Score test for random changepoint in a mixed model

Transcription

1 Score test for random changepoint in a mixed model Corentin Segalas and Hélène Jacqmin-Gadda INSERM U1219, Biostatistics team, Bordeaux GDR Statistiques et Santé October 6, 2017 Biostatistics 1 / 27

2 Introduction Model Score test Simulations Application Conclusion 2 / 27

3 Alzheimer s Disease (AD) A major public health issue today and tomorrow A very long pre-diagnostic phase Heterogeneous and non-linear decline trajectories 3 / 27

4 Different profiles? Figure: Estimated mean BVRT score according to age for 2 subjects demented at 90 with low or high educational level (Jacqmin-Gadda et al., 2006) 4 / 27

5 Objective Propose a test for the existence of a random changepoint in a mixed model for longitudinal data. 5 / 27

6 The mixed model with random changepoint Y (t ij ) = Y ij = β 0i + β 1i t ij + β 2 (t ij τ i ) 2 + γ + ε ij (1) with β ki = βk T X ki + α ki for k = 0, 1, α i = (α 0i, α 1i ) T N (0, B), τ i = µ τ + σ τ τ i with α i independent from τ i and τ i N (0, 1), γ = 0.1 a fixed smoothness parameter. β 1i is the mean slope and β 2 half the difference of the slopes. 6 / 27

7 The mixed model with random changepoint marker Y(t) t= time (t) 7 / 27

8 Estimation Model estimated by MLE and integral computed by gaussian quadrature (15 nodes) l n (Y ; θ) = n i=1 log n i f (Y ij α i, τ i )f (α i )f ( τ i )dα i d τ i. j=1 No known methods to test the existence of a random CP 8 / 27

9 Classic score test (H 0 ) : β 2 = β2 0 vs. (H 1) : β 2 β2 0 test statistic: S n = U n(β2 0, ˆθ 0 ) 2 Var(U n (β2 0, ˆθ 0 ) 2 ) with U n(β2, 0 ˆθ 0 ) = l n(y ; β 2, ˆθ 0 ) β 2 with ˆθ 0 the MLE of nuisance parameters under the null null distribution: χ 2 (1) β2 =β / 27

10 Identifiability issue Y ij = β 0i + β 1i t ij + β 2 (t ij µ τ σ τ τ i ) 2 + γ + ε ij Hypotheses: (H 0 ) : β 2 = 0 vs. (H 1 ) : β 2 0 nuisance parameters : β 0, β 1, σ, σ 0, σ 1, σ 01, µ τ, σ τ µ τ and σ τ unidentifiable under the null: we can t use the classic score test statistic S n which depends on them. 10 / 27

11 U n (0; θ) = The score under the null (β 2 = 0) N [ i=1 f ( τ i ) f (α i )f ( τ i )( 2πσ) n i n i 1 f (α i ) exp 2πσ n i j=1 j=1 [ 1 σ 2 exp (Y ij β 0i β 1i t ij ) (t ij µ τ σ τ τ i ) 2 + γ exp k j { 1 } ] 1 2σ 2 (Y ij β 0i β 1i t ij ) 2 dα i d τ i { 1 } 2σ 2 (Y ij β 0i β 1i t ij ) 2 { 1 } ] 2σ 2 (Y ik β 0i β 1i t ik ) 2 dα i d τ i How to circumvent this problem? 11 / 27

12 Score test with identifiability issue Classic problem when testing homogeneity on mixture models. Two main approaches : replace µ τ and σ τ by the MLE under the alternative (Conniffe, 2001) consider the supremum in (µ τ, σ τ ) of the score test statistic (Hansen, 1996) 12 / 27

13 The sup score test (H 0 ) : β 2 = 0 vs. (H 1 ) : β 2 0 test statistic: T n = sup S n (0; µ τ, σ τ, ˆθ 0 ) (µ τ,σ τ ) with S n (0; µ τ, σ τ, ˆθ 0 ) = U n(0; µ τ, σ τ, ˆθ 0 ) 2 Var(U n (0; µ τ, σ τ, ˆθ 0 )) with ˆθ 0 the MLE of identifiable nuisance parameters under the null null distribution: approached by MC perturbation algorithm or multiplier bootstrap (van der Vaart and Wellner, 1996). 13 / 27

14 The complete procedure 1. estimation of the null model (linear mixed model) using nlme 2. computing the observed test statistic (optimization via quasi-newton and integral via pseudo-adaptive gaussian quadrature) Tn obs U n (0; µ τ, σ τ, ˆθ 0 ) 2 = sup (µ τ,σ τ ) Var(U ˆ n (0; µ τ, σ τ, ˆθ 0 )) where U n (0; µ τ, σ τ, ˆθ 0 ) = n i=1 u i(0; µ τ, σ τ, ˆθ 0 ) and the variance is estimated by n u i (0; µ τ, σ τ, ˆθ 0 ) 2. i=1 14 / 27

15 The complete procedure 4. perturbation algorithm: for k = 1,..., K = 500 generate n r.v. ξ (k) i N (0, 1) compute T (k) n = sup (µ τ,σ τ ) ( n 5. compute the empirical p value i=1 u i(0; µ τ, σ τ, ˆθ 0 )ξ (k) i n i=1 u i(0; µ τ, σ τ, ˆθ 0 ) 2 K ) 2 p K = 1 K k=1 1 (k) T n >T n (obs) 15 / 27

16 Tests for the variability of β 2 If we reject the null hypothesis, we can test if there is a 1. random effect for the difference of slope: β 2i = β 2 + α 2i with α i = (α 0i, α 1i, α 2i ) N (0, B) corrected LR test for variance component (Stram and Lee, 1994) 2. dependance on covariates: Wald test β 2i = β 20 + β 21 X 2i 16 / 27

17 Simulation scenarios with Y ij = β 0i + β 1i t ij + β 2 (t ij τ i ) 2 + γ + ε ij β 0i = 20 + α 0i and β 1i = α 1i (( ) ( )) α i = (α 0i, α 1i ) T N,, τ i = τ i with α i independant from τ i and τ i N (0, 1), γ = 0.1, σ ε = 1, t ij = 0, 3, 6, 9, 12, 15, 18, 21 for all i, β 2 = 0, 0.05, 0.075, 0.1, 0.2, Probability of drop-out at each visit: 0.1 around 50% of the sample remaining at t = / 27

18 Simulation Scenarios M1 M2 marker marker time time M3 M4 marker marker time time 18 / 27

19 Results N drop-out no yes no yes size M power M M M M Table: Size and power of the test computed on 1000 replicates of each scenarios with K = 500 perturbations. 19 / 27

20 The PAQUID cohort 3777 subjects older than 65 from the french departments of Gironde and Dordogne, 25 years follow-up Marker : Isaac 15s score sample selection: incident case of dementia between year 1 and 25 High education sample 522 subjects with at least 1 measure 1 to 12 measures by subject (mean = 5.8) Low education sample 358 subjects with at least 1 measure 1 to 12 measures by subject (mean = 4.6) model (1) with β ki = β k + α ki for k = 0, 1 (no covariate) 20 / 27

21 Score test results obs. statistic test* p-value High education Low education Table: Score test results with K = 1000 For the high education subjects, we clearly reject the null hypothesis of no random changepoint. 21 / 27

22 Estimation (nq = 15) PAQUID demented sample High education Low education N Log-lik Est sd Est sd β β β µ τ σ σ σ σ τ σ slope 1/ / / / 27

23 Estimation of the mixed model with random CP Isaac 15s High education Low education Delay Figure: Mean estimation trajectory of the mixed model with random changepoint on the two educational level subsamples. 23 / 27

24 Variability of β 2 : random effect? On high education subsample (H 0 ) : σ 2 = 0 vs. (H 1 ) : σ 2 0 where β 2i = β 2 + α 2i with α 2i N (0, σ 2 2 ). LRS = p < We need to add a random effect on β 2 24 / 27

25 Next steps simulations with varying σ τ extension to : joint models joint multi-state models for interval censored data models for multiple markers etc. 25 / 27

26 References 1. Conniffe, D., Score Tests When a Nuisance Parameter Is Unidentified under the Null Hypothesis. Journal of Statistical Planning and Inference (2001) 2. Hansen, Bruce E., Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesi. Econometrica (1996) 3. Jacqmin-Gadda, H., Commenges, D. and Dartigues, J.-F., Random changepoint model for joint modeling of cognitive decline and dementia. Biometrics (2006) 4. Stram, D. O., and Lee J.W., Variance Components Testing in the Longitudinal Mixed Effects Model. Biometrics (1994) 5. van der Vaart, A. W. and Wellner, J.A., Weak Convergence and Empirical Processes. Chapter 2.9, Springer Series in Statistics (1996). 26 / 27

27 Thank you for your attention! 27 / 27