Repeat the analysis from Exercise 11 using a proportional odds regression model. Note that this will require programming a model-fitting function for use in glmulti(), because the proportional odds model is not normally fit using glm(). The code for doing this is available in the program glmultiFORpolr.R. This code needs to be run before glmulti() is called on the polr-class model fit.

(a) For each variable, report the posterior probability that it belongs in the model and interpret the results.

(b) Obtain model-averaged estimates of the odds ratios corresponding to the effect of each explanatory variable. Interpret these values.

(c) Compute 95% confidence intervals for each odds ratio and interpret. In particular, which variables appear to be most important in describing countries' use of the death penalty?

(d) Repeat this analysis on the odds ratios from a single proportional odds model using all explanatory variables. Compare the results to the model-averaged values.

Exercise 11

Continuing Exercise 10, the researcher's main interest was to determine which, if any, of the explanatory variables are associated with the country's use of the death penalty. Complete the items below with this in mind.

(a) Explain why a BMA approach would be more appropriate than a stepwise approach given the context of this problem.

(b) Treating the response as nominal multinomial, carry out the BMA analysis using main effects only. Report the results and draw conclusions. Note that this will require programming a model-fitting function for use in glmulti(), because the multinomial model is not normally fit using glm(). The code for doing this is available in the program glmulti FOR multinom.R. This code needs to be run before glmulti() is called on the multinom-class model fit.

Exercise 10

A criminologist studying capital punishment was interested in identifying whether certain social, economic, and political attributes of a country related to its use of the death penalty. She gathered data from public sources on 194 countries, recording the following variables:

• COUNTRY: The name of the country;

• STATUS: The country's laws on death penalties, coded as a = Public executions; b = Private executions; c = Executions allowed but none carried out in last 10 years; d = No death penalty;

• LEGAL: The basis for the country's legal system, coded as a = Islamic; b = Civil; c = Common; d = Civil/Common; e = Socialist; f = Other;

• HDI: Human Development Index, a numerical measure ranging from 0 (very low human development) to 1 (very high human development);

• GINI: GINI Index of income inequality, a numerical measure ranging from 0 (perfect equality) to 1 (perfect inequality);

• GNI: Gross National Income per capita (US$);

• LITERACY: Literacy rate (% of adults aged 15 and above who are literate);

• URBAN: Percentage of total population living in urban areas;

• POL: Political Instability Index for the level of threat posed to governments by social protest, a numerical measure ranging from 0 (no risk) to 10 (high risk);

• CONFLICT: Level of conflict experienced in the country, coded as a = No conflict, b = latent (non-violent) conflict, c = manifest conflict, d = crisis, e = severe crisis, f = war.

The data for the 141 countries with complete records are available in the DeathPenalty.csv file.11 The response variable is the country's death penalty status, while all other variables except COUNTRY are explanatory.

(a) Fit a multinomial (nominal response) regression model to these data using all available explanatory variables as linear terms. Compute the AIC and the BIC for this model. Also compute AICc by using the formula on page 267. Note that the model degrees of freedom that are needed for computing AICc can be found from the extractAIC() function.

(b) Explain why STATUS can be viewed as an ordinal variable.

(c) Fit a proportional odds regression model (see Section 3.4) to these data using all available explanatory variables. Compute AIC, BIC, and AICc as in the multinomial regression model.

(d) Note that the proportional odds model assumes both the ordinality of the response and equal coefficients across different logits. Based on the information criteria, is there evidence to suggest that these assumptions lead to a poor model fit? Explain.