Stratified Simple Random Sampling: Multiple Populations

Suppose you take two simple random samples, 50 men and 50 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 8.2.

Table 8.2: Two-Way Contingency Table: Sex by Color

	Favorite Color
Sex	Red	Blue	Green	Total
Male	30	10	10	50
Female	20	10	20	50
Total	50	20	30	100

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

$p_{ij} = \frac{n_{ij}}{n_ i}$

where $n_{ij}$ denotes the frequency for the ith population and the jth response and $n_ i$ is the total frequency for the ith population. For this contingency table, the sample proportions are shown in Table 8.3.

Table 8.3: Table of Sample Proportions by Sex

	Favorite Color
Sex	Red	Blue	Green	Total
Male	0.60	0.20	0.20	1.00
Female	0.40	0.20	0.40	1.00

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

$\Pr (n_{11},n_{12},n_{13},n_{21},n_{22},n_{23}) = \frac{n_1! n_2! \pi _{11}^{n_{11}} \pi _{12}^{n_{12}} \pi _{13}^{n_{13}} \pi _{21}^{n_{21}} \pi _{22}^{n_{22}} \pi _{23}^{n_{23}} }{n_{11}! n_{12}! n_{13}! n_{21}! n_{22}! n_{23}!}$

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 the modeling procedures when there is more than one population. The product multinomial distribution is used 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 this result is used 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.