[R] Looking to maximize a conditional likelihood

Ross Boylan ross at biostat.ucsf.edu
Thu Jul 17 23:10:01 CEST 2003

```I want to maximize a conditional likelihood function that is basically
logistic conditional on the number of successes within strata.  What
would be a good starting place for this?  A complication is that the
denominator includes a term that is the sum over all permutations.

Although there is no time dimension to the problem, it's possible a
degenerate use of the Cox proportional hazards model (in the survival
package of 1.7) will do what I want.

Here's a little more detail on the function, though this is still quite
terse.  Pseudo Tex notation:

We have cases in clusters.  y_ij is the outcome (0 or 1) for the j'th
case in the i'th cluster  It has vector covariates X_ij. c_ij is a
transform of y_ij and is in (0, 1) (it's actually the probability of
being in cluster j).  It may be specified a priori or to be estimated.

Maximize the product over i of

c_ij exp(sum_j y_ij X_ij b)
----------------------------
sum_K c_ij exp(sum_K y_ij X_ij b)

Where sum_K means we are summing over all possible subsets of the strata
that contain the observed number of successes.
b, and perhaps c, are the parameters to estimate.

The more vanilla case has no c_ij terms.  That says we maximize the
probability of the observed outcomes, given the total number of
successes within strata--i.e., we condition out on strata-specific
effects.  I kind of expected there would already be a routine that does
this, but I can't find it.