Comparison of Several Treatments with a Control Using Multiple Contrasts

Abstract

Abstract A problem frequently encountered in the practice of statistics is the comparison of several treatments with a control or standard. We consider an experimental situation where prior knowledge indicates that all of the treatments are at least as effective as the control and the problem is to determine if any are significantly better than the control. A number of statistical procedures have been proposed for this situation, of which the best known is Dunnett's (1955) multiple comparison procedure. Dunnett's test rejects equality of the treatments and the control for a large value of the maximum contrast of the data vector with several vectors that are located “symmetrically” within the alternative region. We study a large class of such tests, which includes Dunnett's test as a particular case. One of these tests, which is based on the maximum contrast of the data vector with several orthogonal vectors, is very easy to implement and has an uncomplicated, good, and fairly uniform power function over the entire alternative region. In fact, a small Monte Carlo power study suggests that this orthogonal contrast test is approximately “maximin” within this class of tests. Moreover, the simplicity of the power function of the orthogonal contrast test enables the experimenter to determine sample sizes for designed experiments with specific power characteristics. Such sample size determinations can be difficult, if not impossible, using other procedures. Abelson and Tukey (1963) suggested tests for a large class of restricted problems that are based on contrasts of the data vector with a single vector that is “centrally” located within the alternative region. These single contrast tests have the advantage that their distributions under the null hypotheses (a t distribution) and under the alternative (a noncentral t distribution) are rather simple. For the problem studied in this article the Abelson–Tukey contrast test is also a member of the class of tests proposed. Its power function varies considerably over the alternative region, with a high power in the center but very low power near the boundaries of the alternative region. The power of Dunnett's test has similar characteristics with less pronounced extremes. If only one of the treatments is effective, then a single contrast with the appropriate “corner” of the alternative region would have highest power. Of course, such a test would require prior knowledge of which treatment might be effective. Such a single contrast test is not in the class of tests studied here. Within this class of tests, the most powerful test when only one treatment is effective is based on the maximum contrast with all of the corners of the alternative region. Such a test has low power in a “center” of the alternative, that is, if all of the treatments are equally effective. Each test in the class of tests proposed here is associated with a set of vectors that are “strategically” located within the alternative region. The particular “strategy” chosen would depend on which alternatives are most important to protect against. Each such set of vectors can be regarded as a convex combination of the corner vectors of the alternative and the center vector. This center vector corresponds to the single contrast of Abelson and Tukey. The choice of the particular convex combination depends on the experimenter's belief regarding the relative magnitudes of the “common” effectiveness of the treatments and their individual effectiveness. If, for example, in a drug experiment each treatment is a combination of a common drug, known to be effective, and some individual new drug, the experimenter may wish to determine the convex combination based on his belief of the relative effectiveness of the new drugs. In the absence of such prior information we recommend the orthogonal contrast test for its simplicity and uniform power.