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Flexible Discriminant Analysis Using Multivariate Mixed Models D. Hughes Flexible Discriminant Analysis Using Motivation MGLMM Multivariate Mixed Models Discriminant Analysis ISDR Example Conclusions David Hughes 2015 Flexible

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Flexible Discriminant Analysis Using Multivariate Mixed Models

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Motivation MGLMM Discriminant Analysis ISDR Example Conclusions

Flexible Discriminant Analysis Using Multivariate Mixed Models

David Hughes 2015

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Flexible Discriminant Analysis Using Multivariate Mixed Models

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Motivation MGLMM Discriminant Analysis ISDR Example Conclusions

Outline

  • 1. Motivation
  • 2. Multivariate Generalized Linear Mixed Models (MGLMM)
  • 3. Longitudinal Discriminant Analysis
  • 4. ISDR Example
  • 5. Conclusions
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Flexible Discriminant Analysis Using Multivariate Mixed Models

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Motivation MGLMM Discriminant Analysis ISDR Example Conclusions

Motivation

◮ Complex data.

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Flexible Discriminant Analysis Using Multivariate Mixed Models

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Motivation MGLMM Discriminant Analysis ISDR Example Conclusions

Motivation

◮ Complex data.

◮ Longitudinal

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Flexible Discriminant Analysis Using Multivariate Mixed Models

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Motivation MGLMM Discriminant Analysis ISDR Example Conclusions

Motivation

◮ Complex data.

◮ Longitudinal ◮ Multivariate

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Flexible Discriminant Analysis Using Multivariate Mixed Models

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Motivation MGLMM Discriminant Analysis ISDR Example Conclusions

Motivation

◮ Complex data.

◮ Longitudinal ◮ Multivariate ◮ Different types of data

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Motivation

◮ Complex data.

◮ Longitudinal ◮ Multivariate ◮ Different types of data ◮ Complicated correlation structure

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Flexible Discriminant Analysis Using Multivariate Mixed Models

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Motivation MGLMM Discriminant Analysis ISDR Example Conclusions

Motivation

◮ Complex data.

◮ Longitudinal ◮ Multivariate ◮ Different types of data ◮ Complicated correlation structure

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Flexible Discriminant Analysis Using Multivariate Mixed Models

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Motivation MGLMM Discriminant Analysis ISDR Example Conclusions

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 Models 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.

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A more flexible approach

◮ Typical assumption about the random effects distribution can

be relaxed by using a mixture of normal distributions (Kom´ arek et al. (2010)).

◮ This methodology only considers three continuous markers.

◮ Cluster Analysis with continuous, binary and count variables

with mixture distributions for the random effects is possible (Kom´ arek and Kom´ arekov´ a (2013))

◮ In Cluster Analysis the groups are unknown whereas in our case

groups are known beforehand.

◮ Software is available in the mixAK package in R created by

Arnoˇ st Kom´ arek.

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Progress Map

Dataset for Analysis Fitting of the multivariate mixed-effects model (MGLMM) Discriminant model built using parameters of MGLMM Allocate new patients to diagnostic groups

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Definitions

◮ Yi,r,j is the j‘th observation of the r‘th marker for patient i and

is measured at time ti,r,j.

◮ We consider r = 1, . . . , R markers on i = 1, . . . , N patients. ◮ Yi,r is a vector containing all observations of marker r for

patient i.

◮ Yi 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.

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Multivariate Generalized Linear Mixed Models

◮ To allow for different types of marker we model each marker

using a generalised linear mixed model h−1

r

[E(Yi,r|αr, bi,r)] = Xi,rαr + Zi,rbi,r (1)

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Multivariate Generalized Linear Mixed Models

◮ To allow for different types of marker we model each marker

using a generalised linear mixed model h−1

r

[E(Yi,r|αr, bi,r)] = Xi,rαr + Zi,rbi,r (1)

◮ hr is a link function used depending on the type of longitudinal

marker.

◮ αr is a vector of fixed parameters for marker r. ◮ bi,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.

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Joint Distribution of the random effects

◮ The dependence between markers is captured by the joint

distribution of the random effects bi = (bi,1, . . . , bi,R), i = 1, . . . , N.

◮ The most common assumption is that the random effects

follow a Normal distribution. bi ∼ N(µ, D) (2)

◮ This assumption can be difficult to verify and additional

flexibility can be achieved by allowing a mixture of Normal distributions. bi ∼

K

  • k=1

wkN(µk, Dk) (3)

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Parameter Estimation

◮ We need to estimate the following parameters.

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Parameter Estimation

◮ We need to estimate the following parameters.

◮ Fixed effects α = (α1, . . . , αR) ◮ Possible dispersion parameters φ = (φ1, . . . , φR) ◮ Mixture weights w = (w1, . . . , wK) ◮ Mean vector of random effects µ = (µ1, . . . , µK) ◮ Covariance matrix of random effects (vec(D1), . . . , vec(DK))

◮ In all, we need to estimate,

θ = (α, φ, w, µ, vec(D1), . . . , vec(DK)) (4)

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MCMC estimates

◮ 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.

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Progress Map

Dataset for Analysis Fitting of the multivariate mixed-effects model (MGLMM) Discriminant model built using parameters of MGLMM Allocate new patients to diagnostic groups

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Longitudinal Discriminant Analysis

◮ Fit MGLMM to data in each diagnostic group g, g = 1, . . . , G

to obtain MCMC parameter estimates, ˆ θg.

◮ Use the fitted GLMM model to derive the discriminant rule

that assigns the patients into two (or more) diagnostic groups.

◮ Let ˆ

Pg,new be the probability that a new observation Yi, is from group g.

◮ The prior probability of being in group g is denoted πg. ◮ Using Bayes rule it can be seen that

ˆ Pg,new = πgˆ fg,new G−1

h=0 πhˆ

fh,new (5)

◮ Assign new patients to disease group if ˆ

Pdisease,new is greater than a specified value. If not assign to the group for which ˆ Pg,new is largest.

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Specifying the predictive density fg,new

◮ Marginal Prediction

f marg

g,new = p(ynew|θg)

(6)

◮ Conditional Prediction

f cond

g,new = p(ynew|bnew = ˜

bg,new, θg) (7)

◮ Random Effects Prediction

f rand

g,new = p(˜

bg,new|θg) (8)

◮ These values are calculated using numerical integration

methods such as Gauss Quadrature since they involve complex integrals that cannot be solved analytically.

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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

  • bservation period.

Figure: Left: Image of diabetic eye without retinopathy. Right: Image of diabetic eye with late stage diabetic retinopathy (Kindly provided by Dr. Yalin Zheng).

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Example: ISDR data

◮ We consider two groups, 600 patients with a positive screening

event (indicating STDR) and 12068 patients without.

◮ 80% of the patients in each group to train MGLMMs (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.)

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Example: ISDR data

We fit the following models: E[log(HbA1c)] = α1Sex + α2Age + bi,0 + bi,1time (9) E[log(Cholesterol)] = α3Sex + α4Age + α5time + bi,2 (10) logE[Visit] = α6Sex + α7Age + α8time + bi,3 (11) logitE[Grading] = α9Sex + α10Age + bi,4 + bi,5time (12)

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Example: ISDR data

Posterior Mean Standard Error Posterior Median 95% Credible Interval No STDR Group α1

  • 2.74e-03

1.48e-06

  • 2.74e-03

(-3.03e-03,-2.46e-03) α2

  • 5.13e-03

3.73e-05

  • 5.14e-03

(-1.25e-02,2.07e-03) ) α3

  • 3.27e-03

1.84e-06

  • 3.27e-03

(-3.65e-03,-2.93e-03) ) α4

  • 8.76e-02

4.35e-05

  • 8.77e-02

(-9.61e-02,-7.9e-02) α5

  • 2.01e-05

2.24e-08

  • 2e-05

(-2.45e-05,-1.57e-05) α6 3.66e-03 3.97e-06 3.67e-03 (2.86e-03,4.44e-03) α7

  • 1.66e-02

1.01e-04

  • 1.65e-02

(-3.64e-02,3.03e-03) α8 2.99e-04 9.87e-08 2.99e-04 (2.8e-04,3.18e-04) α9 9.24e-03 2.6e-05 9.2e-03 (4.2e-03,1.45e-02) α10 1.2e-01 6.4e-04 1.2e-01 (-5.64e-03,2.45e-01) STDR Group α1

  • 6.76e-03

8.62e-06

  • 6.77e-03

(-8.4e-03,-5.05e-03) α2

  • 3.34e-02

2.45e-04

  • 3.33e-02

(-8.03e-02,1.49e-02) α3

  • 3.1e-03

7.47e-06

  • 3.12e-03

(-4.54e-03,-1.59e-03) α4

  • 8.25e-02

2.3e-04

  • 8.23e-02

(-1.27e-01,-3.79e-02) α5

  • 2.35e-05

1.71e-07

  • 2.33e-05

(-5.7e-05,1.05e-05) α6 9.07e-03 2.21e-05 9.1e-03 (4.77e-03,1.33e-02) α7

  • 2.65e-02

6.13e-04

  • 2.54e-02

(-1.46e-01,9.26e-02) α8 4.78e-04 5.97e-07 4.78e-04 (3.61e-04,5.95e-04) α9

  • 4.72e-03

1.09e-04

  • 4.51e-03

(-2.63e-02,1.58e-02) α10

  • 1.29e-01

3.43e-03

  • 1.24e-01

(-8.16e-01,5.33e-01)

Table: Posterior summary statistics for the fixed effects α in our MGLMM.

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Example: ISDR data

Posterior Mean Standard Error Posterior Median 95% Credible Interval No STDR Group E[b0] 4.15 1.02e-04 4.15 (4.13,4.17) E[b1] 6.07e-06 3.37e-08 6.11e-06 (-5.18e-07,1.26e-05) E[b2] 1.68 1.24e-04 1.68 (1.65,1.7) E[b3] 5.1e-01 2.77e-04 5.1e-01 (4.55e-01,5.65e-01) E[b4]

  • 2.96

1.97e-03

  • 2.96

(-3.35,-2.56) E[b5]

  • 3.35e-04

6.69e-07

  • 3.36e-04

(-4.65e-04,-2.03e-04) SD[b0] 2.71e-01 4.39e-05 2.71e-01 (2.63e-01,2.8e-01) SD[b1] 2.2e-04 5.85e-08 2.2e-04 (2.09e-04,2.32e-04) SD[b2] 1.83e-01 1.76e-05 1.83e-01 (1.8e-01,1.87e-01) SD[b3] 2.27e-01 7.16e-05 2.27e-01 (2.13e-01,2.41e-01) SD[b4] 2.54 1.27e-03 2.54 (2.3e,2.79) SD[b5] 9.46e-04 1.3e-06 9.5e-04 (6.91e-04,1.2e-03) STDR Group E[b0] 4.62 5.79e-04 4.62 (4.51,4.73) E[b1] 3.61e-05 1.88e-07 3.61e-05 (-8.23e-07,7.26e-05) E[b2] 1.65 4.96e-04 1.65 (1.55,1.74) E[b3] 3.05e-01 1.45e-03 3.01e-01 (2.17e-02,5.96e-01) E[b4] 3.81 7.94e-03 3.76 (2.38,5.44) E[b5] 8.66e-04 2.09e-06 8.88e-04 (2.82e-04,1.3e-03) SD[b0] 3.05e-01 2.11e-04 3.04e-01 (2.64e-01,3.47e-01) SD[b1] 1.75e-04 2.55e-07 1.75e-04 (1.24e-04,2.24e-04) SD[b2] 2.06e-01 1.05e-04 2.05e-01 (1.85e-01,2.26e-01) SD[b3] 3.15e-01 3.69e-04 3.16e-01 (2.4e-01,3.86e-01) SD[b4] 2.75 3.63e-03 2.71 (2.16,3.59) SD[b5] 8.94e-04 1.59e-06 8.59e-04 (6.43e-04,1.3e-03)

Table: Posterior summary statistics for the means and standard deviations of the random effects bi in our MGLMM.

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Example: ISDR data

−2000 −1500 −1000 −500 3.5 4.0 4.5 5.0 Time (days) Log(HbA1c)

No STDR Group

−2000 −1500 −1000 −500 3.5 4.0 4.5 5.0 Time (days)

STDR Group

−2000 −1500 −1000 −500 0.5 1.0 1.5 2.0 2.5 Time (days) Log(Cholesterol) −2000 −1500 −1000 −500 0.5 1.0 1.5 2.0 2.5 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.

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Example: ISDR data

−2000 −1500 −1000 −500 5 10 15 Time (days) Number of GP Visits

No STDR Group

−2000 −1500 −1000 −500 5 10 15 Time (days)

STDR Group

−2000 −1500 −1000 −500 0.0 0.2 0.4 0.6 0.8 1.0 Time (days) Grading −2000 −1500 −1000 −500 0.0 0.2 0.4 0.6 0.8 1.0 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.

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Example: ISDR data

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

ROC Plot for Methods of Group Prediction

1−Specificity Sensitivity Marginal Conditional Random Effects LDA QDA

Figure: ROC curve to compare the predictive abilities of the three longitudinal methods of group membership prediction and the simple LDA and QDA techniques.

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Example: ISDR data

Marginal Conditional Random effects LDA QDA Cutoff 0.06 0.05 0.02 0.01 0.01 Sensitivity 0.79 0.78 0.50 0.55 0.52 Specificity 0.81 0.74 0.65 0.52 0.49 PCC 0.81 0.75 0.64 0.52 0.50 AUC 0.87 0.78 0.57 0.52 0.51 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 Discriminant Analysis. QDA = Quadratic Discriminant Analysis.

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Conclusions

◮ There is a definite advantage to using longitudinal information

in comparison to simply applying LDA (or QDA) to the last

  • bservations for each patient.
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Conclusions

◮ There is a definite advantage to using longitudinal information

in comparison to simply applying LDA (or QDA) to the last

  • bservations for each patient.

◮ The marginal prediction method gives the best classification for

the ISDR data (on all measures).

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Motivation MGLMM Discriminant Analysis ISDR Example Conclusions

Conclusions

◮ There is a definite advantage to using longitudinal information

in comparison to simply applying LDA (or QDA) to the last

  • bservations 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.

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Further work

◮ Can we make more use of the credible intervals that are readily

available from the MCMC procedure?

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Further work

◮ 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?

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Further work

◮ 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?

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Acknowledgements

◮ Joint work with Arnoˇ

st Kom´ arek (Charles University in Prague), Gabriela Czanner, Christopher P. Cheyne, Simon Harding and Marta Garc´ ıa-Fi˜ nana.

◮ We are grateful for the support of the ISDR team. ◮ We acknowledge support from the Medical Research Council

(Research project MR/L010909/1).

◮ Garc´

ıa-Fi˜ nana M, Czanner G, Cox T, Bonnett L, Harding S, Marson T. Discriminant Function Analysis for Longitudinal Data: Applications in Medical Research (2014–2017) funded by MRC MRP (£ 334,170)

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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):419–431

◮ Kom´

arek, A., Hansen, B.E., Kuiper, E.M.M., van Buuren, H.R., and Lesaffre, E. (2010) Discriminant analysis using a multivariate linear mixed model with a normal mixture in the random effects distribution. Statistics in medicine, 29(30):3267–3283.

◮ Kom´

arek A. and Kom´ arekov´ a, L. (2013) Clustering for multivariate continuous and discrete longitudinal data. The Annals of Applied Statistics, 7(1):177–200.

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References

◮ Lix, L.M., and Sajobi, T.T. (2010)

Discriminant analysis for repeated measures data: a review. Frontiers in psychology, 1, Article 146

◮ Marshall, G., De la Cruz-Mes´

ıa, R., Quintana, F.A., and Baron, A.E. (2009) Discriminant Analysis for Longitudinal Data with Multiple Continuous Responses and Possibly Missing Data. Biometrics 65:69–80.

◮ 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):1151–1175.

◮ Tomasko, L., Helms, R.W. and Snapinn, S.M. (1999)

A discriminant analysis extension to mixed models. Statistics in medicine, 18(10):1249–1260.

◮ 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):246–254.