T) – h(t)),t.e-,I (X ii n1 tt),A(s)h(s)K 1 (s)K 2 (s) dR(s) (h(s)e-2 – 1) . ^ ^ (1 + R(s))(1 + R(s; ))K (s) P(s; )From Theorem A2 of Yang and Prentice (2005) and some algebra,0 i>n 1Q( ) =i n1 dMi +2 dMi + o p (1),358 where 1 (t) = -S. YANG AND R. L. P RENTICE^ ^ P- (t; )(1 + R(t; )) A(t)K 2 (t)h(t) + (t), K (t) K ^ ^ P- (t; )(e-1 + e-2 R(t; )) K 1 (t) + (t), K (t) K (t)t2 (t) = A(t)(2.4)Mi (t) = i I (X it) -I (X is)e-1 Z idR(s) , + e-2 Z i R(s)i = 1, . . . , n.^ ^ Now for R(t; ), from Lemma A3 in Yang and Prentice (2005) and some algebra, t t 1 ^ n( R(t; ) – R(t)) = 1 dMi + 2 dMi , ^ n P(t; ) 0i n1 i>n(2.5)where1 (t) = Let^ n P- (t; ) (1 + R(t)) , K (t) ^ R(t; ) ,2 (t) =^ n P- (t; ) -1 (e + e-2 R(t)). K (t)-D(t; ) =U= -1 Q( ) n,B(t) = h(t)A(t) + C(t) = For t , define the process Wn (t) = B T (t)U ne-1 – e-2 D(t; ), (e-1 + e-2 R(t))e-1 – e-2 1 . -1 + e-2 R(t))2 ^ (e P(t; ) (2.6)^ With the representations for Q( ) and n( R(t; )- R(t)), in Appendix B of the Supplementary Material available at Biostatistics online, we show that Wn is asymptotically equivalent to Wn which converges weakly to a zero-mean Gaussian process W . The weak convergence of Wn thus follows. The limiting covariance function (s, t) of W involves the derivative D(t; ) and the derivative matrix in U . Although ARA290 biological activity analytic forms of these derivatives are available, they are quite complicated and cumbersome. Here, we approximate them by purchase MGCD516 numerical derivatives. For the functions B(t), C(t), 1 (t), 2 (t), 1 (t), and 2 (t), ^ ^ ^ ^ ^ define corresponding B(t), C(t),. . . , by replacing with , R(t) with R(t; ) and D(t; ) with the ^ numerical derivatives. Similarly, let U be the numerical approximation of U . Simulation studies showC(t) + ni n1 0 t1 dMi +i>n 1 0 t2 dMii n11 dMi +i>n 12 dMi .Estimation of the 2-sample hazard ratio function using a semiparametric modelthat the results are fairly stable with respect to the choice of the jump size in the numerical derivatives, and that the choice of n -1/2 works well. With these approximations, we can estimate (s, t), s t , by ^ ^ (s, t) = B T (s)U ^1 [1 (w)1 T (w)K 1 (w) ^ ^ ^ ^ n(1 + R(w; )) ^ ^ ^ ^ ^ + 2 (w)2 T (w)K 2 (w)h(w)] R(dw, ) U T B(t) ^ ^^ ^ + C(s)C(t)s1 [^1 2 (w)K 1 (w) ^ ^ n(1 + R(w; ))t^ ^ ^ + ^2 2 (w)K 2 (w)h(w)] R(dw, )^ ^ ^ + C(t) B T (s)U1 [1 (w)^1 (w)K 1 (w) ^ ^ ^ n(1 + R(w; ))^ ^ ^ + C(s) B T (t)Us^ ^ ^ + 2 (w)^2 (w)K 2 (w)h(w)] R(dw, ) ^1 [1 (w)^1 (w)K 1 (w) ^ ^ ^ n(1 + R(w; ))^ ^ ^ + 2 (w)^2 (w)K 2 (w)h(w)] R(dw, ). ^(2.7)^ results in the asymptotic 100(1 – ) confidence interval h(t0 ) exp 100(1 – /2) percentile of the standard normal distribution., from the above results, confidence intervals for h(t0 ) can be obtained from the Now for a fixed t0 ^ asymptotic normality of h(t0 ) and the estimated variance (t0 , t0 ). The usual logarithm transformation ^ z /(t0 ,t0 ) ^ ^ n h(t0 ), where z /2 is the3. S IMULTANEOUS CONFIDENCE BANDS To make simultaneous inference on h(t) over a time interval I = [a, b] [0, ], consider Vn (t) = ^ h(t) ^ n (ln(h(t)) – ln(h(t))), s(t)where s(t) converges in probability, uniformly in t over I , to a bounded function s (t) > 0. From the weak convergence of Wn to W and the functional delta method, we have the weak convergence of Vn to W /s . Thus, if c is the upper th percentile of suptI |W /s |, an asymptotic 100(1-) simultaneous confidence band for h(t), t I, can be obtained as ^ h(t) exp c s(t) . ^ n h(t)It is dif.T) – h(t)),t.e-,I (X ii n1 tt),A(s)h(s)K 1 (s)K 2 (s) dR(s) (h(s)e-2 – 1) . ^ ^ (1 + R(s))(1 + R(s; ))K (s) P(s; )From Theorem A2 of Yang and Prentice (2005) and some algebra,0 i>n 1Q( ) =i n1 dMi +2 dMi + o p (1),358 where 1 (t) = -S. YANG AND R. L. P RENTICE^ ^ P- (t; )(1 + R(t; )) A(t)K 2 (t)h(t) + (t), K (t) K ^ ^ P- (t; )(e-1 + e-2 R(t; )) K 1 (t) + (t), K (t) K (t)t2 (t) = A(t)(2.4)Mi (t) = i I (X it) -I (X is)e-1 Z idR(s) , + e-2 Z i R(s)i = 1, . . . , n.^ ^ Now for R(t; ), from Lemma A3 in Yang and Prentice (2005) and some algebra, t t 1 ^ n( R(t; ) – R(t)) = 1 dMi + 2 dMi , ^ n P(t; ) 0i n1 i>n(2.5)where1 (t) = Let^ n P- (t; ) (1 + R(t)) , K (t) ^ R(t; ) ,2 (t) =^ n P- (t; ) -1 (e + e-2 R(t)). K (t)-D(t; ) =U= -1 Q( ) n,B(t) = h(t)A(t) + C(t) = For t , define the process Wn (t) = B T (t)U ne-1 – e-2 D(t; ), (e-1 + e-2 R(t))e-1 – e-2 1 . -1 + e-2 R(t))2 ^ (e P(t; ) (2.6)^ With the representations for Q( ) and n( R(t; )- R(t)), in Appendix B of the Supplementary Material available at Biostatistics online, we show that Wn is asymptotically equivalent to Wn which converges weakly to a zero-mean Gaussian process W . The weak convergence of Wn thus follows. The limiting covariance function (s, t) of W involves the derivative D(t; ) and the derivative matrix in U . Although analytic forms of these derivatives are available, they are quite complicated and cumbersome. Here, we approximate them by numerical derivatives. For the functions B(t), C(t), 1 (t), 2 (t), 1 (t), and 2 (t), ^ ^ ^ ^ ^ define corresponding B(t), C(t),. . . , by replacing with , R(t) with R(t; ) and D(t; ) with the ^ numerical derivatives. Similarly, let U be the numerical approximation of U . Simulation studies showC(t) + ni n1 0 t1 dMi +i>n 1 0 t2 dMii n11 dMi +i>n 12 dMi .Estimation of the 2-sample hazard ratio function using a semiparametric modelthat the results are fairly stable with respect to the choice of the jump size in the numerical derivatives, and that the choice of n -1/2 works well. With these approximations, we can estimate (s, t), s t , by ^ ^ (s, t) = B T (s)U ^1 [1 (w)1 T (w)K 1 (w) ^ ^ ^ ^ n(1 + R(w; )) ^ ^ ^ ^ ^ + 2 (w)2 T (w)K 2 (w)h(w)] R(dw, ) U T B(t) ^ ^^ ^ + C(s)C(t)s1 [^1 2 (w)K 1 (w) ^ ^ n(1 + R(w; ))t^ ^ ^ + ^2 2 (w)K 2 (w)h(w)] R(dw, )^ ^ ^ + C(t) B T (s)U1 [1 (w)^1 (w)K 1 (w) ^ ^ ^ n(1 + R(w; ))^ ^ ^ + C(s) B T (t)Us^ ^ ^ + 2 (w)^2 (w)K 2 (w)h(w)] R(dw, ) ^1 [1 (w)^1 (w)K 1 (w) ^ ^ ^ n(1 + R(w; ))^ ^ ^ + 2 (w)^2 (w)K 2 (w)h(w)] R(dw, ). ^(2.7)^ results in the asymptotic 100(1 – ) confidence interval h(t0 ) exp 100(1 – /2) percentile of the standard normal distribution., from the above results, confidence intervals for h(t0 ) can be obtained from the Now for a fixed t0 ^ asymptotic normality of h(t0 ) and the estimated variance (t0 , t0 ). The usual logarithm transformation ^ z /(t0 ,t0 ) ^ ^ n h(t0 ), where z /2 is the3. S IMULTANEOUS CONFIDENCE BANDS To make simultaneous inference on h(t) over a time interval I = [a, b] [0, ], consider Vn (t) = ^ h(t) ^ n (ln(h(t)) – ln(h(t))), s(t)where s(t) converges in probability, uniformly in t over I , to a bounded function s (t) > 0. From the weak convergence of Wn to W and the functional delta method, we have the weak convergence of Vn to W /s . Thus, if c is the upper th percentile of suptI |W /s |, an asymptotic 100(1-) simultaneous confidence band for h(t), t I, can be obtained as ^ h(t) exp c s(t) . ^ n h(t)It is dif.
