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# Marginal Models

If we focus on the trial-level surrogacy, we can conduct an analysis based solely on the marginal models (7.1) and (7.2), with the latter specified without assuming any particular form of the baseline hazards. Toward this aim, model (7.1) can be fitted by using PROC MIXED and (7.2) by using PROC PHREG. In particular, a possible call to PROC MIXED is:

PROC MIXED data=dataset;

CLASS trial;

MODEL contsurr = trial treat(trial) / noint;

REPEATED / group = trial;

RUN;

Variable trial is the trial identifier (trial-by-country group in our example), while treat is the treatment indicator (with 0 for the control group and 1 for the experimental one). The response variable contsurr is the logarithm of PSA at day 28. Of interest are the trial-specific treatment-effect estimates, provided by the nested effect treat(trial). The REPEATED statement includes the group=trial option, which implies the use of trial-specific residual variances.

On the other hand, (7.2) can be fitted by using PROC PHREG, as described in Section 6.3.1.1.

After fitting the models, the variance-covariance of the estimated coefficients of the models can be estimated while taking into account the association between the negative-log-PSA and OS (see Appendix 7.5). Toward this aim, a dedicated SAS macro can be used (see Section 12.6.3).

Figure 7.4 presents the estimated mean differences and log-hazard-ratios. As was the case in the copula-based estimates (see Figure 7.3), the association between the treatment effects is weak. A simple linear regression model yields the value of Rrial(r) equal to 0.0007 (95% CI: [-0.025, 0.026]). The resulting regression equation is

with the standard errors of the intercept and slope estimated to be equal to 0.212 and 0.229, respectively.

Analysis adjusted for the estimation error leads to an estimate of the variance-covariance matrix D that is singular. The analysis weighted by the group sample size, conducted with the help of the dedicated SAS macro (see Section 12.6.3), yields the following estimate of the variance-covariance matrix D of the (random) treatment effects:

for which the condition number is equal to 1.9. The corresponding estimate of Rrial(r) is equal to 0.017 (95% CI: [-0.108,0.141]), with the linear regression equation given by

and standard errors of the intercept and slope estimated equal to 0.155 and 0.185, respectively. The line corresponding to equation (7.9) is indicated in Figure 7.4 as “predicted.”

## Using R

Currently, there are no specific tools aimed at evaluation of a continuous (normally distributed) surrogate for a failure-time true endpoint.

FIGURE 7.4

Trial-level association between the marginal treatment effects on negative-log- PSA and OS in advanced prostate cancer (vertical axis on a log scale).

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