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MIXED EFFECTS MODELS FOR CATEGORICAL DATA
J. Richard Landis Department of Biostatistics. School of Public Health, University of Michigan, Ann Arbor, MI 48109, U.S.A.
James M. Lepkowski Sampling Section of the Survey Research Center, Institute for Social Research and Department of Biostatistics, School of Public Health,
University of Michigan, Ann Arbor, MI 48109, U.S.A. Paula J. Beitler
Ortho Pharmaceutical Corp., Route 202 South, Raritan, NJ 08869, U.S.A.
Abstract Several mixed effects models for categorical data from unbalanced designs are proposed. The analogy between these models and a clustered sampling model is discussed. Variance components are estimated by an extension of the fitting constants method for the mixed ANOVA models and through the estimation of design effects for the cluster sampling model. These methods are illustrated using health research data from two different contexts. First, the results from a multi-center randomized clinical trial are analyzed under the assumptions that the treatment effects are fixed and the clinics are presumed to be random. Secondly, the application of these methods to data obtained by cluster sampling is illustrated with data from the second National Health and Nutrition Examination Survey(NHANES II). In particular, the relationship between elevated blood lead levels(2 15 Mg/dl) and elevated diastolic blood pressure^ 90 mm Hg) is modeled across the 64 sampling sites for 3,181 males ages 12-74.
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Notation for Binomial Data (Two-way Model)
Notation a = 1, 2, .... A i = 1, 2,I k = 1,' 2, ... nai.
Role
indexes the 1st stage clusters(clinics)
indexes the fixed effects(treatments)'-
indexes the subjects within the ath clinic assigned to the ith treatment
Indicator Variables
yaik = '
1, if the &th subject in the ath clinic assigned to the ith treatment responds favorably
0, otherwise
Sample Sizes and Proportions
Notation
Description
ni ~ ^anai Pai _ ^kyaik^nai Pi = UlrVaik/ni
total number of subjects assigned to the ith treatment
sample proportions responding favorably to the ith treatment within the ath clinic
overall sample proportion responding favorably to the ith treatment
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Means & Variances: Indicator Variables
Notation
Description
E^aik} = V Varb-aikl *
overall probability of favorable response on ith treatment
v.% "element" variance for the indicator variables associated with the ith treatment
Covariances & Correlations: Indicator Variables
Notation
Description
Same Clinic, Same Treatment, Different Subjects
ii = cov {yaik,yaik,}
"element" covariance for the ith treatment
= 0../0? 'ii U 1
(within clinic) intraclass correlation for the ith treatment
Same Clinic, Different Treatments, Different Subjects *
- " Cov !yaik,yai'k'^
"element" covariance between treatments i and i'
II
(within clinic) intraclass cross-correlation between treatments i and i'
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4 Landis, J.R.: Mixed Effects Models Variances & Covariances: Sample Proportions Within Clinics (conditional on fixed sample sizes nai.)
Variances & Covariances: Overall Sample Proportions (conditional on fixed sample sizes n.)
%
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Matrix Notation for I = 2 Treatments
Notation
Description
JTa
'al wa2
vector of probabilities of favorable response within the ath clusters(clinics)
jr = *1 *2
vector of overall probabilities of favorable response for each treatment
vector of sample proportions of favorable response within ath cluster(clinic)
Sample Proportion Vector p' = (pir p12' pal' pa2'PA1' PA2^'
Covariance Matrix for Sample Proportions in ath Cluster(Clinic)
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6 Landis, J.R.: Mixed Effects Models Mixed Effects ANOVA Model With Interaction "aik * M + Ca + 0K.i + g6ai. + eftilr
aej.Ina.i + q a + og,' a=a ', t=i cov(Pai' P; .T* ~ aQl a=a', i^i'
0 otherwise.
2 el nal V(pa) =
pa2
oa2 a2a
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Landis, J.R.: Mixed Effects Models Mixed Effects ANOVA Model Without Interaction
7
Estimated Covariance Matrix Based on Fixed Effects Multinomial Model (Unrestricted Covariance Matrix for Sample Proportions Within Clinic)
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8 Landis, J.R.: Mixed Effects Models Estimated Covariance Matrix Based on Fixed Effects Multinomial Model (Hq Covariance Matrix for Sample Proportions Within Cli
Fixed Effects Models (A = 8 Clinics) (Reduced Model (Main Effects))
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Quantity
Description
Design Effect
Var{p.} Deff.i = a-jCl--jr)
n.i
ratio of the variance of the proportion favorable under clustered design to the variance under simple random sampling assumptions.
design effect for response to ith treatment due to regarding clinics as random sample.
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10 Landis, J.R.: Mixed Effects Models
Variances & Covariances: Overall Sample Proportions (conditional on fixed sample sizes n.)
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Estimated Covariance Matrix Based on Complex Sample Estimators
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Table 1
Distribution of Favorable Response to Active Drug and Control Treatment From Multicenter Randomized Clinical Trial
Clinic No. Treatment
Response Proportion
Favorable Unfavorable Total Favorable
1 Drug Control
11 10
25 36 0.306 27 37 0.270
2 Drug Control
16 22
4 20 0.800 10 32 0.688
3 Drug Control
14 7
5 19 0.737 12 19 0.368
4 Drug Control
2 1
14 16 0.125 16 17 0.059
5 Drug Control
6 0
11 17 0.350 12 12 0.000
6 Drug Control
1 0
10 11 0.091 10 10 0.000
7 Drug Control
1 1
4 5 0.200 8 9 0.110
8 Drug Control
4 6
2 6 0.667
1 7 0.857 - --.. * - --
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Table 2
Distribution of Favorable Response to Active Drug and Control Treatment Pooled Across Clinics 1 -- 8
Treatment Drug Control
Distribution of Response
Favorable
Unfavorable
55 75
47 96
Total 130 143
Proportion Favorable
0.423
0.329
Total 102 171 273 0.374
Table 3 Chi-square Statistics with df = 1 and Fisher's Exact Test
Criterion Pearson'sCUncorrected) Yates'(Corrected) Randomization Model Likelihood Ratio Criterion
Test Statistic
2.593
2.206 2.584 2.594
Significance Level
0.1073
0.1375 0.1080 ,, 0.1073
Fisher Exact Probability(Total) Primary Tail Secondary Tail
Observed Table
0.13271 0.06875 0.06396
0.02746
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Table 4 Chi-square Statistics with df = 1 Directed at Average Partial Association
Criterion
Hypergeometric Model
Uncorrected (Birch) Corrected(Mantel-HaenszeI)
Product Binomial Model
Uncorrected(Cochran) Corrected
Woolf WLS Test(Modified)
Test Statistic
6.384 5.672
6.559 5.827 4.257
Significance Level
0.01 0.02
0.01 0.02 0.04
Table 5
Estimates of Variance Components Under Alternative Mixed Model Assumptions for Clinical Trial Data
Mixed Effects Models
Variance Components
22 5el ae2
Interaction
No Interaction/ No Pooling
No Interaction/' Pooling
0.1901 0.1901 0.1866
0.1576 0.1576 0.1623
0.0700 0.0710 0.0709
0.0019
... ...
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Table 6
Estimates of Homogeneity and CrossHomogeneity Coefficients for Clinical Trial Data -
Cluster Sampling Model
Homogeneity Coefficients
roh 11
roh 22
roh 12
Unconstrained
0.2160
0.2750
0.2401
Table 7
Weighted Least Squares Analysis of Multicenter Randomized Clinical Trial Data
Covariance Structure
Model-Based Test Statistics
Lack of fit
Treatment(d.f. = 1)
Qt Significance LOF d.f. Level
Qp Significance Level
Fixed Effects Model
Unrestricted Multinomial
8.76 7
0.27
Mixed Effects (Interaction)
Mixed Effects Model
14.75 14
0.40
Mixed Effects (No Interaction' No Pooling)
15.99 14
0.31
Mixed Effects (No Interaction/ Pooling)
15.93 14
0.32
Cluster Sampling Model
Variable Homogeneity 16.21 14 Coefficients
0.30
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7.65
0.01
l 5.35
0.02
6.37
0.01
6.34
0.01
5.52
0.02
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Table 11
Distribution of Elevated Diastolic Blood Pressure by Level of Blood Lead-by NHANES II Sampling Site: Males, Ages 12-74, United States, 1976-80
Sampling Blood Lead
Site No.
vg/dl
Diastolic Blood Pressure
<90
a90
Proportion
Total
a 90
1 <15 16
a 15
35
1 17 0.059 10 45 0.222
2
<15
21
al5
13
5 26 0.192 6 19 0.316
3 <15 a 15
3 6
2 5 0.400 1 7 0.143
4 <15
6
S 15
14
2 8 0.250 2 16 0.125
63 <15 32
a 15
24
64 <15 42
a 15
16
9 41 0.220 8 32 0.250
8 50 0.160 5 21 0.238
*
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Table 12
Distribution of Elevated Diastolic Blood Pressure by Level of Blood Lead Pooled Across 64 NHANES II Sampling Sites: Males, Ages 12-74, United States, 1976-80
Blood Lead yg/dl
Diastolic Blood Pressure
............. - ------ - ............
<90
2:90
x Proportion
Total
2:90
<15
1,134
328
1,462
0.224
& 15
1,222
497
1,719
0.289
Total
2,356
825
3,181
0.259
Table 13
Chi-square Statistics with df = 1 for NHANES II Data Pooled Across 64 Sampling Sites
Criterion Pearson's(Uncorrected) Yates'(Corrected) Randomization Model . Likelihood Ratio Criterion
Test Statistic
17.255 16.920 17.250
17.365
Significance Level
<0.001 <0.001
* <0.001
<0.001
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Table 14
Chi-square Statistics with df = 1 Directed at Average Partial Association For NHANES EE Data, Adjusting for 64 Sampling Sites
Criterion
Hypergeometric Model
Uncorrected (Birch) Corrected (Mantel-Haenszel)
Product Binomial Model
Uncorrected(Cochran) Corrected
Woolf WLS Test(Modified)
Test Statistic
10.355 10.073
10.566 10.279
7.341
Significance Level
<0.01 <0.01
<0.01 <0.01 <0.01
Table 15
Estimates of Variance Components Under Alternative Mixed Model Assumptions for NHANES II Data
Mixed Effects Models
Interaction
2 el
0.1651
Variance Components
22 ae2 *c
0.2013
0.0065
2 g
* 0.0001
No Interaction/ No Pooling
No Interaction/ Pooling
0.1651 0.1672
0.2013 0.1996
0.0066 0.0066
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Table 16
Estimates of Homogeneity and CrossHomogeneity Coefficients for NHANES II Data -
Cluster Sampling Model
Homogeneity Coefficients
rohu
roh22
'' roh12
Unconstrained
0.056445
0.020558
0.031679
Table 17 Weighted Least Squares Analysis of NHANES II Data
Covariance Structure
Model-Eased Test Statistics
Lack of fit
Treatment(d.f. = 1)
9Lor if. Significance Level
Qc Significance Level
Mixed Effects Model
Mixed Effects (Interaction)
127.24 126
0.45
Mixed Effects (No Interaction/ No Pooling)
128.36 126
0.42
Mixed Effects (No Interaction/ Pooling)
128.04 126
0.43
Cluster Sampling Model
Variable Homogeneity 121.43 126 Coefficients
0.60
13.29 13.47
* 13.43
12.31
<0.001 <0.001 <0.001
<0.001
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References Beider, P.J. (1981). Mixed effects models for categorical response
variables. Ph.D. dissertation, University of Michigan. Beitler, P.J. and Landis, J.R. (1985). A Mixed .Effects Model for
Categorical Data. Submitted to Biometrics. Kleinman. J.C. (1973). Proportions with extraneous variancersingie and
independent samples. Journal of the American Statistical Association 68, 46-54. Kleinman, J.C. (1975). Proportions with extraneous variance: two dependent samples. Biometrics 31, 737-743. Landis, J.R. and Koch, G.G. (1977). A one-way components of variance model for categorical data. Biometrics 33, 671-679. Lepkowski, J.M. (1980). Design effects for multivariate categorical interactions. Ph.D. dissertation, University of Michigan.
*
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