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Introduction to Categorical Data Analysis Procedures

Stratified Simple Random Sampling: Multiple Populations

Suppose you take two simple random samples, fifty men and fifty women, and ask the same question as before. You are now sampling two different populations that may have different response probabilities. The data can be tabulated as shown in Table 5.2.

Table 5.2: Two-Way Contingency Table: Sex by Color
  Favorite Color  
 SexRedBlueGreenTotal 
 Male30101050 
 Female20102050 
 Total502030100 

Note that the row marginal totals (50, 50) of the contingency table are fixed by the sampling design, but the column marginal totals (50, 20, 30) are random. There are six probabilities of interest for this table, and they are estimated by the sample proportions

pij = [(nij)/(ni)]
where nij denotes the frequency for the ith population and the jth response, and ni is the total frequency for the ith population. For this contingency table, the sample proportions are shown in Table 5.3.

Table 5.3: Table of Sample Proportions by Sex
  Favorite Color  
 SexRedBlueGreenTotal 
 Male0.600.200. 201.00 
 Female0.400. 200.401.00 
 

The probability distribution of the six frequencies is the product multinomial distribution

\Pr(n_{11},n_{12},n_{13},n_{21},n_{22},n_{23}) =
\frac{n_1! n_2! \pi_{11}^{n_{11...
 ...}^{n_{22}}
\pi_{23}^{n_{23}} }
{n_{11}! n_{12}! n_{13}! n_{21}! n_{22}! n_{23}!}
where {\pi}_{ij} is the true probability of observing the jth response level in the ith population. The product multinomial distribution is simply the product of two or more individual multinomial distributions since the populations are independent. This distribution can be generalized to any number of populations and response levels.

Stratified simple random sampling is the type of sampling required by PROC CATMOD when there is more than one population. PROC CATMOD uses the product multinomial distribution to estimate a probability vector and its covariance matrix. If the sample sizes are sufficiently large, then the probability vector is approximately normally distributed as a result of central limit theory, and PROC CATMOD uses this result to compute appropriate test statistics for the specified statistical model. The statistics are known as Wald statistics, and they are approximately distributed as chi-square when the null hypothesis is true.

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