Other applications

Repeated measurements within visits

Often done when data collection methods are inaccurate/imprecise. Can just use summary statistics from each visit, but does not deal well with missing data or testing interaction. Same basic approach as normal repeated measurements: covariance pattern or random coefficients model.

Covariance pattern models

Several ways to model covariance, five reasonable model presented below:

• constant covariances: constant correlation between all observations on
• the same patient
• extra covariance for observations at the same visit: would probably
• expect measurements taken on same visit to be more correlated, so fit
• separate covariance between observations on same visit, and observations
• on different visits
• covariance pattern structured by visits: correlation bewteen observations
• different for each pair of visits
• covariance pattern structured by reps: possible correlation between
• observations at same visit depends differs depending on rep number
• extra covariance for observations on same reps

Many options – how to choose? If interested in pattern itself, start with simplest and build up. If interested in estimates of mean and standard errors, then second option probably adequate.

Random coefficients models

Again, more appropriate if interested in relationship with time, but now have two time scales, within and between visits, and can fit slopes to either or both of these.

Visit time: take reps as categorical fixed effects, not usually used for cross-over data Rep time: Both: not usually used for cross-over data

Matched case-control studies

Match group of subjects with disease to those without on age, sex, etc and determine which other factors differ between groups. Usually, take candidate risk factors as outcome variables, and matched set as random effect. Similar to cross-over trial.

Will be same as fixed effect model, when same number of controls for every case and positive matched set variance component. In fixed effects model information from group effects with all case or all control is completely lost. Another fixed effects approach is to fit the matching variables as covariates but otherwise ignore matching – this can cause bias if matching variables are associated with group.

With binary variables

If a variable is binary, then likely that several uniform categories for matched set effects. This may cause bias in random effects model and should be reparameterised as a covariance pattern model with compound symmetry. Preferable to conditional logistic regression where information lost on all tied matching sets.

Different variances for patient groups

In simple between-patient trial sometimes treatment groups have different variances – allowing for this will give more appropriate standard errors, and variance estimates may be useful for understanding treatment mechanisms. Easy to do – just allow $\mathbf{R}$ to have difference variances for each group (use repeated statement in SAS, eg. repeated int / subject=patient group=treat).

Estimating variance components

Sometimes interested in just estimating variance components so that they can be used to help design future trials. Set $\Delta = (Z_{\alpha/2} + and then use standard variance formula $Var(t_i - t_j) = \sum_{\text{error strata}} \sigma^2 / n$ . Will usually need to determine via grid search.

Small sample sizes

Mixed models useful, because shrinkage properties to reduce small sample variability.