MATH 6643 Summer 2012 Applications of Mixed Models/Submitted sample exam questions

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Question 1:

Consider the following output:

> head(hs)
  school mathach    ses    Sex Minority Size   Sector PRACAD DISCLIM
1   1317  12.862  0.882 Female       No  455 Catholic   0.95  -1.694
2   1317   8.961  0.932 Female      Yes  455 Catholic   0.95  -1.694
3   1317   4.756 -0.158 Female      Yes  455 Catholic   0.95  -1.694
4   1317  21.405  0.362 Female      Yes  455 Catholic   0.95  -1.694
5   1317  20.748  1.372 Female       No  455 Catholic   0.95  -1.694
6   1317  18.362  0.132 Female      Yes  455 Catholic   0.95  -1.694
> fit <- lme( mathach ~  ses * cvar(ses,school), hs, 
+             random = ~ 1 + ses|school)
> summary(fit)
Linear mixed-effects model fit by REML
 Data: hs 
       AIC      BIC    logLik
  12846.85 12891.54 -6415.423

Random effects:
 Formula: ~1 + ses | school
 Structure: General positive-definite, Log-Cholesky parametrization
            StdDev    Corr  
(Intercept) 1.6293867 (Intr)
ses         0.6614903 -0.469
Residual    6.1109156       

Fixed effects: mathach ~ ses * cvar(ses, school) 
                          Value Std.Error   DF  t-value p-value
(Intercept)           12.681917 0.3054760 1935 41.51526  0.0000
ses                    2.243374 0.2416545 1935  9.28339  0.0000
cvar(ses, school)      3.687892 0.7699000   38  4.79009  0.0000
ses:cvar(ses, school)  0.873953 0.5771829 1935  1.51417  0.1301
 Correlation: 
                      (Intr) ses    cv(,s)
ses                   -0.188              
cvar(ses, school)      0.022 -0.261       
ses:cvar(ses, school) -0.258  0.065  0.014

Standardized Within-Group Residuals:
       Min         Q1        Med         Q3        Max 
-3.2291287 -0.7433282  0.0306118  0.7770370  2.6906899 

Number of Observations: 1977
Number of Groups: 40 

  • Identify the expression of the model in mathematical formula and the usual model assumption.
  • Find out the variances of the group effects, residuals and the response.
  • Sketch the estimated response function for a school with mean ses of 0.
  • For what value of ses is the variance of mathach estimated to be minimized.

Question 2:

Longitudinal data analysis with mixed models: Consider a mixed model with random intercept and slope with respect to time, T. Suppose that the G matrix is


\begin{bmatrix}
\tau_{00}      &  \tau_{01}      \\
\tau_{10}      & \tau_{11} 
\end{bmatrix}
  • Find the value of T for which the variance of Y is minimized and the minimum variance.
  • Show that recentering T on this value (if known) turns the G matrix into one with only two free parameters.
  • Sketch a data plot to show the location and value of the minimum standard deviation of lines.

Question 3:

Explain why would would want to add \bar{X}_j or SDj to a multilevel model.

Question 4:

Discuss the interpretation/ramifications of extreme ICC values in the design of a multilevel study.

Question 5:

Explain the similarities and differences between within effects, between effects and clusters.

Question 5:

Discuss/explain the relationship between longitudinal data and hierarchical data with appropriate examples and theory.

Question 6:

Let Σ be symmetric. Show that Σ is positive-definite if and only there exists a non-singular matrix A such that Σ = AA'

Question 7:

Why do we study longitudinal and mixed models?

Ryan's Questions

Early Chapters/Course Content:

What is Simpson's Paradox and how is it related to HLM? - Please describe the necessary relationships and sketch them accordingly.

Under what two conditions might someone be completely unconcerned with the possible problems associated with this paradox?

Later Chapters/Course Content:

In Chapter 11 Snijders & Bosker discuss the problem of optimal sample size in order to obtain accurate estimates of the ICC. Explain the reasoning behind the process of optimization for the hierarchical sample (assuming equal cluster sizes with a sample of size M with N clusters of size n).

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