Looking at the 955 counties in the US that have a population over 50,000 people (total of 955 counties).

Model how the county voted in the 2016 presidential election.

The response variable how the county voted (variable is Vote_Ordered), which has 4 values:

Value Label Vote Percent for Democrat

1 Rep Strong Less than 40%

2 Rep Weak 40% - 50%

3 Dem Weak 50%-60%

4 Dem Strong More than 60%

The two explanatory variables we will use are:

Region of the US (z): (MW/NE/S/W, 4 levels)

College (x): Percentage of people that attended at least some college - 5 # summary = {26, 52, 59, 65, 86}

1. Since the data are ordinal, we can use cumulative logistic regression to estimate the probabilities of the counties falling into one of the 4 voting types: RS, RW, DW, DS.

a. Write out the cumulative logit links that our model will predict in terms of js. No linear predictors required.

b. The model below shows the model estimates assume proportional odds for the changes in the predictors, including the interaction terms between region and education. Interpret estimate of "College" in the output on how the state votes for president. Keep in mind this is for a model with interaction!

call:

polr(formula = Vote_Type ~ Region * college, data = counties2, Hess = T)

coeffients :

value std. Error t value

RegionW 4.1347 1.4631 2 . 826

RegionMW 1. 3162 1. 3935 0.945

RegionS 1.2543 1. 3005 0.964

College 0.1263 0.0179 7 .040

RegionW: college -0.0702 0.0237 -2.968

RegionMW: College -0.0381 0.0229 -1. 665

Regions : College -0.0429 0.0217 -1.977

Intercepts :

value std. Error t value

Rep_strong | Rep_weak 6. 6921.072 6.240

Rep_weak | Dem_weak 7.806 1.081 7.220

Dem_weak I Dem_Strong 8. 890 1.091 8. 148

Residual Deviance: 2088. 544

AIC: 2108.544

Repeat part 2b), but compare the odds of a county in the NE vs S when the college percentage is 50% for both counties.

d. Calculate the probability a county in the West is in the "Weak Dem" category if it has a college percentage of 58%.

e. The deviance for the model without interaction terms is 2097.67. Calculate the test statistic, df, p-value, and state the conclusion if the interaction term should be kept in the model.

f. An additive model (no interaction) was fit with non-parallel slopes and has a deviance of 2066.6. Is there evidence that the proportional odds assumption is not true for the data? Calculate the test statistic, df, p-value, and state the conclusion.

3. For any cumulative logistic regression model with proportional odds, 1 < 2 < < J1. Why?