Sometimes the observed data do not come from a random sample but instead represent a complete set of observations on some population. For example, suppose a class of 100 students is classified according to sex and favorite color. The results are shown in Table 8.4.

In this case, you could argue that all of the frequencies are fixed since the entire population is observed; therefore, there is no sampling error. On the other hand, you could hypothesize that the observed table has only fixed marginals and that the cell frequencies represent one realization of a conceptual process of assigning color preferences to individuals. The assignment process is open to hypothesis, which means that you can hypothesize restrictions on the joint probabilities.

Table 8.4: Two-Way Contingency Table: Sex by Color

Favorite Color |
||||
---|---|---|---|---|

Sex |
Red |
Blue |
Green |
Total |

Male |
16 |
21 |
20 |
57 |

Female |
12 |
20 |
11 |
43 |

Total |
28 |
41 |
31 |
100 |

The usual hypothesis (sometimes called *randomness*) is that the distribution of the column variable (Favorite Color) does not depend on the row variable (Sex). This implies
that, for each row of the table, the assignment process corresponds to a simple random sample (without replacement) from the
finite population represented by the column marginal totals (or by the column marginal subtotals that remain after sampling
other rows). The hypothesis of randomness implies that the probability distribution on the frequencies in the table is the
*hypergeometric distribution*.

If the same row and column variables are observed for each of several populations, then the probability distribution of all
the frequencies can be called the *multiple hypergeometric distribution.* Each population is called a *stratum*, and an analysis that draws information from each stratum and then summarizes across them is called a *stratified analysis* (or a *blocked analysis* or a *matched analysis*). PROC FREQ does such a stratified analysis, computing test statistics and measures of association.

In general, the populations are formed on the basis of cross-classifications of independent variables. Stratified analysis is a method of adjusting for the effect of these variables without being forced to estimate parameters for them. Note that PROC LOGISTIC can perform analyses on stratified tables as well, using the usual modeling procedure assumptions, by using conditional or exact conditional logistic regression.

The multiple hypergeometric distribution is the one used by PROC FREQ for the computation of Cochran-Mantel-Haenszel statistics.
These statistics are in the class of *randomization model test statistics*, which require minimal assumptions for their validity. PROC FREQ uses the multiple hypergeometric distribution to compute
the mean and the covariance matrix of a function vector in order to measure the deviation between the observed and expected
frequencies with respect to a particular type of alternative hypothesis. If the cell frequencies are sufficiently large, then
the function vector is approximately normally distributed as a result of central limit theory, and PROC FREQ uses this result
to compute a quadratic form that has a chi-square distribution when the null hypothesis is true.