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
Complete cure
Partial cure
No improvement
Side-effect
Severity
None
Mild
Moderate
severe
Duration
Temporary
Permanent
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
I.
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
log ijk    i   j   k ,
i  1,2,3; j  1,2,3,4; k  1,2.
1
The model,
log ijk    ij   k ,
corresponds to C being independent of A, B, jointly.
The model,
log  ijk    ij    jk ,
corresponds to the independence of A and C conditionally on B.
II.
Canonical correlation models
For a model with two responses A and B with k A and k B levels,
respectively, there are k A  k B  1 parameters for the additive
terms  i   j and k AkB parameters for the interaction terms
 ij . For example, in the pharmaceutical example, suppose there
are only two responses, target effect and severity (side-effect). Then,
there are 3+4-1=6 parameters for  i   j and 3•4=12 parameters
for  ij . 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 s1 , s2 , , sk A
and t1 , t 2 ,  , t k B
are
available for the response categories of A and B, respectively, then for
i  1,2,, k A ; j  1,2,, kB , the following models can be used,
log  ij    i   j  r si t j 
log  ij    i   j  ri t j
log  ij    i   j  ri t j   j si
In the absence of score, we may entertain the single-root canonical
covariance model
2
log  ij    i   j   i j ,
where
  1 ,  2 ,,  k

t
A
  1 ,  2 ,,  k
and
kA
unknown unit vectors satisfying
kB
   
i 1
i
j 1
j

t
B
are
 0 and   0 is
unknown.
Note:
The above model is not of the log-linear type.
Note:
The likelihood equation for   0 satisfies
kA
kB
 ˆ ˆ y
i 1 j 1
i
j
kA
ij
kB
  ˆiˆ j ˆ ij
i 1 j 1
.
Note:
The likelihood ratio statistic for testing independence H 0 :   0
against the above model does not have an asymptotic
2
distribution. The correct asymptotic distribution is the distribution of
the largest root of a certain Wishart matrix.
3