endpoint should be fully captured by surrogate endpoint. Symbolically, let and be random Tj Sj variables of true (DAS28 at Day 56) and surrogate endpoints (DAS28 at Day 14), respectively, for the subject and let be an indicator variable for treatment, so method (1989) can be jth Zj Prentice’s illustrated using below formulas (Buyse 1998, 2000), Sj = µS + + 7.7 αZj εSj Tj = µT + + 7.8 βZj εTj Tj = µ + + 7.9 ϒSj εj where and are the slopes for the treatment Z on surrogate endpoint S and true endpoint T. α β ϒ is the slope of the true endpoint (T) on surrogate endpoint S as linear regression. and Zj are Sj, Tj surrogate response, true response, and treatment in jth patients. µS, µT, and µ are the intercepts for the linear regressions from (7.7), (7.8), and (7.9) respectively. ej is error term for effect of true effect on surrogate endpoint. Both eSj and are Normally distributed correlated errors which εTj, satisfy , and Lastly, the true response is examined using linear regression model with both treatment and surrogate endpoint as shown below. Tj = µT + + + 7.10 βsZj ϒzSj εTj where S is the estimate of the effect of on after adjusting surrogate endpoint S. is the Z T s parameter estimate for treatment effect Z; is the slope estimate of effect of surrogate endpoint ϒz S on true endpoint T after adjusting treatment Z. To achieve surrogacy using criteria, Prentice’s
endpoint should be fully captured by surrogate endpoint. Symbolically, let Tj and Sj be random variables of true (DAS28 at Day 56) and surrogate endpoints (DAS28 at Day 14), respectively, for the jth subject and let Zj be an indicator variable for treatment, so Prentice’s method (1989) can be illustrated using below formulas (Buyse 1998, 2000), Sj = µS + αZj + εSj 7.7 Tj = µT + βZj + εTj 7.8 Tj = µ + ϒSj + εj 7.9 where α and β are the slopes for the treatment Z on surrogate endpoint S and true endpoint T. ϒ is the slope of the true endpoint (T) on surrogate endpoint S as linear regression. Sj, Tj and Zj are surrogate response, true response, and treatment in jth patients. µS, µT, and µ are the intercepts for the linear regressions from (7.7), (7.8), and (7.9) respectively. ej is error term for effect of true effect on surrogate endpoint. Both eSj and εTj, are Normally distributed correlated errors which satisfy , and Lastly, the true response is examined using linear regression model with both treatment and surrogate endpoint as shown below. Tj = µT + βsZj + ϒzSj + εTj 7.10 where S is the estimate of the effect of Z on T after adjusting surrogate endpoint S. s is the parameter estimate for treatment effect Z; ϒz is the slope estimate of effect of surrogate endpoint S on true endpoint T after adjusting treatment Z. To achieve surrogacy using Prentice’s criteria,
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