Quantitative estimates of the harmful or beneficial effect of exposure to an agent such as cigarette smoke, ionizing radiation or a new vaccine are frequently based on comparisons of "failure rates" (e. g. mortality or incidence) or of "success rates" (e.g. survival) in several groups. The increased or decreased risk associated with the agent is often measured either as a "failure ratio''" (e.g. the mortality rate among smokers divided by the mortality rate among nonsmokers) or as the difference between 2 failure or success rates. The underlying biological assumptions indicate that a relevant method of making such estimates is determined by the nature of the effect in question. In a comparison with a control group, beneficial effects may be estimated as proportional to the "failures" in the control group; and conversely, harmful effects may be measured as proportional to the "successes" in the control group. Mathematical models are developed for several situations in which two or more samples are compared. Maximum likelihood solutions are derived for the estimates and for their variances and covariances.