ABSTRACT

We have so far been concerned entirely with the dependence of our ob­ servations Y on a single explanatory variable t. Whilst this is sufficient to deal with a number of problems of interest, there are many situations in practice where observed responses are influenced simultaneously by several variables. Statistical analysis of the dependence on explanatory variables then usually leads to the use of multiple regression. The ex­ planatory variables may be either quantitative (numerical) or qualitative (categorical) and the most well known general framework is provided by the linear model

Yj = x j0 + error. (4.1) Here x, is a vector of explanatory variables for the ith observation, and

/3 the corresponding vector of regression coefficients, to be estimated. In general, the vector x, may include a constant entry 1 for an intercept term, indicator variables to model categorical explanatory variables, and products of other components so that interactions can be assessed. The simple case (1.1) of univariate linear regression is given by setting x,- = (1 Ji)T and (3 = (a,b)T. It is not appropriate here to give a complete discussion of the linear model: the interested reader should consult one of the many standard textbooks on the subject.