Problems and answers, R studio only
STA 542 Final Exam Name (Print): Question: 1 2 3 4 Total Points: 30 15 20 35 100 Score: Some formulas 1. Simple logistic regression: logit[π(x)] = α+ βx π(x) = exp(α+ βx) 1 + exp(α+ βx) 2. Likelihood ratio test LR statistic = −2(`0 − `1) = Deviance0−Deviance1, df = df0 − df1 3. Baseline-category logit model has form log ( πi πJ ) = αj + βjx, j = 1, 2, . . . , J − 1. 4. Cumulative logit model has form logit[Pr(Y ≤ j)] = αj + βx, j = 1, . . . , J − 1 odds(Y ≤ j|x+ 1) odds(Y ≤ j|x) = eβ 1. Let π denote the probability that a randomly selected respondent supports current 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 party affiliation (P1 = 1, Democrat, 0 otherwise; P2 = 1, Republican, 0 otherwise, P1 = P2 = 0, Independent). The logit model with main effects has prediction equation logit(π̂) = 0.11 + 0.16G− 0.57R1 − 0.66R2 + 0.47P1 − 1.67P2 (a) (7 points) Controlling for religious affiliation and political party affiliation, are Females estimated to be more likely than males to support legalized abortion? Why? (b) (7 points) Controlling for gender and religious affiliation, what is ratio between the estimated odds that a Democrat supports legalized abortion and the estimated odds that a Republican supports legalized abortion? (c) (8 points) What is the estimated probability that a male Jewish Independent supports legalized abortion? (d) (8 points) Which group has the highest estimated probability of supporting legalized abortion? Page 2 2. Consider the loglinear model of independence for a two-way contingency table. This has equation for expected frequencies {µij} in an I × J contingency table, logµij = λ+ λ X i + λ Y j (a) (5 points) Motivate this model, by showing how the definition of statistical independenceof two categorical variables implies that a loglinear model of this form holds. (b) (10 points) To allow for association between X and Y, this model is extended to logµij = λ+ λ X i + λ Y j + λ XY ij For a 2 × 2 contingency table, express the log odds ratio in terms of expected frequencies, and use it to show that the odds ratio for this model equalsexp ( λXY11 + λ XY 22 − λXY12 − λXY21 ) . (Hence the two-factor interaction parameters provide information about the XY association.) 3. A model fit predicting preference for President (Democrat, Republican Independent) suing x = annual income (in $10.000 dollars) is log(π̂D/π̂I) = 3.0− 0.3x and log(π̂R/π̂I) = 1.0 + 0.28x (a) (6 points) State the prediction equation for log(π̂R/π̂D). Interpret its slope. (b) (7 points) Find the range of x for which π̂R > π̂D (c) (7 points) State the prediction equation for π̂I Page 3 4. Let Y = political ideology (on an ordinal scale from 1 = very liberal to 5 = very conservative), x1 = gender (1 = female, 0 = male), x2 = political party (1 = Democrat, 0 = Republican). (a) (7 points) A main effects model with a cumulative logit link gives the output shown. Explain why the output reports four intercepts. ## Estimate Std. Error z value Pr(>|z|) ## (Intercept):1 -2.532 0.150 -16.934 2.53e-64 ## (Intercept):2 -1.539 0.130 -11.879 1.53e-32 ## (Intercept):3 0.175 0.117 1.496 1.35e-01 ## (Intercept):4 1.009 0.124 8.116 4.83e-16 ## PartyDem 0.964 0.129 7.449 9.39e-14 ## GenderFemale 0.117 0.127 0.921 3.57e-01 (b) (7 points) Explain how to describe gender effect on political ideology with an odds ratio. (c) (7 points) Beloew is the results of LRT of the two explanotory vairables. What is the hypotheses to which the LRT for gender refers? And explain how to interpret the result of the test. ## Single term deletions ## ## Model: ## cbind(VLib, SLib, Mod, SCon, VCon) ~ Party + Gender ## Df Deviance AIC LRT Pr(>Chi) ##
15.1 107 ## Party 1 71.9 162 56.8 4.7e-14 *** ## Gender 1 15.9 106 0.8 0.36 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 (d) (7 points) 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 on political ideology for Republicans. ## Estimate Std. Error z value Pr(>|z|) ## (Intercept):1 -2.6743 0.166 -16.111 2.14e-58 ## (Intercept):2 -1.6772 0.148 -11.316 1.09e-29 ## (Intercept):3 0.0424 0.135 0.313 7.54e-01 ## (Intercept):4 0.8790 0.141 6.255 3.97e-10 ## PartyDem 1.2653 0.197 6.419 1.38e-10 ## GenderFemale 0.3661 0.180 2.037 4.16e-02 ## PartyDem:GenderFemale -0.5091 0.254 -2.004 4.51e-02 (e) (7 points) Using the interaction model, show how to find the estimated probability that a female Republican is in the first category (very liberal). Page 4