Resolving paradoxes involving surrogate end points

We define a surrogate end point as a measure or indicator of a biological process that is obtained sooner, at less cost or less invasively than a true end point of health outcome and is used to make conclusions about the effect of an intervention on the true end point. Prentice presented criteria for valid hypothesis testing of a surrogate end point that replaces a true end point. For using the surrogate end point to estimate the predicted effect of intervention on the true end point, Day and Duffy assumed the Prentice criterion and arrived at two paradoxical results: the estimated predicted intervention effect by using a surrogate can give more precise estimates than the usual estimate of the intervention effect by using the true end point and the variance is greatest when the surrogate end point perfectly predicts the true end point. Begg and Leung formulated similar paradoxes and concluded that they indicate a flawed conceptual strategy arising from the Prentice criterion. We resolve the paradoxes as follows. Day and Duffy compared a surrogate-based estimate of the effect of intervention on the true end point with an estimate of the effect of intervention on the true end point that uses the true end point. Their paradox arose because the former estimate assumes the Prentice criterion whereas the latter does not. If both or neither of these estimates assume the Prentice criterion, there is no paradox. The paradoxes of Begg and Leung, although similar to those of Day and Duffy, arise from ignoring the variability of the parameter estimates irrespective of the Prentice criterion and disappear when the variability is included. Our resolution of the paradoxes provides a firm foundation for future meta-analytic extensions of the approach of Day and Duffy.