LogisticNormal Model with Binomial Data
This example analyzes the data from Beitler and Landis (1985), which
represent results from a multicenter clinical trial investigating the
effectiveness of two topical cream treatments (active drug, control)
in curing an infection. For each of eight clinics, the number of
trials and favorable cures are recorded for each treatment. The SAS
data set is as follows.
data infection;
input clinic t x n;
datalines;
1 1 11 36
1 0 10 37
2 1 16 20
2 0 22 32
3 1 14 19
3 0 7 19
4 1 2 16
4 0 1 17
5 1 6 17
5 0 0 12
6 1 1 11
6 0 0 10
7 1 1 5
7 0 1 9
8 1 4 6
8 0 6 7
run;
Suppose n_{ij} denotes the number of trials for the ith clinic and
the jth treatment (i = 1, ... ,8 j = 0,1), and
x_{ij} denotes the corresponding number of favorable cures. Then a
reasonable model for the preceding data is the following logistic
model with random effects:
and
The notation t_{j} indicates the jth treatment, and the u_{i} are
assumed to be iid .The PROC NLMIXED statements to fit this model are as follows:
proc nlmixed data=infection;
parms beta0=1 beta1=1 s2u=2;
eta = beta0 + beta1*t + u;
expeta = exp(eta);
p = expeta/(1+expeta);
model x ~ binomial(n,p);
random u ~ normal(0,s2u) subject=clinic;
predict eta out=eta;
estimate '1/beta1' 1/beta1;
run;
The PROC NLMIXED statement invokes the procedure, and the PARMS
statement defines the parameters and their starting values. The
next three statements define p_{ij}, and the MODEL statement
defines the conditional distribution of x_{ij} to be binomial.
The RANDOM statement defines U to be the random effect with subjects
defined by the CLINIC variable.
The PREDICT statement constructs predictions for each observation in
the input data set. For this example, predictions of and approximate standard errors of prediction are output to a
SAS data set named ETA. These predictions include empirical Bayes
estimates of the random effects u_{i}.
The ESTIMATE statement requests an estimate of the reciprocal of
.
The output for this model is as follows.
Specifications 
Data Set 
WORK.INFECTION 
Dependent Variable 
x 
Distribution for Dependent Variable 
Binomial 
Random Effects 
u 
Distribution for Random Effects 
Normal 
Subject Variable 
clinic 
Optimization Technique 
Dual QuasiNewton 
Integration Method 
Adaptive Gaussian Quadrature 

The "Specifications" table provides basic information about the
nonlinear mixed model.
Dimensions 
Observations Used 
16 
Observations Not Used 
0 
Total Observations 
16 
Subjects 
8 
Max Obs Per Subject 
2 
Parameters 
3 
Quadrature Points 
5 

The "Dimensions" table provides counts of various variables.
You should check this table to make sure the data set and model have
been entered properly. PROC NLMIXED selects five quadrature points
to achieve the default accuracy in the likelihood calculations.
Parameters 
beta0 
beta1 
s2u 
NegLogLike 
1 
1 
2 
37.5945925 

The "Parameters" table lists the starting point of the
optimization.
Iteration History 
Iter 

Calls 
NegLogLike 
Diff 
MaxGrad 
Slope 
1 

2 
37.3622692 
0.232323 
2.882077 
19.3762 
2 

3 
37.1460375 
0.216232 
0.921926 
0.82852 
3 

5 
37.0300936 
0.115944 
0.315897 
0.59175 
4 

6 
37.0223017 
0.007792 
0.01906 
0.01615 
5 

7 
37.0222472 
0.000054 
0.001743 
0.00011 
6 

9 
37.0222466 
6.57E7 
0.000091 
1.28E6 
7 

11 
37.0222466 
5.38E10 
2.078E6 
1.1E9 
NOTE: GCONV convergence criterion satisfied. 

The "Iterations" table indicates successful convergence in
seven iterations.
Fit Statistics 
2 Log Likelihood 
74.0 
AIC (smaller is better) 
80.0 
BIC (smaller is better) 
80.3 
Log Likelihood 
37.0 
AIC (larger is better) 
40.0 
BIC (larger is better) 
40.1 

The "Fitting Information" table lists some useful statistics
based on the maximized value of the log likelihood.
Parameter Estimates 
Parameter 
Estimate 
Standard Error 
DF 
t Value 
Pr > t 
Alpha 
Lower 
Upper 
Gradient 
beta0 
1.1974 
0.5561 
7 
2.15 
0.0683 
0.05 
2.5123 
0.1175 
3.1E7 
beta1 
0.7385 
0.3004 
7 
2.46 
0.0436 
0.05 
0.02806 
1.4488 
2.08E6 
s2u 
1.9591 
1.1903 
7 
1.65 
0.1438 
0.05 
0.8554 
4.7736 
2.48E7 

The "Parameter Estimates" table indicates marginal significance
of the two fixedeffects parameters. The positive value of the
estimate of indicates that the treatment significantly
increases the chance of a favorable cure.
Additional Estimates 
Label 
Estimate 
Standard Error 
DF 
t Value 
Pr > t 
Alpha 
Lower 
Upper 
1/beta1 
1.3542 
0.5509 
7 
2.46 
0.0436 
0.05 
0.05146 
2.6569 

The "Additional Estimates" table displays results from the
ESTIMATE statement. The estimate of equals 1/0.7385 =
1.3541 and its standard error equals 0.3004/0.7385^{2} = 0.5509 by
the delta method (Billingsley 1986). Note this particular
approximation produces a tstatistic identical to that for the
estimate of .Not shown is the ETA data set, which contains the original 16
observations and predictions of the .
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.