Semiparametric Mixed Effects Models with Flexible Random Effects Distribution

Transcription

1 Semiparametric Mixed Effects Models with Flexible Random Effects Distribution Marie Davidian North Carolina State University davidian Joint work with A. Tsiatis, J. Chen, X. Song, and D. Zhang

2 Outline 1. Introduction and motivation 2. Semiparametric mixed models 3. The class H and the SNP representation 4. Implementation 5. Simulations 6. Examples, revisited 7. Discussion

3 1. Introduction and motivation Longitudinal studies: Clinical trials, epidemiological investigations Repeated measures on some variable on each subject intermittently over time Survival endpoint (e.g., death, time to disease progression) Objectives: Inference on Within-subject patterns of change of repeated measurement variable, association with covariates Relationship between repeated measurement variable and survival, association with covariates

4 Example 1: Framingham study Cholesterol measurements every 2 years for 2634 participants over 10-year period Objectives: Change in cholesterol over time and association with age at baseline, gender Consider a subset of 200 subjects for illustrative purposes Individual profiles: Approximate straight line

5 Time in years Cholesterol level Cholesterol levels over time

6 Standard model: Linear mixed effects model For subject i at time t ij, j = 1,..., n i, i = 1,..., m Y ij = β 1 age i + β 2 gender i + β 3 age i t ij + β 4 gender i t ij + b 0i + b 1i t ij + e ij Usual assumptions: e i = (e i1,..., e ini ) T N(0, σ 2 I) b i = (b 0i, b 1i ) T N(µ b, D) Relevance of assumptions: Individual regression fits Pooled residuals, subject-specific slopes appear normal However, subject-specific intercepts do not...

7 Percentage Estimates of subject specific intercepts

8 Example 2: Six cities study Binary indicator of respiratory infection recored annually for 537 Ohio children at ages 7 10 Objectives: Within-subject change in respiratory status, association with baseline maternal smoking Standard model: Generalized linear mixed effects model For subject i at age a ij, j = 1,..., n i, i = 1,..., m E(Y ij b i ) = exp(β 1smoke + β 2 a ij + b i ) 1 + exp(β 1 smoke + β 2 a ij + b i ), var(y ij b i ) = E(Y ij b i ){1 E(Y ij b i )} Usual assumption: b i N(µ b, σ 2 b )

9 Example 3: ACTG 175 Clinical trial with 2467 subjects to compare 4 antiretroviral regimens Main objective: Compare on basis of time to AIDS or death Also, CD4 counts approximately every 12 weeks Subsequent objective: Characterize within-subject patterns of CD4 change complicated by informative censoring Subsequent objective: Characterize relationship between features of CD4 profiles and survival

10 log CD week Death/progression, censoring throughout

11 Standard model: Joint mixed effects-proportional hazards model Longitudinal data model: For subject i at weeks t i = (t i1,..., t ini ) T W ij = X i (t ij ) + e ij, X i (u) = b 0i + b 1i u X i (u) = inherent trajectory e i = (e i1,..., e ini ) T N(0, σ 2 I) Survival data model: For subject i, observe V i = min(t i, C i ), i = I(T i C i ), covariates S i λ i (u) = lim du 0 du 1 P {u T i < u + du T i u, α i, S i, C i, e i, t i } = λ 0 (u) exp{γx i (u) + η T S i } Usual assumption: b i = (b i0, b i1 ) T N(µ b, D)

12 intercept slope

13 2. Semiparametric mixed models Theme: The foregoing examples suggest that A simple parametric model may be adequate to describe subject-specific profiles in terms of random effects b i However, the relevance of the usual normality assumption on random effects is questionable Concern: Sensitivity of inferences to departures from normality

14 Needed: Relax the normality assumption on b i Semiparametric model Completely nonparametric (e.g., Mallet, 1986; Butler and Louis, 1992) includes unusual, discrete distributions Alternatively: Impose some realistic yet not overly restrictive conditions Restrict to a smooth class (e.g., Davidian and Gallant, 1993; Madger and Zeger, 1996; Verbeke and Lesaffre, 1996; Tao et al., 1999) Here: Assume b i have distribution with density in a smooth class H

15 3. The class H and the SNP representation Assume: b i = g(µ, S i ) + RZ i, Z i has density h H, R lower triangular with distinct elements θ E.g., for ACTG 175, g(µ, S i ) = µ 0 (1 S i ) + µ 1 S i S i = I(Trt=ZDV) H is a class of smooth densities studied by Gallant and Nychka (1987) Densities in H are sufficiently differentiable to rule out kinks, jumps, violent oscillation But can be skewed, multi-modal, fat- or thin-tailed relative to the normal (and the normal is H)

16 Formally: h H for Z (q 1) may be written as h(z) = P (z)ϕ 2 q (z) + small lower bound for tail behavior P (z) is an infinite-dimensional polynomial ϕ q (z) is q-variate standard normal density Practically speaking: Suggests approximating h H by truncation h K (z) = PK(z)ϕ 2 q (z) P K (z) is Kth order polynomial; e.g., for K = 2 P K (z) = a 00 + a 10 z 1 + a 01 z 2 + a 20 z a 02 z a 11 z 1 z 2 Vector of coefficients a must satisfy h K (z) dz = 1 K = 0 is standard normal b i N{g(µ, S i ), RR T }

17 Imposing h K (z) dz = 1: a (d 1), d depends on K h K (z) dz = 1 E{P 2 K(U)} = 1, U N(0, I) E{P 2 K (U)} = at Aa (A p.d.) = a T BBa = c T c = 1, c = Ba Polar coordinate transformation c 1 = sin(φ 1 ), c 2 = cos(φ 1 ) sin(φ 2 ),. c d 1 = cos(φ 1 ) cos(φ 2 ) cos(φ d 2 ) sin(φ d 1 ), c d = cos(φ 1 ) cos(φ 2 ) cos(φ d 2 ) cos(φ d 1 ), π/2 < φ r π/2, r = 1,..., d 1. Parameterize h K (z) in terms of φ = (φ 1,..., φ d 1 ) T.

18 Result: For fixed K, may represent density of b i in terms of (µ T, θ T, φ T ) T Likelihood for Ω = (µ T, θ T, φ T ) T plus any other model parameters (e.g., β, γ, η) is usual, finite-dimensional problem In principle, can use standard optimization methods to estimate Ω (coming up... ) Lingo: Seminonparametric Choosing tuning parameter K: K controls degree of flexibility and departure from normality (like a bandwidth )

19 Adaptive choice of K based on information criteria: If l K ( Ω) is maximized loglikelihood for fixed K, N = total number of observations, Ω (p 1), minimize { l K ( Ω) + pc(n)}/n AIC, C(N) = 1; BIC, C(N) = log N/2; Hannan-Quinn (HQ), C(N) = log log N AIC prefers larger models, BIC smaller, HQ intermediate Confidence intervals fixing K at choice achieve nominal coverage (Eastwood and Gallant, 1991)

20 4. Implementation Linear mixed effects model: For normal e i, Ω = (µ T, θ T, φ T, β T, σ) T, can write loglikelihood l K (Ω; Y ) in a closed form Maximize in Ω using standard optimization routines, e.g., SAS nlpqn Starting values chosen by grid search or penalized loglikelihood SEs, confidence intervals usual inverse of observed information for chosen K Zhang and Davidian (2001, Biometrics)

21 Generalized linear mixed effects model: Other models (e.g., binomial, Poisson), Ω = (µ T, θ T, φ T, β T ) T, cannot write l K (Ω; Y ) in a closed form Gallant and Tauchen (1992) provide efficient rejection sampling algorithm from estimated h K (z) (acceptance rate > 50%) Facilitates use of MCEM algorithm (e.g., McCulloch, 1997; Booth and Hobert, 1999) SEs, confidence intervals MC approx observed information for chosen K Chen, Zhang, and Davidian (2002, Biostatistics)

22 Joint longitudinal-survival model: Ω = (µ T, θ T, φ T, γ, η T, λ 0 ) T Under assumptions, l K (Ω; V,, W, t, Z) = log L(Ω; V,, W, t, Z) n m i L(Ω; V,, W, t, S) = p(v i, i b i, S i, γ, η, λ 0 ) p(w ij b i, σ 2, t ij ) i=1 p(b i Z i, µ, θ, φ)db i j=1 p(w ij b i, σ 2, t ij ) = { 1 exp (W ij b i0 b i1 t ij ) 2 } 2πσ 2 2σ 2, p(v i, i b i, S i, γ, η, λ 0 ) = [λ 0 (V i ) exp{γ(b i0 + b i1 V i ) + ηz i }] i [ exp Vi 0 λ 0 (u) exp{γ(α i0 + α i1 u) + ηs i }du, p(b i S i, µ, θ, φ) = h K [R 1 {b i g(µ, S i )}] R 1 ]

23 EM algorithm: L(Ω; V,, W, t, Z) is maximized when λ 0 (u) is non-zero only at death times, and Ω maximizing L(Ω; V,, W, t, Z) exists E-step: Intractable integration carried out via Gauss-Hermite quadrature M-step: Maximization in (µ T, θ T, φ T ) T and (γ, η T, σ, λ 0 ) T separates One-step Newton-Raphson update for (γ, η T ) T SEs and confidence intervals: Profile likelihood Song, Davidian, and Tsiatis (2002, Biometrics)

24 5. Simulations Linear mixed model: 100 data sets, each fit with K = 0, 1, 2 Y ij = b i + β 1 t ij + β 2 w i + e ij, i = 1,..., 100, j = 1,..., 5 t ij = j 3, β 1 = 2, w i = I(i 50), β 2 = 1, e ij N(0, ) Case 1: b i 0.7N( 3, 1) + 0.3N(2, 1) (mixture of normals) AIC preferred K = 1, 2 35%, 65% of time (BIC: 76%, 24%; HQ: 56%, 44%) Case 2: b i N( 1.5, 6.25) AIC preferred K = 0, 1, 2 84%, 7%, 9% (BIC: 97%, 3%, 0%; HQ: 89%, 5%, 6%)

25 K = 0 (Normality) Preferred by HQ MC Ave. MC SD Ave. SE MC Ave. MC SD Ave. SE RE Case 1: Mixture Scenario β 1 (2) β 2 (1) E(b) ( 1.5) var(b) (6.25) σ (0.5) Case 2: Normal Scenario β 1 (2) β 2 (1) E(b) ( 1.5) var(b) (6.25) σ (0.5)

26 (a) (b) Densities Densities x x (a) Average of 100 estimates (K = 0 and AIC, BIC, HQ) and truth (b) 100 estimates chosen by HQ

27 Joint model: 200 data sets, each fit with K = 0, 1, 2, 2-point quadrature in E-step W ij = b 0i + b 01 t ij + e ij, i = 1,..., 200, e ij N(0, 0.60) t ij = 0, 2, 4, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 10% missingness {var(b i0 ), cov(b i0, b i1 ), var(b i1 )} = (4.96, , 0.012) λ i (u) λ 0 (u) exp{γx i (u)}, λ 0 (u) 1 for u 16, = 0 ow; γ = 1.0 Exponential (110) censoring (53% censored data) Case 1: b i bivariate mixture of normals Case 2: b i bivariate normal

28 Estimation of b i distributions: Similar to linear mixed model Estimation of hazard parameter γ: K = 0 AIC HQ BIC Case 1: Mixture Scenario γ SD SE CP %K = (0, 1, 2) (0.0,70.5,29.5) (0.0,93.5,6.5) (0.0,100.0,0.0) Case 2: Normal Scenario γ SD SE CP %K = (0, 1, 2) (91.5,5.0,3.5) (99.0,1.0,0.5) (100.0,0.0,0.0) Amazing robustness to distribution of b i

29 density density b b b b

30 density b0 density b1

31 density b0 density b1

32 6. Examples, revisited Example 1: Framingham cholesterol data Y ij = β 1 age i + β 2 gender i + β 3 age i t ij + β 4 gender i t ij + b 0i + b 1i t ij + e ij e i N(0, σ 2 I), b i = µ + RZ i, h H Subset of 200 subjects, each every 2 years Fit using K = 0, 1, 2 All criteria select K = 1

33 0 2 4 Density Slope Intercept Estimated joint density for K = 1

34 Density Intercept Estimated marginals for intercepts, K = 0 and 1

35 Example 3: ACTG 175 joint longitudinal-survival model, W ij = X i (t ij ) + e ij, X i (u) = b 0i + b 1i u, e i N(0, σ 2 ) b i = µ 0 (1 S i ) + µ 1 S i + RZ i, S i = I(Trt=ZDV) h H λ i (u) = lim du 0 du 1 P {u T i < u + du T i u, α i, S i, C i, e i, t i } = λ 0 (u) exp{γx i (u) + η T S i } 2467 subjects Fit for K = 0, 1, 2, 3, 4; K = 3 or 4 chosen

36 800 0 density b1 Estimated joint density for K = b0

37 density b1 Estimated marginals for slope for K = 0, 2, 3, 4

38 7. Discussion Potential to gain efficiency in parameters associated with subject-level covariates in longitudinal models; other parameters robust Remarkable robustness of inference on hazard parameters to misspecification of random effects distribution in joint model Insight on population heterogeneity, possible omitted covariates Implementation only mildly more difficult than assuming normal random effects