An Examination of Some Methods of Comparing Several Rates or Proportions

Abstract

The investigation of such problems as the association between smoking and lung cancer, between congenital malformations and exposure to radiation, or between coronary disease and obesity, usually requires quantitative estimates of the increase in the of the condition under study (Sheps [1958]). Several rather different methods of measuring changed risks have been used by investigators. In the field trials of poliomyelitis vaccine (Francis [19551), for example, the incidence of paralytic poliomyelitis per 100,000 was 16 (12) among vaccinated children and 57 (Ii) among those who had received placebo injections. The effectiveness of the vaccine was estimated as 100(1 12/I1) = 100(1 16/57) = 71.9 per cent protection. Hammond and Horn [1954] compared death rates of smokers (D18) with the mortality rates observed in their sample of nonsmokers (DO), by deriving the ratio D8/Do . Cornfield developed a method (Cornfield [1956], Neyman [1955]) for estimating an analogous from retrospective data on the smoking history of men with and without cancer. He defined relative risk as C8/Co where C. and C0 represent the prevalence of lung cancer among smokers and nonsmokers respectively. On the other hand, Berkson [1958] estimated the effect of smoking on mortality as the difference between two mortality rates, i.e., D. Do. For example, the standardized mortality rates from lung cancer observed by Doll and Hill [1956] were 1.66 and 0.07 per 1000 for heavy smokers and nonsmokers respectively. Rather than consider this a of 1.66/0.07 = 23.7, Berkson calculated the additional rate of 1.66 0.07 = 1.59 per 1000 smokers. He contended that his comparison was more appropriate since it treated the increased risk as proportional to those who would have survived if they had not smoked, rather than treating the risk as proportional to the deaths. As will be shown below, the meaning of the described statistics