Time-Varying Coefficient Model with Time-Varying Coefficient Model with Linear Smoothing Function for Linear Smoothing Function for Longitudinal Data in Clinical Trial Longitudinal Data in Clinical Trial Masanori Ito, Toshihiro Misumi and Hideki Hirooka Biostatistics Group, Data Science Dept., Astellas Pharma Inc.
Introduction Introduction In clinical trials, the treatment period and the number of scheduled visits for efficacy evaluation are predetermined by design. Baseline Visit 1 Visit 2 ・・・ Visit 8 [Ex.] each patient would be treated for eight weeks (baseline and at the end of each week)
Introduction Introduction ☻ Patients are evaluated at a number of time points ☻ “primary end time point” at which efficacy of test drug would be evaluated ☻ The primary end time point is taken as the last time point of the predetermined treatment period ☻ If a patient withdrew from the trial before completion, some observations posterior to the discontinuation would be missed last time point completer withdrawal Baseline Visit 1 Visit 2 ・・・ Visit J
Last Observation Carried Forward: LOCF Last Observation Carried Forward: LOCF Missing data is often stored into carrying the last observation forward (LOCF). primary end time point completer withdrawal Baseline Visit 1 Visit 2 ・・・ Visit J However, the LOCF approach assumes that 1) missing data are MCAR (missing completely at random), 2) subject’s responses are constant from the last observed value to the endpoint of the trial. Both of the assumptions are often unrealistic unrealistic in clinical trials, so these conditions are seldom seen (Verbeke and Molenberghs, 2000).
LOCF ANCOVA (analysis of covariance) Model LOCF ANCOVA (analysis of covariance) Model For subjects and repeated observations per visit = i 1 ,..., I = (end of study visit), LOCF ANCOVA model is j 1 ,..., J i = β + β + β + ε Y Y x iJ 0 1 i 0 2 i ij i Y : change from baseline ( ) of outcome Y ij i 0 measurement at the j th time point for the i th subject β : intercept 0 β : effect of baseline measurement ( ) Y i 0 1 β : effect size at time J 2 x : dummy coded covariate for subject i i (ex. for placebo group and for treatment group) = = x 1 x 0 i i ε : assumed to be independently distributed from a ij univariate normal distribution = * If is missing, then (where j =1,…, J -1) Y Y Y iJ iJ ij
Mixed-effects Model Repeated Measures: MMRM Mixed-effects Model Repeated Measures: MMRM Several authors propose likelihood-based mixed effects models to analyze incomplete data from longitudinal clinical trials. same meaning as Missing At Random (MAR) In general, when dropouts are ignorable, the parameters of dropout and outcome processes are assumed to be distinct, and hence likelihood-based methods can be used on the marginal distribution of the observed data for statistical inferences. Mixed-effects Model Repeated Measures approach
MMRM analysis MMRM analysis For subjects and repeated observations per visit = i 1 ,..., I , MMRM model can be described as = j 1 ,..., J i = + + Y β b ε X Z i i i i i Y : dimensional vector of outcome measurement for the i th subject J i i β : dimensional vector containing the fixed effects (e. g. baseline, p treatment effect and time) : and dimensional design matrices of known J i × J i × X , i Z ( p ) ( q ) i covariates : dimensional vector containing the random effects b b i ～ q ( N ( 0 , D )) i ε : dimensional vector of residual components ∑ ε ～ ( N ( 0 , )) J i i i i q × = D : general covariance matrix with ( i , j ) element ( q ) d d ij ji ∑ : covariance matrix which depends on i only through J × ( J ) i i i its dimension J i
Issues Issues MMRM is quite flexible and powerful parametric model approach for a longitudinal data in clinical trials. As is well known, parametric approaches are often too restrictive and unrealistic for the clinical trials data. While parametric approaches are useful, questions will always arise about the adequacy of the model assumptions and the potential impact of model misspecifications on the analysis Hoover et al. , 1998
Time-Varying Coefficient Model: TVCM Time-Varying Coefficient Model: TVCM The useful model for studying the association between the covariates and response for the longitudinal data in clinical trial is the time-varying coefficient model, = + β ε T Y X ( t ) ( t ) ij ij ij i ij where are smooth function of t and ′ = β β β ( t ) ( ( t ),..., ( t ) ) 0 p is zero mean stochastic process. ε ( t ) i Estimation of β ( t ) - smoothing spline method (Hastie and Tibshirani, 1993) - locally weighted polynomial (Hoover et al ., 1998) - investigated the cross validation criteria for selecting smoothing parameters
How to select the regression models and Knots How to select the regression models and Knots Efficacy score β ˆ 1 β ˆ β ˆ 3 2 Baseline Visit 1 Visit 2 Visit 3 We focus on the analysis for the clinical trial data of chronic condition . In general, the subjects visit the hospital according to the scheduled time for a chronic disease study, therefore subjects data are concentrated visit by visit. Linear smoothing spline function with visits as knots is enough to express the longitudinal variation of treatment effects
Linear smoothing spline Linear smoothing spline function function Time-varying coefficient model allows the intercept and slope coefficients to be arbitrary smooth functions of t ij . The penalized linear spline version of this model is K K ∑ ∑ ( ) α β = α + α + − κ + β + β + − κ + ε Y t b ( t ) t b ( t ) x , + + ij 0 1 ij k ij k 0 1 ij k ij k i ij = = k 1 k 1 κ ,..., κ : knots (visits) over the range of the t ij values 1 J : the number of the knots K α 0 , α : parameters of the intercept 1 α = : random effects of the intercept b k ( k 1 ,..., K ) β 0 , β : parameters of the slope coefficients 1 β = : random effects of the slope coefficients b k ( k 1 ,..., K ) : positive part of the function t ij – κ k − κ ( t ) + ij k (It is zero for those values of t ij where t ij – κ k is negative)
Representation of Mixed-Effect Model Representation of Mixed-Effect Model From the equation of MMRM and Linear spline function, the mixed-effects model representation is written as = + + Y β b ε . X Z i i i i It is obtained by setting = X [ 1 t x t x ] , ≤ ≤ ij i ij i 1 j J i = α α β β β T [ ] , 0 1 0 1 = − κ − κ Z [ ( t ) x ( t ) ] , + + ≤ ≤ i ij k i ij k 1 j J i ≤ ≤ ≤ ≤ 1 k K 1 k K = α α β β b [ b ,..., b , b ,..., b ], 1 K 1 K = σ σ b 1 1 2 2 Cov ( ) diag { , }. α × β × K 1 K 1
Example: Sample Study Data Example: Sample Study Data Design Randomized, double blinded parallel dose finding study Dose Placebo, Low dose, Middle dose and High dose Duration 12 weeks Primary Efficacy QOL (Quality Of Life) variable score change from baseline ( negative direction means improvement ) Assumable Chronic disease disease area (ex. CNS=Central Nerve System disease) Sample size 100 patients per dose group
Mean response prediction of MMRM (linear model) 0 投与群 Dose プラセボ 600 mg M 900 mg 1200 mg H P L -2 -4 -6 Efficacy score change from baseline -8 IRLSス コ ア 変 化 量 -10 -12 -14 -16 -18 -20 -22 0 10 20 30 40 50 60 70 80 90 日数 （日） Days 症例数：プラセボ＝116例，600mg＝120例，900mg＝119例，1200mg＝113例
Mean response prediction of time-varying coefficient model (linear smoothing function) 0 投与群 Dose Placebo 600mg M 900mg 1200mg H P L -2 -4 -6 Efficacy score change from baseline -8 IRLSス コ ア 変 化 量 -10 -12 -14 -16 -18 -20 -22 0 7 14 21 28 35 42 49 56 63 70 77 84 91 日数 （日） Days 症例数：プラセボ＝116例，600mg＝120例，900mg＝119例，1200mg＝113例
Results Results Least square means for efficacy score change from baseline difference between placebo and each treatment group (p-values are adjusted by Dunnett test) Low dose Middle dose High dose LOCF ANCOVA -2.17 -1.42 -2.44 P=0.0738 P=0.3411 P=0.0413 MMRM -2.04 -1.93 -2.33 (first order time P=0.0147 P=0.0236 P=0.0049 effect) MMRM -2.31 -1.92 -2.31 (second order time P=0.0161 P=0.0246 P=0.0054 effect) Time-Varying -1.01 -1.94 -2.07 Coefficient P=0.269 P=0.0078 P=0.0044
Plots of the predictions for the time-varying coefficient Plots of the predictions for the time-varying coefficient -2 Placebo -3 BETA -4 Low dose Middle dose -5 High dose -6 0 7 14 21 28 35 42 49 56 63 70 77 84 91 日数 （日） Days XKE☆キーコード Placebo 600mg 900mg 1200mg 症例数：プラセボ＝116例，600mg＝120例，900mg＝119例，1200mg＝113例
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