The NLMIXED Procedure

Hierarchical Model Specification

PROC NLMIXED supports multiple RANDOM statements to accommodate nested multilevel nonlinear mixed models, starting with SAS/STAT 13.2. If you use multiple RANDOM statements, PROC NLMIXED assumes that the SUBJECT= variable from each RANDOM statement forms a containment hierarchy. In the containment hierarchy, each SUBJECT= variable is contained by another SUBJECT= variable, and the SUBJECT= variable that is contained by all SUBJECT= variables is considered "the" SUBJECT= variable. For example, consider the following three-level nested model that has three SUBJECT= variables:

A

B

C

1

1

1

1

1

2

1

2

1

1

2

2

2

1

1

2

1

2

2

1

3

2

2

1

2

2

2

Suppose you specify a three-level nested model by using the following three RANDOM statements: 

random r11 ~ normal(0,sd1) subject = A;
random r21 ~ normal(0,sd2) subject = B(A);
random r31 ~ normal(0,sd3) subject = C(A*B);

Then PROC NLMIXED assumes the containment hierarchy as follows. The first-level hierarchy is defined using "the" SUBJECT= variable A. Similarly, the second-level hierarchy is defined using SUBJECT= variable B, which is nested within first level. Finally, SUBJECT= variable C defines the third-level hierarchy that is nested within both the first and second levels. In short, the SUBJECT= variable A is "the" SUBJECT= variable. B is contained in A, and C is contained in both A and B. In this example, there are two first-level subjects that are determined using "the" SUBJECT= variable A.

Based on the preceding hierarchy specification, PROC NLMIXED’s indexing of nested subjects for each first-level subject can be visualized as follows:

\[ \begin{array}{c c} \overbrace{\overbrace{C_{1(11)} C_{2(11)}}^{B_{1(1)}} \overbrace{C_{1(12)} C_{2(12)}}^{B_{2(1)}}}^{A_1} & \overbrace{\overbrace{C_{1(21)} C_{2(21)} C_{3(21)}}^{B_{1(2)}} \overbrace{C_{1(22)} C_{2(22)}}^{B_{2(2)}}}^{A_2} \end{array} \]

The ith subject from the first level is denoted as A$_ i$, the second-level nested subjects are denoted as B$_{j(i)}$, and the third-level nested subjects are denoted as C$_{k(ij)}$.

You can specify any nested structure by using the SUBJECT= syntax in PROC NLMIXED. For example, using the following three RANDOM statements, PROC NLMIXED fits a different model: 

random r11 ~ normal(0,sd1) subject = C;
random r21 ~ normal(0,sd2) subject = B(C);
random r31 ~ normal(0,sd3) subject = A(C*B);

In this case, PROC NLMIXED processes the subjects by using SUBJECT= variable C as "the" SUBJECT= variable, and the containment hierarchy is changed as follows:

A

B

C

1

1

1

1

1

2

1

2

1

1

2

2

2

1

1

2

1

2

2

1

3

2

2

1

2

2

2

 $\Rightarrow $

C

B

A

1

1

1

1

1

2

1

2

1

1

2

2

2

1

1

2

1

2

2

2

1

2

2

2

3

1

2

Again, PROC NLMIXED’s indexing of the nested subjects in this containment hierarchy can be visualized as follows:

\[ \begin{array}{c c c} \overbrace{\overbrace{A_{1(11)} A_{2(11)}}^{B_{1(1)}} \overbrace{A_{1(12)} A_{2(12)}}^{B_{2(1)}}}^{C_1} & \overbrace{\overbrace{A_{1(21)} A_{2(21)}}^{B_{1(2)}} \overbrace{A_{1(22)} A_{2(22)}}^{B_{2(2)}}}^{C_2} & \overbrace{\overbrace{A_{1(31)}}^{B_{1(3)}}}^{C_3} \end{array} \]

Here, PROC NLMIXED assumes that C is "the" SUBJECT= variable. B is contained in C, and A is contained in C and B. In this case, there are three first-level subjects that are determined using "the" SUBJECT= variable C.

As explained before, in this case, the ith subject from the first level is denoted as C$_ i$, the second-level nested subjects are denoted as B$_{j(i)}$ and for third level, the nested subjects are denoted as A$_{k(ij)}$.

Note that the containment hierarchy could potentially create more subjects than the unique number of subjects. For example, consider the following table, in which A is "the" SUBJECT= variable and B is nested within the subject A:

A

B

a

1

a

2

b

1

b

2

c

1

c

2

Even though the SUBJECT = B variable has only two unique subjects (1 and 2), when the containment hierarchy that is specified along with B is nested within A, PROC NLMIXED creates six nested B subjects. These nested subjects can be denoted as 1(a), 2(a), 1(b), 2(b), 1(c), and 2(c).

PROC NLMIXED does not support noncontainment hierarchy (or non-nested) models. For example, the following statements are not supported, because subject C is not nested within B and A: 

random r11 ~ normal(0,sd1) subject = A;
random r21 ~ normal(0,sd2) subject = B(A);
random r31 ~ normal(0,sd3) subject = C;