# S and T: Ordinal-Binary or Ordinal-Ordinal

## Information-Theoretic Approach

### Individual-Level Surrogacy: Ordinal-Binary or Ordinal- Ordinal Settings

For the extension to the case of an ordinal S and binary *T,* we replace (11.1) and (11.2) with two logistic regression models, for each trial i, as the response variable is binary:

For the situation of ordinal *S* and ordinal *T*, the proportional odds models of (11.1) and (11.2) would again be used. In each of these, the ordinal surrogate may be modeled in one of two ways. The first, as represented in (11.10), is to treat S as if it were continuous. The alternative would be to model S as a factor using dummy variables, but this ignores the ordering of the categories of the surrogate and is not parsimonious. Hence, modeling the ordinal surrogate as an interval scale variable is preferable.

At the individual level we apply the *likelihood reduction factor* (LRF) as before, basing Gj on the difference in — 2 x log-likelihood of (11.9) and (11.10) for ordinal S, binary T, and (11.1) and (11.2) for ordinal S and ordinal T. As T is discrete, the individual-level R is bounded above by 1 — e^{-2H(T}) and should again be rescaled (Alonso and Molenberghs, 2007) using (11.3).

### Trial-Level Surrogacy: Ordinal-Binary or Ordinal-Ordinal Settings

The only substantial difference in modeling the ordinal-binary and ordinal- ordinal settings, compared to the binary-ordinal setting, is in the way intercepts are handled. With an ordinal surrogate, the first-stage models (11.11)- (11.12) return one intercept for each cut-point on the ordinal surrogate. These intercepts are then averaged for use in the penalized likelihood approach of the second stage, as described in detail in Section 11.3.2.