David Hughes. Flexible Discriminant Analysis Using. Multivariate Mixed Models. D. Hughes. Motivation MGLMM. Discriminant. Analysis.

Transcription

1 Using Using David Hughes 2015

2 Outline Using Multivariate Generalized Linear Mixed () 3. Longitudinal 4. 5.

3 Using Complex data.

4 Using Complex data. Longitudinal

5 Using Complex data. Longitudinal Multivariate

6 Using Complex data. Longitudinal Multivariate Different types of data

7 Using Complex data. Longitudinal Multivariate Different types of data Complicated correlation structure

8 Using Complex data. Longitudinal Multivariate Different types of data Complicated correlation structure

9 Available Methods Univariate models using a classical linear mixed model (e.g Brant et al. (2003), Lix and Sajobi (2010), Tomasko et al. (1999) and Wernecke et al. (2004)). Fails to account properly for the dependence between markers in our case. Multivariate for continuous markers using multivariate mixed models (eg Morrell et al. (2012) using linear mixed models and Marshall et al. (2009) using non-linear mixed models). Not applicable if some of the markers are not continuous. Pairwise models for continuous and binary markers (Fieuws et al. (2008)). This method in principle is suitable for our purposes but in this talk we outline a more flexible approach. Using

10 A more flexible approach Using Typical assumption about the random effects distribution can be relaxed by using a mixture of normal distributions (Komárek et al. (2010)). This methodology only considers three continuous markers. Cluster with continuous, binary and count variables with mixture distributions for the random effects is possible (Komárek and Komáreková (2013)) In Cluster the groups are unknown whereas in our case groups are known beforehand. Software is available in the mixak package in R created by Arnošt Komárek.

11 Progress Map Using Dataset for Fitting of the multivariate mixed-effects model () model built using parameters of Allocate new patients to diagnostic groups

12 Definitions Using Y i,r,j is the j th observation of the r th marker for patient i and is measured at time t i,r,j. We consider r = 1,..., R markers on i = 1,..., N patients. Y i,r is a vector containing all observations of marker r for patient i. Y i is a stacked vector containing all the observations of all markers for patient i. Distribution of each marker may depend on additional covariates such as time, Age, Gender. It is possible for each marker to be measured at different time points and a different number of times.

13 Multivariate Generalized Linear Mixed Using To allow for different types of marker we model each marker using a generalised linear mixed model h 1 r [E(Y i,r α r, b i,r )] = X i,r α r + Z i,r b i,r (1)

14 Multivariate Generalized Linear Mixed Using To allow for different types of marker we model each marker using a generalised linear mixed model h 1 r [E(Y i,r α r, b i,r )] = X i,r α r + Z i,r b i,r (1) h r is a link function used depending on the type of longitudinal marker. α r is a vector of fixed parameters for marker r. b i,r is a vector of random effects for patient i for marker r (i.e subject specific parameters). X and Z are matrices containing covariate information for each patient.

15 Joint Distribution of the random effects Using The dependence between markers is captured by the joint distribution of the random effects b i = (b i,1,..., b i,r ), i = 1,..., N. The most common assumption is that the random effects follow a Normal distribution. b i N(µ,D) (2) This assumption can be difficult to verify and additional flexibility can be achieved by allowing a mixture of Normal distributions. K b i w k N(µ k,d k ) (3) k=1

16 Parameter Estimation Using We need to estimate the following parameters.

17 Parameter Estimation Using We need to estimate the following parameters. Fixed effects α = (α1,..., α R ) Possible dispersion parameters φ = (φ1,..., φ R ) Mixture weights w = (w1,..., w K ) Mean vector of random effects µ = (µ1,..., µ K ) Covariance matrix of random effects (vec(d1),..., vec(d K )) In all, we need to estimate, θ = (α, φ, w, µ, vec(d 1 ),..., vec(d K )) (4)

18 MCMC estimates Using Full maximum likelihood estimates are difficult to obtain due to the complexity of the likelihood. We instead use a Bayesian approach based on MCMC. We utilise weakly informative priors and a block Gibbs sampler. A benefit of this method, not explored in this talk is that credible intervals for the group membership probabilities are readily available. These could be incorporated into a classification procedure in some cases.

19 Progress Map Using Dataset for Fitting of the multivariate mixed-effects model () model built using parameters of Allocate new patients to diagnostic groups

20 Longitudinal Fit to data in each diagnostic group g, g = 1,..., G to obtain MCMC parameter estimates, ˆθ g. Use the fitted GLMM model to derive the discriminant rule that assigns the patients into two (or more) diagnostic groups. Let ˆP g,new be the probability that a new observation Y i, is from group g. The prior probability of being in group g is denoted π g. Using Bayes rule it can be seen that Using ˆP g,new = π gˆfg,new G 1 h=0 π hˆf h,new (5) Assign new patients to disease group if ˆP disease,new is greater than a specified value. If not assign to the group for which ˆP g,new is largest.

21 Specifying the predictive density f g,new Marginal Prediction Conditional Prediction f marg g,new = p(y new θ g ) (6) Using Random Effects Prediction f cond g,new = p(y new b new = b g,new, θ g ) (7) f rand g,new = p( b g,new θ g ) (8) These values are calculated using numerical integration methods such as Gauss Quadrature since they involve complex integrals that cannot be solved analytically.

22 Diabetic Retinopathy example Our motivation comes from the ISDR cohort study. We consider 12,628 patients with diabetes who were screened between 2009 and 2013 for diabetic retinopathy. Various markers measured over time, HbA1c and Cholesterol (continuous markers), retinopathy grading (treated as binary marker), and number of GP visits (count variable). 600 patients had positive screening event within the observation period. Using Figure: Left: Image of diabetic eye without retinopathy. Right: Image of diabetic eye with late stage diabetic retinopathy (Kindly provided by Dr. Yalin Zheng).

23 Example: ISDR data Using We consider two groups, 600 patients with a positive screening event (indicating STDR) and patients without. 80% of the patients in each group to train s (one for each group). 20% of patients to test the classification accuracy. End goal is to identify patients who will have a positive screening event in one years time (so only consider data gathered up to one year before final visit.)

24 Example: ISDR data Using We fit the following models: E[log(HbA1c)] = α 1 Sex + α 2 Age + b i,0 + b i,1 time (9) E[log(Cholesterol)] = α 3 Sex + α 4 Age + α 5 time + b i,2 (10) loge[visit] = α 6 Sex + α 7 Age + α 8 time + b i,3 (11) logite[grading] = α 9 Sex + α 10 Age + b i,4 + b i,5 time (12)

25 Example: ISDR data Posterior Mean Standard Error Posterior Median 95% Credible Interval No STDR Group α e e e-03 (-3.03e-03,-2.46e-03) α e e e-03 (-1.25e-02,2.07e-03) ) α e e e-03 (-3.65e-03,-2.93e-03) ) α e e e-02 (-9.61e-02,-7.9e-02) α e e-08-2e-05 (-2.45e-05,-1.57e-05) α e e e-03 (2.86e-03,4.44e-03) α e e e-02 (-3.64e-02,3.03e-03) α e e e-04 (2.8e-04,3.18e-04) α e e e-03 (4.2e-03,1.45e-02) α e e e-01 (-5.64e-03,2.45e-01) STDR Group α e e e-03 (-8.4e-03,-5.05e-03) α e e e-02 (-8.03e-02,1.49e-02) α 3-3.1e e e-03 (-4.54e-03,-1.59e-03) α e e e-02 (-1.27e-01,-3.79e-02) α e e e-05 (-5.7e-05,1.05e-05) α e e e-03 (4.77e-03,1.33e-02) α e e e-02 (-1.46e-01,9.26e-02) α e e e-04 (3.61e-04,5.95e-04) α e e e-03 (-2.63e-02,1.58e-02) α e e e-01 (-8.16e-01,5.33e-01) Table: Posterior summary statistics for the fixed effects α in our. Using

26 Example: ISDR data Posterior Mean Standard Error Posterior Median 95% Credible Interval No STDR Group E[b 0] e (4.13,4.17) E[b 1] 6.07e e e-06 (-5.18e-07,1.26e-05) E[b 2] e (1.65,1.7) E[b 3] 5.1e e e-01 (4.55e-01,5.65e-01) E[b 4] e (-3.35,-2.56) E[b 5] -3.35e e e-04 (-4.65e-04,-2.03e-04) SD[b 0] 2.71e e e-01 (2.63e-01,2.8e-01) SD[b 1] 2.2e e e-04 (2.09e-04,2.32e-04) SD[b 2] 1.83e e e-01 (1.8e-01,1.87e-01) SD[b 3] 2.27e e e-01 (2.13e-01,2.41e-01) SD[b 4] e (2.3e,2.79) SD[b 5] 9.46e e e-04 (6.91e-04,1.2e-03) STDR Group E[b 0] e (4.51,4.73) E[b 1] 3.61e e e-05 (-8.23e-07,7.26e-05) E[b 2] e (1.55,1.74) E[b 3] 3.05e e e-01 (2.17e-02,5.96e-01) E[b 4] e (2.38,5.44) E[b 5] 8.66e e e-04 (2.82e-04,1.3e-03) SD[b 0] 3.05e e e-01 (2.64e-01,3.47e-01) SD[b 1] 1.75e e e-04 (1.24e-04,2.24e-04) SD[b 2] 2.06e e e-01 (1.85e-01,2.26e-01) SD[b 3] 3.15e e e-01 (2.4e-01,3.86e-01) SD[b 4] e (2.16,3.59) SD[b 5] 8.94e e e-04 (6.43e-04,1.3e-03) Table: Posterior summary statistics for the means and standard deviations of the random effects b i in our. Using

27 Example: ISDR data No STDR Group STDR Group Using Log(HbA1c) Time (days) Time (days) Log(Cholesterol) Time (days) Time (days) Figure: Observed longitudinal profiles (in light blue) of log(hba1c) and log(cholesterol) for patients without positive screening events (left column) and patients with positive screening events (right column). The average profile over time of a male with median age is shown in each group by the red and green lines respectively.

28 Using Example: ISDR data STDR Group Number of GP Visits No STDR Group Time (days) Grading 1500 Time (days) 1.0 Time (days) 500 Time (days) Figure: Observed longitudinal profiles (in light blue) of number of GP visits and retinopathy grading for patients without positive screening events (left column) and patients with positive screening events (right column). The average profile over time of a male with median age is shown in each group by the red and green lines respectively.

29 Example: ISDR data Sensitivity ROC Plot for Methods of Group Prediction Marginal Conditional Random Effects LDA QDA Using Specificity Figure: ROC curve to compare the predictive abilities of the three longitudinal methods of group membership prediction and the simple LDA and QDA techniques.

30 Example: ISDR data Using Marginal Conditional Random effects LDA QDA Cutoff Sensitivity Specificity PCC AUC Table: The precision of the prediction of diagnostic groups for three longitudinal methods and the classical LDA and QDA methods. PCC = Probability of Correct classification. AUC = Area Under Curve. LDA = Linear. QDA = Quadratic.

31 Using There is a definite advantage to using longitudinal information in comparison to simply applying LDA (or QDA) to the last observations for each patient.

32 Using There is a definite advantage to using longitudinal information in comparison to simply applying LDA (or QDA) to the last observations for each patient. The marginal prediction method gives the best classification for the ISDR data (on all measures).

33 Using There is a definite advantage to using longitudinal information in comparison to simply applying LDA (or QDA) to the last observations for each patient. The marginal prediction method gives the best classification for the ISDR data (on all measures). Our methodology is able to obtain promising classification results by incorporating markers of different types.

34 Further work Using Can we make more use of the credible intervals that are readily available from the MCMC procedure?

35 Further work Using Can we make more use of the credible intervals that are readily available from the MCMC procedure? Can we identify the ideal timing of the next screening interval?

36 Further work Using Can we make more use of the credible intervals that are readily available from the MCMC procedure? Can we identify the ideal timing of the next screening interval? Can we include categorical longitudinal outcomes within this framework?

37 Acknowledgements Using Joint work with Arnošt Komárek (Charles University in Prague), Gabriela Czanner, Christopher P. Cheyne, Simon Harding and Marta García-Fiñana. We are grateful for the support of the ISDR team. We acknowledge support from the Medical Research Council (Research project MR/L010909/1). García-Fiñana M, Czanner G, Cox T, Bonnett L, Harding S, Marson T. Function for Longitudinal Data: Applications in Medical Research ( ) funded by MRC MRP ( 334,170)

38 References Brant, L.J., Sheng S.L., Morrell, C.H., Verbeke, G. N., Lesaffre, E. and Carter, H. B. (2003) Screening for prostate cancer by using random-effects models. Journal of the Royal Statistical Society: Series A, 166(1):51 62 Fieuws, S., Verbeke, G., Maes, B., and Vanrenterghem, Y. (2008) Predicting renal graft failure using multivariate longitudinal profiles. Biostatistics, 9(3): Komárek, A., Hansen, B.E., Kuiper, E.M.M., van Buuren, H.R., and Lesaffre, E. (2010) analysis using a multivariate linear mixed model with a normal mixture in the random effects distribution. Statistics in medicine, 29(30): Komárek A. and Komáreková, L. (2013) Clustering for multivariate continuous and discrete longitudinal data. The Annals of Applied Statistics, 7(1): Using

39 References Lix, L.M., and Sajobi, T.T. (2010) analysis for repeated measures data: a review. Frontiers in psychology, 1, Article 146 Marshall, G., De la Cruz-Mesía, R., Quintana, F.A., and Baron, A.E. (2009) for Longitudinal Data with Multiple Continuous Responses and Possibly Missing Data. Biometrics 65: Morrell, C.H., Brant, L.J., Sheng, S.L., and Metter, E. J. (2012) Screening for prostate cancer using multivariate mixed-effects models. Journal of applied statistics, 39(6): Tomasko, L., Helms, R.W. and Snapinn, S.M. (1999) A discriminant analysis extension to mixed models. Statistics in medicine, 18(10): Wernecke, K-D., Kalb, G., Schink T., and Wegner, B. (2004) A mixed model approach to discriminant analysis with longitudinal data. Biometrical journal, 46(2): Using