Spring 2024 Assignment 2: SOC386L/SDS385 Briefly answer the following questions. Please include selected segments of computer code you used in your write-up if needed. You may consult with other students,

but the work you hand in should reflect your own thinking. Background. The data set (RiesbyWide.dta) posted on Canvas comes from a study by Riesby et al. (1977) in which hospitalized depressed patients were treated for 5 weeks with imipramine. Clinical rating of depressive symptoms was carried out once weekly by Hamilton's Rating Scale (HRS). The data are in wide form as follows: id HamDO HamD5, EndoG, where id is the subject identifier and HamD0-HamD5 are depression measures (HamDO is the baseline or initial measurement) and Endog identifies subjects whose depression, as classified on the WHO Depression Scale, was "endogenous" or "non-endogenous" depression, with "endogenous" roughly pertaining to clinical depression in current practice. 10 points each. 1. Convert these data from the individual-level format to the "person-period" format (i.e., from wide to long). Carry out a basic exploratory analysis by graphing the raw trajectories for each subject (or randomly selected subjects) along with the linear fit through each individual's measurements in side-by-side plots (e.g., spaghetti plots). Comment on whether a linear growth curve model would seem appropriate given the observed and fitted patterns. 2. Fit an unconditional means model to these data and assess whether this model is an improve- ment over a simple regression model and why? 3. Fit an unconditional growth model to these data using a full model specification for the variance components (i.e., a model that estimates the variance of the random slope and intercept as well as their covariance). Use a likelihood ratio test to determine if this model offers an improvement over the unconditional means model. 4. Using a likelihood ratio test, test whether a more parsimonious unconditional growth model fits these data as well as the model used in Question 3. Use the preferred model as the "working" model. Interpret the fixed effects and variance components from your working unconditional growth model. 5. Under the assumptions of normality of the MLEs (i.e., asymptotic normality) we can determine some aspects of the distribution of random slopes in the depressed population. Using the slope fixed effect and the estimated variance of the random slope as a gauge of variability of the individual slopes around the fixed effect, about what proportion of the random slopes in the depressed population would you expect to be greater than 0? 6. Using the properties of MLEs we can determine some aspects of the distribution of random intercept (i.e., initial levels) in the depressed population. Using the intercept fixed effect and the estimated variance of the random intercept as gauge of variation in initial depression levels, about what proportion of the random intercepts in the depressed population would be expected be less than 20? 7. Obtain estimates of the random slope by predicting the random effect U₁; for each subject and adding these to the fixed effect ₁. Obtain estimates of the random intercept by predicting the random effect Uo; for each subject and adding these to the fixed effect Bo. Provide a histogram of the random slopes and intercepts. 1 8. Using your preferred model from Question 4, assess whether there is evidence that the "endoge- nous depression" classification is able to statistically differentiate between depression patients' initial levels of depression and also whether there is evidence that this classification moderates the change in depression scores over time. Extra Credit Problems. (2 points) Consider the OLS estimates of the individual-level slopes obtained from the marital adjustment data (see handouts 1-2). Use the information in the handout to obtain estimates of the slopes and their standard errors. We can follow the steps spelled out in the Bryk and Raudenbush article (available on Canvas) to improve upon these estimates as follows: • Denote the slope for the ith subject by bį. The total variance in the individual slope is a function of the sampling variance and the parameter variance. • The parameter variance is simply the squared standard error of the slope from the OLS model, denoted by se(bi)² = vi. • The parameter variance is the variance in the slopes, estimated as 7 = Σ¿(bi − b)²/(n − 1). • Define the weight function Wi=T/(t + vi) An improved estimate can be obtained as a weighted average of the OLS slope and the average slope. That is, b* = bįWį + (1 − W;)b 1. Use the individual slope estimates and their std. errors from handouts 1-2 to obtain the improved estimates and plot them against the original OLS estimates. Explain in what sense these estimates are "improved." ● 2. A simplified growth model can be fit using “centered” data, where the centered dependent variable is y¡ = Yij − Yi and tåj = tij – tį. Consider the following model Yij = Bitij + ε Bi = ß + Ui Discuss the differences between this model and the standard linear growth model. Fit this model as a linear mixed model. and 3. Compare the slope estimate (fixed effect) to the one from the centered model. Explain why they are similar or different. 4. Obtain the empirical Bayes estimates of the level-2 residuals, the Û¿. Compute the EB esti- mates of the slope as +₂. Compare these to the ones computed in the first extra-credit problem. 2