Stats 112/203: Sample Midterm Exam Solutions

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1 Stats 112/203: Sample Midterm Exam Solutions 1. Consider the following linear mixed model: Y ij = β 0 + b 0i + (β 1 + b 1i )t ij + ɛ ij, where Y ij is the response measurement for the ith individual s jth occasion taken at time t ij. The coefficients β 0 and β 1 are fixed effects. Assume and ( b 0i ( b0i b 1i ) iid N N(0, σ 2 ), (( ) ( 0 σ 2, 0 σ 01 0 σ 01 σ1 2 ɛ ij iid b 1i ) and ɛij are independent. )), (a) Derive the marginal variance V ar(y ij ) and covariance Cov(Y ij, Y ik ) (j k). (b) What is the conditional variance V ar(y ij b 0i, b 1i )? (c) Suppose the observed responses for the first individual are Y 11 = 32, Y 12 = 24, Y 13 = 25, and Y 14 = 28, observed at days t 11 = 0, t 12 = 8, t 13 = 16, and t 14 = 30. Write the linear mixed model for this observation in matrix notation: Y i = X i β + Z i b i + ɛ i. (d) Suppose the true values of the fixed effects are β 0 = 20 and β 1 = 5. Draw a time plot that demonstrates Solution: (a) i the overall population mean response trajectory, ii a hypothetical random sample of three individual mean response trajectories, and iii the hypothetical observed response measurements for each of the three chosen individuals under each of the following conditions: 1. σ 2 0 = 0 and 0 < σ2 < σ σ 2 1 = 0 and 0 < σ2 < σ σ 2 = 0, σ 2 0 > 0, and σ2 1 > 0. V ar(y ij ) = Cov(β 0 + b 0i + (β 1 + b 1i )t ij + ɛ ij, β 0 + b 0i + (β 1 + b 1i )t ij + ɛ ij ) = Cov(b 0i + b 1i t ij + ɛ ij, b 0i + b 1i t ij + ɛ ij ) = V ar(b 0i ) + t 2 ijv ar(b 1i ) + V ar(ɛ ij ) + 2t ij Cov(b 0i, b 1i ) + 2Cov(b 0i, ɛ ij ) + 2t ij Cov(b 1i, ɛ ij ) = σ t 2 ijσ σ 2 + 2t ij σ = σ t 2 ijσ t ij σ 01 + σ 2 since b 0i and ɛ ij are independent, and b 1i and ɛ ij are independent. 1

2 Similarly, Cov(Y ij, Y ik ) = Cov(β 0 + b 0i + (β 1 + b 1i )t ij + ɛ ij, β 0 + b 0i + (β 1 + b 1i )t ik + ɛ ik ) = Cov(b 0i + b 1i t ij + ɛ ij, b 0i + b 1i t ik + ɛ ik ) = V ar(b 0i ) + t ij t ik V ar(b 1i ) + Cov(ɛ ij, ɛ ik ) + (t ij + t ik )Cov(b 0i, b 1i ) + Cov(b 0i, ɛ ij ) + Cov(b 0i, ɛ ik ) + t ij Cov(b 1i, ɛ ij ) + t ik Cov(b 1i, ɛ ik ) = σ0 2 + t ij t ik σ1 2 + (t ij + t ik )σ 01 since b 0i, ɛ ij and ɛ ik are independent; and b 1i, ɛ ij and ɛ ik are independent. (b) V ar(y ij b 0i, b 1i ) = V ar(ɛ ij ) = σ 2. (c) = [ β0 β 1 ] [ b01 b 11 ] ɛ 11 + ɛ 12 ɛ 13 ɛ 14 (d) σ 2 0 = 0 and 0 < σ2 < σ 2 1 : Problem 2(b)1. Y Pop. Mean Indiv. 1 Indiv. 2 Indiv Time 2

3 σ 2 1 = 0 and 0 < σ2 < σ 2 0 : Problem 2(b)2. Y Pop. Mean Indiv. 1 Indiv. 2 Indiv Time σ 2 = 0, σ 2 0 > 0, and σ2 1 > 0: Problem 2(b)3. Y Pop. Mean Indiv. 1 Indiv. 2 Indiv Time 3

4 2. Data were collected on N = 22 students by the Minneapolis Public School District in Minnesota to comply with federal accountability requirements, namely Title X of the No Child Left Behind Act of Reading achievement scores (Read) were measured in grades 5, 6, 7, and 8 for each student in the study. The predictor of interest is called Risk, which takes on the value DADV if the student was designated as disadvantaged (based on poverty or homeless measures), and ADV if the student was designated as advantaged. R code, output, and summary plots are in the Appendix. (a) Write out the equation of the mod1 fitted model for the subject-specific (conditional) mean response. Define any variables used. (b) Give an interpretation of the estimated Grade coefficient in mod1 in context of the problem. (c) Does the intercept for mod1 have a useful interpretation? If yes, give the interpretation. If no, explain why. (d) What is the estimated variance of the random slopes using mod1.reml? Provide an interpretation of the magnitude of this value in context of the problem. (e) Here are the predicted random effects (EBLUPs) for individual 1, who was classified as disadvantaged: > ranef(mod1.reml) (Intercept) Grade You may leave your answers to the following three questions unsimplified. i. What is the predicted subject-specific slope for individual 1? ii. What is the predicted mean reading score for individual 1 in grade 9? iii. What is the estimated marginal mean reading score for disadvantaged students in grade 9? (f) The R output includes a likelihood ratio test of mod2.reml vs. mod1.reml. State the null and alternative hypotheses being tested (in mathematical symbols), the p-value (report at least four digits), and state a conclusion of the test in context of the problem. (g) Write a short paragraph addressing the effect of being advantaged or disadvantaged on the changes in reading scores across grades. Include both an estimate of the effect as well as results from an appropriate test assessing statistical significance of the effect. 4

5 Solution: (a) where ˆ Y ij ˆb i = D i t ij t ij D i + ˆb 0i + ˆb 1i t ij Ŷij is the fitted mean reading achievement score of the ith individual in grade t ij, t ij is the grade level of individual i at the time of occasion j, and { 1 individual i is disadvantaged D i = 0 individual i is advantaged (b) For advantaged students, the estimated mean reading score increases by per grade level. (c) No. The intercept is the mean reading level at grade zero for advanted students, which is an extrapolation beyond the observed data (grades 5-8). (d) Vˆar(b 1i ) = (2.8595) 2 = Approximately 95% of advantaged students have an estimated mean change in reading score between a 1.15 decrease and a increase per grade level (4.570± 2(2.860)). (e) i = ii ( )(9) = iii ( )(9) = (f) We are testing the set of hypotheses H 0 : σ 2 1 = 0 H a : σ 2 1 > 0 where σ 2 = V ar(b 1i ). The p-value is (from a χ 2 (2) and χ 2 (1) mixture distribution), which is less than a significance level of 0.05, so we reject H 0. There is statistically significant evidence of individual variation in the mean change in read reading score per grade level. (g) Disadvantaged students have an estimated increase in mean reading score that is larger than the estimated increase in mean reading score for advantaged students per grade level. However, this difference between the two groups is not statistically significant (p-value = 0.710). 5

6 3. Consider a three-year longitudinal study of binge drinking among 100 college students, with a weekly indicator variable Y ij as the response variable (1 for any binge drinking episode that week; 0 otherwise). Covariates are time t ij in weeks, and membership in organized group housing X i (1 for yes, such as fraternity, sorority, or student cooperative; 0 otherwise). (Note that membership in organized group housing for an individual does not change over time.) One of the goals of the study is to assess changes in binge drinking behavior over time, and particularly whether changes over time are influenced by membership in group housing. (a) For the goal of this study, would a model for population-averaged effects or subject-specific effects be more appropriate? Explain. (b) Write out an appropriate generalized linear mixed model for this study, including any model assumptions. (c) Using your model from part (a), write out the null and alternative hypotheses that match the study goal of determining whether changes over time are influenced by membership in group housing in terms of the model parameters. Solution: (a) To assess within-subject changes in binge drinking behavior over time, a subjectspecific model would be more appropriate. For the question of whether changes over time are influenced by membership in group housing, a population-averaged model would be more appropriate since group membership does not change over time within an individual in this study. (b) Assume Y ij b i Bin(1, π ij ) where we model the conditional probability of any binge drinking episode in week j for individual i, π ij = E(Y ij b i ), as ( ) πij log = β 0 + b i + β 1 X i + β 2 t ij + β 3 X i t ij, 1 π ij where we assume b i iid N(0, σ 2 b ). (Note that the model must include the interaction term between X i and t ij in order to assess whether changes over time are influenced by membership in group housing.) (c) H 0 : β 3 = 0 vs. H a : β

7 Appendix: R code, output, and plots for Problem 2 Individual Reading Score Trajectories Mean Reading Score Trajectories by Group Reading Score Advantaged Disadvantaged Mean Reading Score Advantaged Disadvantaged Grade Grade > mod1 = lme(read ~ Risk*Grade, random = ~ 1+Grade ID, method="ml",...) > summary(mod1) Linear mixed-effects model fit by maximum likelihood... Fixed effects: Read ~ Risk * Grade Value Std.Error DF t-value p-value (Intercept) RiskDADV Grade RiskDADV:Grade > mod1.reml = lme(read ~ Risk*Grade, random = ~ 1+Grade ID, method="reml",...) > summary(mod1.reml) Linear mixed-effects model fit by REML... Random effects: Formula: ~1 + Grade ID Structure: General positive-definite, Log-Cholesky parametrization StdDev Corr (Intercept) (Intr) Grade Residual > mod2.reml = lme(read ~ Risk*Grade, random = ~ 1 ID, data=dat.long, method="reml", na.action=na.omit) > anova(mod2.reml,mod1.reml) Model df AIC BIC loglik Test L.Ratio p-value mod2.reml mod1.reml vs > pchibarsq( , 2, lower.tail=false) [1] > pchibarsq( , 1, lower.tail=false) [1]

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