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Individual-level surrogacy

To assess the quality of the surrogate at the individual level, a measure of association between Sj and Tj, calculated while adjusting the marginal distributions of the two endpoints for both the trial and treatment effects, is needed. In the copula model (5.1), the strength of the association depends on the copula parameter в. Hence, в could be used as the individual-level surrogacy measure. However, the value of the parameter is generally hard to interpret. Instead, one can use Kendall’s concordance-coefficient т, which can be obtained from в by the following transformation (Nelsen, 2006):

Alternately, one can use Spearman’s rank-correlation coefficient p (Nelsen, 2006):

The relationship (5.8) is especially simple for the Clayton and Hougaard copulas, defined in (5.4) and (5.5), respectively. In particular, for the former,

whereas for the latter

On the other hand, for the Plackett copula (5.6)-(5.7), the relationship (5.9) between в and p is explicit:

Trial-level surrogacy

The quality of the surrogate at the trial level can be evaluated by considering the correlation coefficient between the treatment effects a* and в*. The coefficient can be estimated by using the estimates a* and в* of the treatment effects obtained from (5.2)-(5.3). Note that, in this step, the adjustment for the estimation error, present in a* and в*, should be made. Toward this aim, as suggested by Burzykowski and Cortinas Abrahantes (2005), assume that the estimated treatment effects Sj and (3i follow the model

where the estimation errors eai and ?bi are normally distributed with mean zero and variance-covariance matrix

and (ai,ei)T follows a bivariate-normal distribution with mean (a, в)т and variance-covariance matrix D given by (4.13). Consequently, (ai, /3i)T follows a normal distribution with mean (a, в)т and variance-covariance matrix D + Qi. One can then fit model (5.13)-(5.14) while fixing matrices Hi at their values estimated in the first-stage model, and obtain an estimate of D.

If the individual-level association is not of immediate interest, one may base analysis on the marginal models (5.2)-(5.3) without specifying the baseline hazards. When fitting the models, it is important to use the “robust” estimator of the variance-covariance matrix of the estimated treatment effects ai and (3i (see, e.g., Lin, 1994). This is because the estimator takes into account the correlation between the surrogate and true endpoints and provides a correct estimate of the variance-covariance matrix. The latter can then be used to adjust the estimation of the correlation coefficient of ai and pi for the estimation error present in ai and (3i.

 
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