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 The CATMOD Procedure

## POPULATION Statement

POPULATION variables ;
The POPULATION statement specifies that populations are to be formed on the basis of cross-classifications of the specified variables. If you do not specify the POPULATION statement, then populations are formed on the basis of cross-classifications of the independent variables in the MODEL statement. The POPULATION statement has two major uses:
• When you enter the design matrix directly, there are no independent variables in the MODEL statement; therefore, the POPULATION statement is the only way of inducing more than one population.
• When you fit a reduced model, the POPULATION statement may be necessary if you want to induce the same number of populations as there are for the saturated model.
To illustrate the first use, suppose that you specify the following statements:

```   data one;
input A \$ B \$ wt @@;
datalines;
yes yes 23   yes no 31   no yes 47   no no 50
;

proc catmod;
weight wt;
population B;
model A=(1 0,
1 1);
run;
```

Since the dependent variable A has two levels, there is one response function per population. Since the variable B has two levels, there are two populations. Thus, the MODEL statement is valid since the number of rows in the design matrix (2) is the same as the total number of response functions. If the POPULATION statement is omitted, there would be only one population and one response function, and the MODEL statement would be invalid.

To illustrate the second use, suppose that you specify

```   data two;
input A \$ B \$ Y wt @@;
datalines;
yes  yes  1  23       yes  yes  2  63
yes  no   1  31       yes  no   2  70
no   yes  1  47       no   yes  2  80
no   no   1  50       no   no   2  84
;

proc catmod;
weight wt;
model Y=A B A*B / wls;
run;
```

These statements induce four populations and produce the following design matrix and analysis of variance table.

 Source DF Chi-Square Pr > ChiSq Intercept 1 48.10 <.0001 A 1 3.47 0.0625 B 1 0.25 0.6186 A*B 1 0.19 0.6638 Residual 0

Since the B and A*B effects are nonsignificant (p>0.10), you may want to fit the reduced model that contains only the A effect. If your new statements are

```    proc catmod;
weight wt;
model Y=A / wls;
run;
```

then only two populations are induced, and the design matrix and the analysis of variance table are as follows.

 Source DF Chi-Square Pr > ChiSq Intercept 1 47.94 <.0001 A 1 3.33 0.0678 Residual 0

However, if the new statements are

```   proc catmod;
weight wt;
population A B;
model Y=A / wls;
run;
```

then four populations are induced, and the design matrix and the analysis of variance table are as follows.

 Source DF Chi-Square Pr > ChiSq Intercept 1 47.76 <.0001 A 1 3.30 0.0694 Residual 2 0.35 0.8374

The advantage of the latter analysis is that it retains four populations for the reduced model, thereby creating a built-in goodness-of-fit test: the residual chi-square. Such a test is important because the cumulative (or joint) effect of deleting two or more effects from the model may be significant, even if the individual effects are not.

The resulting differences between the two analyses are due to the fact that the latter analysis uses pure weighted least-squares estimates with respect to the four populations that are actually sampled. The former analysis pools populations and therefore uses parameter estimates that can be regarded as weighted least-squares estimates of maximum likelihood predicted cell frequencies. In any case, the estimation methods are asymptotically equivalent; therefore, the results are very similar. If you specify the ML option (instead of the WLS option) in the MODEL statements, then the parameter estimates are identical for the two analyses.

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