Variation in Response to Stress Treatment Program: A Study of Men and Women

Assignment Task

The data set (anxietyL.raw, anxietyL.dta) (posted on Canvas) is a set of 11 clinical measurements on a widely used anxiety scale taken over a period of 5 years for a sample of 20 individuals in a stress disorder treatment program. The order of the variables is as follows: id coded 1-20, wave coded 1-11, y, time coded 0-60 months, female coded 1 if respondent is female, 0 otherwise. The general relationship between the anxiety measure, time, and sex is shown in the figure below.

1. Make some tentative observations about possible differences between men and women on their response to the stress treatment program. For example, what can you learn about the variance in the intercept, slope, and possible differences in fixed effects from the figure above?

2. Determine the better fitting unconditional growth curve model for these data by using likelihood ratio tests of two alternative unconditional growth models - one of which is a constrained version of the other. Note: Ignore gender for the time being.

3. Use the results from the model in question 2 for the following:

(a) Graph the residual variance at 6 months, 1 year, 3 years, and 5 years using the formula for the composite variance given in Singer and Willett and in a previous handout (use the entire range if you prefer) and comment on the nature of heteroskedasticity in the residual variance measurements.

(b) How is the pattern of heteroskedasticity over time affected by the relative value of σ _²1 ?

4. Determine the residual autocorrelation between the composite residuals at 6 months, 1 year, 3 years, and 5 years using the formulas given by Singer and Willett and in a previous handout (use/graph the entire range if you prefer).

5. Suppose that a “heterogeneous 1st-order autoregressive” error covariance structure fit to a 3-wave panel study yielded the following variances: σ ²₁ = 1000, σ ²₂ = 1100, σ ² ₃ = 1500, with ρ = .8. Use this information to produce the fitted composite error covariance matrix ΣbR (hint: see page 259 in Singer and Willett).

6. Expand on your linear mixed model fit in question 2 to test if separate fixed effects by sex are warranted. Let F denote a dummy variable for “female” and let TF = F ×T denote the interaction of “female” with time. Compare the fit of the model in question 2 to the following model:

y_ij = β₀ + β_{F 0}F_i + β₁T_ij + β_{F 1}F_iTF _ij + U_0i + U_1iT_ij + ϵ_ij .

7. Let F denote a dummy variable for “female” and let M denote a dummy variable for “male.” Let TF = F × T and TM = M × T denote the respective interactions with time.

a) Fit a model that includes fixed effects of male and female, in addition to male × time and female × time interactions as well as random intercepts and slopes that vary by sex (heterogeneous random effects by sex). In this case, the model would be,

y_ij = β_M0M_i + βF 0Fi + β_M1T_{M ij} + β_{F 1}T_{F ij} + U₀M _i + U₀F _i + U_{1M i}T_{M ij} + U_{1F i}T_{F ij} + ϵ_ij ,

b) Determine if heterogeneous variance components by sex are supported by testing against your preferred model thus far based on question 6.

8. For paired data, such as what one might find when examining traits of twins or siblings, the correlation of traits within a pair will range from near perfect association (identical twins) to no correlation (adopted siblings). Use the twin/pair simulation programs (either in R or Stata) on Canvas to demonstrate the effect on the level-1 and level-2 variance components of varying the correlation between siblings using selected values of ρ in the range (0, 1). Describe—and show by a graph if helpful—the relationship between σb 2 ϵ and σb 2 0 based on the output of the unconditional means model yij = β0i + εij , (j = 1, 2).

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