Section 13.2 Comparing Two Distributions
Up until this point in the book we have been considering tools for the investigation of the characteristics of the distribution of a single measurement. In most applications, however, one is more interested in inference regarding the relationships between several measurements. In particular, one may want to understand how the outcome of one measurement effects the outcome of another measurement.
A common form of a mathematical relation between two variables is when one of the variables is a function of the other. When such a relation holds then the value of the first variably is determined by the value of the second. However, in the statistical context relations between variables are more complex. Typically, a statistical relation between variables does not make one a direct function of the other. Instead, the distribution of values of one of the variables is affected by the value of the other variable. For a given value of the second variable the first variable may have one distribution, but for a different value of the second variable the distribution of the first variable may be different. In statistical terminology the second variable in this setting is called an explanatory variable and the first variable, with a distribution affected by the second variable, is called the response.
As an illustration of the relation between the response and the explanatory variable consider the following example. In a clinical trial, which is a precondition for the marketing of a new medical treatment, a group of patients is randomly divided into a treatment and a control sub-groups. The new treatment is anonymously administered to the treatment sub-group. At the same time, the patients in the control sub-group obtain the currently standard treatment. The new treatment passes the trial and is approved for marketing by the Health Authorities only if the response to the medical intervention is better for the treatment sub-group than it is for the control sub-group. This treatment-control experimental design, in which a response is measured under two experimental conditions, is used in many scientific and industrial settings.
In the example of a clinical trial one may identify two variables. One variable measures the response to the medical intervention for each patient that participated in the trial. This variable is the response variable, the distribution of which one seeks to investigate. The other variable indicates to which sub-group, treatment or control, each patient belongs. This variable is the explanatory variable. In the setting of a clinical trial the explanatory variable is a factor with two levels, “treatment” and “control”, that splits the sample into two sub-samples. The statistical inference compares the distribution of the response variable among the patients in the treatment sub-sample to the distribution of the response among the patients in the control sub-group.
The analysis of experimental settings such as the treatment-control trial is a special case that involves the investigation of the effect an explanatory variable may have on the response variable. In this special case the explanatory variable is a factor with two distinct levels. Each level of the factor is associate with a sub-sample, either treatment or control. The analysis seeks to compare the distribution of the response in one sub-sample with the distribution in the other sub-sample. If the response is a numeric measurement then the analysis may take the form of comparing the response’s expectation in one sub-group to the expectation in the other. Alternatively, the analysis may involve comparing the variance. In a different case, if the response is the indicator of the occurrence of an event then the analysis may compare two probabilities, the probability of the event in the treatment group to the probability of the same event in the control group.
In this chapter we deal with statistical inference that corresponds to the comparison of the distribution of a numerical response variable between two sub-groups that are determined by a factor. The inference includes testing hypotheses, mainly the null hypothesis that the distribution of the response is the same in both subgroups versus the alternative hypothesis that the distribution is not the same. Another element in the inference is point estimation and confidence intervals of appropriate parameters.
In the next chapter we will consider the case where the explanatory variable is numeric and in the subsequent chapter we describe the inference that is used in the case that the response is the indicator of the occurrence of an event.
