Reconstruction of individual patient data for meta analysis via Bayesian approach

Transcription

1 Reconstruction of individual patient data for meta analysis via Bayesian approach Yusuke Yamaguchi, Wataru Sakamoto and Shingo Shirahata Graduate School of Engineering Science, Osaka University Masashi Goto Biostatistical Research Association, NPO 1

2 Outline Introduction Individual patient data and aggregate data 2 2 contingency tables in which margins only are observed Proposed method A method based on simulated individual patient data Simulation study Concluding remarks and future problems 2

3 Introduction Meta analysis based on aggregate data (AD) Trial Trial Trial Summary statistics (Effect size) Summary statistics (Effect size) Summary statistics (Effect size) A combine d result Meta regression (MR) : model for searching patient characteristic factors 3

4 Introduction Individual patient data (IPD) The scheme of sampling IPD is ignored. Original IPD Original IPD Original IPD Trial Trial Trial AD AD AD A combined result Patient-specific covariates are expediently replaced by trial-specific covariates. 4

5 Introduction Criticisms of MR models (Riley & Steyerberg, 2010; Berlin et al., 2002; Thompson & Higgins, 2002) MR assumes that the across trial relationship between summary estimates and mean covariate values reflects the within trial relationship between individual response and covariate values; however, this may not be true in practice. The relationship described by MR is an observational association, so this suffers from the bias by confounding. The power of MR to detect patient level covariates that truly modify response is generally low. IPD should be used to explore the patient characteristic factors. But, collecting IPD may often be impossible. 5

6 Proposed method: a method based on simulated IPD (SIPD method) Pseudo IPD from each trial are generated, and then their simulated IPD are combined. A combined result Pseudo IPD Pseudo IPD Pseudo IPD Statistical Simulation Trial Trial Trial AD AD AD 6

7 Situation: a binary outcome and a binary covariate (controlled trial with two groups) IPD (individual binary variables for each patient) 1 y ij : outcome, x ij : group indicator, z ij : covariate ( zi = ni ) j = z 1 ij i: the number of trials (i = 1,,I ) j: the number of patients ( j = 1,, n i ) IPD model: Fixed effect logistic regression model (Riley et al., 2008) y ij Bernoulli( q ) log( q 1 q ) = α + β x + β z + γ x z + γ x ( z z ) ij ij i 1 ij 2 ij B ij i W ij ij i AD (number of patients partially classified by each variable) Fixed effect MR model ij 2 log OR i = a+ bzi + εi, εi N(0, σi ) m ( n + n m ) between trial effect ( ) i 1i 10i 00i 0i 2 OR i =, σ i = Var log OR m0i( n11 i + n01 i m1 i) n i within trial effect Trial i X = 0 X = 1 Z = 0 n 00i n 01i Z = 1 n 10i n 11i Y = 1 m 0i m 1i 7

8 Notation and problem For group k in i th trial Y = 0 Y = 1 Z = 0 m 0ki n 0ki Z = 1 m 1ki n 1ki Not available Observed n ki m ki m ki n ki Y NA-IPD = {(m 0ki, m 1ki, n 0ki, n 1ki ): i = 1,,I, k = 0,1} Y AD = h(y NA-IPD ) = {(m ki, n 0ki, n 1ki ): i=1,,i, k = 0,1} Synthesis for 2 2 contingency tables in which margins only observed 8

9 Overview of SIPD method Assumption For all values of Y NA-IPD that is consistent with the observed data Y AD, the conditional distribution of Y AD given Y NA-IPD takes the same values (coarsening at random; Heitjan & Rubin, 1991). Observed data likelihood function: parameter θ py ( θ ) = py ( θ ) dy AD Y = h( Y ) NA-IPD NA-IPD AD NA-IPD Bayesian model specification (King, 1999; Wakefield, 2004) py (, θ Y ) = py ( Y, θ) π( θ Y ) NA-IPD AD NA-IPD AD AD πθ ( Y ) p( Y θπθ ) ( ) AD AD 9

10 Implementing procedure STEP 1. Modeling and simulation STEP 2. Analysis STEP 3. Combining SIPD trials estimat e Observed AD trials SIPD trials estimat e combined result SIPD trials estimat e 10

11 STEP 1. Observed data likelihood function Underlying distribution M p Bin ( p, n ) 0ki 0ki 0ki 0ki M p Bin ( p, n ) 1ki 1ki 1ki 1ki p0ki = Pr( Y = 1 Z = 0, k, i) p1 ki = Pr( Y = 1 Z = 1, k, i) Z Y = 0 Y = 1 0 M 0ki n 0ki 1 M 1ki n 1ki n ki m ki m ki n ki Convolution likelihood (Wakefield, 2004; McCullagh & Nelder, 1989) pm (, n, n p, p ) ki 0ki 1ki 0ki 1ki n n = p0ki (1 p0ki ) p1 ki (1 p1 ki ) m0 ki = l m ki 0ki mki m0ki l = max(0, m n ), u = min( n, m ) u ki 0ki 1ki m0ki n0ki m0ki mki m0ki n1ki mki + m0ki ki ki 1ki ki 0ki ki 11

12 STEP 1. Bayesian model specification Joint posterior distribution of unobserved data and parameters given Y AD(ki) = (m ki,n 0ki,n 1ki ) p( M, p, p Y ) = p( M Y, p, p ) π ( p, p Y ) 0ki 0ki 1ki AD( ki) 0ki AD( ki) 0ki 1ki 0ki 1ki AD( ki) M p, p 0ki 0ki 1ki (1 p ) 0ki 1ki Noncentral-hypergeometric, p p (1 p ) 1ki 0ki Y AD( ki) log p0ki (1 p0ki ) = α + β1k+ ( γb γw) kzi log p (1 p ) = α + β k+ β + ( γ γ ) kz + γ k 1ki 1ki 1 2 B W i W I 1 π ( θ Y ) π( θ) p( Y θ), θ = ( α, β, β, γ, γ ) AD AD( ki) 1 2 B W i= 1 k= 0

13 STEP 1. Draw from posterior distributions Random sampling from the posterior distribution of θ = (α, β 1, β 2, γ B, γ W ) by Markov chain Monte Carlo method (Gelman et al., 1995) Single component Metropolis Hastings algorithm α π( α Y, β, β, γ, γ ) [ t] [ t 1] [ t 1] [ t 1] [ t 1] AD 1 2 B W [] t [] t [ t 1] [ t 1] [ t 1] 1 ( 1 YAD,, 2, B, W ) β π β α β γ γ M γ π( γ Y, α, β, β, γ ) [] t [] t [] t [] t [] t W W AD 1 2 B Get T sets of parameters θ = ( α, β, β, γ, γ ), t = 1,..., T [] t [] t [] t [] t [] t [] t 1 2 B W [] t [] t 0ki 1ki {( p, p ) : i= 1,..., I, k = 0,1}, t = 1,..., T iteration 13

14 STEP 1. Generate SIPD Random sampling from the posterior predictive distribution of M 0ki pm ( Y ) = 1 T 0ki AD( ki) pm ( Y, p, p ) π ( p, p Y ) dp dp T t= 1 0ki AD( ki) 0ki 1ki 0ki 1ki AD( ki) 0ki 1ki pm ( Y, p, p ) [] t [] t 0ki AD( ki) 0ki 1ki Get R sets of SIPD ( m = m m ) 1ki ki 0ki [ r] [ r] 0ki 1ki = = = [ r] [ r] [ r] {( ij, ij, ij ) : = 1,..., ; = 1,..., i}, = 1,..., {( m, m ) : i 1,..., I, k 0,1}, r 1,..., R y x z i I j n r R 14

15 STEP 2. Fit a standard model to SIPD Fixed effect logistic regression model Y [ r] ij Get a point estimate of the parameters of interest and its standard error from r th set of SIPD e.g. Bernoulli( q ) ij log( q 1 q ) = α + β x + β z + γ x z + γ x ( z z ) [ r] [ r] [ r] [ r] [ r] ij ij i 1 ij 2 ij B ij i W ij ij i [ r] [ r] ( ˆ ˆ W W ) γ,se( γ ), r = 1,..., R 15

16 STEP 3. Combine the results from each SIPD Rubin s (1987) combining rule (e.g. inference on γ W ) Point estimate and standard error for γ W ˆ γ W 1 = R R r= 1 ˆ γ [ r] W 1 R R 1+ R [ r] 2 1 [ r] SE( ˆ γ ˆ ˆ ˆ W) = ( γw γw) + Var( γw ) R 1 r= 1 R r= 1 Reference distribution ( ˆ γ γ ) SE( ˆ γ ) W W W ( ) 1 [ ] 2 ˆ R r R ˆ ˆ W r= 1 W W ν = ( R 1) 1 + γ ( 1) ( γ γ ) t ν 1/2 16

17 Simulation study Objective To compare the estimates of treatment covariate interaction effect (b or γ W ), which are obtained by three methods. Methods Fit a fixed logistic regression model to original IPD Fit an MR model to AD summarized from IPD Apply SIPD method to AD summarized from IPD (R = 100 sets of SIPD) 17

18 Designs True IPD model (Lambert et al, 2002) Z = 0 Z = 1 Pr (Y = 1 Z = 0) odds Pr (Y = 1 Z = 1) odds k = ratio ratio - k = Each patient in each trial and group was allocated to Z = 1 by the probabilities 0.3, 0.4, 0.5, 0.6 or 0.7. These probabilities were used with two, four or six trials, respectively. (I, n ki ) {(10,100), (20,100), (30,100)}, i = 1,, I, k = 0,1 Generate 100 sets of original IPD for each scenario 18

19 Result No. of trials No. of patients Using original IPD p<0.05 IPD SIPD MR Mean (SD) of estimates p<0.05 Mean (SD) of estimates p<0.05 Mean (SD) of estimates % 0.56 (0.26) 29% 0.60 (0.33) 9% 0.42 (0.84) % 0.53 (0.21) 52% 0.55 (0.25) 12% 0.43 (0.63) % 0.56 (0.15) 82% 0.58 (0.18) 16% 0.47 (0.47) *SD: Standard Deviation, p<0.05: percentage of significant results γ W Using AD only IPD represents the analysis for the original IPD, which cannot be performed in the case of general meta analysis The SIPD method provided the results on treatment covariate interaction effect closer to those obtained by fitting the IPD model than fitting the MR model γ W b 19

20 Concluding remarks We presented the SIPD method for meta analyzing binary data from multiple trials. The SIPD method provided more appropriate estimates for treatment covariate interaction effect. Moreover, it was shown that the SIPD method preserved remarkably higher power to detect these effects than those obtained by fitting the MR model. 20

21 Future problems It will be necessary to verify the benefit of the SIPD method thought various simulation studies and practical case studies. We are particularly interested in whether this improves publication bias which is one of the most important research topics in metaanalysis. Although we considered only Bayesian approach for the model specification in SIPD method, we would be able to take a frequentist approach for this. 21

22 References Berlin J.A., Santanna J., Scmid C.H., Szczech L.A. & Feldman H.I. (2002). Individual patient versus group level data meta regression for the investigation of treatment effect modifiers: Ecological bias rears its ugly head. Statistics in Medicine, 21, Gelman A., Carlin J.B., Stern H.S. & Rubin D.B. (1995). Bayesian Data Analysis. Chapman and Hall. Heitjan D.F. & Rubin D.B. (1991). Ignorability and coarse data. The Annals of Statistics, 19, King G., Rosen O. & Tanner M.A. (1999). Binomial beta hierarchical models for ecological inference. Sociological Methods and Research, 28, Lambert P.C., Sutton A.J., Abrams K.R. & Jones D.R. (2002). A comparison of summary patient level covariates in meta regression with individual patient data meta analysis. Journal of Clinical Epidemiology, 55, McCullagh H. & Nelder J.A. (1989). Generalized Linear Models, 2nd edition, London: Chapman and Hall. Riley R.D., Lambert P.C., Staessen J.A., Wang J., Gueyffier F., Thijs L. & Boutitie F. (2008). Meta analysis of continuous outcomes combining individual patient data and aggregate data. Statistics in Medicine, 27,

23 References Riley R.D. & Steyerberg E.W. (2010). Meta analysis of a binary outcome using individual participant data and aggregate data. Research Synthesis Methods, 1, Rubin D.B. (1987). Multiple Imputation for Nonresponse in Surveys. New York: John Wiley & Sons. Thompson S.G. & Higgins J.P.T. (2002). How should meta regression analyses be undertaken and interpreted? Statistics in Medicine, 21, Tsiatis A.A. (2006). Semiparametric Theory and Missing Data. Springer. Wakefield J. (2004). Ecological inference for 2*2 tables. Journal of the Royal Statistical Society: Series A, 167, Wang N. & Robins J.M. (1998). Large sample theory for parametric multiple imputation procedure. Biometrika, 85,

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