The fixed effects model

All fixed effects models can be specified in the general form yt = j + aiXii + a2Xi2 +-----+ apxip + e¡, var(ei) = a2.

For example, in Section 1.2, Model B was presented as yij = j + Pi + tj + eij.

This model used a subscript i to denote results from the ith patient, and a subscript j to denote results on the jth treatment, in the context of a cross-over trial. In the general model notation, however, every observation is denoted separately with a single subscript. Thus, y 1 and y2 could represent the observations from patient 1, y3 and y4 the observations from patient 2, etc. The a terms in the general model will correspond to p1, p2, p3, p4, ps and p6 and to t1 and t2 and are constants giving the size of the patient and treatment effects. The terms xi1, xi2, ..., xi8 are used in this example to indicate the patient and treatment to which the observation yi belongs and here will take the values one or zero. If y1 is the observation from patient 1 who receives treatment 1, x11 then will equal one (corresponding to a1, which represents the first patient effect), x12 to x16 will equal zero (as this observation is not from patients 2 to 6), x17 will equal one (corresponding to a7, representing the first treatment effect) and x18 will equal zero. A further example to follow shortly should clarify this notation further.

The model described above fits p + 1 fixed effects parameters, a1 to ap, and an intercept term, /. Ifthere are n observations, then these maybe written as y1 = j + a1xn + a2x12 +-----+ apx1p + e1, y2 = j + a1x21 + a2x22 +-----+ apx2p + e2, yn var(e1) var(en) = a2.

These can be expressed more concisely in matrix notation as y = Xa + e,

= a where y = (yi, y3, ■ ■ ■, yn)' = observed values, a = (/¿, a1,a2, ■ ■ ■, ap)' = fixed effects parameters, e = (e1, e2, e3, ■ ■ ■ , en)' = residuals, a2 = residual variance,

The parameters in a may encompass several variables. In the example above they covered patient effects and treatment effects. Both of these are qualitative or categorical variables and we will refer to such effects as categorical effects. They are also sometimes referred to as factor effects. More generally, categorical effects are those where observations will belong to one of several classes. There may also be several covariate effects (such as age or baseline measurement) contained in a. These relate to variables which are measured on a quantitative scale. Several parameters may be required to model a categorical effect, but just one parameter is needed to model a covariate effect.

X is known as the design matrix and has the dimension n x p (i.e. n rows andp columns). It specifies values of fixed effects corresponding to each parameter for each observation. For categorical effects the values of zero and one are used to denote the absence and presence of effect categories, and for covariate effects the variable values themselves are used in X.

We will exemplify the notation with the following data, which are the first nine observations in a multi-centre trial of two treatments to lower blood pressure.

Centre

Treatment

Pre-treatment systolic BP

Post-treatment systolic BP

1

A

178

176

1

A

168

194

1

B

196

156

1

B

170

150

2

A

165

150

2

B

190

160

3

A

175

150

3

A

180

160

3

B

175

160

The observation vector y is formed from the values of the post-treatment systolic blood pressure:

y = (176, 194, 156, 150, 150, 160, 150, 160, 160)'^

If pre-treatment blood pressure and treatment were fitted in the analysis model as fixed effects (ignoring centres for the moment), then the design matrix would be

ai

a2

a3

178

1

0

168

1

0

196

0

1

170

0

1

165

1

0

190

0

1

175

1

0

180

1

0

175

0

1

where the columns of the design matrix correspond to the parameters H = intercept, a1 = pre-treatment blood pressure parameter, a2, a3 = parameters for treatments A and B.

We note here that the design matrix, X, is overparameterised. This means that there are linear dependencies between the columns, e.g. we know that a3 will be zero if a2 = 1, and one if a2 = 0. X could alternatively be specified omitting the a3 column to correspond with the number of parameters actually modelled. However, the overparameterised form is used here since it is used for specifying contrasts by sas procedures such as proc mixed (this procedure will be used to analyse most of the examples in this book).

V is a matrix containing the variances and covariances of the observations. In the usual fixed effects model, variances for all observations are equal and no observations are correlated. Thus, V is simply a 21.

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