# Marginal Models

If we focus on the trial-level surrogacy, we can conduct an analysis based solely on the marginal models (5.2)-(5.3) specified without assuming any particular form for the baseline hazards. Toward this aim, we can use PROC PHREG. In particular, the data have to be transformed to the “long” format, with two records per patient, one providing the time and censoring indicator for PFS, and the other the time and censoring indicator for OS. An illustration of the first few records in such a dataset is:

treat idpat endp time status

- 1 219 1 2.35729 1
- 1 219 2 0.75838 1
- 0 220 1 1.79877 1
- 0 220 2 0.39151 1
- 1 221 1 0.71458 1
- 1 221 2 0.71458 1

Then, for each trial, treatment effects on PFS and OS can be estimated by the following SAS syntax:

PROC PHREG data=joined covs(aggregate) covout outest=outests; MODEL time*status(0)=treat1 treat2 / ties=efron;

STRATA endp;

ID idpat;

treat1 = treat * (endp=1); treat2 = treat * (endp=2);

RUN;

Note the use of the covs (aggregate) option and the ID idpat statement that allow estimating the variance-covariance of the estimated treatment effects by using the “robust” estimator, which takes into account the association between PFS and OS. Variables treat1 and treat2 define treatment indicators specific for, respectively, the surrogate and true endpoints.

Figure 5.2 presents the estimated log-hazard ratios. As was the case in the copula-based estimates (see Figure 5.1), the association between the treatment effects is only moderate. A simple linear regression model, fitted without any adjustment for the estimation error present in the estimated treatment effects, yields the value of RLiaip) equal to 0.452 (95% CI: [0.130, 0.775]), somewhat smaller than the estimate obtained from the copula model with Weibull marginal hazards. The value is based on the following estimate of matrix D:

for which the condition number is equal to 5.4. The resulting regression equation is

with the standard errors of the intercept and slope estimated to be equal to 0.052 and 0.144, respectively. The slope is markedly smaller than it was in the case of the “naive” regression model (5.15) corresponding to the copula model. with the standard errors of the intercept and slope estimated to be equal to 0.052 and 0.144, respectively. The slope is markedly smaller than it was in the case of the “naive” regression model (5.15) corresponding to the copula model.

After adjusting for the estimation error, R2r_{ial}(r) is estimated to be equal to 0.224 (95% CI: [-0.638, 1.085]), based on the following estimate of the variance-covariance matrix D:

for which the condition number is equal to 4.0. The estimated value of R2_{rial}(_{r}) is much smaller than the estimates obtained for the copula model or for the “naive” regression model (5.15). However, the CI for the adjusted R_{rial}(_{r}) is very wide; it essentially covers the entire admissible range [0,1]. In fact, it extends beyond the range; this is due to the fact that the CI was obtained as (Rtriai(r))^{2} ± 1.96 x SE{(i?triai(r))^{2}}, where SE{(Rtrial(r))^{2}} was computed by applying the delta method to the standard error of i?_{trial}(_{r}), i.e.,

The linear regression model obtained with an adjustment for estimation errors is

with the standard errors of the intercept and slope estimated to be equal to 0.101 and 0.379, respectively. There is a marked difference between the estimated values of the intercept and slope as compared to the “naive” regression (5.20); however, given the (im)precision of the estimates, the difference is not significant. The standard errors of the coefficients of the equation (5.22) are larger than the corresponding estimates for (5.20).

FIGURE 5.2

*Advanced Gastric Cancer Data. Trial-level association between marginal-PH- model-based treatment effects on PFS and OS (both axes are on a log scale). The circle surfaces are proportional to trial size.*

Regression line (5.22) is labeled “predicted” in Figure 5.2. The 95% prediction limits, presented in the plot, indicate the range of effects on OS that can be expected for a given effect on PFS. Note that the limits do not cross the horizontal line corresponding to *HR* = 1. Hence, it is not possible to estimate the surrogate threshold effect STE.