Web-based Supplementary Materials for Inference for the Effect of Treatment. on Survival Probability in Randomized Trials with Noncompliance and

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1 Bometrcs 000, DOI: Web-based Supplementary Materals for Inference for the Effect of Treatment on Survval Probablty n Randomzed Trals wth Noncomplance and Admnstratve Censorng by Ne, Cheng and Small Hu Ne,, Jng Cheng 2,, and Dylan S. Small, Department of Statstcs, The Wharton School, Unversty of Pennsylvana, Phladelpha, PA 904, U.S.A. 2 Dvson of Oral Epdemology & Dental Publc Health, UCSF School of Dentstry, San Francsco, CA 9443, U.S.A. emal: nehu@wharton.upenn.edu emal: Jng.Cheng@ucsf.edu emal: dsmall@wharton.upenn.edu

2 Web Appendx. Web Appendx A: Proof of Theorem 7. Proof. We prove ths theorem by showng that t s a convex optmzaton problem. The maxmzaton problem (2) under constrants (3) - (7) n the paper s equvalent to mnmzng subect to log L obs = m +m 2 +N =m +m 2 + log (ˆπ c L + ( ˆπ c )L 2 ) () n 3 p nt V ) = Ŝnt(V ) (2) n (3) + + Here, L and L 2 are the same as before. n (3) Let x = (p c0,, p c0 n 3, p nt,, p nt n 3 ). Then note that p c0 0 (3) p nt 0 (4) p c0 0, =,, n 3 (5) p nt 0, =,, n 3 (6) ˆπ c L + ( ˆπ c )L 2 s a lnear functon of x and hence s a concave functon log(z) s strctly convex and decreasng Summaton preserves convexty Then accordng to Boyd and Vandenberghe (2004, Chapter 3), we have that our obectve functon () s strctly convex. Moreover, for our constrants (2) - (6), we have that The nequalty constrant functons n (3) - (6) are all lnear functons of x and hence are convex functons The equalty constrant functon (2) s a lnear combnaton of x and thus s affne Therefore, our maxmzaton problem s a strctly convex optmzaton problem. Hence, accordng to Boyd and Vandenberghe (2004, Chapter 4), t has a unque global maxmum.

3 2 Bometrcs, Web Appendx B: Proof of Theorem 7.2 Proof. The proof for ths theorem s smlar to that of Lemma n Cheng, Small, Tan and Ten Have (2009b). There are two maor dfferences. One s that our optmzaton problem s gven by () - (6), however, lke Cheng et al. (2009b), our problem s also a convex optmzaton problem. The other s that our constrant parameter space s gven by Θ = {p c0, p nt, =,, n 3 : q = ˆπ c p c0 + ( ˆπ c )p nt, p nt I(T (3) V ) = S ˆ nt (V ), p nt, p c0, p nt 0, p c0 0}. The set Θ, as n Cheng et al. (2009b), s a convex set because the feasble set of a convex optmzaton problem s also convex. Because our optmzaton problem s convex and the constrant parameter space s a convex set, we follow the same steps as n Cheng et al. (2009b) to prove the theorem. 3. Web Appendx C: Proof of Theorem 7.3 Proof. Step From Kaplan and Meer (958), under the condtons r () v and r (2) v that, as m + m 2 + N, By the Law of Large Numbers, as m + m 2 + N, Step 2 Let = dg r G, we know Ŝ c (V ) P S c (V ) (7) Ŝ nt (V ) P S nt (V ) (8) ˆπ c a.s. π c (9) ( dg ), whch s the Kaplan-Meer estmator of the mxture n the r G control group. Consder the condtons (3) - (7) n the paper along wth ˆπ c p c0 + ( ˆπ c )p nt = (0) Here, notce that wth (4) - (7) n the paper and (0), sometmes (3) n the paper cannot

4 Web Appendx 3 be satsfed. Gven (4) - (7) n the paper and (0), we want to fnd the lower bound and upper bound Defne l = sup{ : ˆ S M nt (V ) of n 3 p nt V ). ˆπ c }, then = ˆπ c, =,, l l q KP ˆπ c, = l ˆ S L nt(v ) 0, = l +,, n 3 and Defne m = nf{ : n 3 =+ ˆ S L nt(v ) = ˆπ c p nt,m = n 3 }, then V ) 0, =,, m n 3 q KP =m+ ˆπ c, = m ˆπ c, = m +,, n 3 and ˆ S M nt (V ) = n 3 p nt,m V ) Therefore, n 3 p nt V ) [ S ˆ nt(v L ), SM ˆ nt (V )] Smlarly, defne l = sup{ : = π c }, then π c, =,, l l q KP π c, = l 0, = l +,, n 3 and S L nt(v ) = n 3 V )

5 4 Bometrcs, and Defne m = nf{ : n 3 =+ π c nt, M p = }, then S M nt (V ) = 0, =,, m n 3 q KP = m+ π c, = m π c, = m +,, n 3 n 3 We want to show that, as m + m 2 + N, nt, M p V ) Sˆ nt(v L ) S nt(v L ) a.s. 0 Sˆ nt M (V ) S nt M (V ) a.s. 0 Snce S ˆ nt(v L ) S nt(v L ) n 3 = mn{l, l} + V ) n 3 mn{l, l} V ) =mn{l, l}+ V ) V ) V ) =mn{l, l}+ V ) where c =b a = 0 f b > c. Note that f ˆπ c π c, then l l. We get that mn{l, l} mn{l, l} V ) = ( ) ˆπ c π c +( ˆπ c l l V ) )I(T (3) l V ) V )

6 Web Appendx 5 Smlarly, f ˆπ c < π c, then l l. We also get that mn{l, l} mn{l, l} V ) = ( ) ˆπ c π c ( π c l l V ) q KP )I(T (3) V ) l V ) Therefore, combnng these two stuaton, we get that mn{l, l} mn{l, l} V ) = max{, ˆπ c π c +( max{, ˆπ c ( ˆπ c mn{l, l} ) π c nt, L p V ) mn{l, l} } π c ˆπ c mn{l, l} } π c q KP )I(T (3) V ) mn{l, l} mn{l, l} q KP + V ) max{, ˆπ c π c } From (9), we know that ˆπ c π c a.s. 0. And from the fact that mn{l, l}, ( ˆπ c mn{l, l} ) π c q KP a.s. 0 For the second part, from defntons, t s easy to verfy that Thus, mn{ ˆπ c, π c } max{ ˆπ c, π c } max{, ˆπ c Therefore, mn{l, l} mn{l, l} max{ ˆπ c, π c } mn{l, l} } π c max{, ˆπ c π c } max{, ˆπ c π c } 0

7 6 Bometrcs, max{ ˆπ c, π c } max{, } ˆπ c π c = mn{ ˆπ c, π c } max{ ˆπ c, π c } Agan, from (9) and the condton that 0 < π c <, we know that Hence, mn{l, l} mn{l, l} max{, ˆπ c mn{l, l} V ) Next, f ˆπ c π c, then l l. We also get that =mn{l, l}+ V ) π c } a.s. 0 nt, L p V ) a.s. 0 =mn{l, l}+ V ) = l =l+ V ) l =l+ = l π c From the defntons of l, l and the fact that ˆπ c π c, we get that ˆπ c l π c Therefore, =mn{l, l}+ l π c ˆπ c π c V ) =mn{l, l}+ V )

8 Web Appendx 7 If ˆπ c < π c, then l l. We get that = =mn{l, l}+ l = l+ l = l+ = V ) V ) l ˆπ c =mn{l, l}+ V ) Agan, from the defntons of l, l and the fact that ˆπ c π c, we get that π c l ˆπ c Therefore, =mn{l, l}+ l ˆπ c π c ˆπ c V ) =mn{l, l}+ V ) Combnng these two case, we get that =mn{l, l}+ V ) mn{ π c, ˆπ c } max{ π c, ˆπ c } = max{π c, ˆπ c } mn{π c, ˆπ c } =mn{l, l}+ From (9) and the condton that 0 < π c <, we know that =mn{l, l}+ To sum up, we verfy that V ) =mn{l, l}+ ˆ S L nt(v ) S L nt(v ) a.s. 0 V ) V ) a.s. 0

9 8 Bometrcs, Smlarly, we can also prove that Step 3 Sˆ nt M (V ) S nt M (V ) a.s. 0 In ths step, we would lke to show that, as m + m 2 + N, As m + m 2 + N, Smlarly, Step 4 S L nt(v ) = S L nt(v ) P S G(V ) π c π c I V G ( π c) () S nt M (V ) P + S G(V ) ( π c ) I V G π (π c ) (2) c = n 3 l V ) q KP I(T (3) (3) V ) + π p li(t V ) l c P ( π c) ( S G (V )) I (V G π ( π c )) c = S G(V ) π c π c I (V G ( π c)) S nt M (V ) P + S G(V ) ( π c ) I (V G π (π c )) c From (2) n the paper, whch s gven by S G (V ) π c π c I (V G ( π c)) < S nt < + S G(V ) ( π c ) π c I (V G (π c)) (3), along wth (8), () and (2), we verfy that, as m + m 2 + N, S G (V ) π c π c I (V G ( π c)) n 3 p nt V ) + S G(V ) ( π c ) π c I (V G (π c)) s asymptotcally vald n probablty. Hence, (3) s asymptotcally satsfed n probablty under maxmzaton constrants (4) - (7) n the paper and (0). Therefore, the maxmzaton problem (2) under constrants (3) - (7) n the paper s asymptotcally equvalent to the maxmzaton problem (2) under constrants (4) - (7) n the paper and (0) n probablty.

10 Web Appendx 9 Note that n (0) s actually the Kaplan-Meer estmator of the mxture n the control group. Therefore, from Kaplan and Meer (958), snce r G v, as m + m 2 + N, we have that Ŝ G (V ) P S G (V ) (4) Step 5 From (8), (9), (4) and the fact that ŜG(V ),Ŝnt(V ),ˆπ c are all bounded, we get that Hence, along wth (7), we verfy that Ŝ c0 (V ) = ŜG(V ) ˆπ c Ŝ nt (V ) ˆπ c P S G(V ) π c S nt (V ) π c = S c0 (V ) Ŝ c (V ) Ŝc0(V ) P S c (V ) S c0 (V ) 4. Web Appendx D: Standard Errors Calculaton through Delta Method The dfference of the RMSEs of two estmators δ and δ 2 are defned as: T = E((δ θ) 2 ) E((δ 2 θ) 2 ) Therefore, we can get the varance of T as below: V ar(t ) = σδ 2 ( T ) 2 + σδ 2 δ 2 ( T ) 2 + 2σ δ δ δ 2 ( T )( T ) 2 δ δ 2 = σδ 2 E(δ θ) ( E((δ θ) 2 ) )2 + σδ 2 E(δ 2 θ) E(δ θ) 2 ( E((δ 2 θ) 2 ) )2 + 2σ δ δ 2 ( E((δ θ) 2 ) )( E(δ 2 θ) E((δ 2 θ) 2 ) ) From Delta method, we can estmate the standard error of T as

11 0 Bometrcs, where covarance, V ar(t ) = V ˆ Bas(δ ˆ ) ar(δ )( RMSE(δ ˆ ) )2 + V ˆ Bas(δ ˆ 2 ) ar(δ 2 )( RMSE(δ ˆ 2 ) )2 + 2Cov(δ ˆ Bas(δ ˆ ), δ 2 )( RMSE(δ ˆ ) )( Bas(δ ˆ 2 ) RMSE(δ ˆ 2 ) ) ˆ V ar(δ ) and ˆ Bas(δ ) and ˆ V ar(δ 2 ) are sample varances of δ and δ 2, ˆ Bas(δ 2 ) are sample bases, as well as ˆ Cov(δ, δ 2 ) s the sample ˆ RMSE(δ ) and ˆ RMSE(δ 2 ) are sample RMSEs. All of the above quanttes are avalable through the smulaton studes. 5. Web Appendx E: Parametrc Maxmum Lkelhood Estmaton under Welbull Assumpton The maxmum lkelhood approach s a powerful tool under a parametrc model. We use the EM-algorthm to fnd the maxmum lkelhood estmators of the parameters n our model, and then get the maxmum lkelhood estmator of the dfference between the survval probablty of the complers n the treatment group and the survval probablty of the complers n the control group at a specfc tme V. We llustrate ths method through an example wth assumptons of Webull dstrbutons. However, our method can be easly appled under assumptons of other dstrbutons. Example Assume that the complers n the treatment group, the complers n the control group and the never-takers have Webull dstrbutons wth parameters ρ c, κ c, ρ c0, κ c0 and ρ nt, κ nt, respectvely. The lkelhood functon can be wrtten as: where m L obs = = π c L c m +m 2 =m + ( π c )L nt m +m 2 +N =m +m 2 + (π c L c0 + ( π c )L nt ) L c = (κ c ρ c (ρ c Y ) κ c ) exp ( (ρ c Y ) κ c ), =,, m L c0 = (κ c0 ρ c0 (ρ c0 Y ) κ c0 ) exp ( (ρ c0 Y ) κ c0 ), = m + m 2 +,, m + m 2 + N

12 Web Appendx L nt = (κ nt ρ nt (ρ nt Y ) κnt ) exp ( (ρ nt Y ) κnt ), = m +,, m + m 2 + N Vewng the complance class as mssng data, the complete data lkelhood functon s m L c = = π c L c m +m 2 =m + ( π c )L nt m +m 2 +N =m +m 2 + (π c L c0 ) I (( π c )L nt ) I (5) We drectly fnd the MLEs for ρ c and κ c by solvng the two equatons below m = m + log Y = κ c = ρ c = ( m = m = Y κ c log Y m = Y κ c m = m = Y κ c ) κ c For the other parameters, we use the EM-algorthm to fnd ther MLEs. E-step Let Î denote E(I ˆθ (t) ), where ˆθ (t) s the vector of estmates of π c, ρ c0, κ c0, ρ nt and κ nt at the tth teraton of the EM-algorthm. For = m + m 2 +,, m + m 2 + N, we have where L c0, L nt Î = π c L c0 π c L c0 + ( π c )L nt (6) and π c are evaluated at ˆθ (t). For = m +,, m + m 2, Î = 0. Because the complete data log-lkelhood functon s lnear n I, the expected value of the complete data log-lkelhood gven ˆθ (t) s obtaned by substtutng Î nto (5). M-Step After substtutng (6) nto (5), we get the maxmzers of π c, ρ c0, κ c0, ρ nt and κ nt by settng the frst dervatves as zero and solvng the equatons below. m +m 2 +N =m + m +m 2 +N =m +m 2 + Î κ c0 + ( Î) + κ nt m +m 2 +N =m +m 2 + m +m 2 +N =m + π c = Î log Y = ( Î) log Y = m m + +m 2 +N =m +m 2 + Î m + m 2 + N m +m 2 +N m +m 2 +N =m +m 2 + Î m +m 2 +N =m +m 2 + Î ρ c0 = ( m +m 2 +N ρ nt = ( =m +m 2 + ÎY κ c0 log Y m +m 2 +N =m +m 2 + ÎY κ c0 =m +m 2 + ÎY κ c0 m +m 2 +N =m + ) κ c0 ( Î) m +m 2 +N =m + m +m 2 +N ( =m + Î)Y κ nt m +m 2 +N ( =m + Î) m +m 2 +N ( =m + Î)Y κ nt ) κ nt ( Î)Y κ nt log Y

13 2 Bometrcs, We run ths EM-algorthm untl the parameters converge. We choose the startng values as π c = m /(m + m 2 ), ρ nt and κ nt whch maxmze m +m 2 =m + L nt m +m 2 +N =m +m 2 +(L nt ) πc as well as ρ c0 and κ c0 whch maxmze m +m 2 +N =m +m 2 +(L c0 ) π c. Then the estmated dfference between the survval probablty of the complers n the treatment group and the survval probablty of the complers n the control group at a specfc tme V s gven by Ŵ (V ) = Ŝc(V ) Ŝc0(V ) = exp ( (ˆρ c V )ˆκ c ) exp ( (ˆρ c0 V )ˆκ c0 ) 6. Web Appendx F: Detals of Extenson to Trals under Assumptons - 6 In ths secton, we extend our PNEMLE approach to more general trals under Assumptons - 6 n whch the control group has access to the treatment. For such trals, we have one more complance class, the always-takers, n addton to the complers and the never-takers. If R = and A =, we know that the subect s ether a compler or an always-taker; f R = and A = 0, the subect s a never-taker; f R = 0 and A =, the subect s an always-taker; and f R = 0 and A = 0, the subect s ether a compler or a never-taker. The always-takers are dentfable n the control group and the never-takers are dentfable n the treatment group. Let S at (V ), S nt (V ), S c (V ) and S c0 (V ) be the survval probablty of the always-takers, the never-takers, the complers n the treatment group and the complers n the control group at tme pont V, respectvely. We also use π at, π nt and π c to denote the proporton of the always-takers, the never-takers and the complers. Smlar to the approach n Secton 5 of our paper, we follow fve steps below to estmate the dfference W (V ) = S c (V ) S c0 (V ) for trals under Assumptons - 6. Step I: Estmate S at (V ) and π at from the always-takers (R=0, A=) n the control group (R=0). We estmate S at (V ) by the Kaplan-Meer estmator Ŝat(V ), and we use the observed fracton of complers n the treatment group ˆπ at = Ê(I = R = 0) to estmate π at.

14 Web Appendx 3 Step II: Estmate S nt (V ) and π nt from the never-takers (R=, A=0) n the treatment group (R=). We estmate S nt (V ) by the Kaplan-Meer estmator Ŝnt(V ), and we use the observed fracton of complers n the treatment group ˆπ nt = Ê(I = 0 R = ) to estmate π nt. Step III: Estmate S c (V ) from the mxture of the complers and the always-takers (R=, A=) n the treatment group (R=). We get our estmator Ŝc(V ) by applyng the nonparametrc emprcal lkelhood approach to model the dstrbuton of the mxture of the complers and the always-takers n the control group wth constrants S at (V ) = Ŝat(V ), π at = ˆπ at and π c = ˆπ at ˆπ nt. Step IV: Estmate S c0 (V ) from the mxture of the complers and the never-takers (R=0, A=0) n the control group (R=0). We get our estmator Ŝc0(V ) by applyng the nonparametrc emprcal lkelhood approach to model the dstrbuton of the mxture of the complers and the never-takers n the control group wth constrants S nt (V ) = Ŝnt(V ), π nt = ˆπ nt and π c = ˆπ at ˆπ nt. Step V: Estmate W (V ) = S c (V ) S c0 (V ) by Ŵ (V ) = Ŝc(V ) Ŝc0(V ), where Ŝc(V ) and Ŝc0(V ) are obtaned n Step III and Step IV. 7. Web Appendx G: Detaled Table of Table 2 [Table about here.] 8. Web Appendx H: Detaled Table of Table 3 [Table 2 about here.] 9. Web Appendx I: Test for Parametrc Webull Assumpton We wll gve the detaled dscusson on whether the potental falure tmes of complers and never-takers n our lmted mortalty analyss of the HIP data set follow the Webull

15 4 Bometrcs, dstrbuton. From Cox and Oakes (984), f T follows the Webull dstrbuton wth densty functon ρκ(ρx) κ exp ( (ρx) κ ), t s easy to verfy that log(h(t)) = κ log(ρ) + κ log(t), where H(t) s the cumulatve hazard functon of T. Therefore, log(h(t)) and log(t) should have a lnear relatonshp when T follows a Webull dstrbuton. We estmate log(h(t)) by log(ĥ(t)) through the Kaplan-Meer estmator. Fgure n Web Appendx shows us the result for complers n the treatment arm. The mean squared error (MSE) for the regresson between log(ĥ(t)) and log(t) s We use ths MSE as a test statstc for testng that T follows a Webull dstrbuton and use the parametrc Bootstrap method to perform the test. If we assume that the falure tmes of complers n the treatment arm follow a Webull dstrbuton wth parameters κ c and ρ c, the MLEs for κ c and ρ c are gven by ˆκ c =.939 and ˆρ c = We generate random samples for Webull dstrbuton wth parameters ˆκ c =.939 and ˆρ c = 0.579, calculate log(ĥ(t)) through the Kaplan-Meer method, ft the lnear regresson between log(ĥ(t)) and log(t), and get the MSE for ths regresson. We repeat ths procedure 0,000 tmes. The estmated p-value s gven by Therefore, there s not strong evdence for us to reect the null hypothess that the falure tmes of complers n the treatment group follow the Webull dstrbuton. However, from Fgure 2 n Web Appendx whch shows us the result for never-takers n the treatment arm, t does not reveal a strong lnear pattern between log(ĥ(t)) and log(t). The MSE for the regresson between log(ĥ(t)) and log(t) s If we assume that the falure tmes of complers n the treatment arm follow a Webull dstrbuton wth parameters κ nt and ρ nt, the MLEs for κ nt and ρ nt are gven by ˆκ nt =.202 and ˆρ nt = We follow the same approach as above to conduct the hypothess testng, and the estmated p-value s gven by Thus, there s evdence for us to cast doubts on the valdty of the Webull model for never-takers n the treatment arm.

16 Web Appendx 5 [Fgure about here.] [Fgure 2 about here.] 0. Web Appendx J: Detals of BC a method Besdes Bootstrap percentle method, Efron & Tbshran (994) suggested usng BC a method to obtan the confdence nterval for censored data sets {(Y, )} m +m 2 +N =. We can construct the BC a confdence nterval followng the steps below. Step I: Draw a Bootstrap sample {(Y, )} m +m 2 +N =. For complers n the treatment group {(Y, )} m =, we sample wth replacement by puttng mass m at each pont (Y, ) n order to get Bootstrap sample {(Y, )} m =. For never-takers n the treatment group {(Y, )} m +m 2 =m +, we sample wth replacement by puttng mass m 2 at each pont (Y, ) n order to get Bootstrap sample {(Y, )} m +m 2 =m +. For mxtures n the control group {(Y, )} m +m 2 +N =m +m 2 +, we sample wth replacement by puttng mass N at each pont (Y, ) n order to get Bootstrap sample {(Y, )} m +m 2 +N =m +m 2 +. We on three Bootstrap samples together to get a Bootstrap sample {(Y, )} m +m 2 +N =. Step II: Estmate PNEMLE Ŵ (V ) for ths Bootstrap sample followng the procedures n Secton 5. Step III: Independently repeat steps I and II B tmes and obtan {Ŵ b (V )} B b=. Fnd the lower α percentle Ŵ LOW (V ) and the upper α 2 percentle Ŵ UP (V ), where α and α 2 are gven as α = Φ(ẑ 0 + α 2 = Φ(ẑ 0 + ẑ 0 + z (α) ˆα(ẑ 0 + z (α) ) ) ẑ 0 + z ( α) ˆα(ẑ 0 + z ( α) ) ) The value of ẑ 0 s obtaned drectly from the proporton of Bootstrap replcatons less than

17 6 Bometrcs, the orgnal estmate Ŵ (V ), that s, ẑ 0 = Φ ( No. of {Ŵ b (V ) < Ŵ (V )} ) B Let Ŵ(.)(V ) = B b= Ŵ b (V ) B. The value of ˆα s obtaned from Bb= (Ŵ(.)(V ) ˆα = Ŵb(V )) 3 6( B b= (Ŵ(.)(V ) Ŵb(V )) 2 ) 3 2 Step IV: The ( 2α) confdence nterval s gven by (Ŵ LOW (V ), Ŵ UP (V )).. Web Appendx K: Detaled results from the HIP study [Table 3 about here.]

18 Web Appendx 7 Complers n the Treatment Arm Log(H(t)) Log(t) Fgure. Complers n the Treatment Arm

19 8 Bometrcs, Never takers n the Treatment Arm Log(H(t)) Pont Estmate Regresson Lne Log(t) Fgure 2. Never-takers n the Treatment Arm

20 Web Appendx 9 Table Estmates of the dfference of the survval probabltes between the complers n the treatment group and those n the control group at tme pont V when π c = 0.5 Relatve bas wth RMSE wth Group V C 0 C K True PNEMLE IV Para PNEMLE IV Para E W LN LL G

21 20 Bometrcs, Table 2 Estmates of the dfference of the survval probabltes between the complers n the treatment group and those n the control group at tme pont V when π c = 0.2 Relatve bas wth RMSE wth Group V C 0 C K True PNEMLE IV Para PNEMLE IV Para W LN

22 Web Appendx 2 Table 3 Results from the HIP study: estmates and 95% bootstrap percentle confdence ntervals of the dfference between survval probablty between the survval probablty of the complers n the treatment group and that of the complers n the control group at every half year Tme Pont PNEMLE IV Para Webull ITT (-0.032,0.043) ( ,0.043) ( ,0.0292) ( ,0.002) ( ,0.0884) (0.0036,0.0884) (0.0066,0.076) ( , ) (0.0432,0.74) 0.06 (0.0432,0.74) (0.0256,0.26) (0.0307,0.24) (0.078,0.74) (0.078,0.74) 0.09 (0.044,0.78) (0.026,0.24) (0.0408,0.206) 0.23 (0.0408,0.206) 0.37 (0.0507,0.222) (0.0290,0.46)