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

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'''Early Chapters/Course Content:'''
'''Early Chapters/Course Content:'''
-
We learned about a simulation conducted in which the estimate of the slope from a mixed model (without contextual effect) was obtained varying the within-cluster variance.  What are the implications of this simulation study?
+
We learned about a simulation conducted in which the estimate of the slope from a mixed model (without contextual effect) was obtained varying the within-cluster variance (see figure below).  What are the implications of this simulation study?
[[File:MixedModelSimResult.png|350px]]
[[File:MixedModelSimResult.png|350px]]
'''Later Chapters/Course Content:'''
'''Later Chapters/Course Content:'''
 +
Data is obtained from on the effects of two treatments (A and B).  Data was recorded on symptoms weekly throughout 12 weeks of active treatment and then for 8 more weeks following termination of the treatment.  A linear spline model is fit to the data and the following results were obtained:
Data is obtained from on the effects of two treatments (A and B).  Data was recorded on symptoms weekly throughout 12 weeks of active treatment and then for 8 more weeks following termination of the treatment.  A linear spline model is fit to the data and the following results were obtained:
::
::

Revision as of 23:14, 16 July 2012

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).

Carrie's Questions

Early Chapters/Course Content:

We learned about a simulation conducted in which the estimate of the slope from a mixed model (without contextual effect) was obtained varying the within-cluster variance (see figure below). What are the implications of this simulation study?

MixedModelSimResult.png

Later Chapters/Course Content:

Data is obtained from on the effects of two treatments (A and B). Data was recorded on symptoms weekly throughout 12 weeks of active treatment and then for 8 more weeks following termination of the treatment. A linear spline model is fit to the data and the following results were obtained:

> fit <- lme( Symptom ~ sp(Weeks)*tx, random=~1+Weeks|id, data = d)
> summary(fit)
Linear mixed-effects model fit by REML
 Data: d 
       AIC      BIC    logLik
  15099.44 15153.18 -7539.719

Random effects:
 Formula: ~1 + Weeks | id
 Structure: General positive-definite, Log-Cholesky parametrization
            StdDev    Corr  
(Intercept) 37.921194 (Intr)
Weeks        2.395329 -0.218
Residual    23.285420       

Fixed effects: Symptom ~ sp(Weeks) * tx 
                         Value Std.Error   DF   t-value p-value
(Intercept)          131.84484  6.450470 1516  20.43957  0.0000
sp(Weeks)D1(0)       -11.21889  0.508022 1516 -22.08346  0.0000
sp(Weeks)C(10).1      18.24584  0.571104 1516  31.94835  0.0000
txB                   -2.61621  9.122343   78  -0.28679  0.7750
sp(Weeks)D1(0):txB     1.55017  0.718452 1516   2.15765  0.0311
sp(Weeks)C(10).1:txB  -4.83791  0.807664 1516  -5.99001  0.0000
 Correlation: 
                     (Intr) sp(W)D1(0) sp(W)C(10).1 txB    s(W)D1(0):
sp(Weeks)D1(0)       -0.371                                          
sp(Weeks)C(10).1      0.256 -0.604                                   
txB                  -0.707  0.262     -0.181                        
sp(Weeks)D1(0):txB    0.262 -0.707      0.427       -0.371           
sp(Weeks)C(10).1:txB -0.181  0.427     -0.707        0.256 -0.604    

Standardized Within-Group Residuals:
         Min           Q1          Med           Q3          Max 
-4.109811205 -0.593593851 -0.002546668  0.654503092  3.122385555 

Number of Observations: 1600
Number of Groups: 80 
  • Interpret the coefficients in the model.
  • Sketch the predicted trajectories.
  • Bonus: How would you test whether there is a significant difference in the predicted symptom score at week 10 or week 18?
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