Adjusting for possibly misclassified causes of death in cause-specific survival analysis

Transcription

1 Adjusting for possibly misclassified causes of death in cause-specific survival analysis Bart Van Rompaye Els Goetghebeur Department of Applied Mathematics and Computer Sciences Ghent University, BE 1 / 21

2 Problem description Drugs developed against disease-specific mortality exposure affects cause-specific mortality interest in reduction of cause-specific mortality need good assessment of cause of death ICD-guidelines death registries: little control, little motivation to do it right? clinical trials: more motivation, but sometimes practical constraints 2 / 21

3 Problem description Drugs developed against disease-specific mortality exposure affects cause-specific mortality interest in reduction of cause-specific mortality need good assessment of cause of death ICD-guidelines death registries: little control, little motivation to do it right? clinical trials: more motivation, but sometimes practical constraints 2 / 21

4 Problem description Motivating study: Jaffar et al., I.J.E Effects of misclassification of causes of death on the power of a trial to assess the efficacy of a pneumococcal conjugate vaccine in The Gambia phase III randomized, placebo controlled trial impact of a vaccine on mortality from airway infections in Gambian children need for post-mortem questionnaires or verbal autopsy misclassification of causes dilution of treatment effect power loss 3 / 21

5 Problem description Motivating study: Jaffar et al., I.J.E CODs: λ doc = py, λ dfd = py expected treatment effect: 31.5% reduction of λ dfd misclassification of causes p 1 =0.6 sensitivity=0.4=1 p 1 p 0 =0.1 specificity=0.9=1 p 0 power loss? 4 / 21

6 Impact of misclassification: Jaffar et al. - naive logrank p 0 = 0.1 p 1 = 0.6 power = 21% 5 / 21

7 Problem description Motivating study: Jaffar et al., I.J.E cause-specific mortality: power loss all-cause mortality: sample size radiological pneumonia ITT-result: significant reduction (Cutts et al., Lancet 2005) 6 / 21

8 Problem description Motivating study: Jaffar et al., I.J.E cause-specific mortality: power loss all-cause mortality: sample size radiological pneumonia ITT-result: significant reduction (Cutts et al., Lancet 2005) misclassification determines design and analysis! 6 / 21

9 Our approach derive adapted logrank test model true and observed cause-specific hazards acknowledging misclassification rates input this in likelihood-based weighting evaluate the power that is regained compare efficiencies for endpoint selection: all-cause vs. naive vs. improved cause-specific testing derive adjusted sample size formula estimate effect of treatment (on cause-specific hazard): ongoing 7 / 21

10 Our approach derive adapted logrank test model true and observed cause-specific hazards acknowledging misclassification rates input this in likelihood-based weighting evaluate the power that is regained compare efficiencies for endpoint selection: all-cause vs. naive vs. improved cause-specific testing derive adjusted sample size formula estimate effect of treatment (on cause-specific hazard): ongoing 7 / 21

11 Design assumptions treatment Z: 0 or 1 (1= treatment) observation time T failure indicator δ: 0=censored, 1=failure true cause-of-death COD true : doc or dfd misclassification indicator M: 0 or 1 (1=misclassified) observed COD = COD obs observed data: (Z, T, δ, COD obs ) Z T,δ, COD obs t 8 / 21

12 Design assumptions Assumptions: similar to Goetghebeur and Ryan, Biometrika 1990 misclassification probabilities depend on true COD and on time: p 0 (t) and p 1 (t) failure patterns from proportional cause-specific hazards: COD other cause disease control treatment Null H 0 : φ = 0 h dfd (t) h dfd (t)e φ 9 / 21

13 Design assumptions Assumptions: similar to Goetghebeur and Ryan, Biometrika 1990 misclassification probabilities depend on true COD and on time: p 0 (t) and p 1 (t) failure patterns from proportional cause-specific hazards: COD other cause disease control treatment Null H 0 : φ = 0 h dfd (t) h dfd (t)e φ 9 / 21

14 Design assumptions Assumptions: similar to Goetghebeur and Ryan, Biometrika 1990 misclassification probabilities depend on true COD and on time: p 0 (t) and p 1 (t) failure patterns from proportional cause-specific hazards: COD other cause disease control treatment Null H 0 : φ = 0 h dfd (t) h dfd (t)e φ 9 / 21

15 Design assumptions Assumptions: similar to Goetghebeur and Ryan, Biometrika 1990 misclassification probabilities depend on true COD and on time: p 0 (t) and p 1 (t) failure patterns from proportional cause-specific hazards: COD other cause disease control h doc (t) = h dfd (t)e ξ h dfd (t) treatment h doc (t) = h dfd (t)e ξ h dfd (t)e φ Null H 0 : φ = 0 9 / 21

16 Partial likelihood no misclassification: L p (observed data) = i:cod true,i =dfd h dfd (t i )e φz i j R i h dfd (t i )e φz j 10 / 21

17 Partial likelihood no misclassification: L p (observed data) = misclassification: i:cod true,i =dfd h dfd (t i )e φz i j R i h dfd (t i )e φz j L p (observed data) h dfd (t i )(e ξ (1 p 0 (t i )) + e φz i p 1 (t i )) = j R i h dfd (t i )(e ξ (1 p 0 (t i )) + e φz j p 1 (t i )) i:cod obs =doc i:cod obs =dfd h dfd (t i )(e φz i (1 p 1 (t i )) + e ξ p 0 (t i )) j R i h dfd (t i )(e φz j (1 p 1 (t i )) + e ξ p 0 (t i )) 10 / 21

18 Partial likelihood no misclassification: L p (observed data) = misclassification: i:cod true,i =dfd h dfd (t i )e φz i j R i h dfd (t i )e φz j L p (observed data) h dfd (t i )(e ξ (1 p 0 (t i )) + e φz i p 1 (t i )) = j R i h dfd (t i )(e ξ (1 p 0 (t i )) + e φz j p 1 (t i )) i:cod obs =doc i:cod obs =dfd h dfd (t i )(e φz i (1 p 1 (t i )) + e ξ p 0 (t i )) j R i h dfd (t i )(e φz j (1 p 1 (t i )) + e ξ p 0 (t i )) 10 / 21

19 Partial likelihood no misclassification: L p (observed data) = misclassification: i:cod true,i =dfd h dfd (t i )e φz i j R i h dfd (t i )e φz j L p (observed data) h dfd (t i )(e ξ (1 p 0 (t i )) + e φz i p 1 (t i )) = j R i h dfd (t i )(e ξ (1 p 0 (t i )) + e φz j p 1 (t i )) i:cod obs =doc i:cod obs =dfd h dfd (t i )(e φz i (1 p 1 (t i )) + e ξ p 0 (t i )) j R i h dfd (t i )(e φz j (1 p 1 (t i )) + e ξ p 0 (t i )) 10 / 21

20 Partial likelihood no misclassification: L p (observed data) = misclassification: i:cod true,i =dfd h dfd (t i )e φz i j R i h dfd (t i )e φz j L p (observed data) h dfd (t i )(e ξ (1 p 0 (t i )) + e φz i p 1 (t i )) = j R i h dfd (t i )(e ξ (1 p 0 (t i )) + e φz j p 1 (t i )) i:cod obs =doc i:cod obs =dfd h dfd (t i )(e φz i (1 p 1 (t i )) + e ξ p 0 (t i )) j R i h dfd (t i )(e φz j (1 p 1 (t i )) + e ξ p 0 (t i )) 10 / 21

21 Score statistic with: w i = and variance: 1 1+e ξ 1 p 0 (t i ) p 1 (t i ) 1 1+e ξ p 0 (t i ) 1 p 1 (t i ) V n = δ i =1 T n = δ i =1 w i (t i, COD obs )(Z i Z i ) w 2 COD obs = doc 1 negative predictive value COD obs = dfd i (t i, COD obs ) positive predictive value Zj 2 /n i j R i Z 2 i 11 / 21

22 Score statistic with: w i = and variance: 1 1+e ξ 1 p 0 (t i ) p 1 (t i ) 1 1+e ξ p 0 (t i ) 1 p 1 (t i ) V n = δ i =1 T n = δ i =1 w i (t i, COD obs )(Z i Z i ) w 2 COD obs = doc 1 negative predictive value COD obs = dfd i (t i, COD obs ) positive predictive value Zj 2 /n i j R i Z 2 i 11 / 21

23 Score statistic with: w i = and variance: 1 1+e ξ 1 p 0 (t i ) p 1 (t i ) 1 1+e ξ p 0 (t i ) 1 p 1 (t i ) V n = δ i =1 T n = δ i =1 w i (t i, COD obs )(Z i Z i ) w 2 COD obs = doc 1 negative predictive value COD obs = dfd i (t i, COD obs ) positive predictive value Zj 2 /n i j R i Z 2 i 11 / 21

24 T Some features of n V n standard normal under the null no misclassification: standard logrank test time-constant misclassification T n V n = w doct n doc + w dfdt n dfd w 2 doc V n doc + w 2 dfd V n dfd 12 / 21

25 Impact of misclassification: Jaffar et al. - corrected logrank p 0 = 0.1 p 1 = 0.6 power = 27% 13 / 21

26 Impact of misclassification: Jaffar et al. - corrected logrank p 0 = 0.1 p 1 = 0.6 power = 27% 27% 21% = / 21

27 Impact of misclassification Asymptotic relative efficiencies General statistic: three possible tests w doc Tdoc n + w dfd Tdfd n (w doc ) 2 Vdoc n + (w dfd ) 2 Vdfd n the all-cause mortality logrank test: w doc = 1,w dfd = 1 the naive statistic (observed-cause-specific logrank test): w doc = 0,w dfd = 1 the corrected statistic (time-constant weighted logrank test): w doc = w doc,w dfd = w dfd 14 / 21

28 Impact of misclassification Asymptotic relative efficiencies n Pitman efficiencies: compare are(1, 2) = lim φ,2 φ n φ,1 naive 1 naive corrected all-cause corrected 1 + w doc p 1 w dfd (1 p 1 ) 1 all-cause 1 (1+e ξ )w dfd (1 p 1 ) 1 (1+e ξ )(w doc p 1 +w dfd (1 p 1 )) 1 15 / 21

29 are(all-cause, naive) = 1 (1+e ξ )w dfd (1 p 1 ) favour naive > 1 at low p 0, p 1 < 1 at high p 0, p 1 favour allcause 16 / 21

30 are(naive, corrected) = ( ) w docp 1 w dfd (1 p 1 ) equal 1 = 1 when p 1 = 0 favour corrected 17 / 21

31 are(all-cause, corrected) = 1 (1+e ξ )(w doc p 1 +w dfd (1 p 1 )) equal < 1 when p 0 and p 1 extreme 1 when p 0 + p 1 1 w doc w dfd weighted test equals all-cause test never > 1 favour corrected 18 / 21

32 are(all-cause, corrected) = 1 (1+e ξ )(w doc p 1 +w dfd (1 p 1 )) equal < 1 when p 0 and p 1 extreme 1 when p 0 + p 1 1 w doc w dfd weighted test equals all-cause test never > 1 favour corrected 18 / 21

33 are(all-cause, corrected) = 1 (1+e ξ )(w doc p 1 +w dfd (1 p 1 )) equal + p 0 = 0.1 p 1 = 0.6 are = 0.68 favour corrected 18 / 21

34 Impact of misclassification Asymptotic relative efficiencies example: Jaffar et al., I.J.E remember: p 0 = 0.1, p 1 = 0.6 naive 1 naive corrected all-cause corrected all-cause Sample size implications? naive: 84,841 corrected: 60,171 all-cause: 88,487 n = 1 w doc p 1 + w dfd (1 p 1 ) n standard 19 / 21

35 Impact of misclassification Asymptotic relative efficiencies example: Jaffar et al., I.J.E remember: p 0 = 0.1, p 1 = 0.6 naive 1 naive corrected all-cause corrected all-cause Sample size implications? naive: 84,841 corrected: 60,171 all-cause: 88,487 n = 1 w doc p 1 + w dfd (1 p 1 ) n standard 19 / 21

36 Impact of misclassification Asymptotic relative efficiencies example: Jaffar et al., I.J.E remember: p 0 = 0.1, p 1 = 0.6 naive 1 naive corrected all-cause corrected all-cause Sample size implications? naive: 84,841 corrected: 60,171 all-cause: 88,487 n = 1 w doc p 1 + w dfd (1 p 1 ) n standard 19 / 21

37 Impact of misclassification Asymptotic relative efficiencies example: Jaffar et al., I.J.E remember: p 0 = 0.1, p 1 = 0.6 naive 1 naive corrected all-cause corrected all-cause Sample size implications? naive: 84,841 corrected: 60,171 all-cause: 88,487 n = 1 w doc p 1 + w dfd (1 p 1 ) n standard 19 / 21

38 What to remember misclassification occurs in many settings it leads to loss of power under some general assumptions, we can partly correct the logrank test for this this can lead to much smaller sample sizes choice of endpoint can be affected, cause-specific analysis becomes more attractive! fundamental correction, otherwise bias (estimation!) 20 / 21

39 What to remember misclassification occurs in many settings it leads to loss of power under some general assumptions, we can partly correct the logrank test for this this can lead to much smaller sample sizes choice of endpoint can be affected, cause-specific analysis becomes more attractive! fundamental correction, otherwise bias (estimation!) 20 / 21

40 What to remember misclassification occurs in many settings it leads to loss of power under some general assumptions, we can partly correct the logrank test for this this can lead to much smaller sample sizes choice of endpoint can be affected, cause-specific analysis becomes more attractive! fundamental correction, otherwise bias (estimation!) 20 / 21

41 Goetghebeur E. and Ryan L. A Modified Log Rank Test for Competing Risks with Missing Failure Types, Biometrika, 77: , 1990 Jaffar S., Leach A., Smith P.G., Cutts F. and Greenwood B. Effects of misclassification of causes of death on the power of a trial to assess the efficacy of a pneumococcal conjugate vaccine in The Gambia, International Journal of Epidemiology, 32: , 2003 Cutts F.T., Zaman S.M.A., Enwere G., Jaffar S., Levine O.S., Okoko J.B. et al. Efficacy of nine-valent pneumococcal conjugate vaccine against pneumonia and invasive pneumococcal disease in The Gambia: randomised, double-blind, placebo-controlled trial, Lancet, 365: , / 21