Transcript Statistical Analysis of Repeated Measures Data Using SAS

Lecture 7
Model Checking for Linear
Mixed Models for
Longitudinal Data
Ziad Taib
Biostatistics, AZ
MV, CTH
May 2009
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Outline of lecture 7
1. Introduction to model checking
2. Model checking for the linear model
3. Model checking for the linear mixed models for
longitudinal data
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1. Introductio to model checking
 The process of statistical analysis might take the form
Select
Model Class
Data
Summarize
Some Models
Conclusions
Stop
 In the above process, however, even after a careful selection of
model class, the data themselves may indicate that the particular
model is unsuitable. Thus, it seems to be reasonable to introduce
model checking to the original process. The news process of
statistical analysis is
Select
Model Class
Some Models
Conclusions
Summarize
Data
Model Checking
Stop
 The inadequacy indicated by model checking could
take two forms and is part of the technique of model
checking.
1. The detection of systematic discrepancies. It may be e.g. that
the data as a whole show some systematic departure from
the fitted model. An example of this type is informal
checking using residuals.
2. The detection of isolated discrepancies. It may be that a few
data values are discrepant from the rest. This can be done
using measures of leverage or measures of influence
2. The linear model
Yij  X i   Zi bi   ij , i  1, ,2,...,n
Model checking for linear models uses mainly the following statistics:
 The fitted values:
ˆ n1  X n p ˆ p1
 The mean residual sum of square:

e  Y  ˆ
 The residual:


t
ˆ
Y  X Y  Xˆ
2
s 
n p

Residual checking
0
-1
-2
-3
Residuals
1
2
3
 Plot residuals against mean
0
20
40
60
80
100
Checks for Isolated Departures from the
Model
 In the case of the standard linear model, the Cook’s
distance can be used to assess the influence of
observation i, by considering the parameter estimate
without the contribution from the i’th observation:

ˆ  ˆ  X X ˆ  ˆ  ˆ  ˆ  Vaˆr ˆ  ˆ  ˆ 
C 

t
(i )
(i )
i
(i )
ps2

Yˆ  Yˆ  Yˆ  Yˆ  Yˆ  Yˆ


(i )
(i )
2
ps
(i )
p
t
(i )
1
t
t
2
ps
2
…, n
, i  1,
3. Model checking in linear mixed
models
3.1 Model selection: likelihood
 When choosing between different models we want to be
able to decide which model fits our data best. If the models
compared are nested within each other it is possible to do
a likelihood ratio test where the test statistic has an
approximate distribution. The test statistic for the likelihood
statistic is,
2
2log(L1 )  log(L2 ) ~  DF
 where DF are the degrees of freedom which is the
difference in number of parameters for the models and L1
and L2 are the likelihoods for the first and second model
respectively.
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
If the two models compared are not nested with each other but
contain the same number of parameters they can be compared
directly by looking at the log likelihood and
the model with the biggest likelihood value wins

If the two models are not nested and contain different number of
parameters the likelihood can not be used directly. It is still possible
to compare these models with some of the methods described below.
 The bigger the likelihood is the better the model fits data and we use this
when we compare different models

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Since we are interested in getting as simple models as possible we also
have to consider the number of parameters in the structures. A model
with many parameters usually fits data better than a model with less
number of parameters
Information.
3.2 Model selection: Information criteria
 It is possible to compute so called information criteria and there are
different ways to do that and here we show two of these, Akaikes
information criteria (AIC) and Bayesian information criteria (BIC). The
idea with both of these is to punish models with many parameters in
some way. We present the information criteria the way they are
computed in SAS.
 The AIC value is computed as below where q is the number of parameters
in the covariance structure. Formulated this way, a smaller value of AIC
indicates a better model.
AIC  2 LL  2q
 The BIC value is computed using the following formula where q is the
number of parameters in the covariance structure and n is the number of
effective observations, which means the number of individuals. Like for
AIC a smaller value of BIC is better than a larger.
BIC  2 LL 
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q
2 log(n)
Model fit
 It is possible to define a goodness of fit measure similar to, R, the
coefficient of determination often used for linear models. It is called
Concordance Correlation coefficient (CCC). Unlike the AIC or the BIC,
the CCC does not compare the model at hand to other models, thus it
does not require that other models be fitted.
 For simple linear regression we have
3.3 Residuals for linear mixed models
 In model selection, we accept the model with the best
likelihood value in relation to the number of parameters
but we still do not know if the model chosen is a good
model or even if the normality assumption we have made
is realistic.To check this we can look at two types of plots
for our data,
 normal plots
 residual plots to check
1. normality of the residuals and the random effects
2. if the residuals seem to have a constant variance
3. outliers
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 The predicted values and residuals can be computed in
many different ways. Some of these are accounted for in
the in what following.
 Recall that the general linear mixed model is of the form:
Yij  X i   Zi bi   ij , i  1, ,2,...,n
 Assuming we have ML estimates of the fixed parameters
and EB predictions of the random parameters
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 We can “estimate” the residuals according to the following
three methods:
 Each type of residual is useful to evaluate some of the
assumptions of the model.
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Marginal residuals
 Can be used to assess linearity of the response wrt
explanatory variables. A random behviour around zero is
a sign of linearity.
Conditional residuals
 Plots of /s against Y can be used to assess
homogeneity of the variances as well as normality.
EBLUP
 Plots of bi against subject indices can be used to find
outliers. Plot elements in bi b to assess normality and
check for outliers.
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3.4 An example
To illustrate the above procedures, we analyze data from a
study conducted at the School of Dentistry of the University
of So Paulo, Brazil, designed to compare a low cost
toothbrush (monoblock) with a conventional toothbrush
with respect to the maintenance of the capacity to remove
bacterial plaque under daily use. The data in the table
correspond to bacterial plaque indices obtained from 32
children aged 4 to 6 before and after tooth brushing in four
evaluation sessions.
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Following Singer et al. (2004) who analyze a different data
set from the same study, we considered fitting models of
the form
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i subject
d session
j type of toothbrush
Three possible models
(3.1)
(3.2)
(3.3)
post
pre
The model reduction procedure can be based on likelihood
ratio tests (LRT) and AIC and BIC:
 The LRT p-values corresponding to the reduction of (3.1) to (3.2)
and of (3.2) to (3.3) were, respectively 0.3420 and 0.1623.
 The AIC (BIC) for the three models are
(3.1)
(3.2)
(3.3)
AIC
95.0
102.8
105.6
(BIC)
68.6
86.7
92.1
Based on these results, we adopt (3.3) to illustrate the use
of the proposed diagnostic procedures.
To check for the linearity of effects, we plot the marginal
residuals versus the logarithms of the pretreatment
bacterial plaque index in Figure 2. The figure supports the
regression model for the transformed response (log of the
bacterial plaque index)
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Figure 2
 The figure suggests something is wrong with
observations #12.2 and #29.4
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References
1. Atkinson, C. A. (1985). Plots, transformations, and regression: an
introduction to graphical methods of diagnostic regression analysis.
Oxford University Press, Oxford.
2. Cook, R. D. and Weisberg, S. (1982). Residuals and influence
regression. Chapman & Hall, New York.
3. Cox, D. R. and Snell, E. J. (1968). A general definition of residuals
(with discussion). Journal Royal Statistical Society B 30, 248–275.
4. Fei, Y. and Pan, J. (2003). Influence assessments for longitudinal
data in linear mixed models. In 18th international workshop on
Statistical Modelling. G. Verbeke, G. Molenberghs, M. Aerts and S.
Fieuws (eds.). Leuven: Belgium, 143–148.
5. Grady, J. J. and Helms, R.W. (1995). Model selection techniques for
the covariance matrix for incomplete longitudinal data. Statistics in
Medicine 14, 1397–1416.
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References
6. Jiang, J. (2001). Goodness-of-fit tests for mixed model diagnostics.
The Annals of Statistics 29, 1137–1164.
7. Lange, N. and Ryan, L. (1989). Assessing normality in random effects
models. The Annals of Statistics 17, 624– 642.
8. Longford, N. T. (2001). Simulation-based diagnostics in randomcoefficient models. Journal of the Royal Statistical Society A 164,
259–273.
9. Nobre, J. S. and Singer, J. M. (2006). Fixed and random effects
leverage for influence analysis in linear mixed models. (Submitted;
http://www.ime.usp.br/jmsinger).
10. Oman, S. D. (1995). Checking the assumptions in mixed-model
analysis of variance: a residual analysis approach. Computational
Statistics and Data Analysis 20, 309–330.
References
11. Verbeke, G. and Lesaffre, E. (1997). The effect of misspecifying the
random-effects distributions in linear mixed models for longitudinal
data. Computational Statistics and Data Analysis 23, 541–556.
12. Waternaux, C., Laird, N. M., and Ware, J. H. (1989). Methods for
analysis of longitudinal data: blood-lead concentrations and cognitive
development. Journal of the American Statistical Association 84, 33–
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13. Weiss, R. E. and Lazaro, C. G. (1992). Residual plots for repeated
measures. Statistics in Medicine 11, 115–124.
14. Wolfinger, R. (1993). Covariance structure selection in general mixed
models. Communications in Statistics-Simulation 22, 1079–1106.
Any Questions
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?