen Shot 2019-05-07 at 3.39.48 PM

Now, what does all this mean?

Well, the word great:

  • DOESN’T appear in 93% of negative reviews but DOES appear in the other 7% of negative reviews
  • DOESN’T appear in 77.1% of positive reviews but DOES appear in the other 22.9% of positive reviews

These results are a little surprising since I thought the word great would appear in a majority of positive reviews. Then again, great is more prevalent in positive reviews than in negative reviews, which isn’t surprising, since negative reviews aren’t likely to use the word great.

And the word disappoint:

  • DOESN’T appear in 93.5% of negative reviews but DOES appear in the other 6.5% of negative reviews
  • DOESN’T appear in 98.8% of positive reviews but DOES appear in the other 1.2% of positive reviews

Just as with great, these results are and aren’t surprising. The surprising thing is that disappoint only appears in 6.5% of negative reviews, when I thought (and probably you did too) that disappoint would be found in more negative reviews. Then again, disappoint is more common in negative reviews than in positive reviews, which isn’t surprising.

Last but not least, let’s create a confusion matrix, which evaluates the performance of this classifier using our testing dataset (which is the variable test). Remember to install the gmodels package:

Screen Shot 2019-05-07 at 3.55.27 PM

The confusion matrix contains 787 observations-the amount of observations in our testing set. The column total for column 1 represents the amount of correctly classified observations-729. The column total for column 0 represents the amount of incorrectly classified observations-58. In other words, 729 reviews were correctly classified while 58 reviews were incorrectly classified. The overall accuracy of the classifier is 93% which is excellent, but for the misclassified reviews, it could complicate customer feedback analysis, in which case a more sophisticated model would be needed.

Thanks for reading.

Michael

 

R Analysis 3: K-Means vs Hierarchical Clustering

Hello everybody,

It’s Michael, and I will be doing an R analysis on this post. More specifically, I will be doing a comparative clustering analysis, which means I’ll take a dataset and perform both k-means and hierarchical clustering analysis with that dataset to analyze the results of each method. However, this analysis will be unique, since I will be revisiting of the earliest datasets I used for this blog-TV shows-which first appeared in R Lesson 4: Logistic Regression Models on July 11, 2018 (exactly nine months ago!) In case you forgot what this dataset was about, it basically gives 85 shows that aired during the 2017-18 TV season and whether or not they were renewed for the 2018-19 TV season along with other aspects of those shows (such as the year they premiered and the network they air on). I’ll admit I chose this dataset because I wanted to analyze one of my old datasets in a different way (remember I performed linear and logistic regression the first time I used this dataset).

So, as always, let’s load the file and get a basic understanding of our data:

Screen Shot 2019-04-07 at 10.27.47 PM

As you can see, we have 85 observations of 10 variables. Here’s a detailed breakdown of each variable:

  • TV.Show-The name of the show
  • Genre-The genre of the show
  • Premiere.Year-The year the show premiered; for revivals like Roseanne, I used the original premiere year (1988) as opposed to the revival premiere year (2018)
  • X..of.seasons..17.18.-How many seasons the show had aired as of the end of the 2017-18 TV season
  • Network-The network (or streaming service) the show airs on
  • X2018.19.renewal.-Whether or not the show was renewed for the 2018-19 TV season; 1 denotes renewal and 0 denotes cancellation
  • Rating-The content rating for the show. Here’s a more detailed breakdown:
    • 1 means TV-G
    • 2 means TV-PG
    • 3 means TV-14
    • 4 means TV-MA
    • 5 means not applicable
  • Usual.Day.of.Week-The usual day of the week the show airs its new episodes. Here’s a more detailed breakdown:
    • 1 means the show airs on Mondays
    • 2 means the show airs on Tuesdays
    • 3 means the show airs on Wednesdays
    • 4 means the show airs on Thursdays
    • 5 means the show airs on Fridays
    • 6 means the show airs on Saturdays
    • 7 means the show airs on Sundays
    • 8 means the show doesn’t have a regular air-day (usually applies to talk shows or shows on streaming services)
  • Medium-the type of network the show airs on. Here’s a more detailed breakdown:
    • 1 means the show airs on either one of the big 4 broadcast networks (ABC, NBC, FOX or CBS) or the CW (which isn’t part of the big 4)
    • 2 means the show airs on a cable channel (AMC, Bravo, etc.)
    • 3 means the show airs on a streaming service (Hulu, Amazon Prime, etc.)
  • Episode Count-the new variable I added for this analysis; this variable shows how many episodes a show has had overall since the end of the 2017-18 TV season. For certain shows whose seasons cross the 17-18 and 18-19 seasons, I will count how many episodes each show has had as of September 24, 2018 (the beginning of the 2018-19 TV season)

Now that we’ve learned more about our variables, let’s start our analysis. But first, I convert the final four variables into factors, since I think it’ll be more appropriate for the analysis:

Screen Shot 2019-04-07 at 10.28.52 PM

Ok, now onto the analysis. I’ll start with k-means:

Screen Shot 2019-04-08 at 3.18.33 PM

Here, I created a data subset using our third and tenth columns (Premiere.Year and Episode.Count respectively) and displayed the head (the first six observations) of my cluster.

Now let’s do some k-means clustering:

Screen Shot 2019-04-08 at 3.26.48 PM

I created the variable tvCluster to store my k-means model using the name of my data subset-cluster1-the number of clusters I wanted to include (4) and nstart, which tells the models to start with 35 random points then select the one with the lowest variation.

I then type in tvCluster to get a better idea of what my cluster looks like. The first thing I see is “K-means clustering with (X) clusters of sizes”, before mentioning the amount of observations in each cluster (which are 17, 64, 1 and 3, respectively). In total, all 85 observations were used since I didn’t have any missing data points.

The next thing that is mentioned is cluster means, which gives the mean for each variable used in the clustering analysis (in this case, Episode.Count and Premiere.Year). Interestingly enough, Cluster 2 has the highest mean Premiere.Year (2015) but the lowest mean Episode.Count (49. rounded to the nearest whole number).

After that, you can see the clustering vector, which shows you which observations belong to which cluster. Even though the position of the observation (e.g. 1st, 23rd) isn’t explicitly mentioned, you can tell which observation you are looking at since the vector starts with the first observation and works its way down to the eighty-fifth observation (and since there is no missing data, all 85 observations are used in this clustering model). For instance, the first three observations all correspond to cluster 1 (the first three shows listed in this dataset are NCISBig Bang Theory, and The Simpsons). Likewise, the final three observations all correspond to cluster 2 (the corresponding shows are The Americans, Baskets, and Comic Book Men).

Next you will see the within cluster sum of squares for each cluster, which I will abbreviate as WCSSBC; this is a measurement of the variability of the observations in each cluster. Remember that the smaller this amount is, the more compact the cluster. In this case, 3 of the 4 WCSSBC are above 100,000, while the other WCSSBC is 0 (which I’m guessing is cluster 3, which has only one observation).

Last but not least is between_SS/total_SS=94.5%, which represents the between sum-of-squares and total sum-of-squares ratio, which as you may recall from the k-means lesson is a measure of the goodness-of-fit of the model. 94.5% indicates that there is an excellent goodness-of-fit for this model.

Last but not least, let’s graph our model:

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In this graph, the debut year is on the x-axis, while the episode count is on the y-axis. As you can see, the 2 largest clusters (represented by the black and red dots) are fairly close together while the 2 smallest clusters (represented by the blue and green dots) are fairly spread out (granted, the two smallest clusters only have 1 and 3 observations, respectively). An observation about this graph that I wanted to point out is that the further back a show premiered doesn’t always mean the show has more episodes than another show that premiered fairly recently (let’s say anytime from 2015 onward). This happens for several reasons, including:

  • Revived series like American Idol (which took a hiatus in 2017 before its 2018 revival) and Roseanne (which had been dormant for 21 years before its 2018 revival)
  • Different shows air a different number of episodes per season; for instance, talk shows like Jimmy Kimmel live have at least 100 episodes per season while shows on the Big 4 networks tend to have between 20-24 episodes per season (think Simpsons, The Big Bang Theory, and Grey’s Anatomy). Cable and streaming shows usually have even less episodes per season (between 6-13, like how South Park only does 10 episode seasons)
  • Some shows just take long breaks (like how Jessica Jones on Netflix didn’t release any new episodes between November 2015 and March 2018)

Now time to do some hierarchical clustering on our data. And yes, I plan to use all the methods covered in the post R Lesson 12: Hierarchical Clustering.

Let’s begin by scaling all numerical variables in our data (don’t include the ones that were converted into factor types):

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Now let’s start off with some agglomerative clustering (with both the dendrogram and code):

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After setting up the model using Euclidean distance and complete linkage, I then plot my dendrogram, which you can see above. This dendrogram is a lot neater than the ones I created in R Lesson 12, but then again, this dataset only has 85 observations, while the one in that post had nearly 7,200. The names of the shows themselves aren’t mentioned, but each of the numbers displayed correspond to a certain show. For instance, 74 corresponds to The Voice, since it is the 74th show listed in our dataset. Look at the spreadsheet to figure out which number corresponds to which show.

You may recall that I mentioned two general rules for interpreting dendrograms. They are:

  • The higher the height of the fusion, the more dissimilar two items are
  • The wider the branch between two observations, the more dissimilar they are

Those rules certainly apply here, granted, the highest height is 8 in this case, as opposed to 70. For instance, since the brand between shows 7 and 63 is fairly narrow, these two shows have a lot in common according to the model (even though the two shows in question are Bob’s Burgers and The Walking Dead-the former being a cartoon sitcom and the latter revolving around the zombie apocalypse). On the other hand, the gap between shows 74 and 49 is wider, which means they don’t share much in common (even though the two shows are The Voice and Shark Tank, which both qualify as reality shows, though the former is more competition-oriented than the latter). All in all, I think it’s interesting to see how these clusters were created, since the shows that were grouped closer together seem to have nothing in common.

Now let’s try AGNES (remember that stands for agglomerative clustering):

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First of all, remember to install the package cluster. Also, remember that the one important result is the ac, or agglomerative coefficient, which measures the strength of clustering structure. As you can see, our ac is 96.1%, which indicates very strong clustering structure (I personally think any ac at east 90% is good).

Now let’s compare this ac (which used complete linkage) to the ac we get with other linkage methods (not including centroid). Remember to install the purrr package:

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Of the four linkage method’s, Ward’s method gives us the highest agglomerative coefficient (98.2%), so that’s what we’ll use for the next part of this analysis.

Using ward’s method and AGNES, here’s a dendrogram of our data (and the corresponding code):

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Aside from having a greater maximum height than our previous dendrogram (the latter had a maximum height of 8 while this diagram has a maximum height of presumably 18), the observations are also placed differently. For instance, unlike in the previous dendrogram, observations 30 and 71 are side by side. But just as with the last dendrogram, shows that have almost nothing in common are oddly grouped together; for instance, the 30th and 71st observations correspond to The Gifted and Transparent; the former is a sci-fi show based off the X-Men universe while the latter is a transgender-oriented drama. The 4th and 17th observations are another good example of this, as the corresponding shows are The Simpsons and Taken; the former is a long-running cartoon sitcom while the latter is based off of an action movie trilogy.

The last method I will use for this analysis is DIANA (stands for divisive analysis). Recall that the main difference between DIANA and AGNES is that DIANA works in a top-down manner (objects start in a single supercluster and are divided into smaller clusters until single-element clusters are created) while AGNES works in a bottom-up (objects start in single-element clusters and are morphed into progressively larger clusters until a single supercluster is created). Here’s the code and dendrogram for our DIANA analysis:

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Remember that the divisive coefficient is pretty much identical to the agglomerative coefficient, since both measure strength of clustering structure and the closer each amount is to 1, the stronger the clustering structure. Also, in both cases, a coefficient of .9 (or 90%) or higher indicates excellent clustering structure. In this case, the dc (divisive coefficient) is 95.9%, which indicates excellent clustering structure.

Just as with the previous two dendrograms, most of the observation pairs still have nothing in common. For instance, the 41st and 44th observations have nothing in common, since the corresponding shows are House of Cards (a political drama) and Brooklyn 99 (a sitcom), respectively. An exception to this would be the 9th and 31st observations, since both of the corresponding shows-Designated Survivor and Bull respectively-are dramas and both are on the big 4 broadcast networks (though the former airs on ABC while the latter airs on FOX).

Now, let’s assign clusters to the data points. I’ll go with 4 clusters, since that’s how many I used for my k-means analysis (plus I think it’s an ideal amount). I’m going to use the DIANA example I just mentioned:

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Now let’s visualize our clusters in a scatterplot (remember to install the factoextra package):

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As you can see, cluster 3 has the most observations while cluster 4 has the least (only one observation corresponding to Jimmy Kimmel Live). Some of the observations in each cluster have something in common, like the 1st and 2nd observations (NCIS and The Big Bang Theory in cluster 1, both of which air on CBS) and the 42nd and 80th observations in cluster 3 (Watch What Happens Live! and The Chew-both talk shows).

Now, let’s visualize these clusters on a dendrogram (using the same DIANA example):

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The first line of code is exactly the same line I used when I first plotted my DIANA dendrogram. The rect.hclust line draws the borders to denote each cluster; remember to set k to the amount of clusters you created for your scatterplot (in this case, 4). Granted, the coloring scheme is different from the scatterplot, but you can tell which cluster is which judging by the size of the rectangle (for instance, the rectangle for cluster 4 only contains the 79th observation, even though it is light blue on our dendrogram and purple on our scatterplot). Plus, all the observations are in the same cluster in both the scatterplot and dendrogram.

Thanks for reading.

Michael