4.6 Multiple Responses

Motivating example:

In a pharmaceutical trial, a drug is designed with a particular target effect in mind but invariably there are side-effects of varying duration and severity. The target response and supplementary responses are summarized in the following table:

Target
effect / Side-effect
Severity Duration
Complete cure / None / Temporary
Partial cure / Mild / Permanent
No improvement / Moderate
severe

The following lines of inquiry would often be considered worth pursuing:

Model construction for the dependence of each response marginally on covariates x.

Model construction for the joint distribution of all responses.

Model construction for the joint dependence of all response variables on covariates x.

(a)Joint Dependence of Response Variables

  1. Independence and conditional independence

Suppose we have 3 responses, A, B, and C. For example, in the previous section, A is the target effect, B is the severity of side-effects and C is the duration of side effects. Then, mutual independence of the 3 responses A, B, and C corresponds to the log-linear model

The model,

corresponds to C being independent of A, B, jointly.

The model,

corresponds to the independence of A and C conditionally on B.

  1. Canonical correlation models

For a model with two responses A and B with and levels, respectively, there are parameters for the additive terms and parameters for the interaction terms . For example, in the pharmaceutical example, suppose there are only two responses, target effect and severity (side-effect). Then, there are 3+4-1=6parameters for and 3•4=12parameters for . It is natural to explore the intermediate ground where the nature of the interaction is described by a small number of parameters. If the scores and are available for the response categories of A and B, respectively, then for the following models can be used,

In the absence of score, we may entertain the single-root canonical covariance model

where and are unknown unit vectors satisfying and is unknown.

Note:

The above model is not of the log-linear type.

Note:

The likelihood equation for satisfies

.

Note:

The likelihood ratio statistic for testing independence against the above model does not have an asymptotic distribution. The correct asymptotic distribution is the distribution of the largest root of a certain Wishart matrix.

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