MATH 6643 Summer 2012 Applications of Mixed Models/Snijders and Bosker: Discussion Questions

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Chapter 5

Alex

Matthew

At the bottom of page 83, Snijders and Bosker outline the process for probing interactions between two level one variables, and how there can be four possibilities for how to model it. If a researcher was to include all four, discuss how each would be interpreted. What might a good selection strategy be if our model had substantially more than two variables?

Qiong

If we do not have any information about the data set, how to choose a level - two variable to predict the group dependent regression coefficients? After we choose the level - two variable z, how to explain the cross - level interaction term.

Carrie

A client arrives with a random slope and intercept model using IQ as a predictor. IQ was measured on the traditional scale with a mean of 100 and standard deviation of 15. What should the client keep in mind about the interpretation of the variance of the intercept and covariance of the slope-intercept?

This raises the interesting question of how the variance of the random intercept and the covariance of the random intercept with the random slope are changed under a recentering of IQ. Let
Var\left[ \begin{matrix}
   {{u}_{0j}}  \\
   {{u}_{1j}}  \\
\end{matrix} \right]=\left[ \begin{matrix}
   \tau _{0}^{2} & {{\tau }_{01}}  \\
   {{\tau }_{10}} & \tau _{1}^{2}  \\
\end{matrix} \right] for raw IQ.
If we recenter IQ with: \tilde{\text{IQ}}=\text{IQ}-c then:
\left[ \begin{matrix}
   {{{\tilde{u}}}_{0j}}  \\
   {{{\tilde{u}}}_{1j}}  \\
\end{matrix} \right]=\left[ \begin{matrix}
   1 & c  \\
   0 & 1  \\
\end{matrix} \right]\left[ \begin{matrix}
   {{u}_{0j}}  \\
   {{u}_{1j}}  \\
\end{matrix} \right]
and
Var\left[ \begin{matrix}
   {{{\tilde{u}}}_{0j}}  \\
   {{{\tilde{u}}}_{1j}}  \\
\end{matrix} \right]=\left[ \begin{matrix}
   1 & c  \\
   0 & 1  \\
\end{matrix} \right]\left[ \begin{matrix}
   \tau _{0}^{2} & {{\tau }_{01}}  \\
   {{\tau }_{10}} & \tau _{1}^{2}  \\
\end{matrix} \right]\left[ \begin{matrix}
   1 & 0  \\
   c & 1  \\
\end{matrix} \right]=\left[ \begin{matrix}
   \tau _{0}^{2}+2c{{\tau }_{01}}+{{c}^{2}}\tau _{1}^{2} & {{\tau }_{01}}+c\tau _{1}^{2}  \\
   {{\tau }_{10}}+c\tau _{1}^{2} & \tau _{1}^{2}  \\
\end{matrix} \right]
If c=-{{\tau }_{01}}/\tau _{1}^{2}, then the variance of the intercept is minimized:
Var\left[ \begin{matrix}
   {{{\tilde{u}}}_{0j}}  \\
   {{{\tilde{u}}}_{1j}}  \\
\end{matrix} \right]=\left[ \begin{matrix}
   \tau _{0}^{2}-\tau _{01}^{2}/\tau _{1}^{2} & 0  \\
   0 & \tau _{1}^{2}  \\
\end{matrix} \right]=\left[ \begin{matrix}
   \tau _{0}^{2}\left( 1-\rho _{01}^{2} \right) & 0  \\
   0 & \tau _{1}^{2}  \\
\end{matrix} \right]

Gilbert

In chapter 5, they talk about hierarchical linear model where fixed effects and random effects are taken into consideration. Discuss a clear simple example in class which shows both effects and give interpretations of each of the coefficients and their use in real life.

Daniela

In chapter 5, they talked about mostly about the two-level nesting structure. Can we have a bigger example with at least 4 levels that includes the random intercept and slope and how to apply this into R coding?

In 'lme', multilevel nesting is handled with. e.g.
        fit <- lme( Y ~ X * W, dd, random = ~ 1 | idtop/idmiddle/idsmall)
Contextual variables present an ambiguity. Assuming that the id variables 'idsmall' and 'idmiddle' are coded uniquely overall, then the the higher level, say 'idmiddle', contextual variables could be coded as either:
        cvar(X,idmiddle)
or
        capply( dd , ~ id, with, mean( c(tapply( X, idsmall, mean))))
Here is a table prepared for SPIDA showing how to handle multilevel nesting and crossed structures in a selection of R functions:
Function Notes
lme

in package nlme

Linear mixed effects: normal response

G side and R side modelling
Model syntax:

 Y ~ X * W, random = ~ 1 + X | id

For nested effect:

 Y ~ X * W, random = ~ 1 + X | higher/lower

or, to have different models at different levels:

 Y ~ X * W, random = list(higher = ~ 1, lower = ~ 1 + X )
lmer

in package lme4

Linear mixed models for gaussian response with Laplace approximation

G side modeling only, R = σ2I
Model syntax:

 Y ~ X * W +(1+X|id)

For nested effect:

 Y ~ X * W +(1+X|higher) + (1+X|higher:lower)

For crossed effect:

 Y ~ X * W +(1+X|id1) + (1+X|id2)
glmer

in package lme4

Generalized linear mixed models with adaptive Gaussian quadrature
  • family: binomial, Gamma, inverse.gaussian, poisson, gaussian

G side only, no R side
Model syntax:

 Y ~ X * W +(1+X|id)

For nested effect:

 Y ~ X * W +(1+X|higher) + (1+X|higher:lower)

For crossed effect:

 Y ~ X * W +(1+X|id1) + (1+X|id2)
glmmPQL

in packages MASS/nlme

Generalized linear mixed models with Penalized Quasi Likelihood
  • family: binomial, Gamma, inverse.gaussian, poisson, gaussian

G side and R side as in lme
Model syntax:

 Y ~ X * W, random = ~ 1 + X | id

For nested effect:

 Y ~ X * W, random = ~ 1 + X | higher/lower
MCMCglmm

in package MCMCglmm

Generalized linear mixed models with MCMC
  • family: poisson, categorical, multinomial, ordinal, exponential, geometric, cengaussian, cenpoisson,

cenexponential, zipoisson, zapoisson, ztpoisson, hupoisson, zibinomial (cen=censored, zi=zero-inflated, za=zero-altered, hu=hurdle
G side and R side, R side different from 'lme': no autocorrelation but can be used for multivariate response
Note: 'poisson' potentially overdispersed by default (good), 'binomial' variance for binary variables is unidentified.
Model syntax:

 Y ~ X * W, random = ~ us(1 + X):id  [Note: id should be a factor, us=unstructured]

For nested effect:

 Y ~ X * W, random =  ~us(1 + X):higher + us(1 + X):higher:lower

For crossed effect:

 Y ~ X * W, random =  ~us(1 + X):id1+ us(1 + X):id2

Phil

Exluding fixed effects that are non-significant is common practice in regression analyses, and Snijders and Bosker follow this practice when simplifying the model in Table 5.3 to the model found in Table 5.4. While this practice is used to help make the model more parsimonious it can ignore the joint effect that these variables have on the model as a whole. Discuss alternative criteria that one should explore when determining whether a predictor should be excluded from the model.

Ryan

When using random slopes it is generally the case that the level 1 model contains a fixed effect for what will also be the level 2 random effect. The random effect is then an estimate of the group/cluster/individual departure from the fixed effect. However, non-significant level 1 variables are not determinable as different from zero. Are there cases where a non-significant fixed effect can be excluded from the model while retaining the random effect at level 2. What would be the consequence of this and what might it reveal about the level 1 variable? Would this help control for the error of excluding a non-significant but confounding variable?

This is a very interesting question. It would be interesting to create a simulated data set illustrating the issue so we could consider the consequences of having random effects for a confounding factors whose within cluster effect changes sign from cluster to cluster. Can we think of a confounding factor that would do that?

and others

Chapter 6

Alex

Matthew

At the beginning of the chapter, S&J present two models (Table 6.1). They note that "The variable with the random slope is in both models the grand-mean centered variable IQ" (in R: the random side would look like: random = ~ 1 + (IQ - IQbar) | school), even though (IQ - IQbar) isn't in the fixed part of the second model. Does this have any ramifications for interpretation? Would the interpretation of any of the coefficients change if IQ had been used on the random side instead of the grand mean deviation?

Qiong

The author introduce the t - test to test fixed parameters. We can use summary(model) to get the p - value directly in R. In practice, we use wald test to test fixed parameter. On page 95, they mentioned that "for 30 or more groups, the Wald test of fixed effects using REML standard errors have reliable type I error rates." Is it the only reason we use the wald test in practice?

Carrie

Gilbert

On page 104 , the author discuss different approaches for model selection including working upward from level one, joint consideration of both level one and two. Are there other methods to be used If both methods are not providing a satisfactory model?

Daniela

On page 97 and 99, the authors showed us how the tests for random intercept and random slope independently. What test would you use if you wanted to test for both random slope and intercept in the same model? and what would you test it against? ... the linear model or a model with just a random slope or a random intercept?

Phil

Ryan

and others

Chapter 7

Alex

Matthew

Qiong

Carrie

Gilbert

Daniela

Phil

Ryan

and others

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