Helene Jacqmin-Gadda

Transcript

1. Quantile regression for incomplete longitudinal data with follow-up truncated by death Hélène Jacqmin-Gadda, Anaïs Rouanet, Viviane Philipps, Marion Medeville Inserm Research Center Bordeaux Population Health, Biostatistics team SAM Conference July 2017 - Liverpool
2. Introduction WIEE for quantile regression weights computation Application Conclusion Quantile regressions Qτ (Y |X) = X βτ Possible τ values : 0.05, 0.10, 0.25, 0.50, .... Factors associated with change in population quantiles Avoid distribution assumption (no Gaussian assumption) Characterize the entire distribution by modelling several quantiles (= mean regression) → Useful for estimating cognitive norms in the elderly Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 2 / 16
3. Introduction WIEE for quantile regression weights computation Application Conclusion Estimating cognitive norms from elderly cohorts Estimation method should account for : Correlation among repeated measures Follow-up truncated by death Dropout Intermittent missing data Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 3 / 16
4. Introduction WIEE for quantile regression weights computation Application Conclusion Objective & outline Objective : Propose a weighted estimating equation approach to estimate quantile regression using incomplete longitudinal data due to death, dropout and intermittent missing data. Outline : Weighted IEE for quantile regression Weight computation given type of missing data Application for cognitive norm estimation in the Paquid cohort Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 4 / 16
5. Introduction WIEE for quantile regression weights computation Application Conclusion Notations Yij cognitive test for subject i, i = 1, ..., N, at visit j, j = 1, ..., ni < J Xij explanatory variable for i at visit j (fixed or known at all visits) Qτ (Yij |Xij ) = Xij βτ Ti time of death Rij = 1 if Yij observed, Rij = 0 if Yij missing HY ij measures of Y for subject i before visit j ˜ Yi complete trajectory of Y including unobserved values Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 5 / 16
6. Introduction WIEE for quantile regression weights computation Application Conclusion Missingness mechanisms with death P(Rij = 1 | Ti > tij , Xi , ˜ Yi , Ti ) = P(Rij = 1|Ti > tij , Xij ) MCAR = P(Rij = 1|Ti > tij , Xij , HY ij ) MAR = P(Rij = 1|Ti > tij , Xij , HY ij , Ti ) MAR − S (Kurland and Heagerty, 2005) In the following, we assume P(Ri1 = 1) = 1 Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 6 / 16
7. Introduction WIEE for quantile regression weights computation Application Conclusion WIEE for incomplete longitudinal data U(ˆ βτ ) = N i=1 ni j=1 1 πij Xij τ − 1(Yij − Xij ˆ βτ < 0) = 0 and πij = P(Rij = 1|Ti > tij , Xij , HY ij ) (or stabilized weights) Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 7 / 16
8. Introduction WIEE for quantile regression weights computation Application Conclusion WIEE for incomplete longitudinal data U(ˆ βτ ) = N i=1 ni j=1 1 πij Xij τ − 1(Yij − Xij ˆ βτ < 0) = 0 and πij = P(Rij = 1|Ti > tij , Xij , HY ij ) (or stabilized weights) Estimated with R package quantreg with the "weights" option Variance estimated by bootstrapping subjects (Yi , Xi ) −→ WIEE estimates change in the population quantiles among subjects currently alive Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 7 / 16
9. Introduction WIEE for quantile regression weights computation Application Conclusion Weights computation for monotone MAR-S data πij = P(Rij = 1|Ti > tij , Xij , HY ij ) = j k=2 P(Rik = 1|Rik−1 = 1, Ti > tij , Xij , HY ij ) = j k=2 P(Rik = 1|Rik−1 = 1, Ti > tij , Xik, Yik−1) = j k=2 λijk Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 8 / 16
10. Introduction WIEE for quantile regression weights computation Application Conclusion Weights computation for monotone MAR-S data πij = P(Rij = 1|Ti > tij , Xij , HY ij ) = j k=2 P(Rik = 1|Rik−1 = 1, Ti > tij , Xij , HY ij ) = j k=2 P(Rik = 1|Rik−1 = 1, Ti > tij , Xik, Yik−1) = j k=2 λijk Dropout model under MAR-S : logit (λijk) = γ0k(j−k) + γ1(j−k) Xik + γ2(j−k) Yik−1 Estimated for each interval j − k ≥ 0 by pooling samples of subjects alive at visit j and observed at visit k − 1 for k = 2, J. Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 8 / 16
11. Introduction WIEE for quantile regression weights computation Application Conclusion Weights computation for monotone MAR data πij = j k=2 P(Rik = 1|Rik−1 = 1, Ti > tij , Xij , HY ij ) = j k=2 P(Rik = 1|Rik−1 = 1, Ti > tik, Xik, Yik−1) = j k=2 λik Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 9 / 16
12. Introduction WIEE for quantile regression weights computation Application Conclusion Weights computation for monotone MAR data πij = j k=2 P(Rik = 1|Rik−1 = 1, Ti > tij , Xij , HY ij ) = j k=2 P(Rik = 1|Rik−1 = 1, Ti > tik, Xik, Yik−1) = j k=2 λik Dropout model under MAR : logit (λik) = γ0k + γ1Xik + γ2Yik−1 Estimated by a unique regression pooling samples of subjects alive at visit k = 2, ..., J and observed at visit k − 1. (Dufouil et al, 2004) Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 9 / 16
13. Introduction WIEE for quantile regression weights computation Application Conclusion Weights computation for intermittent MAR data HR ij = (Ri1, ..., Rij−1) and ηij = P(Rij = 1|Ti > tij , HR ij , Xij , HY ij ) πij = P(Rij = 1|Ti > tij , Xij , HY ij ) = HR ij ηij j−1 k=2 ηrik ik (1 − ηik)(1−rik ) Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 10 / 16
14. Introduction WIEE for quantile regression weights computation Application Conclusion Weights computation for intermittent MAR data HR ij = (Ri1, ..., Rij−1) and ηij = P(Rij = 1|Ti > tij , HR ij , Xij , HY ij ) πij = P(Rij = 1|Ti > tij , Xij , HY ij ) = HR ij ηij j−1 k=2 ηrik ik (1 − ηik)(1−rik ) Intermittent missingness model under MAR : logit(ηij ) = γ0j + γ1Xij + γ2(1 − Rij−1) + γ3Y ∗ ij−1 with Y ∗ ij−1 = Yij−1 if Rij−1 = 1 and Y ∗ ij−1 = µ if Rij−1 = 0. Estimated by pooling samples of subjects alive at visit j = 2, ..., J (whatever the observation pattern). Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 10 / 16
15. Introduction WIEE for quantile regression weights computation Application Conclusion Application to gender difference in cognitive decline Paquid cohort : 3421 subjects followed every 2/3 years during 25 years Y : Isaacs test of verbal fluency (score at 60sec, ceiling effect) X : gender, education (CEP) and age (for Y and missingess model) Sample A : include all measurements (med :3, IQR=2-6) Sample B : follow-up truncated at the first missing value (med :1, IQR=1-4, 2516 dropouts, 819 deaths before dropout) Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 11 / 16
16. Introduction WIEE for quantile regression weights computation Application Conclusion Estimates for sample B (monotone) : MCAR, MAR & MAR-S Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 12 / 16
17. Introduction WIEE for quantile regression weights computation Application Conclusion Estimates for sample A & B : MCAR & MAR Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 13 / 16
18. Introduction WIEE for quantile regression weights computation Application Conclusion Estimates for sample A under MCAR & MAR τ=0.10 τ=0.50 IEE ˆ β se p value ˆ β se p value Intercept 36.02 0.88 <0.0001 39.50 0.25 <0.0001 age65 -0.71 0.86 0.406 -0.55 0.25 0.027 age652 -1.86 0.22 <0.0001 -0.32 0.06 <0.0001 Education 4.05 0.85 <0.0001 0.30 0.25 0.219 Male -2.25 0.31 <0.0001 0.15 0.05 0.001 Baseline visit -1.87 0.17 <0.0001 -0.03 0.008 <0.0001 age65 :Educ 0.58 0.63 0.354 1.11 0.22 <0.0001 age65 :Male 1.34 0.38 0.0003 -0.27 0.09 0.002 WIEE (MAR) ˆ β se p value ˆ β se p value Intercept 33.86 1.15 <0.0001 39.35 0.24 <0.0001 age65 1.52 1.51 0.323 -0.30 0.32 0.336 age652 -2.82 0.53 <0.0001 -0.56 0.09 <0.0001 Education 6.20 1.09 <0.0001 0.36 0.23 0.112 Male -2.41 0.63 0.0003 0.11 0.08 0.149 Baseline visit -1.06 0.24 <0.0001 -0.04 0.012 0.0006 age65 :Educ -1.28 0.94 0.173 1.18 0.23 <0.0001 age65 :Male 1.76 0.78 0.02 -0.23 0.12 0.05 Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 14 / 16
19. Introduction WIEE for quantile regression weights computation Application Conclusion Conclusion Make possible to estimate quantile accounting for death, dropout and intermittent missingness Little difference between estimates under MAR and MAR-S Censoring at first intermittent missing data has more impact WIEE with the same weights available for standard regression R functions available soon Hélène Jacqmin-Gadda SAM 2017, Liverpool July 4th 2017 15 / 16
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