Issues with the Relative Effect
It can be argued that a good surrogate endpoint should be able to predict the unobserved effect of Z on T in a future clinical trial i = 0 (i.e., во). The RE allows for such a prediction (provided that RE is sufficiently precisely estimated), but doing so requires a strong and unverifiable assumption. To illustrate this issue, consider Figure 3.2. The small circle in the figure depicts the estimated treatment effects on S and T in the ARMD trial (i.e., a = -0.8893 and /3 = -1.4562; see Section 3.4.2). Obviously, based on this information alone it will not be possible to predict the effect of Z on T in a future clinical trial i = 0 using a0 unless one is willing to make an additional assumption. Indeed, an infinite number of lines can be drawn through the single point (3, a) in Figure 3.2, and each of these lines would result in a different prediction of 3o given a particular value of a0 (except in the trivial case when a0 = a). Often, it is assumed that the relationship between 3 and a is multiplicative, which comes down to a regression line through (0, 0) and (3, a). In other words, it is assumed that RE remains constant across clinical trials. This constant RE assumption is depicted by the solid line in Figure 3.2. Based on the point estimate for RE and the constant RE assumption, the effect of Z on T can now be predicted in a straightforward way. For example, suppose that it is observed that a3 0 = -1 . 1 in a new clinical trial where the effect of interferon-a on visual acuity after 24 weeks is examined, then 30 = -1.8013, i.e., 30 = RE*S0, under the constant RE assumption, i.e., RE0 = RE = 1.6375. Of course, the validity of the constant RE assumption cannot be verified in the single-trial setting.
Second, in contrast to pZ, which has an interpretation in terms of the strength of the treatment-corrected association between S and T, the RE has no direct interpretation in terms of the strength of the association between 3 and a. This issue arises from the fact that a single clinical trial replicates patients — and thus a basis is provided for inference regarding patient-related characteristics — but not characteristics of the trial itself. Thus, even when the unverifiable constant RE assumption would not be an issue, there still is
Age-Related Macular Degeneration (ARMD) Trial. Graphical depiction of the Relative Effect (RE). The small circle shows the estimated treatment effects on T and S, i.e., (j3, cxj ) and the solid line depicts the constant RE assumption. The regression line passes through the origin (0, 0).
a problem (Molenberghs et al., 2013). To see this more clearly, rewrite (3.11) as в = RE * a and include an intercept so that в = p + RE * a is obtained. The question is now how accurate this relationship is. To study this in proper statistical terms, a final rewrite is necessary by adding an error term e in the previous expression: в = P + RE * a + e, where it is assumed that e follows a normal distribution with mean zero and variance a2. Obviously, a2 can only be estimated when there is appropriate replication of the pair (в, a). Put differently, the data of multiple clinical trials are necessary to quantify the accuracy with which в can be predicted. Thus, the regression model can be finalized as:
where в* and a* are the treatment effects on T and S in the i = 1, ..., N trials. If the regression model is correctly specified and a2 is sufficiently small, the treatment effect on T in a new trial can be accurately predicted based on the treatment effect on S in that trial.