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Analysis of a Case StudyIn this section, we illustrate the application of the methods, mentioned in the previous section, by the individualpatientdata metaanalysis of patients with advanced (metastatic) prostate cancer (see Section 2.2.7). The goal of the analysis is to evaluate the logarithm of PSA, measured at about 28 days, as a FIGURE 7.1 Histograms of PSA and logPSA at 28 days. surrogate for overall survival (OS). The data were analyzed before by Buyse et al. (2003), but without considering the value of PSA at a particular time point as a candidate surrogate for OS. For analysis purposes, patients were grouped by trial and by country. Treatment with flutamide or cyproterone acetate is considered the control treatment, while liarozole is regarded as the experimental treatment. Using SASCopulaBased ModelsAmong the 596 patients included in the dataset, 421 had a PSA measurement obtained at about 28 days (±6 days). There are 19 trialbycountry groups containing between two and 55 patients per group. Two groups (one with two and one with seven patients) have to be eliminated from the analysis, because at least one of the treatment arms within the group does not contain any deaths. Consequently, the analysis includes 412 patients spread across 17 groups. Figure 7.1 presents histograms of PSA (lefthandside panel) and logPSA values for the 412 patients. Note that, for legibility purposes, the histogram of PSA has been obtained for PSA values smaller than 2000 ng/mL. The histogram on the righthand side of Figure 7.1 shows that the logarithmic transformation makes the distribution of the data more symmetric. Figure 7.2 presents the scatter plot of logPSA at 28 days and OS. Note that we ignore censoring in the latter variable. The scatter plot indicates a negative association between the two variables. Given that Clayton and Hougaard copula models assume a positive association (see Section 5.2), in what follows we will consider the negative of logPSA as a surrogate. We first conduct a twostage analysis using the Clayton, Hougaard, and Plackett copula with the marginal models (7.1) and (7.2); the latter is speci FIGURE 7.2 Scatter plot of overall survival and logarithmic PSA at 28 days. fied by assuming the Weibull hazard functions. Toward this aim, we use the dedicated SAS macros (see Section 12.4.1). We use the copulabased likelihood function to estimate the copula parameter and all the coefficients of the marginal models. The estimated value of the copula parameter в for the Clayton copula is equal to 1.61 (95% CI: [1.40, 1.83]). Given the relationship (5.10), the value of т is estimated to be equal to 0.235 (95% CI: [0.171, 0.298]). Spearman’s rankcorrelation coefficient p, computed from (5.9) by numerical integration, is equal to 0.344 (95% CI: [0.255, 0.433]). For the Hougaard copula, the estimated value of в is equal to 0.724 (95% CI: [0.662, 0.790]). Following (5.11), the estimated value of т is equal to 0.276 (95% CI: [0.212, 0.340]). Spearman’s p, computed by numerical integration, is equal to 0.400 (95% CI: [0.305, 0.495]). Finally, for the Plackett copula, the estimated value of в is equal to 3.56 (95% CI: [2.49, 4.64]). Note that it can interpreted in the following way (see Sections 5.2 and 6.2): the odds of surviving beyond time t given a value of negativelogPSA at 28 days larger than x, say, are about 3.6 times higher than the odds of surviving beyond time t given a value of negativelogPSA smaller than or equal to x. As — ln(PSA) > x = PSA < e^{x}, we can equivalently conclude that the odds of surviving beyond time t given a lower value of PSA at 28 days are about 3.6 times higher than the odds of surviving beyond time t given a higher value of PSA. From (5.12) it follows that the estimated value FIGURE 7.3 Triallevel association between the marginal treatment effects on negativelog PSA and OS in advanced prostate cancer (vertical axis on a log scale). of p is equal to 0.402 (95% CI: [0.317, 0.488]). Kendall’s т, computed from (5.8) by numerical integration, is equal to 0.276 (95% CI: [0.214, 0.339]). Thus, all three copula models suggest a weak association between the negativelogPSA at 28 days and OS at the individualpatient level. It is worth noting that the value of the maximum loglikelihood for the Clayton, Hougaard, and Plackett copulas are equal to 1047.8, 1042.2, and 1043.6, respectively. Thus, one could conclude that the Hougaardcopulabased model offers the best fit to the data. Figure 7.3 presents the estimated trialspecific treatment effects, i.e., logarithms of the Weibullmodelbased hazard ratios for OS and mean differences for negativelogPSA, obtained by using the Hougaardcopulabased model; each trial is represented by a circle of the size proportional to the group sample size. The figure indicates almost no association between the treatment effects. It seems to suggest that the larger the mean difference in negativelogPSA (the larger mean reduction of PSA for the experimental treatment), the smaller corresponding loghazard ratio for OS (the larger reduction in hazard of death for the experimental treatment). A simple linear regression model, fitted without any adjustment for the estimation error present in the estimated treatment effects, yields the following regression equation: where ^i_{n}PSA is the mean difference in negativelogPSA and the standard errors of the intercept and slope are estimated to be equal to 0.253 and 0.282, respectively. The corresponding value of Rriagr) is equal to 0.003 (95% CI: [0.046, 0.052]); the negative lower limit of the CI is due to the use of the normal approximation to the distribution of (R_{trial}(_{r}))^{2} (see Section 5.3.1.1). The analysis weighted by the sample size, using the tools described in Section 6.3.1.1, yields the estimate of R2_{rial}(_{r}) equal to 0.007 (95% CI: [0.072, 0.086]), with the underlying weighted linear regression model
with the standard errors of the intercept and slope estimated to be equal to 0.183 and 0.221, respectively. As discussed in Section 6.3.1.1, weighting by sample size is not optimal, as the precision of the estimation of the treatment effect on OS depends rather on the number of deaths observed in the trial, not on the total sample size. To properly adjust the analysis for the estimation error, model (5.13)(5.14) can be used. Applying it, using PROC MIXED code described in Section 5.3.1, yields the estimate of R2_{rial}(_{r}) equal to 0.0001 (95% CI: [0.054,0.055]). The value is based on the following estimate of the variancecovariance matrix D of the (random) treatment effects:
for which the condition number is equal to 9.6. Thus, the obtained estimate of R_{t}^{2}rial(_{r}) can be regarded as numerically stable. The underlying linear regression model for the treatment effect is
with the standard errors of the intercept and slope estimated to be equal to 0.202 and 0.372, respectively. This is the line indicated in Figure 7.3 as “predicted.” 
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