~Extra Juicy~

In design of experiments, a covariate is an independent variable not manipulated by the experimenter but still affecting the response. See Variables (in design of experiments) for an explanatory example.

Variables (in design of experiments):

Many statistical methods rest on a statistical model which states a relationship

Y = f(X₁,..,X_N)

between a dependent variable (Y) and independent variable(s) X₁,...,X_N. In designed experiments, the dependent variable is often named "response", independent variables manipulated by the experimenter "factors", independent variables not manipulated by the experimenter (but still affecting the response) "covariates". The values which a factor can take on are named "levels" of this factor. Tested combinations of levels of all factors are called "treatments".

Consider an example of a clinical trial of drugs. The question addressed by the trial is how combinations of two drugs affect the survival rate. Two drugs have been tested: "Drug 1" and "Drug 2". Drug 1 has been administered in three ways - "None", "Orally (pills)", "Injection"; drug 2 has been administered in a single way, but in 4 different doses "None", "Low", "Moderate", "High". All the 12 (3x4) possible combinations of administration of the two drugs have been tested. The following model is used to interpret outcomes of the trial:

Y = T_ij + B X + E ;    i=1,...,3;    j=1,...,4;

where Y is the survival rate, indices i and j correspond to the methods of administration of the drug 1 and drug 2 respectively, X is age, and E is random variation in survival rate. Coefficients T_ij, which characterize the effect of the two drugs on the survival rate, are of primary interest.

The response here is the survival rate. We have two factors - "drug 1" and "drug 2". The first factor has three levels ("none", "orally", "injection"), the second factor - four levels ("none", "low dose", "moderate", "high"). (Note: levels are not necessarily doses, as for factor 2; for factor 1, for example, levels are related to methods of administration). We have here 12 treatments being tested - all the possible combinations of the three levels of factor 1 and the four levels of factor 4 (specified above). Coefficients T_ij are "effects" of the corresponding treatments. For example, T₂₃ is the effect of the treatment by drug 1 orally plus drug 2 in moderate dose. Age here is a covariate - it is not manipulated by the experimenter but still may have effect on the survival rate.