STA 4504/5503Sample questions for exam 21. True-False questions.(a)For General Social Survey data on Y = political ideology (cat-egories liberal, moderate, conservative), X1= gender (1 = female, 0 =male), and X2= political party (1 = Democrat, 0 = Republican), the MLfit of the cumulative logit model is logit[ˆP (Y ≤ j)] = ˆαj+ .12x1+ .96x2.Hence, for each gender, according to this model fit the estimated odds thata Demo crat’s response is liberal rather than moderate or conser vative, andthe estimated odds that a Democrat’s response is liberal or moderate ratherthan conservative, is e.96= 2.6 times the corresponding estimated odds fora Republican’s response. This odds ratio estimate indicates that in thissample Democrats tended to be more liberal than Republicans.(b)Subj ects suffering from mental depression a re measured after 1week of treatment, 2 weeks of treatment, and 4 weeks of treatment interms of a ( nor mal, a bnormal) response outcome. Covar ia t es are severityof condition at original diagnosis (1 = severe, 0 = mild) and treatmentused (1 = new, 0 = standard). Since each subject contributes three ob-servations to the analysis, we can use the GEE (generalized estimatingequations) method to fit the model. To use this method, we must choosea “working” correlation matrix for the form of the dependence among thethree responses, but the method is robust in the sense that it still givesappropriate estimates and standard errors for large n even if the actualcorrelation structure is somewhat different from the one we assumed.(c)A difference between logit and loglinear models is that the logitmodel is a generalized linear model assuming a binomial random compo-nent whereas the loglinear model is a generalized linear model assuminga Poisson random component. Hence, when both are fitted to a contin-gency table having 50 cells, the logit model treats the cell counts as 25binomial observatio ns whereas the loglinear model treats the cell counts as50 Poisson o bservations.(d) The cumulative logit model assumes that the response variable Yis ordinal; it should not be used with nominal variables. By contrast, thebaseline-category logit model treats Y as nominal. It can be used withordinal Y , but it then ignor es the ordering information.(e)The cumulative logit model for J r esponse categories correspondsto a logistic regression model holding for each of the J − 1 cumulativeprobabilities, such that the curves for each cumulative probability haveexactly the same shape (i.e., the same β parameter); t hat is, they increase1or decrease at the same rate, so one can useˆβ to describe effects that applyto all J − 1 of the cumulative probabilities.(f)If X a nd Y are binary, and Z has K categories, so the data canbe summarized in a 2 × 2 × K contingency table, one can test condi-tional independence of X and Y , controlling for Z, using a Wald test or alikelihood-ratio test of H0: β = 0 in the modellogit[P (Y = 1)] = α + βx + β1z1+ ···+ βK−1zK−1,where zi= 1 fo r observations in category i of Z and zi= 0 otherwise.(g)For a sample o f retired subjects in Florida, a contingency tableis used to relate X = cholesterol (8 ordered levels) to Y = whether thesubject has symptoms of heart disease (yes = 1, no = 0). For the linearlogit model logit[P (Y = 1)] = α + βx fitted to the 8 binomials in the8 × 2 contingency table by assigning scores to the 8 cholesterol levels, thedeviance statistic equals 6.0. Thus, this model provides a poor fit to thedata.(h) In the example j ust mentioned, at the lowest cholesterol level,the observed number of heart disease cases equals 31. The standardizedresidual equals 1.35. This means that the model predicted 29.65 cases (i.e.,1.35 = 31 - 29.6 5).2. Multiple choice question. Circle the letter(s) for the correct response(s). Morethan one response may be correct.Let π denote the probability that a r andomly selected respondent supportscurrent laws legalizing abortion, predicted using gender of respondent (G = 0,male; G = 1, female), religious affiliation (R1= 1, Protestant, 0 otherwise;R2= 1, Catholic, 0 otherwise; R1= R2= 0, Jewish), and political partyaffiliation (P1= 1, Democrat, 0 otherwise; P2= 1, Republican, 0 otherwise,P1= P2= 0, Independent). The logit model with main effects has predictionequationlogit(ˆπ) = .11 + .16G − .57R1− .66R2+ .47P1− 1.67P2For this prediction equation,a. Females are estimated to be more likely than males to support legalizedabortion, controlling for religious affiliation and political party affiliation.b. Controlling for gender and religious affiliation, the estimated odds that aDemocrat supports legalized abortio n equal e.47−(−1.67)times the estimatedodds that a Republican supports legalized abor t io n.2c. The estimated probability that a male Jewish Independent supports legal-ized abortion equals e.11/(1 + e.11).d. The estimated probability of supporting legalized abortion is highest forfemale Jewish Independents.3. Let Y = political ideology (on an ordinal scale from 1 = very liberal to 5 =very conservative), x1= g ender (1 = female, 0 = male), x2= political party (1= D emocrat, 0 = Republican).(a) A main effects model with a cumulative logit link gives the output shown.Explain why the output reports four intercepts.Standard Wald 95% ConfidenceParameter DF Estimate Error LimitsIntercept1 1 -2.5322 0.1489 -2.8242 -2.2403Intercept2 1 -1.5388 0.1297 -1.7931 -1.2845Intercept3 1 0.1745 0.1162 -0.0533 0.4023Intercept4 1 1.0086 0.1232 0.7672 1.2499gender female 1 0.1169 0.1273 -0.1327 0.3664gender male 0 0.0000 0.0000 0.0000 0.0000party democ 1 0.9636 0.1297 0.7095 1.2178party repub 0 0.0000 0.0000 0.0000 0.0000LR Statistics For Type 3 AnalysisChi-Source DF Square Pr > ChiSqgender 1 0.84 0.3586party 1 56.85 <.0001(b) Explain how to describe gender effect on political ideology with an oddsratio.(c) Give the hypotheses to which the LR statistic for gender refers, and explainhow to int erpret the result of the test.(d) When we add an interaction term to the model, we get the output shown.Explain how to find the estimated odds ratio for the gender effect onpolitical ideology for Republicans.3StandardParameter DF Estimate ErrorIntercept1 1 -2.6743 0.1655Intercept2 1 -1.6772 0.1476Intercept3 1 0.0424 0.1338Intercept4 1 0.8790 0.1389gender female 1 0.3661 0.1784gender male 0 0.0000 0.0000party democ 1 1.2653 0.1995party repub 0 0.0000 0.0000gender*party female democ 1 -0.5091 0.2550gender*party