## The random coefficients model covariance structure

The statistical properties of random coefficients models were described in the repeated measures example introduced in Section 1.4. Here, we define their covariance structure in terms of the general matrix notation we have just introduced for mixed models. Random coefficients models will be discussed in more detail in Section 6.5.

The following data will be used to illustrate the covariance structure. They represent measurement times for the first three patients in a repeated measures trial of two treatments.

 Patient Treatment Time (days) 1 A ill 1 A tl2 1 A tl3 1 A tl4 2 B t21 2 B i22 3 A i31 3 A i32 3 A i33

If patient and patient-time effects were fitted as random coefficients, then there would be six random coefficients. We will change notation from Chapter 1 for ease of reading to define these as ^p1, ^pti1, ^p,2, ^pt,2, Pp,3 and ^pt,3 allowing an intercept (patient) and slope (patient-time) to be calculated for each of the three patients. The Z matrix would then be

Z =

Pp,1

Ppt,1

Pp,2

Ppt,2

Pp,3

Ppt,3

1

i11

0

0

0

0

\

1

i12

0

0

0

0

1

i13

0

0

0

0

1

i 14

0

0

0

0

0

0

1

i21

0

0

0

0

1

i22

0

0

0

0

0

0

1

i31

0

0

0

0

1

i32

V0

0

0

0

1

i33

As in random effects models the residuals are uncorrelated and the residual covariance matrix is

(a 2

0

0

0

0

0

0

0

0

0

a2

0

0

0

0

0

0

0

0

0

a2

0

0

0

0

0

0

0

0

0

a2

0

0

0

0

0

0

0

0

0

a2

0

0

0

0

0

0

0

0

0

a2

0

0

0

0

0

0

0

0

0

a2

0

0

0

0

0

0

0

0

0

a2

0

0

0

0

0

0

0

0

0

In a random coefficients model the patient effects (intercepts) are correlated with the random patient-time effects (slopes). Correlation occurs only for coefficients on the same patient (i.e. between ftp,j and ยก3ptj) and coefficients on different patients are uncorrelated. Thus, the G matrix would be

where ap2 and apt are the patient and patient-time variance components and appt is the covariance between the random coefficients. Note that G has dimension 6 x 6 because the model includes six random coefficients.

The V matrix

Again, V is obtained as V = ZGZ' + R. ZGZ' specifies the covariance due to the random coefficients and for our data has the form

 f Op2 ap,pt 0 0 0 0 ap,pt ap2t 0 0 0 0 0 0 ap2 ap,pt 0 0 0 0 ap,pt ap2t 0 0 0 0 0 0 ap2 ap,pt 0 0 0 0 ap,pt ap2t
 Vl,12 V1,13 V1,14 0 0 0 0 0 Vl,12 v1,22 v1,23 V1,24 0 0 0 0 0 vi,13 V1,23 V1,33 V1,34 0 0 0 0 0 Vl,14 V1,24 V1,34 V1,44 0 0 0 0 0 0 0 0 0 V2,11 V2,12 0 0 0 0 0 0 0 V2,12 V2,22 0 0 0 0 0 0 0 0 0 V3,11 v3,12 v3,13 0 0 0 0 0 0 V3,12 V3,22 V3,23 0 0 0 0 0 0 V3,13 V3,23 V3,33 /

Thus, ZGZ' has a block diagonal form with the size of blocks corresponding to the number of observations on each patient. It is added to the diagonal R = a 21 to form the total covariance matrix, V = ZGZ' + R, which will also have a block diagonal form. It may appear that covariances will increase with time and that a different origin for time would lead to different results. However, V is invariant to time origin and although the covariance parameters alter, we still obtain the same overall results (see further discussion in Section 6.5 and examples in Section 6.6).

Note that the covariance structure in random coefficients models is induced by the random coefficients. This differs from covariance pattern models (below) where covariance parameters in the R (or occasionally G) matrix are chosen to reflect a particular pattern in the data.