Separation causes the same problems for ordinal S as for binary S and ordinal T, but the penalized likelihood technique again offers a solution. As the currently available penalized likelihood software for ordinal outcomes cannot handle the large datasets and hierarchical data structures we need to use, for each trial i, separate models of S and T are regressed on treatment Z in stage one of trial-level estimation:

where v = 1,..., V — 1; V is the number of categories in the ordinal true outcome; p_{Sv}. is the set of intercept parameters for each of the V — 1 cut- points on the ordinal surrogate outcome in trial i; and all other parameters are the same as in the continuous case.

For the information-theoretic approach, the second-stage model (11.6) can be replaced with the following model, using all the surrogate intercept variables of (11.11):

Here, usu, ? ? ?, Aare the intercept variables for each cut-point modeled alongside a treatment effect estimate, cp. However, we can see that in the case of few trials, this model would lead to over-fitting. One way to resolve this would be to calculate the mean intercept value for each trial over all cut-points to give a single estimate of the intercept for each trial, цз„.. Then the stage 2 model of trial-level surrogacy would become:

The difference in — 2 x log-likelihood between (11.14) and a null (intercept only) model can be used to determine the LRF (11.7) and estimate Rr

Summary: Ordinal-Binary or Ordinal-Ordinal Setting

Only two additional considerations are required where S is ordinal. First, for individual-level surrogacy, one must define how the surrogate is handled as a covariate in a generalized linear model. We recommend that treating it as a continuous measure is preferable to comparing a series of dummy variables to a reference category. The second change, which is required for evaluating trial-level surrogacy, is to derive a single intercept estimate /TyT which summarizes the multiple intercepts generated in the proportional odds modeling for the various cut points in the ordinal outcome. As in Section 11.2, penalized likelihood may be used to correct for any bias due to occurrences of separation.

Concluding Remarks

We have outlined the extension of the information-theoretic approach to categorical surrogate and true outcomes. This was done without using joint modeling of the surrogate and true outcomes, thus avoiding additional computational complexities. The penalized likelihood approach provides an effective solution to prevent problems of bias due to separation or quasi-separation of categorical outcomes. As penalized likelihood does not readily handle large or hierarchically structured datasets, it may be implemented instead by separate modeling of the data for each trial.