Month: June 2019

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:

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:

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
NM-AK 21.06666667 1.8254695 40.307863868 0.0019407
NV-AK 26.66666667 7.4254695 45.907863868 0.0000062
NY-AK 19.76666667 0.5254695 39.007863868 0.0063352
OH-AK 19.10000000 -0.1411972 38.341197201 0.0112786
OK-AK 24.00000000 4.7588028 43.241197201 0.0001069
OR-AK 32.00000000 12.7588028 51.241197201 0.0000000
PA-AK 20.33333333 1.0921361 39.574530535 0.0038167
RI-AK 20.50000000 1.2588028 39.741197201 0.0032794
SC-AK 32.90000000 13.6588028 52.141197201 0.0000000
SD-AK 7.70000000 -11.5411972 26.941197201 0.9996197
TN-AK 25.93333333 6.6921361 45.174530535 0.0000138
TX-AK 36.00000000 16.7588028 55.241197201 0.0000000
UT-AK 16.70000000 -2.5411972 35.941197201 0.0731639
VA-AK 27.16666667 7.9254695 46.407863868 0.0000036
VT-AK 7.56666667 -11.6745305 26.807863868 0.9997438
WA-AK 30.73333333 11.4921361 49.974530535 0.0000001
WI-AK 7.83333333 -11.4078639 27.074530535 0.9994447
WV-AK 21.60000000 2.3588028 40.841197201 0.0011709
WY-AK 10.26666667 -8.9745305 29.507863868 0.9423542
AR-AL -7.43333333 -26.6745305 11.807863868 0.9998304
AZ-AL 7.83333333 -11.4078639 27.074530535 0.9994447
CA-AL 8.70000000 -10.5411972 27.941197201 0.9954754
CO-AL -17.46666667 -36.7078639 1.774530535 0.0418707
CT-AL -15.53333333 -34.7745305 3.707863868 0.1572874
DE-AL -9.26666667 -28.5078639 9.974530535 0.9866689
FL-AL 14.46666667 -4.7745305 33.707863868 0.2858501
GA-AL -1.16666667 -20.4078639 18.074530535 1.0000000
HI-AL 26.76666667 7.5254695 46.007863868 0.0000056
IA-AL -24.10000000 -43.3411972 -4.858802799 0.0000964
ID-AL -21.23333333 -40.4745305 -1.992136132 0.0016592
IL-AL -20.50000000 -39.7411972 -1.258802799 0.0032794
IN-AL -17.53333333 -36.7745305 1.707863868 0.0398126
KS-AL -16.20000000 -35.4411972 3.041197201 0.1029146
KY-AL -12.10000000 -31.3411972 7.141197201 0.7075040
LA-AL 4.80000000 -14.4411972 24.041197201 1.0000000
MA-AL -16.26666667 -35.5078639 2.974530535 0.0984460
MD-AL -10.43333333 -29.6745305 8.807863868 0.9297498
ME-AL -20.83333333 -40.0745305 -1.592136132 0.0024125
MI-AL -19.06666667 -38.3078639 0.174530535 0.0116018
MN-AL -29.00000000 -48.2411972 -9.758802799 0.0000005
MO-AL -14.13333333 -33.3745305 5.107863868 0.3368657
MS-AL 1.93333333 -17.3078639 21.174530535 1.0000000
MT-AL -22.00000000 -41.2411972 -2.758802799 0.0007960
NC-AL -6.30000000 -25.5411972 12.941197201 0.9999977
ND-AL -36.76666667 -56.0078639 -17.525469465 0.0000000
NE-AL -22.70000000 -41.9411972 -3.458802799 0.0003998
NH-AL -22.46666667 -41.7078639 -3.225469465 0.0005039
NJ-AL -11.03333333 -30.2745305 8.207863868 0.8693993
NM-AL -13.26666667 -32.5078639 5.974530535 0.4888757
NV-AL -7.66666667 -26.9078639 11.574530535 0.9996549
NY-AL -14.56666667 -33.8078639 4.674530535 0.2715191
OH-AL -15.23333333 -34.4745305 4.007863868 0.1880399
OK-AL -10.33333333 -29.5745305 8.907863868 0.9375203
OR-AL -2.33333333 -21.5745305 16.907863868 1.0000000
PA-AL -14.00000000 -33.2411972 5.241197201 0.3585928
RI-AL -13.83333333 -33.0745305 5.407863868 0.3867198
SC-AL -1.43333333 -20.6745305 17.807863868 1.0000000
SD-AL -26.63333333 -45.8745305 -7.392136132 0.0000065
TN-AL -8.40000000 -27.6411972 10.841197201 0.9976614
TX-AL 1.66666667 -17.5745305 20.907863868 1.0000000
UT-AL -17.63333333 -36.8745305 1.607863868 0.0368937
VA-AL -7.16666667 -26.4078639 12.074530535 0.9999296
VT-AL -26.76666667 -46.0078639 -7.525469465 0.0000056
WA-AL -3.60000000 -22.8411972 15.641197201 1.0000000
WI-AL -26.50000000 -45.7411972 -7.258802799 0.0000075
WV-AL -12.73333333 -31.9745305 6.507863868 0.5900517
WY-AL -24.06666667 -43.3078639 -4.825469465 0.0000998
AZ-AR 15.26666667 -3.9745305 34.507863868 0.1844181
CA-AR 16.13333333 -3.1078639 35.374530535 0.1075485
CO-AR -10.03333333 -29.2745305 9.207863868 0.9571836
CT-AR -8.10000000 -27.3411972 11.141197201 0.9988710
DE-AR -1.83333333 -21.0745305 17.407863868 1.0000000
FL-AR 21.90000000 2.6588028 41.141197201 0.0008771
GA-AR 6.26666667 -12.9745305 25.507863868 0.9999981
HI-AR 34.20000000 14.9588028 53.441197201 0.0000000
IA-AR -16.66666667 -35.9078639 2.574530535 0.0748907
ID-AR -13.80000000 -33.0411972 5.441197201 0.3924661
IL-AR -13.06666667 -32.3078639 6.174530535 0.5265762
IN-AR -10.10000000 -29.3411972 9.141197201 0.9532685
KS-AR -8.76666667 -28.0078639 10.474530535 0.9948059
KY-AR -4.66666667 -23.9078639 14.574530535 1.0000000
LA-AR 12.23333333 -7.0078639 31.474530535 0.6835500
MA-AR -8.83333333 -28.0745305 10.407863868 0.9940553
MD-AR -3.00000000 -22.2411972 16.241197201 1.0000000
ME-AR -13.40000000 -32.6411972 5.841197201 0.4641051
MI-AR -11.63333333 -30.8745305 7.607863868 0.7858266
MN-AR -21.56666667 -40.8078639 -2.325469465 0.0012088
MO-AR -6.70000000 -25.9411972 12.541197201 0.9999875
MS-AR 9.36666667 -9.8745305 28.607863868 0.9841955
MT-AR -14.56666667 -33.8078639 4.674530535 0.2715191
NC-AR 1.13333333 -18.1078639 20.374530535 1.0000000
ND-AR -29.33333333 -48.5745305 -10.092136132 0.0000003
NE-AR -15.26666667 -34.5078639 3.974530535 0.1844181
NH-AR -15.03333333 -34.2745305 4.207863868 0.2108675
NJ-AR -3.60000000 -22.8411972 15.641197201 1.0000000
NM-AR -5.83333333 -25.0745305 13.407863868 0.9999998
NV-AR -0.23333333 -19.4745305 19.007863868 1.0000000
NY-AR -7.13333333 -26.3745305 12.107863868 0.9999373
OH-AR -7.80000000 -27.0411972 11.441197201 0.9994941
OK-AR -2.90000000 -22.1411972 16.341197201 1.0000000
OR-AR 5.10000000 -14.1411972 24.341197201 1.0000000
PA-AR -6.56666667 -25.8078639 12.674530535 0.9999928
RI-AR -6.40000000 -25.6411972 12.841197201 0.9999965
SC-AR 6.00000000 -13.2411972 25.241197201 0.9999995
SD-AR -19.20000000 -38.4411972 0.041197201 0.0103584
TN-AR -0.96666667 -20.2078639 18.274530535 1.0000000
TX-AR 9.10000000 -10.1411972 28.341197201 0.9900922
UT-AR -10.20000000 -29.4411972 9.041197201 0.9469173
VA-AR 0.26666667 -18.9745305 19.507863868 1.0000000
VT-AR -19.33333333 -38.5745305 -0.092136132 0.0092399
WA-AR 3.83333333 -15.4078639 23.074530535 1.0000000
WI-AR -19.06666667 -38.3078639 0.174530535 0.0116018
WV-AR -5.30000000 -24.5411972 13.941197201 1.0000000
WY-AR -16.63333333 -35.8745305 2.607863868 0.0766521
CA-AZ 0.86666667 -18.3745305 20.107863868 1.0000000
CO-AZ -25.30000000 -44.5411972 -6.058802799 0.0000273
CT-AZ -23.36666667 -42.6078639 -4.125469465 0.0002045
DE-AZ -17.10000000 -36.3411972 2.141197201 0.0549574
FL-AZ 6.63333333 -12.6078639 25.874530535 0.9999905
GA-AZ -9.00000000 -28.2411972 10.241197201 0.9917758
HI-AZ 18.93333333 -0.3078639 38.174530535 0.0129826
IA-AZ -31.93333333 -51.1745305 -12.692136132 0.0000000
ID-AZ -29.06666667 -48.3078639 -9.825469465 0.0000004
IL-AZ -28.33333333 -47.5745305 -9.092136132 0.0000010
IN-AZ -25.36666667 -44.6078639 -6.125469465 0.0000254
KS-AZ -24.03333333 -43.2745305 -4.792136132 0.0001033
KY-AZ -19.93333333 -39.1745305 -0.692136132 0.0054663
LA-AZ -3.03333333 -22.2745305 16.207863868 1.0000000
MA-AZ -24.10000000 -43.3411972 -4.858802799 0.0000964
MD-AZ -18.26666667 -37.5078639 0.974530535 0.0224533
ME-AZ -28.66666667 -47.9078639 -9.425469465 0.0000007
MI-AZ -26.90000000 -46.1411972 -7.658802799 0.0000048
MN-AZ -36.83333333 -56.0745305 -17.592136132 0.0000000
MO-AZ -21.96666667 -41.2078639 -2.725469465 0.0008222
MS-AZ -5.90000000 -25.1411972 13.341197201 0.9999997
MT-AZ -29.83333333 -49.0745305 -10.592136132 0.0000002
NC-AZ -14.13333333 -33.3745305 5.107863868 0.3368657
ND-AZ -44.60000000 -63.8411972 -25.358802799 0.0000000
NE-AZ -30.53333333 -49.7745305 -11.292136132 0.0000001
NH-AZ -30.30000000 -49.5411972 -11.058802799 0.0000001
NJ-AZ -18.86666667 -38.1078639 0.374530535 0.0137286
NM-AZ -21.10000000 -40.3411972 -1.858802799 0.0018810
NV-AZ -15.50000000 -34.7411972 3.741197201 0.1605020
NY-AZ -22.40000000 -41.6411972 -3.158802799 0.0005381
OH-AZ -23.06666667 -42.3078639 -3.825469465 0.0002770
OK-AZ -18.16666667 -37.4078639 1.074530535 0.0243236
OR-AZ -10.16666667 -29.4078639 9.074530535 0.9490994
PA-AZ -21.83333333 -41.0745305 -2.592136132 0.0009355
RI-AZ -21.66666667 -40.9078639 -2.425469465 0.0010984
SC-AZ -9.26666667 -28.5078639 9.974530535 0.9866689
SD-AZ -34.46666667 -53.7078639 -15.225469465 0.0000000
TN-AZ -16.23333333 -35.4745305 3.007863868 0.1006599
TX-AZ -6.16666667 -25.4078639 13.074530535 0.9999988
UT-AZ -25.46666667 -44.7078639 -6.225469465 0.0000228
VA-AZ -15.00000000 -34.2411972 4.241197201 0.2148563
VT-AZ -34.60000000 -53.8411972 -15.358802799 0.0000000
WA-AZ -11.43333333 -30.6745305 7.807863868 0.8160859
WI-AZ -34.33333333 -53.5745305 -15.092136132 0.0000000
WV-AZ -20.56666667 -39.8078639 -1.325469465 0.0030852
WY-AZ -31.90000000 -51.1411972 -12.658802799 0.0000000
CO-CA -26.16666667 -45.4078639 -6.925469465 0.0000107
CT-CA -24.23333333 -43.4745305 -4.992136132 0.0000839
DE-CA -17.96666667 -37.2078639 1.274530535 0.0284942
FL-CA 5.76666667 -13.4745305 25.007863868 0.9999998
GA-CA -9.86666667 -29.1078639 9.374530535 0.9659111
HI-CA 18.06666667 -1.1745305 37.307863868 0.0263343
IA-CA -32.80000000 -52.0411972 -13.558802799 0.0000000
ID-CA -29.93333333 -49.1745305 -10.692136132 0.0000002
IL-CA -29.20000000 -48.4411972 -9.958802799 0.0000004
IN-CA -26.23333333 -45.4745305 -6.992136132 0.0000100
KS-CA -24.90000000 -44.1411972 -5.658802799 0.0000417
KY-CA -20.80000000 -40.0411972 -1.558802799 0.0024882
LA-CA -3.90000000 -23.1411972 15.341197201 1.0000000
MA-CA -24.96666667 -44.2078639 -5.725469465 0.0000389
MD-CA -19.13333333 -38.3745305 0.107863868 0.0109637
ME-CA -29.53333333 -48.7745305 -10.292136132 0.0000003
MI-CA -27.76666667 -47.0078639 -8.525469465 0.0000018
MN-CA -37.70000000 -56.9411972 -18.458802799 0.0000000
MO-CA -22.83333333 -42.0745305 -3.592136132 0.0003500
MS-CA -6.76666667 -26.0078639 12.474530535 0.9999838
MT-CA -30.70000000 -49.9411972 -11.458802799 0.0000001
NC-CA -15.00000000 -34.2411972 4.241197201 0.2148563
ND-CA -45.46666667 -64.7078639 -26.225469465 0.0000000
NE-CA -31.40000000 -50.6411972 -12.158802799 0.0000000
NH-CA -31.16666667 -50.4078639 -11.925469465 0.0000000
NJ-CA -19.73333333 -38.9745305 -0.492136132 0.0065239
NM-CA -21.96666667 -41.2078639 -2.725469465 0.0008222
NV-CA -16.36666667 -35.6078639 2.874530535 0.0920448
NY-CA -23.26666667 -42.5078639 -4.025469465 0.0002264
OH-CA -23.93333333 -43.1745305 -4.692136132 0.0001145
OK-CA -19.03333333 -38.2745305 0.207863868 0.0119336
OR-CA -11.03333333 -30.2745305 8.207863868 0.8693993
PA-CA -22.70000000 -41.9411972 -3.458802799 0.0003998
RI-CA -22.53333333 -41.7745305 -3.292136132 0.0004718
SC-CA -10.13333333 -29.3745305 9.107863868 0.9512161
SD-CA -35.33333333 -54.5745305 -16.092136132 0.0000000
TN-CA -17.10000000 -36.3411972 2.141197201 0.0549574
TX-CA -7.03333333 -26.2745305 12.207863868 0.9999560
UT-CA -26.33333333 -45.5745305 -7.092136132 0.0000090
VA-CA -15.86666667 -35.1078639 3.374530535 0.1278119
VT-CA -35.46666667 -54.7078639 -16.225469465 0.0000000
WA-CA -12.30000000 -31.5411972 6.941197201 0.6713740
WI-CA -35.20000000 -54.4411972 -15.958802799 0.0000000
WV-CA -21.43333333 -40.6745305 -2.192136132 0.0013727
WY-CA -32.76666667 -52.0078639 -13.525469465 0.0000000
CT-CO 1.93333333 -17.3078639 21.174530535 1.0000000
DE-CO 8.20000000 -11.0411972 27.441197201 0.9985493
FL-CO 31.93333333 12.6921361 51.174530535 0.0000000
GA-CO 16.30000000 -2.9411972 35.541197201 0.0962725
HI-CO 44.23333333 24.9921361 63.474530535 0.0000000
IA-CO -6.63333333 -25.8745305 12.607863868 0.9999905
ID-CO -3.76666667 -23.0078639 15.474530535 1.0000000
IL-CO -3.03333333 -22.2745305 16.207863868 1.0000000
IN-CO -0.06666667 -19.3078639 19.174530535 1.0000000
KS-CO 1.26666667 -17.9745305 20.507863868 1.0000000
KY-CO 5.36666667 -13.8745305 24.607863868 1.0000000
LA-CO 22.26666667 3.0254695 41.507863868 0.0006135
MA-CO 1.20000000 -18.0411972 20.441197201 1.0000000
MD-CO 7.03333333 -12.2078639 26.274530535 0.9999560
ME-CO -3.36666667 -22.6078639 15.874530535 1.0000000
MI-CO -1.60000000 -20.8411972 17.641197201 1.0000000
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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