1. Summary regression table.
2. Compute the goodness-of-fit measures of the deviance, AIC, and BIC for each model. Add these to your summary regression table.
See Table 1.
3. Based on the goodness-of-fit measures, which model should be adopted as the ‘best fitting’ model? Explain.
Model C has the lowest deviance, AIC, and BIC, and should therefore be adopted as the 'best fitting' model.
4. Compute the proportional reduction in the Level-1 (residual) variance component for Model B and Model C. Add these to your summary regression table.
See Table 1.
5. Interpret the proportional reduction in the Level-1 variance component for Model C.
Compared to the intercept-only model (Model A), including the factors of extraversion, gender, and teacher experience reduces the variation attributed to individuals by 54.7%.
6. Compute the proportional reduction in the Level-2 (intercept, extra) variance components for Model C. Add these to your summary regression table.
See Table 1.
7. Interpret the proportional reduction in the Level-2 variance components for Model C.
When including the factor of teacher experience, the variation in class-wise intercepts is reduced by 81.4%. Additionally, the variation in class-wise slopes is reduced by 86.2%.
8. Compute the squared correlation between the observed and fitted popularity values for all three models. Add these to your summary regression table.
See Table 1.
9. Interpret the squared correlation between the observed and fitted popularity values for Model C.
The squared correlation describes the degree to which the observed popularity scores correlate with the popularity scores predicted by Model C. The correlation (r = 0.727) is quite a strong correlation, suggesting that the model does a pretty reasonable job of estimating the popularity scores.
10. Explain why it is not unexpected that the proportional reduction in the Level-1 (residual) variance for Model B is greater than zero.
It is not unexpected that the proportional reduction in the Level-1 variance for Model B is greater than zero because the factors of gender and extraversion are factors that, intuitively, would be important predictors in determining one's popularity.
11. Explain why it is not unexpected that the proportional reduction in the Level-1 (residual) variance for Model B and Model C are similar.
It is not unexpected that the proportional reduction in the Level-1 (residual) variance for Model B and Model C are similar because the additional factors in Model C account for variation between-classes moreso than between-individuals. Therefore the Level-1 residual variance is nearly identical for both models.
12. Create boxplots of the level-1 residuals conditioned on classroom for the three models. Be sure that the plots are directly comparable by using the same limits or facetting.
13. In 1–2 sentences, comment on what you see in the plot you just created.
Here (and in subsequent related questions), 25 classes were randomly sampled from the 100 classes in the data set for ease of interpretation. The trends are such that the variability in level-1 residuals is quite varied for Model A, and - for the most part - within +/- 2 standard deviations of the mean for Models B and C.
14. Create boxplots of the level-2 residuals conditioned on classroom for Models B and C. Be sure that the plots are directly comparable by using the same limits or facetting.
15. In 1–2 sentences comment on what you see in the plot you just created.
The variation in Level-2 residuals is much greater for Model B than Model C, for both intercepts and slopes. The magnitude of the variation reduction appear to be roughly similar for both intercepts and slopes. Also, the distribution around the mean residual is much more even for Model C for both intercepts and slopes.
16. Create density plots of the level-1 and level-2 residuals from Model C to help you evaluate the assumption of normality for the mixed-effects model.
17. In 1–2 sentences, comment on whether you believe the normailty assumption has been satisfied.
Taking together the density plots and the boxplots from #14, I think it's reasonable to assume normality.
18. Create a single display that you believe is the best visual representation of the results of this anal- ysis. We recognize that there are many viable alternatives; in choosing among them, be sure to consider the substantive points you want to make and create a graph that best allows you to high- light these conclusions.
Mean center each of the continuous predictors. Re-fit Model C using any of the newly created centered predictors rather than their uncentered counterparts.
19. Add the results of fitting the “centered” model to your summary regression table. Call this model, ‘Model D’. Be sure to also compute and include any goodness-of-fit measures for this model in your table as well.
See Table 1.
20. How do the fixed-effects and standard errors from Model D compare to those from Model C?
The interpretation of the mean-centered fixed-effects/standard errors is different for Model D, such that the coefficients are relative to the mean value of each predictor, but the coefficient values are the same because the transformation of mean-centered data is linear, and doesn't change the variation of data.
21. How do the random-effects from Model D compare to those from Model C?
The Level-1 residual variance and Level-2 residual variance for slopes are identical. Because the fixed-effect of intercept is different in Model D, the random-effect (Level-2 intercepts) is also different.
22. How do the goodness-of-fit measures (deviance, AIC, and BIC) from Model D compare to those from Model C?
The goodness-of-fit measures are identical for the two models.
23. How do the pseudo-R2 measures from Model D compare to those from Model C?
The pseudo-R2 measures for Model D are identical to Model C in terms of the proportional reduction in Level-1 residual variance, the Level-2 proportional reduction in slopes, and the squared correlations between observed and predicted values of probability. However, because the fixed-effect and random-effect of intercept is different in Model D, the Level-2 pseudo-R2 for intercept is greater in Model D.