Appedix to: Hypothesis Testig for Multiple Mea ad Correlatio Curves with Fuctioal Data Ao Yua 1, Hog-Bi Fag 1, Haiou Li 1, Coli O. Wu, Mig T. Ta 1, 1 Departmet of Biostatistics, Bioiformatics ad Biomathematics, Georgetow Uiversity Medical Ceter, Washigto DC 0057 USA Office of Biostatistics Research, Natioal Heart, Lug ad Blood Istitute, Natioal Istitutes of Health, Bethesda, MD 089 USA Appedix I. Descriptio of the simulatio. 1 Simulatio To ivestigate the fiite sample properties of the proposed methods, ad compare with the commoly used local liear smoothig (Loess) ad splie methods, we preset here several simulatio studies, which are desiged to mimic the practical situatios with moderate sample sizes. We cosider separately the tests for the equality of two mea curves ad the tests for the correlatio fuctio betwee two stochastic processes. For each case, the simulatio is based o 5000 replicatios, ad the mea values of estimates over the replicatios are reported. 1.1 Testig the Equality of Mea Curves 1.1.1 Simulatio for Test with Two-Sided Alteratives For testig the hypotheses (9), we geerate the observatios { X 1, Y, ( X xy, Y xy )} of { X(t), Y (t) : t T } usig the data structure () with 1 = = 50 ad xy = 0 o k() = 50 equally spaced time poits { t j = j : j = 1,..., 50 }, so that x = y = 30. 1
For subjects with oly X(t) or Y (t) observed, we geerate X i (t j ) + ɛ i (t j ) = µ(t j ) + r i si(8 + t j /10) / 30 + N(0, σ (t j )), µ(t) = (t + Er]) si(8 + t/10) ]/ 30, where the r i s ad r are iid radom itegers uiformly distributed o {1,..., 50} used to make the curves more wiggly lookig; similarly, Y i (t j ) + ξ i (t j ) = η(t j ) + Cr i cos(8 + t j /10) / 100 + N(0, σ (t j )), η(t) = µ(t) + CEr] cos(8 + t j /10) / 100. σ (t) = 0.001 t, r i s ad r are iid radom itegers uiformly distributed o {1,..., 50} used to make the curves more wiggly lookig, ad C is a costat characterizig the differece betwee µ(t) ad η(t). For subjects with ( X(t), Y (t) ) observed, we geerate {( X i (t) + ɛ i (t j ), Y i (t) + ξ i (t j ) ) T (( ) T ) } N µ(t), η(t), Σ(t) : i = x +1,..., 1, where the covariace matrix Σ(t) is composed by the variace σ (t) ad the correlatio coefficiet ρ(t) = 0.01 (t/50). 1.1. Simulatio for Test with Oe-Sided Alteratives For testig the hypotheses (10), we geerate the observatios { X 1, Y, ( X xy, Y xy )} of { X(t), Y (t) : t T } usig the same method as Sectio 5.1.1, except that Y i (t j ) + ξ i (t j ) is replaced with Y i (t j ) + ξ i (t j ) N ( η (t j ), σ (t j ) ), where η (t) = µ(t) + C (t + Er]) cos(8 + t/3) / 100 ]. characterizig the differece betwee µ(t) ad η (t). Here, C, which plays a similar role as C, is a costat 1. Testig Correlatio Fuctios 1..1 Simulatio for Test with Two-Sided Alteratives For simplicity, each of our simulated samples cotais 1 = = xy = 50 subjects observed o k() = 50 time poits { t j = j : j = 1,..., 50 }, so that, i view of (), the sample cotais oly the paired observatios {( X i (t j ) + ɛ i (t j ), Y i (t j ) + ξ i (t j ) ) T : i = 1,..., 50; j = 1,..., 50 }. For testig the hypotheses i (15) usig the test statistic S, we geerate i each sample X i (t j )+ɛ i (t j ) N ( µ(t j ), σ1(t j ) ), where µ(t) = t si(8+t/10)/30
ad σ 1(t) = 0.01 t, ad, coditioal o X i (t j ) = x i (t j ), Y i (t j )+ξ i (t j ) has the coditioal ormal distributio, Y i (t j )+ξ i (t j ) ( xi (t j )+ɛ i (t j ) N µ(t j )+ρ(t j ) σ (t j ) / σ 1 (t j ) ] x i (t j )+ɛ i (t j ) µ(t j ) ], ( 1 ρ ) ) σ(t j ), (9) where σ (t) = 0.01 t ad ρ(t) = ρ si(8 t/10) for some ρ 0. Here, for ay t {1,..., 50}, ρ(t) is the true correlatio coefficiet betwee X i (t) ad Y i (t), ad ρ determies the differece of the correlatio curve R(t) from zero. Appedix II. Proofs. Proof of Theorem 1 (i). Sice, by (3) ad (4), X i,k() ( ) ad Y i,k() ( ) are two stochastic processes o T, we deote by µ k() (t) = EX i,k() (t)] ad η k() (t) = EY i,k() (t)] (A.1) the expectatios of the radom variables X i,k() (t) ad Y i,k() (t), respectively, for each fixed t T. The, we have { µ1(t) µ(t) ] η (t) η(t) ]} 1 = 1 + 1 { µ1(t) µ k() (t) ] η (t) η k() (t) ]} { µk() (t) µ(t) ] η k() (t) η(t) ]}. Note that by (), (4) ad the assumptio Eɛ i (t)] = 0 for all t, µ k() (t) = (t j+1 t)ex i (t j ) + ɛ i (t j+1 )] + (t t j )EX i (t j+1 ) + ɛ i (t j+1 )] t j+1 t j = (t j+1 t)µ(t j ) + (t t j )µ(t j+1 ) t j+1 t j, t t j, t j+1 ]. (A.) Thus, µ k() ( ) is the liear iterpolatio of µ( ) o the t j, t j+1 )] s for j = 0,..., k() + 1 with t 0 = if{t T } ad t k()+1 = sup{t T }. Similarly, η k() ( ) is the liear iterpolatio of η( ) o T. 3
By the assumptios that µ( ) ad η( ) are Lipschitz cotiuous o T which is bouded, it follows that µ k() (t) ad η k() (t) are uiformly cotiuous o T. Thus, we have ad if µ(s) µ k()(t) sup µ(s) s t j,t j+1 ) s t j,t j+1 ) if η(s) η k()(t) sup η(s) s t j,t j+1 ) s t j,t j+1 ) for t t j, t j+1 ), j = 0,..., k() + 1. Let δ k() = max{t j+1 t j : j = 0, 1,..., k()}. The assumptio of first order Lipschitz cotiuity implies there are 0 < c 1, c <, such that sup µ(t) µ(s) c 1 δ k() ad s,t T, t s δ k() sup η(t) η(s) c δ k(). s,t T, t s δ k() Thus by the coditio δ k() 0, we get 1 sup t T µ k() (t) µ(t) ] η k() (t) η(t) ] (c1 + c ) δ k() 0. (A.3) Now, it suffices to show that i l (T ), 1 { µ1( ) µ k() ( ) ] η ( ) η k() ( ) ]} D W ( ). (A.4) To prove (A.4), it suffice to show, i l (T ), 1 µ1 ( ) µ k() ( ) ] D W 1 ( ) ad η ( ) η k() ( ) ] D W 1 ( ). (5) for some Gaussia processes W 1 ( ) ad W ( ). We will use Theorem.11.3 i va der Vaart ad Weller (1996, P.1) to prove (A.5). We oly show the first i (A.5), that for the secod is the same. Deote X i ( ) = X i ( ) + ɛ i ( ), the ˆµ 1 (t) 1 1 1 (t t j ) X i (t j ) + (t j+1 t) X i (t j+1 ) t j+1 t j := 1 1 1 g,t ( X i ), t t j, t j+1 ], 4
where g,t is the liear iterpolatio fuctioal, with kots {t 1,..., t k() }, evaluated at t T, ad µ k() (t) = Eg,t ( X i ) := P g,t ( X i ). Deote P 1 the empirical measure of X 1,..., X 1, the the first i (A.5) is writte as 1/ 1 (P 1 P )g, ( X i ) D W 1 ( ), i l (T ). (A.6) To show the above, we oly eed to check the coditios of Theorem.11.3. For ay s, t T, let ρ(s, t) = t s, the (T, ρ) is a totally bouded semi-metric space. Let X be a iid copy of the X i s, ad defie Ỹi ad Ỹ similarly. Let G = {g,t ( X) : t T }. The G = G = sup t T X (t) + Ỹ (t)] 1/ is a evelope for G. By the give coditio P G <, so P G = P G = O(1), ad P G I(G > δ )] = P G I(G > δ )] 0 for every δ > 0. Also, for every δ 0, by the give codito ( X(t) ] ] ) δ 0 sup t s δ E + ɛ(t) X(s) ɛ(s) + Y (t) + η(t) Y (s) η(s) 0, sup P ( g,s ( X) g,t ( X) ) sup P ( ) X(s) X(t) 0. ρ(s,t)<δ ρ(s,t)<δ Thus (.11.1) i va der Vaart ad Weller (1996, P.0) is satisfied. Let N ] ( ɛ, G, L (P ) ) be the umber of ɛ-brackets eeded to cover G uder the L (P ) metric. Sice for each, there is oe member g,t G, let l,t = u,t = g,t, the l,t ( X) g,t ( X) u,t ( X) (t T ), ad for all ɛ > 0, P ( u,t ( X) l,t ( X) ) = 0 < ɛ G L (P ), i.e., (l,t, u,t ) is a ɛ-bracket of G uder the L (P ) orm. Here we have N ] ( ɛ G L (P ), G, L (P ) ) = 1, thus δ 0 log N ] ( ɛ G L (P ), G, L (P ) ) dɛ 0, for every δ 0. Now, by Theorem.11.3 i va der Vaart ad Weller (1996, P.1), (A.6) is true. Next we idetify the weak limit W ( ). For each fixed iterpolatio g,t G, g,t ( X) is a radom fuctio i t, so W ( ) is a process o T. For each positive iteger k ad fixed (t 1,..., t k ), by cetral limit theorm for double array, (W (t 1 ),..., W (t k )) T is the weak 5
limit of the vector 1 { µ1( ) µ k() (t j ) ] η ( ) η k() (t j ) ] : j = 1,..., k}. So (W (t 1 ),..., W (t k )) T is a mea zero ormal radom vector, ad by the uiform weak covergece showed above, W ( ) is a Gaussia process o T. Clearly EW ( )] = 0. The covariace fuctio R(s, t) = EW (s)w (t)] is give by ( 1 ) R(s, t) = lim Cov{ µ1 (s) µ k() (s) ] η (s) η k() (s) ], µ1 (t) µ k() (t) ] η (t) η k() (t) ]}. Sice { Cov µ1(s) µ k() (s) ] η (s) η k() (s) ], µ 1(t) µ k() (t) ] η (t) η k() (t) ]} we have { 1 x = Cov Xi,k() (s) µ k() (s) ] + 1 1 1 1 1 1 i= x+1 i= x+1 Xi,k() (s) µ k() (s) ] Yi,k() (s) η k() (s) ] 1 Yi,k() (s) η k() (s) ]], i= 1 1 x Xi,k() (t) µ k() (t) ] + 1 1 1 1 1 1 i= x+1 i= x+1 Yi,k() (t) η k() (t) ] 1 Xi,k() (t) µ k() (t) ] i= 1 Yi,k() (t) η k() (t) ]]} = 1 1 Cov X 1,k() (s), X 1,k() (t) ] xy 1 Cov X 1,k() (s), Y 1,k() (t) ] xy 1 Cov Y 1,k() (s), X 1,k() (t) ] + 1 Cov Y 1,k() (s), Y 1,k() (t) ], ( 1 ){ R(s, t) = lim 1 1 Cov X 1,k() (s), X 1,k() (t) ] ( 1 x ) Cov X 1,k() (s), Y 1,k() (t) ] 1 ( 1 x ) 1 = γ R 11 (s, t) γ 1 R 1 (s, t) + R 1 (s, t)] + γ 1 R (s, t). Cov Y 1,k() (s), X 1,k() (t) ] + 1 Cov Y 1,k() (s), Y 1,k() (t) ]} 6
Proof of Theorem. (i). By Theorem 1, we have that, uder H 0 of (9), L D 1 T T W (t)dt. (A.9) Sice R(, ) is almost everywhere cotiuous ad T is bouded, R (, ) is itegrable, that is, T T R (s, t) dsdt <. By Mercer s Theorem (cf. Theorem 5..1 of Shorack ad Weller (1986), page 08), we have that R(s, t) = λ j h j (s)h j (t), (A.10) where λ j 0, j = 1,,..., are the eigevalues of R(, ), ad h j ( ), j = 1,,..., are the correspodig orthoormal eigefuctios. Let { Z 1,..., Z m,... } be the set of idepedet idetically distributed radom variables with Z m N(0, 1). The Z(t) = λj Z j h j (t) is a Gaussia process o T with mea zero ad covariace fuctio R(s, t). W (t) ad Z(t), have the same distributio o T, Thus, the two stochastic processes, W (t) d = Z(t) = λj Z j h j (t) (A.11) ad, by (A.9) ad (A.10), 1 T T W (t)dt = d 1 T T λj Z j h j (t)] dt = 1 T λ j Zj. (A.1) The result of Theorem (i) follows from (A.9), (A.11) ad (A.1). (ii). By Theorem 1, we have that, uder H 0 of (10), D D 1 T t T W (t) dt = U, (A.13) where U has ormal distributio with mea zero. To compute the variace of U, we cosider the partitio { s j, s j+1 ) : j = 1,..., m } of T with δ = max { s j+1 s j : j = 7
1,..., m }. The, it follows from (A.13) that U = lim δ 0 W (s j )(s j+1 s j ). (A.14) Sice E W (s j ) ] = 0 for each fixed j, we have that, by (A.14) ad the cotiuity coditio of R(, ), V ar(u) = lim δ 0 = lim δ 0 E W (s i ) W (s j ) ] (s i+1 s i )(s j+1 s j ) R(s i, s j ) (s i+1 s i )(s j+1 s j ) = s T t T R(s, t) ds dt. The result of Theorem (ii) follows from (A.13) ad V ar(u). Proof of Theorem 3 The proof is similar to derivatio of (A.5) by substitutig µ( ) ad µ ( ) with µ( ) ad µ ( ), respectively. The differece here is that we have order two polyomial iterpolatios for the terms ( X i,k() ( ), Y i,k() ( ), X i,k()( )Y i,k() ( ) ) i additio to the liear iterpolatios for X i,k() ( ) ad Y i,k() ( ) with G playig the role of G i the derivatio of (A.5). The rest of the derivatio is proceeded the same way. The, the delta method leads to the claimed result. To idetify the matrix covariace fuctio Ω(s, t), we ote that µ (t) = 1 ( Xi,k() (t), Y i,k() (t), X i,k()(t), Y i,k()(t), X i,k() (t) Y i,k() (t) ) := 1 Z i (t). The, Ω(s, t) = Cov ( Z(s), Z(t) ) gives the expressio for Ω(s, t). Proof of Theorem 4. 8
The proof here is focused o the derivatio of (8), as the proof of (7) ca be proceeded usig the same approach here ad the results of Theorem 3. We first ote that W (t) = H µ (t) ] ad, uder H 0 of (15) ad (16), H µ(t) ] = 0. It follows that W (t) = { H µ (t) ] H µ(t) ]} = 1 + o p (1) ] Ḣ µ(t) ] µ (t) µ(t) ]. Usig result of Theorem 3, the delta method ad the similar derivatios as the proof of Theorem 1, we have W ( ) P W ( ) i l (T ) uiformly o P, where W ( ) is the mea zero Gaussia process o T with covariace fuctio ] Q(s, t) = Cov {Ḣ µ(s) X(s), Ḣ µ(t) ] Z(t) ]} = Ḣ µ(s) ] Cov Z(s), Z(t) ] Ḣ µ(t) ] = Ḣ µ(s) ] Ω(s, t) Ḣ µ(t) ]. The rest of the proof is the same as i that of Theorem. 9