Covariance modelling for longitudinal randomised controlled trials

Transcription

1 Covariance modelling for longitudinal randomised controlled trials G. MacKenzie 1,2 1 Centre of Biostatistics, University of Limerick, Ireland. 2 CREST, ENSAI, Rennes, France. BSU, Cambridge, UK, March 2014 BSU, Cambridge, March /25

2 Longitudinal Setting: MRC s ARMD Study SFRADS 2.0 Distance Visual Acuity (Y1) 1.5 DVA (Visit - Baseline) Group -1.5 Treated Control Time in Years BSU, Cambridge, March /25

3 First choice model Consider a longitudinal Gaussian response, Y, with µ(t) = x(t) β Σ(t) = Σ(t, θ) where, x(t) is a polynomial in t & possibly baseline covariates θ measures the dependence of Σ(t) on t BSU, Cambridge, March /25

4 Basic Thesis From a statistical standpoint inference for the two parameters µ and Σ should, in principle, be symmetric as the two parameters are only Cox-Reid orthogonal. However, often µ is thought to be more important than Σ - Σ being treated a nuisance parameter!! This latter idea is now rather passe and with it much current statistical practice, and in RCTS. Rather, the goal should be to find the joint (µ, Σ) model which is optimal in the joint mean-covariance model space, {M x C} BSU, Cambridge, March /25

5 A Key Result In general the mles of the regression coefficients, take the form When is ˆβ, Σ invariant? ˆβ Σ = (X Σ 1 X) 1 X Σ 1 Y (1) An obvious case arises when Σ I, i.e., for i.i.d. errors. More importantly, Rao (1965) showed that ˆβ is Σ invariant when Σ = XΓX + QΘQ (2) where Γ of order (p p) and Θ of order ((p m) (p m)) are p.d. and Q is a (p (p m)) matrix orthogonal to X, i.e., Q X = 0. We have assumed exactly m repeated measures over time. BSU, Cambridge, March /25

6 Covariance Model The technique is based on a modified Cholesky decomposition of the marginal covariance Σ(t, θ), viz: T ΣT = D (3) leading to Σ(t, ς, φ), in which the new parameters are the natural logarithms of the innovation variances, ς, and generalized autoregressive coefficients, φ We model these as different polynomial functions of lag and time φ(t) = z(t) γ ς(t) = h(t) λ (4) Here, γ and λ are regression parameters of primary scientific interest while z(t) and h(t) are typically particular polynomials in lag (l) and time(t), respectively (MacKenzie & Pan, 2004). BSU, Cambridge, March /25

7 Structure of the T & D matrices T D = diag ( 2 1,, 2 j,, 2 m) BSU, Cambridge, March /25

8 Regressograms BSU, Cambridge, March /25

9 Joint Mean-Covariance Model We can now put everything together into one general joint mean-covariance model which has the following structure µ(t) = x(t) β φ(t) = z(t) γ ς(t) = h(t) λ which we can view as a basic representation for any set of longitudinal data. All of the scientific and inferential interest is focussed on the regression parameters: β, γ and λ BSU, Cambridge, March /25

10 Major univariate extensions Baseline covariates in the cov. mod. ie, the indicator in a RCT (Pan & MacKenzie, 2006) Linear Mixed Models - modified Cholesky transform to within subject cov. matrix (Pan & MacKenzie, 2007) Change to GEE (Ye and Pan, 2006) Constrained mean cov. mod. (Xu & MacKenzie, 2013) BSU, Cambridge, March /25

11 Bivariate Model - 2 dependent responses Consider a longitudinal bivariate Gaussian response (y 1, y 2 ). Such as DVA and NVA in one eye or DVA in each eye. Consider m repeated measures over time. Covariance structure: within subject over time in y 1 and separately in y 2, cross correlation as in y 1 (t) on say y 2 (t 1) and y 1 (t) on y 2 (t) at time t. Leads to modelling innovation matrices - key point. BSU, Cambridge, March /25

12 Bivariate Response Covariance Modelling Let and T = D = diag(d 1,..., D m ) I Φ 21 I Φ m1 Φ m(m 1) I, where: D l is a 2 2 innovation covariance matrix with innovation variances on the diagonal, I is a 2 2 identity sub-matrix and 0 is a 2 2 zero sub-matrix and Φ jk (k < j, j = 2,..., m) are 2 2 matrices. BSU, Cambridge, March /25

13 Bivariate Covariance Models Now we model the innovation matrices D j via matrix logarithm A j matrices. Let ( ) aj1 a A j = log D j = j2 = C j log(g j )C j. a j2 a j3 Thus A j = log D j is a symmetric matrix. By the definition of the matrix exponential it follows that D j = exp(a j ) and the positive definiteness of the D j and hence Σ is guaranteed. The 2 2 Φ jk (k < j, j = 2,..., m) matrices are asymmetric and can be modelled directly - no time to show. BSU, Cambridge, March /25

14 Bivariate Response, Polynomial Covariance Models As a first step, we illustrate the use of polynomial modelling of the innovation matrices. Let ( aj1 a A j = log D j = j2 Recall that j = 1,..., m whence we have m elements over time, then the polynomial models can be written for each s as a j2 a j3 ). a js = λ (s) 0 + λ (s) 1 j + + λ(s) d s j ds (s = 1, 2, 3) for each element in A j and d = d 1 + d 2 + d is the number of parameters fitted (d=degree). BSU, Cambridge, March /25

15 Bivariate Response - ARMD RCT Data RCT of teletherapy in ARMD 3 UK hospital centres Radiotherapy v Observation Examined at Baseline and 3, 6, 12 & 24 months Two outcomes (y 1, y 2 ) DVA and NVA 100 patients per group Only Treatment group analysed - just to illustrate ideas. BSU, Cambridge, March /25

16 ARMD RCT Data - directed simulation Parameter Truth Ave. Est. St. Dev Root. MSE Coverage freq. β % β % β % β % γ % γ % γ % γ % γ % γ % γ % γ % λ % λ % λ % λ % λ % λ % Simple model: linear 2 x 2 = 4 parameters; linear for covariance model; autoregressive, 4 x 2=8 parameters; log-innovation variances, 3 x 2 parameters. BSU, Cambridge, March /25

17 Modelling Innovation Matrices - Results from ARMD Study (i) Upper curve = element (1,2), (ii) Solid circles = element (1,1), (iii) Empty circles = element(2,2) BSU, Cambridge, March /25

18 Conclusions This paper opens the door on MV covariance modelling in RCTs. It solves the key problem of maintaining a PD covariance matrix. The Gaussian case is merely the exemplar case. But can now also apply all of the Univariate methods - Heterogeneity, GEE, Mixed models, Bayesian methods easily to MV response RCTSs. Bodes well for the impact of covariance modelling in longitudinal RCTs with multiple responses. Forthcoming paper on the ARMD RCT with trivariate response (Contrast Sensitivity). BSU, Cambridge, March /25

19 Key References -1 Daniels M. J. and Pourahmadi, M. (2002). Bayesian analysis of covariance matrices and dynamic models for longitudinal data. Biometrika 89, Jones, R. H. (1993). Longitudinal Data with Serial Correlation: A State-space approach, London: Chapman and Hall. Kenward, M. G. (1987). A method for comparing profiles of repeated measurements. Appl. Statist. 36, Laird, N. M.& Ware, J.J. (1982). Random-effects models for longitudinal data. Biometrics, 38, MacKenzie, G and Pan, J. (2001). Modelling marginal covariance structures in linear mixed models. In: Proceedings of the 16th International Workshop on Statistical Modelling. Editors. B. Klein and L. Korsholm , Odense, Denmark. MacKenzie, G. & Pan J. (2007) Optimal joint-mean covariance modelling. In: Proceedings of the 2nd International Workshop on Correlated Data Modelling. Editors: MacKenzie G. and Gregario D.. BSU, Cambridge, March /25

20 Key References -2 MacKenzie G & Reeves JAF. (2002). Modelling bivariate longitudinal data with serial correlation. In: Correlated Data Modelling. Editors: Dario Gregori, et al. Franco Angeli, 19-26, Milano, Italy. Pan JX & MacKenzie G. (2001). Modelling conditional covariance structure in linear mixed models. In: Proceedings of the 16th International Workshop on Statistical Modelling. Editors. B. Klein and L. Korsholm , Odense, Denmark. Pan JX & MacKenzie G. (2003). On modelling mean-covariance structures in longitudinal studies Biometrika 90,1, Pan JX & MacKenzie G. (2006). Regression models for covariance structures in longitudinal studies Statistical Modelling 6, (1): Pan JX & MacKenzie G. (2007). On modelling conditional mean-covariance structures in longitudinal studies. Statistical Modelling 7, (1): Pourahmadi, M. (1999). Joint mean-covariance models with applications to longitudinal data: Unconstrained parameterisation. Biometrika 86, Pourahmadi, M. (2000). Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix. Biometrika 87, Rao, C. R. (1965). The theory of least squares when the parameters are stochastic and its application to the analysis of growth curves. Biometrika 52,1, BSU, Cambridge, March /25

21 Key References -3 Reeves, JAF. & MacKenzie, G. (1998). A bivariate regression with serial correlation. The Statistician, Vol 47, No. 4, pp Xu, J. & MacKenzie, G. (2012). Modelling covariance structure in bivariate marginal models for longitudinal data. Biometrika, 99, 3, BSU, Cambridge, March /25

22 Joint Model Space BSU, Cambridge, March /25

23 ARMD Study - Cross Correlations Estimated correlation and cross-correlation matrix between Distance visual acuity and Near visual acuity. Estimated correlations are along the main diagonal, estimated cross-correlations at lag k are above the main diagonal, estimated correlations at lag k are below the main diagonal. NVA t DVA BSU, Cambridge, March /25

24 Current SFI Supported Work Science Foundation Ireland s BIO-SI research project Creating R software for our standard methods Developing other classes of covariance models (smoother(s)) Extending to GLMMs Developing multivariate versions (Xu & MacK, 2012) Developing design (MacKenzie & Xu) Developing missingness (Xu & MacK, 2013, forthcoming) BSU, Cambridge, March /25

25 Some History 1999 The Trieste Meeting, WCDM01, November, The Keele Meeting, June 2000 Pourahmadi s 1999 & 2000 Biometrika papers Seminal years: Keele Bayesian ideas - Daniels & Pourahmadi, Biometrika 2002 Pan & MacKenzie, Biometrika 2003 Pan & MacKenzie, 2006, 2007 and Pan The modern era BSU, Cambridge, March /25