# The NLMIXED Procedure

### Example 82.6 Simulated Nested Linear Random-Effects Model

This simulation study example demonstrates how to fit a hierarchical model with PROC NLMIXED by using a simple two-level nested linear model. In this example, the data are generated from a simple two-level nested hierarchical normal model (Littell et al. 2006). Here the conditional distribution of the response variable, given the random effects, is normal. The mean of this conditional normal distribution is a linear function of a fixed parameter and the random effects. In this simulation, only the intercept is used as a fixed parameter. Also, the first-level random effects are assumed to follow a normal distribution. Similarly, the second-level random effects that are nested within the first level follow a normal distribution. The model can be represented as follows:

The simulation code is as follows:

%let na = 100;
%let nb = 5;
%let nr = 2;
data nested;
do A = 1 to &na;
err1 = 3*rannor(339205);
do B = 1 to &nb;
err2 = 2*rannor(0);
do rep = 1 to &nr;
err3 = 1*rannor(0);
resp = 10 + err1 + err2 + err3;
output;
end;
end;
end;
run;


You can use PROC NLMIXED to fit the preceding nested model as follows by using two RANDOM statements:

proc nlmixed data = nested;
bounds vara >=0, varb_a >=0;
mean = intercept + aeffect + beffect;
model resp ~ normal (mean, s2);
random aeffect ~ normal(0,vara) subject = A;
random beffect ~ normal(0,varb_a) subject = B(A);
run;


The MODEL statement specifies the response variable distribution as normal. The linear relationship between the mean of this normal distribution and the fixed and random effects is specified by using a SAS programming statement. The BOUNDS statement specifies the nonnegative constraints on the variance parameters.

PROC NLMIXED with multiple RANDOM statements is used to fit hierarchical (nested) models. In this example, two RANDOM statements are used to fit a two-level nested model. The first RANDOM statement identifies aeffect as the first-level random effect, , and it is followed by the subject variable A. The subject variable A indexes the number of distinct and independent subjects at the first level. Then the second RANDOM statement identifies the beffect, , as the second-level random effect, which is nested within the first level. The subject variable specification in the second RANDOM statement defines the nested structure. In this example, the subject variable B is nested within A.

The results from the analysis follow.

Output 82.6.1: Model Specifications for a Two-Level Nested Model

The NLMIXED Procedure

Specifications
Data Set WORK.NESTED
Dependent Variable resp
Distribution for Dependent Variable Normal
Random Effects aeffect, beffect
Distribution for Random Effects Normal, Normal
Subject Variables A, B
Optimization Technique Dual Quasi-Newton

The "Specifications" table (Output 82.6.1) lists the setup of the nested model. It lists all the random effects and their distribution along with the SUBJECT= variable in the nested sequence. Random effects that are listed in the specifications table are separated by a comma, indicating that aeffect is the first-level random effect, followed by the second-level random effect, beffect, which is nested within the first level. The same scheme applies to the distribution and subject items in the table. In this example, aeffect is the random effect for the first level and is specified using the subject variable A, which follows a normal distribution. Then beffect is the random effect for the second level, which is nested within the first level; it is specified using the subject variable B, which also follows a normal distribution.

The "Dimensions" table (Output 82.6.2) follows the same nested flow to exhibit the model dimensions. It indicates that 1,000 observations are used. There are 100 first-level subjects and 500 second-level subjects, because each first-level subject is contained by five nested second-level subjects. For this model, PROC NLMIXED selects one quadrature point to fit the model.

Output 82.6.2: Dimensions Table for a Two-Level Nested Model

Dimensions
Observations Used 1000
Observations Not Used 0
Total Observations 1000
Subjects [A] 100
Max Obs per Subject 10
Subjects [B] 500
Max Obs per Subject 2
Parameters 4

A total of 15 steps are required to achieve convergence. They are shown in the "Iteration History" table (Output 82.6.3).

Output 82.6.3: Iteration History

Iteration History
Iteration Calls Negative
Log
Likelihood
Difference Maximum
Slope
1 12 2982.0141 2695.598 215.363 -87641.4
2 37 2462.0898 519.9243 518.335 -346.648
3 43 2144.0047 318.0852 60.9954 -2483.38
4 49 2108.7720 35.23267 56.3198 -64.4603
5 61 2098.5627 10.20936 43.0728 -7.79282
6 67 2094.2091 4.353502 34.1536 -20.0831
7 73 2090.2802 3.928941 26.4378 -5.86557
8 80 2089.3443 0.935937 15.0337 -2.41721
9 86 2088.1107 1.233618 20.3762 -0.59126
10 98 2072.7699 15.3408 5.02630 -1.53248
11 105 2068.5424 4.227408 28.6084 -2.25950
12 112 2066.4439 2.09858 9.01353 -2.53321
13 119 2065.8127 0.63118 3.73621 -0.50717
14 126 2065.7590 0.053646 1.02924 -0.07232
15 133 2065.7550 0.00406 0.36114 -0.00578
16 139 2065.7502 0.004798 0.27897 -0.00171

 NOTE: GCONV convergence criterion satisfied.

The "Parameter Estimates" table (Output 82.6.4) contains the maximum likelihood estimates of the parameters. You can see from this table that the intercept and the s2 variables are very close to the simulation parameters. Also, both variances are very close to the simulation parameters.

Output 82.6.4: Parameter Estimates for a Two-Level Nested Model

Parameter Estimates
Parameter Estimate Standard
Error
DF t Value Pr > |t| 95% Confidence Limits Gradient
vara 9.2198 1.4299 498 6.45 <.0001 6.4104 12.0292 0.022612
varb_a 3.6997 0.2973 498 12.44 <.0001 3.1156 4.2839 0.008443
intercept 10.1546 0.3171 498 32.02 <.0001 9.5315 10.7776 -0.04079
s2 0.9661 0.06102 498 15.83 <.0001 0.8462 1.0860 -0.31590