Models for Multivariate Panel Count Data

Transcription

1 Semiparametric Models for Multivariate Panel Count Data KyungMann Kim University of Wisconsin-Madison 2 April 2015

2 Outline 1 Introduction 2 3 4

3 Panel Count Data Motivation Previous Work Observation of the cumulative event counts for an individual at a random number of time points; both the number and the time points may differ across individuals

4 Panel Count Data Motivation Previous Work Observation of the cumulative event counts for an individual at a random number of time points; both the number and the time points may differ across individuals Unobserved: event happens x x x x x 0 x x x x x 0 subject 1 subject 2

5 Panel Count Data Motivation Previous Work Observation of the cumulative event counts for an individual at a random number of time points; both the number and the time points may differ across individuals Unobserved: event happens x x x x x 0 t 11 t 12 t 13 x x x x x 0 t 21 t 22 t 23 t 24 subject 2

6 Panel Count Data Motivation Previous Work Observation of the cumulative event counts for an individual at a random number of time points; both the number and the time points may differ across individuals Unobserved: event happens x x x x x 0 t 11 t 12 t 13 subject 2 2 x x x x x 0 t 21 t 22 t 23 t 24 Observed: n 11 = 2 n 12 = 4 n 13 = 5 0 t 11 t 12 t 13 subject 1 n 21 = 2 n 22 = 4 n 23 = 5 n 24 = 5 subject 0 t 21 t 22 t 23 t 24

7 Panel Count Data Motivation Previous Work Skin Cancer Chemoprevention Trial: Design Randomized, placebo-controlled, double-blind clinical trial Objective: Is difluoromethylornithine (DFMO) effective in preventing recurrence of non-melanoma skin cancers (NMSCs)? 291 subjects with prior skin cancer randomized to either DFMO or placebo Subjects assessed every six months for the new skin cancers until the end of study Primary endpoint: NMSC recurrence rate total number of new NMSCs person-year follow-up

8 Panel Count Data Motivation Previous Work Skin Cancer Chemoprevention Trial: Results Marginal DFMO effect on reducing the NMSC recurrence rate from 0.60 on placebo to 0.43 on DFMO based on a Poisson regression model (p-value=0.062)

9 Deficiencies in the Analysis Panel Count Data Motivation Previous Work Loss of information about longitudinal observations Correlation between basal and squamous cell carcinoma Significant DFMO effect on preventing basal cell carcinoma (BCC) (p-value=0.030) But not for squamous cell carcinoma (SCC) (p-value=0.565)

10 Deficiencies in the Analysis Panel Count Data Motivation Previous Work Loss of information about longitudinal observations n 11 = 0 n 12 = 0 n 13 = 5 subject 1 Rate 1 = 5 T 0 t 11 t 12 t 13 (= T ) n 21 = 4 n 22 = 5 n 23 = 5 n 24 = 5 subject 2 Rate 2 = 5 T 0 t 21 t 22 t 23 t 24 (= T ) Correlation between basal and squamous cell carcinoma Significant DFMO effect on preventing basal cell carcinoma (BCC) (p-value=0.030) But not for squamous cell carcinoma (SCC) (p-value=0.565)

11 Deficiencies in the Analysis Panel Count Data Motivation Previous Work Loss of information about longitudinal observations Correlation between basal and squamous cell carcinoma Significant DFMO effect on preventing basal cell carcinoma (BCC) (p-value=0.030) But not for squamous cell carcinoma (SCC) (p-value=0.565)

12 Deficiencies in the Analysis Panel Count Data Motivation Previous Work Loss of information about longitudinal observations Correlation between basal and squamous cell carcinoma Significant DFMO effect on preventing basal cell carcinoma (BCC) (p-value=0.030) But not for squamous cell carcinoma (SCC) (p-value=0.565)

13 Poisson Analysis Panel Count Data Motivation Previous Work Mean Occurrence Rate overall non!melanoma skin cancer (p!value=0.062) basal cell carcinoma (p!value = 0.030) squamous cell carcinoma (p!value=0.565) Placebo DFMO

14 Alternative Analysis Panel Count Data Motivation Previous Work Panel count data structure + Possibly different effect of DFMO on different skin cancer types Data should be analyzed as the bivariate panel count data

15 Alternative Analysis Panel Count Data Motivation Previous Work Panel count data structure + Possibly different effect of DFMO on different skin cancer types Data should be analyzed as the bivariate panel count data

16 Previous Work Panel Count Data Motivation Previous Work Univariate panel count data Bivariate recurrent event time data

17 Notations Panel Count Data Motivation Previous Work N(t), the number of events observed between time 0 and t as a nonhomogeneous Poisson process Intensity function Mean function 1 λ(t) = lim Pr(N(t + h) N(t) = 1) h 0 h Λ(t) = E[N(t)] = t 0 λ(s)ds Different from hazard function in survival analysis Different from cumulative hazard function in survival analysis

18 Univariate Panel Count Data Inference on the intensity function λ(t) Panel Count Data Motivation Previous Work Parametric Poisson-Gamma frailty model (Thall, 1988) Nonparametric model (Thall and Lachin, 1988) Semiparametric Poisson-Gamma frailty model (Staniswalis et al., 1997)

19 Univariate Panel Count Data Inference on the mean function Λ(t) Panel Count Data Motivation Previous Work Nonparametric maximum pseudo-likelihood estimator (NPMPLE) (Sun and Kalbfleisch, 1995; Wellner and Zhang, 2000) u v=r ˆΛ l = ˆΛ(s l ) = max r l min wv n v u l u v=r wv where w v = n ki i=1 j=1 I{t ij = s v }, n v = 1 n ki w v i=1 j=1 n iji{t ij = s v }, n ij = N(t ij ) is the cumulative event counts up to time t ij of subject i, k i is the number of follow-up visits, and {s v } L v=1 is the unique, ordered {t ij ; j = 1,..., k i, i = 1,..., n} Semiparametric model (Zhang, 2002) Assuming mean function Λ(t) = Λ 0 (t) exp(β Z (t)) Gamma frailty model (Zhang and Jamshidian, 2003) Assuming the conditional mean function Λ(t γ) = γλ 0 (t)

20 Univariate Panel Count Data Inference on the mean function Λ(t) Panel Count Data Motivation Previous Work Nonparametric maximum pseudo-likelihood estimator (NPMPLE) (Sun and Kalbfleisch, 1995; Wellner and Zhang, 2000) Semiparametric model (Zhang, 2002) Assuming mean function Λ(t) = Λ 0 (t) exp(β Z (t)) Gamma frailty model (Zhang and Jamshidian, 2003) Assuming the conditional mean function Λ(t γ) = γλ 0 (t) Nonparametric test for univariate panel count data based on NPMPLE (Sun and Fang, 2003)

21 Panel Count Data Motivation Previous Work Bivariate Recurrent Event Time Data Parametric model for bivariate recurrent event time data (Abu-Libdeh et al. 1990) N 1 (t) and N 2 (t) are two counting processes Two types of random effects: Subject level random effect γ Gamma(α, ν) Process random effects ξ = (ξ 1, ξ 2 ) Dirichlet (υ = (υ 1, υ 2 )) Given (γ, ξ), N 1 (t) and N 2 (t) are independent nonhomogeneous Poisson processes λ p (t γ, ξ p ) = γξ p δt δ 1 exp(β Z ), p = 1, 2 Obtain maximum likelihood estimators of (β, α, ν, υ 1, υ 2 ) Not applicable to the panel count data since event times are unknown

22 Semiparametric Frailty Models Semiparametric Frailty Models Frailty models Estimation procedures Statistical inference on regression parameters

23 General Model Assumptions Semiparametric Frailty Models Frailty variable γ g(η) Subject heterogeneity Dependence between processes comes from the common frailty variable Given γ, N 1 (t) and N 2 (t) are independent nonhomogeneous Poisson processes with conditional mean functions Λ 1 (t γ) = γλ 10 (t) exp(β 1 Z ) and Λ 2(t γ) = γλ 20 (t) exp(β 2 Z )

24 Likelihood Function Semiparametric Frailty Models The complete log-likelihood function of (Λ 10, Λ 20, β 1, β 2, η) is simplified as k n i 2 n l n(λ 10, Λ 20, η) = [n ijp log Λ p0ij + n ijp β pz γ i Λ p0ij e β p Z ] + h i (η) i=1 j=1 p=1 n ijp = N p (t ij ), the cumulative type p event counts up to time t ij Λ p0ij = Λ p0 (t ij ), the baseline mean function of type p event at t ij i=1

25 Estimating Procedures EM algorithm to obtain the MLEs Semiparametric Frailty Models E-step: same as in the nonparametric models M-step: involves the profile-likelihood type estimators of (Λ 10 (t), Λ 20 (t), β 1, β 2 ) and MLE of η step 1: Given (β (0) 1, β(0) (1) 2 ), obtain (ˆΛ 10, ˆΛ (1) 20 ) step 2: Given (ˆΛ (k) 10, ˆΛ (k) 20 ), obtain (β(k) 1, β(k) 2 ) step 3: Repeat steps 1 and 2 until convergence

26 Models Investigated Semiparametric Frailty Models Bivariate Poisson-Gamma frailty models Bivariate Poisson-lognormal frailty models

27 Semiparametric Frailty Models Bivariate Poisson-Gamma Frailty Models EM algorithm: E-step Posterior expectation of γ i given current estimates ( Λ 10, Λ 20, β 1, β 2, α) given by γ i = ki j=1 (n ij1 + n ij2 ) + α ki j=1 ( Λ 10ij e β 1 Z + Λ 20ij e β 2 Z ) + α Posterior expectation of log γ i, log γ i, is calculated by Monte-Carlo method

28 Semiparametric Frailty Models Bivariate Poisson-Gamma Frailty Models EM algorithm: M-step (1) Profile likelihood type estimators of (Λ 10, Λ 20, β 1, β 2 ) Step 1: Given (β (0) 1, β(0) 2 ), ˆΛ (1) 10 (s (1) l) = ˆΛ 10l = max min r l u l ˆΛ (1) 20 (s (1) l) = ˆΛ 20l = max min r l u l n v1 = 1 ω 1v ω 1v = n i=1 j=1 k i ω 2v = n i=1 j=1 n k i i=1 j=1 k i ω 1v n v1 r v u ω 1v r v u ω 2v n v2 r v u ω 2v r v u n ij1 I{t ij = s v } n v2 = 1 ω 2v γ i e β(0) 1 Z I{t ij = s v } γ i e β(0) 2 Z I{t ij = s v } n k i n ij2 I{t ij = s v } i=1 j=1

29 Semiparametric Frailty Models Bivariate Poisson-Gamma Frailty Models EM algorithm: M-step (2) step 2: Given ˆΛ (k) 10l, ˆΛ (k) 20l, obtain MLE of (β(k) maximizing k n i [n ij1 log Λ (1) 10ij + n ij1 β 1Z γ i Λ (1) i=1 j=1 k n i [n ij2 log Λ (1) 20ij + n ij2 β 2Z γ i Λ (1) i=1 j=1 step 3: Solve (ˆΛ (k) (k) 10l, ˆΛ MLE of α: 20l, β (k) β (k) 1, β(k) 10ij eβ 1 Z ] 20ij eβ 2 Z ] 2 ) by ˆ 1, ˆ 2 ) iteratively until convergence n ˆα = arg max [α ( log γ i γ i ) + nα log α n log Γ(α)] α>0 i=1

30 Semiparametric Frailty Models Bivariate Poisson-Lognormal Frailty Models EM algorithm: E-step Posterior expectations of (γ 1i, γ 2i ), denoted by ( γ 1i, γ 2i ), and (log γ 1i, log γ 2i ), denoted by ( log γ 1i, log γ 2i ), are calculated by importance sampling method

31 Semiparametric Frailty Models Bivariate Poisson-Lognormal Frailty Models EM algorithm: M-step Profile likelihood type estimates of (Λ 10, Λ 20, β 1, β 2 ) Similar procedures as in the Poisson-Gamma model MLE of (σ 2 1, σ2 2, ρ) Same estimators as mentioned in the nonparametric Poisson-lognormal frailty model

32 Statistical Inference Semiparametric Frailty Models Bootstrap is used to obtain the estimated variance of the estimators Wald test for the regression parameters based on bootstrap estimated variance

33 Recall: Poisson Analysis Univariate Analyais Nonparametric Methods Semiparametric Methods Mean Occurrence Rate overall non!melanoma skin cancer (p!value=0.062) basal cell carcinoma (p!value = 0.030) squamous cell carcinoma (p!value=0.565) Placebo DFMO

34 Univariate Models Univariate Analyais Nonparametric Methods Semiparametric Methods There was marginal treatment effect of DFMO on prevention the recurrence of NMSC (p-value=0.098 ) >stimated mean function B(CD (ollow!up visit since the randomization date <month= Based on the test for univariate panel count data (Sun and Fang, 2003)

35 Two Separate Univariate Models Univariate Analyais Nonparametric Methods Semiparametric Methods There was significant treatment effect of DFMO on preventing the recurrence of BCC (p-value=0.034 ), but not for SCC (p-value=0.585 ) Basal cell Squamous cell Estimated mean function PlaceBo DFMO Estimated mean function PlaceBo DFMO Follow!up visit since the randomization date (month) Follow!up visit since the randomization date (month) Based on the test for univariate panel count data (Sun and Fang, 2003)

36 Univariate Analyais Nonparametric Methods Semiparametric Methods Mean Functions for BCC and SCC Basal cell carcinoma Squamous cell carcinoma Estimated mean function Shared Gamma frailty Mixed Gamma frailty Lognormal frailty Placebo DFMO Estimated mean function Shared Gamma frailty Mixed Gamma frailty Lognormal frailty Placebo DFMO Follow up visit since the randomization date (year) Follow up visit since the randomization date (year)

37 Nonparametric Tests Univariate Analyais Nonparametric Methods Semiparametric Methods p-values Models NMSC Global BCC SCC Univariate Gamma Lognormal Mixed Gamma p-values are calculated based on the univariate test (Sun and Fang, 2003) p-values are calculated based on the bivariate test

38 Univariate Analyais Nonparametric Methods Semiparametric Methods Bivariate Poisson-Gamma Frailty Models Mean Functions Let N 1 (t) and N 2 (t) represent the number of BCCs and SCCs, respectively, with marginal mean functions E[N 1 (t) Z ] = Λ 10 (t) exp(β 11 Z 1 + β 12 Z 2 ) E[N 2 (t) Z ] = Λ 20 (t) exp(β 21 Z 1 + β 22 Z 2 ) Z 1, the treatment group (placebo/dfmo) Z 2, the logarithm of the previous tumor rate

39 Univariate Analyais Nonparametric Methods Semiparametric Methods Bivaraite Poisson-Gamma Frailty Models Statistical Inference Marginal treatment effect on decreasing recurrence of BCC and SCC after adjusting for the baseline tumor rates Tumor type Covariate Estimate Bootstrap SE p-value BCC Z Z SCC Z >0.5 Z

40 Summary Semiparametric models of the mean functions for a bivariate counting process are proposed Methods of analyzing bivariate panel count data No specific form of mean functions assumed Providing estimation of two mean functions simultaneously Correlation between processes can be derived Event-type specific covariate effects assumed in the semiparametric models

41 Poisson-Gamma vs. Poisson-Lognormal Frailty Gamma frailty One-variable model Easier to implement Shorter computing time Lognormal frailty Two-variable model Frailty variables can be negatively correlated More complicate Longer computing time Easier to genearlized when there are mroe than two event types of interest