6 Two-sample confidence intervals and
Fig. 17 presents the two-sample hypothesis testing situation for proportions.
•
Distribution under H0 Distribution under Ha
0 ~-~
• •
Type I error probability Type II error
probability
•
Reject H
0 Fail to reject H
0 Reject H
0
Example 11.6.1
Suppose two drugs are available for the treatment of a particular type of intestinal parasite. One hundred patients entering a clinic for treatment for this parasite are randomized to one of the two drugs; fifty receiving drug A and fifty receiving drug B. Of the patients receiving drug A, 64% responded favorably; 82% of the patients receiving drug B responded favorably. Are the drugs equally effective or is the observed difference too large to be due to chance alone?
Solution
The problem is set up assuming the two drugs are equally effective and, so long as there is no economic or other reason to prefer one drug over the other, the test can be set up two-sided. That is,
Ho: P1=P2 or Ho: P1-P2=D Ha: Pr;toP2 or Ha: P1-P2 ;eO
Our decision rule, at the 5% level of significance, is to reject Ho if P1-P2 is greater than 0+1.96v'[2(0.73)(0.27)/50] = 0.174 or less than 0-1.96v'[2(0.73)(0.27)/50] = -0.174. The value 0.73 is the average of the proportions responding in each group (previously denoted p). In the actual trial, P1-P2 =0.64-0.82=-0.18.
Therefore, since this falls in the rejection region (i.e.,-0.18<-0.174), the null hypothesis is rejected in favor of the alternative that the drugs are not equally effective.
An alternative approach for testing this hypothesis is to first establish a confidence interval whose end points are given in the following expression:
and then determine whether or not the value 0, i.e., the value which would indicate no difference between the population proportions, falls in the interval. If it does, the null hypothesis is not rejected; if it does not, the null hypothesis is rejected.
Example 11.6.2
In Example 11.6.1, the end points of the confidence interval are:
(0.64-0.82)±1.96v'[(0.64)(0.36)/50 + (0.82)(0.18)/50]
70 Foundations of Sampling and Statistical Theory
-0.18±1.96(0.0869)=-0.18±0.17, yielding the 95% confidence interval,
Notice that 0 does not fall in this interval so that again we would conclude, using a confidence interval approach, that the two drugs are not equally effective.
Note that when establishing a confidence interval the sample proportions are not pooled to obtain an estimate of the variance of the difference since there is no underlying null hypothesis stating that the population parameters are equal.
Confidence intervals have a major advantage over hypothesis tests - more information is obtained about the population parameter than the simple rejection or acceptance of a statement. When testing a null hypothesis we always run a risk of committing an error.
The type-I or a error is possible whenever the null hypothesis is rejected. Fortunately, the magnitude of this error probability is fixed and may be stated in advance by the investigator. Whenever the hypothesis cannot be rejected, there exists the possibility of committing a type-IT or~ error. It is unfortunately true that one never knows how large the
~ error is since one never knows the actual condition of the population.
true difference between the proportions in the two groups or on testing the hypothesis that the proportions in the populations from which the samples were selected are equal. That discussion was presented within the context of the cohort or follow-up study. The key element in the cohort study is that individuals are grouped according to whether or not they have a certain characteristic which is suspected to be related to the outcome of interest.
This characteristic will be called the exposure variable. Individuals in the various exposure groups (usually presence and absence of exposure) are subsequently followed until a determination of the outcome characteristic (e.g., disease or no disease) can be made. For the epidemiologist, however, this type of study design may not be practical when the time between exposure and outcome is lengthy and/or unknown. For example, a cohort study to assess the relationship between consumption of artificial sweeteners and bladder cancer would not be practical for at least two major reasons. First, in order to obtain a sufficiently large group of patients who develop the disease, a huge sample would have to be identified when they are disease-free and then followed for several years to determine subsequent disease status. This is due to the fact that the disease is relatively rare. Second, because exposure is at relatively low levels, a long exposure time would be necessary before the disease could be expected to develop. This presents numerous logistical problems, not the least of which is keeping track of and staying in contact with a large number of study subjects for a long period of time. Finally, from a practical point of view, if sweeteners are suspected of being associated with bladder cancer, we would not want to wait twenty years or so to have confirmatory scientific evidence. Hence, a cohort design is not a realistic option for many modern epidemiologic investigations on chronic diseases.
In a case-control design, subjects are selected on the basis of their outcome status (e.g.
patients with bladder cancer are enlisted into the study, as is a group of "controls" or non- cancer patients) and all subjects are studied with respect to their prior and current exposure to suspected risk factors. From a practical point of view, this type of study may be carried out at relatively low cost and within a relatively short time frame since it is not necessary to wait for the disease to develop in previously disease-free individuals.
There is a third type of epidemiologic study known as a prevalence study. In this type of study, a representative sample is selected from the population in order to estimate the proportion of individuals with a condition of interest at a specific point in time. This condition can relate to either exposure or to disease and is typically presented as a proportion. This proportion is termed the prevalence and represents an instantaneous snapshot of the number of people with the condition at a specified point in time relative to the total number of eligible individuals in the population. The concept of prevalence is distinct from that of incidence which is a measure of the number of new cases occurring in the population in a specified time period. Prevalence may be either greater than or less than incidence depending upon the duration of the condition and the rate at which incident cases die or leave the population. Hence one measure cannot be substituted for the other.