Consider a *randomized experiment* in which patients are assigned to one of two treatment groups according to a randomization process that allocates 50 patients
to each group. After a specified period of time, each patient’s status (cured or not cured) is recorded. Suppose the data
shown in Table 8.5 give the results of the experiment. The null hypothesis is that the two treatments are equally effective. Under this hypothesis,
treatment is a randomly assigned label that has no effect on the cure rate of the patients. But this implies that each row
of the table represents a simple random sample from the finite population whose cure rate is described by the column marginal
totals. Therefore, the column marginals (58, 42) are fixed under the hypothesis. Since the row marginals (50, 50) are fixed
by the allocation process, the hypergeometric distribution is induced on the cell frequencies. Randomized experiments can
also be specified in a stratified framework, and Cochran-Mantel-Haenszel statistics can be computed relative to the corresponding
multiple hypergeometric distribution.

Table 8.5: Two-Way Contingency Table: Treatment by Status

Status |
|||
---|---|---|---|

Treatment |
Cured |
Not Cured |
Total |

1 |
36 |
14 |
50 |

2 |
22 |
28 |
50 |

Total |
58 |
42 |
100 |