# Analysis of Case Studies: The Age-Related Macular Degeneration Trial

The use of the Prentice criteria to evaluate the validity of a candidate surrogate endpoint will be illustrated here using the ARMD dataset (which was described in Section 2.2.1 of Chapter 2, starting on page 16). Recall that in the ARMD dataset, the surrogate and the true endpoints of a patient j (i.e., *Sj* and *Tj*) were the change in visual acuity at 6 and 12 months after starting the treatment, respectively. The binary indicator for the treatment of a patient j (i.e., Zj) was coded as -1: placebo and 1: interferon-a.

In the analyses, it will be assumed that both S and T are normally distributed. In this setting, the first two Prentice criteria (see (3.1)-(3.2)) are examined by fitting the following bivariate linear regression model:

where the error terms *e _{S}j* and

*e*have a joint zero-mean normal distribution with variance-covariance matrix:

_{T}j

Prentice’s third criterion (see (3.3)) can be examined by fitting the following univariate linear regression model (when S and T are continuous normally distributed endpoints):

Finally, the fourth Prentice criterion (see (3.4)) can be examined by fitting the following univariate linear regression model (when S and T are continuous normally distributed endpoints, as in (3.5)-(3.6)):

where

and the variance of eT equals:

When models (3.5)-(3.6), (3.8), and (3.9) are fitted to the ARMD dataset, it is obtained that a = -0.8893 (s.e. 0.9509, *p* = 0.3509), *fi* = -1.4562 (s.e. 1.1771, p = 0.2176), i = 0.9256 (s.e. 0.0617, p < 0.0001), and */3 _{S}* = -0.6362 (s.e. 0.7894, p = 0.4213). Note that a and

*i*are negative, indicating a (albeit non-significant) negative effect of the experimental treatment on visual acuity after 24 and 52 weeks.

To be in line with Prentice’s criteria, the hypotheses H_{0} : *a* = 0, H_{0} : *в* = 0, and Ho : 7 = 0 in models (3.5)-(3.6) and (3.8) should be rejected, whereas the hypothesis H_{0} : *fi _{S}* = 0 in model (3.9) should not be rejected. Thus, in the ARMD dataset, the first and second Prentice criteria were not fulfilled (using the conventional 0.05 level of significance). Hence, based on the Prentice criteria, it is concluded that the change in visual acuity after 6 months is not a good surrogate for the change in visual acuity after 12 months.