# General Linear Model

General Linear Model:

General (or generalized) linear models (GLM), in contrast to linear models, allow you to describe both additive and non-additive relationship between a dependent variable and N independent variables. The independent variables in GLM may be continuous as well as discrete. (The dependent variable is often named “response”, independent variables – “factors” and “covariates”, depending on whether they are controlled or not).

Consider a clinical trial investigating the effect of two drugs on survival time. Each drug is tested at three levels – “not used”, “low dose”, “high dose”, and all the 9 (=3×3) combinations of the three levels of the two drugs are tested. The following general linear model might have been used:

 Yij = A + B X + Ci + Dj + Rij + N; i,j = 1,2,3;

where Y is survival time (response), i and j correspond to the three levels of drug I and drug II respectively, X is age, Ci are additive effects (called “main effects”) of each level of drug I, Dj are main effects of drug II, Rij are non-additive effects (called interaction effects or simply “interactions”) of drugs I and II, N is random deviation.

We have here three independent variables: two discrete factors – “drug I” and “drug II” with three levels each, and a continuous covariate “age”.

In this particular case, because each of the two factors (drugs) has a zero level i,j=1 (“not used”), main effects C1, B1, and interactions R1j, j=1,2,3; Ri1, i=1,2,3 are zeros. The remaining unknown coefficients – A, B, Ci, Dj, Rij – are estimated from the data. The main effects Ci, Dj of the two drugs and their interaction effects Rij are of primary interest. For example, their positive values would indicate a positive effect – longer survival time due to use of the drug(s).

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