Drug Combination Analysis

Drug-combination analysis is discussed and demonstrated. Suppose we have a set of drugs (or other treatments which we misclassify as “drugs” for the purpose of our analysis.) These drugs are supposed to be treatments for some disease, e.g., HIV infection. Our goal is to assess whether, and to what extent, treatment with a particular combination of these drugs is more or less effective than some other such combination treatment. Such combination treatments include the case of treatment with a single drug, so we can also compare the efficacy of two distinct single drugs with our method. For each treatment with a combination of drugs D1, . . . , Dk, the data we have is a set of k + 1 dimensional points of the form (d1, . . . , dk, y), where dj is the dose of drug Dj (measured in any desirable units) and y is the response observed in the subject represented by the point at hand. The observed response is always non-negative, and is generally a value in the interval [0, 1]. Such a value y denotes the relative amount or level of disease present after the drug treatment. Thus a small response value indicates a better response than does a larger response value. Choosing a reasonable and informative response measure is vitally important. A direct measure such as the amount of virus present is generally preferable, (although because this is biology, there are many unknowns and pitfalls in nearly every measure.) Crudely, a response value y should be measured such that 100(1−y) is the percent improvement observed. For example, let w0 denote a subject’s white blood cell count before treatment and let w1 denote that subject’s white