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# Binary Tumor Response

We now conduct the analysis of the binary tumor response (complete/partial vs. stable disease/progression) using the two-stage approach. Note that in the analysis, one of the trials (no. 6) is excluded due to the fact that there were no complete nor partial responses in one of the treatment groups.

For the Plackett copula with the marginal models (6.3) and (6.4), the estimated value of the copula parameter в is equal to 4.91 (95% CI: [4.16, 5.66]). It indicates a substantial association between response and OS at the individual-patient level: the odds for surviving beyond time t for responders are about 5 times higher than the odds of surviving beyond time t for nonresponders. The same result was reported by Burzykowski, Molenberghs, and Buyse (2004). As was the case in the analysis of the four-category response

(see Section 6.3.1.1), we do not attempt to remove the potential length-bias in the estimated value of в; such an analysis is presented in Burzykowski, Molenberghs, and Buyse (2004).

Figure 6.3 presents the estimated trial-specific treatment effects, i.e., logarithms of hazard ratios on OS and log-odds for tumor response; each trial is represented by a bubble whose size is proportional to the trial sample size. The association between the effects appears weak. A simple linear regression model, fitted without any adjustment for the estimation error present in the estimated treatment effects, yields the following regression equation:

with the standard errors of the intercept and slope estimated to be equal to 0.046 and 0.040, respectively. The corresponding value of Rriaii» is equal to 0.442 (95% CI: [0.156, 0.727]). The value is based on the following estimate of the variance-covariance matrix D of the (random) treatment effects:

for which the condition number (the ratio of the largest to the smallest eigenvalue) is equal to 28.0. This value is small enough to regard the obtained estimate as numerically stable.

Attempts to adjust the analysis for the estimation error by using model (5.13)-(5.14) fail due to the fact that the estimated variance-covariance matrix D is not positive-definite. The analysis weighed by the sample size yields the following estimate of the variance-covariance matrix D of the (random) treatment effects:

for which the condition number is equal to 25.5. The resulting estimate of R2rial(r) is equal to 0.390 (95% CI: [0.092, 0.689]) and the weighted linear regression model is

with the standard errors of the intercept and slope estimated to be equal to 0.042 and 0.041, respectively. This is the line indicated in Figure 6.3 as “predicted.” The results are similar to those obtained for the “naive,” unweighted linear regression.

Implementation of the models can be performed as explained on page 87. Figure 6.4 presents the estimated log-odds ratios and log-hazard ratios. 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 R2rial(r) equal to 0.398

FIGURE 6.3

Advanced Colorectal Cancer. Triad-level association between copula-model- based treatment effects on binary tumor response and OS (both axes on a log scale). The circle surfaces are proportional to trial size.

(95% CI: [0.106, 0.690]). The value is based on the following estimate of matrix D:

for which the condition number is equal to 31.3. The resulting regression equation is

with the standard errors of the intercept and slope estimated to be equal to 0.043 and 0.037, respectively.

Analysis adjusted for the estimation error leads to an estimate of variance- covariance matrix D which is singular. The analysis weighed by the sample size yields the estimate of Rrial(r) equal to 0.371 (95% CI: [0.070, 0.671]), based on the estimated variance-covariance matrix D of the (random) treatment effects:

with the condition number equal to 34.0. The weighted linear regression model is

FIGURE 6.4

Advanced Colorectal Cancer. Triad-level association between the marginal- model-based treatment effects on OS and binary tumor response (both axes on a log scale). The circle surfaces are proportional to trial size.

with the estimated standard errors of the intercept and slope equal to 0.036 and 0.035, respectively. This is the line indicated in Figure 6.4 as “predicted.”

## Using R

Currently, there are no specific tools aimed at evaluation of an ordinal surrogate for a failure-time true endpoint, or for the reverse.

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