Statistical Methods in Medical Research 4ed

338 Modelling continuous data
predict which patients have a poor prognosis and to consider alternative
methods of treatment for them (Armitage et al., 1969).
The appropriate technique is called multiple regression. In general, the
approach is to express the mean value of the dependent variable in terms of
the values of a set of other variables, usually called independent variables. The
nomenclature is confusing, since some of the latter variables may be either
closely related to each other logically (e.g. one might be age and another the
square of the age) or highly correlated (e.g. height and arm length). It is preferable
to use the terms predictor or explanatory variables, or covariates, and we
shall usually follow this practice.
The data to be analysed consist of observations on a set of n individuals, each
individual providing a value of the dependent variable, y, and a value of each of
the predictor variables, x1, x2, ..., xp. The number of predictor variables, p,
should preferably be considerably less than the number of observations, n, and
the same p predictor variables must be available for each individual in any one
analysis.
Suppose that, for particular values of x1, x2, ..., xp, an observed value of y is
specified by the linear model:
y ˆ b 0 ‡ b 1x1 ‡ b 2x2 ‡ ...‡ b pxp ‡ e, …11:38†
where e is an error term. The various values of e for different individuals are
supposed to be independently normally distributed with zero mean and variance
s 2 . The constants b 1, b 2, ..., b p are called partial regression coefficients; b 0 is
sometimes called the intercept. The coefficient b 1 is the amount by which y
changes on the average when x1 changes by one unit and all the other xis remain
constant. In general, b 1 will be different from the ordinary regression coefficient
of y on x1 because the latter represents the effect of changes in x1 on the average
values of y with no attempt to keep the other variables constant.
The coefficients b 0, b 1, b 2, ..., b p are idealized quantities, measurable only
from an infinite number of observations. In practice, from n observations, we
have to obtain estimates of the coefficients and thus an estimated regression
equation:
Y ˆ b0 ‡ b1x1 ‡ b2x2 ‡ ...‡ bpxp: …11:39†
Statistical theory tells us that a satisfactory method of obtaining the estimated
regression equation is to choose the coefficients such that the sum of squares of
residuals, P …y Y† 2 , is minimized, that is, by the method of least squares, which
was introduced in §7.2. Note that here y is an observed value and Y is the value
predicted by (11.39) in terms of the predictor variables. A consequence of this
approach is that the regression equation (11.39) is satisfied if all the variables are
given their mean values. Thus,