Multinomial Logistic Regression

STAT 235: Fall 2020Homework 4: Multinomial LogisticRegression

We will be looking at the 955 counties in the US that have a population over 50,000 people (total of 955 counties). We want to 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:ValueLabelVote Percent for Democrat1 Rep StrongLess than 40%2Rep Weak40% -50%3Dem Weak50%-60%4Dem StrongMore than 60%The two explanatory variables we will use are:Region of the US(): (MW/NE/S/W, 4 levels)College(): Percentage of people that attended at least some college–5 # summary= {26, 52, 59, 65, 86}1.We will start with a baseline logistic regression modelwith“Rep Strong” be the baseline category.a.Writeout the three logit models that model will fit, not including the linear predictor component.b.TheRoutput below shows the additive model. Interpret the effect of a county being in the Westrather thanthe South and voting Rep Strong vs Dem Strong

c.Interpret the effect of College between “Dem Weak” vs “RepWeak”d.For a county in the Northeast with a college percentage of 70, find all four estimated probabilities.e.What does the intercept term for Dem_Strong indicate about how a county votes?f.The deviance of a model that includes the interaction terms between College and Regionis 2049.9. Find the test statistic, df, p-value, and state the conclusion of a hypothesis test to determine if the interaction terms improve the fit of the model.g.If we drop Region from the model, the estimates are shown below. Createa plot of the probabilities of howa county votesas College changes (range from 26% to 86%). Make sure the plot clearly indicates which line corresponds to which Vote Type.

2.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: ,,,.a.Write out the cumulative logit links that our model will predict in terms of ′. No linear predictors required.b.The model below shows the model estimates assume proportional odds for the changesin the predictors, including the interaction terms between region and education. Interpret estimate of “College”in the outputon how the state votes for president. Keep in mind this is for a model with interaction!c.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 probabilitya county in the West isin 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 themodel.f.An additivemodel(no interaction) was fit with non-parallel slopes and has a deviance of 2066.6. Is there evidence that theproportional 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<⋯<−1. Why?


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