The impact of covariance misspecification in multivariate Gaussian mixtures on estimation and inference

Transcription

1 The impact of covariance misspecification in multivariate Gaussian mixtures on estimation and inference An application to longitudinal modeling Brianna Heggeseth with Nicholas Jewell Department of Statistics University of California Berkeley August 1, 2012

2 General outline 1 Motivation 2 Finite mixture models 3 Covariance misspecification in mixture models 4 Longitudinal data analysis 5 Concluding remarks 2 / 17

3 Motivating data example How does body mass index (BMI) change with age? BMI (kg/m 2 ) Age (years) National Longitudinal Survey of Youth (NLSY) from / 17

4 Motivating data example How does body mass index (BMI) change with age? BMI (kg/m 2 ) Age (years) National Longitudinal Survey of Youth (NLSY) from / 17

5 Motivating data example How does body mass index (BMI) change with age? BMI (kg/m 2 ) Age (years) National Longitudinal Survey of Youth (NLSY) from / 17

6 Typical longitudinal analysis Use generalized estimation equations (GEE) (Liang & Zeger, 1986) to estimate the mean parameters. Parameter estimators are consistent despite potential covariance misspecification Gain efficiency by using a working correlation matrix Robust sandwich standard error estimators available But, we have a heterogeneous population. BMI of some people does not change much as they age. BMI of other people drastically increases as they age. We don t want to average out all of the interesting relationships. 6 / 17

7 Finite mixture models There are K latent relationships in the population that occur with different frequencies: π 1,..., π K. The mixture density is f (y x, θ) = π 1 f 1 (y x, β 1, Σ 1 ) + + π K f K (y x, β K, Σ K ) where K k=1 π k = 1, θ = (π 1,..., π K 1, β 1, Σ 1,..., β K, Σ K ), and f k (y x, β k, Σ k ) is a multivariate Gaussian density with mean xβ k and covariance Σ k. What working correlation structure should we assume for Σ k? Conditional independence is commonly used (Σ k = σ 2 k I ). 7 / 17

8 Estimation for mixture models Use maximum likelihood estimation for θ. Expectation-Maximization algorithm (Dempster et. al. 1977) Every subject has a estimated posterior probability of belonging to each component Parameter estimators are not consistent under misspecification Robust sandwich standard error estimators available But how bad could the bias be? We didn t have to worry about getting the covariance structure correct with GEE. Let s just assume conditional independence. 8 / 17

9 y y Misspecified covariance structure - separation Well-separated components lead to little bias even when you wrongly assume independence t t ŜE H (µ red ) = 0.02, ŜE R(µ red ) = 0.06 ŜE H (µ red ) = 0.01, ŜE R(µ red ) = 0.01 Black dashed: True means, Solid color: Estimated means 9 / 17

10 y y Misspecified covariance structure - level of dependence Components with little dependence lead to little bias even when you wrongly assume independence t t ŜE H (µ red ) = 0.02, ŜE R(µ red ) = 0.06 ŜE H (µ red ) = 0.03, ŜE R(µ red ) = 0.04 Black dashed: True means, Solid color: Estimated means 10 / 17

11 Bias in parameters and standard error estimates When covariance is misspecified, there is little bias only for well-separated components or when the covariance structure is close to the truth. Estimation differs from GEE because: Each subject has a nonzero probability of being in each component Misspecifying the covariance messes up the probabilities A subject contributes to the estimation of the wrong components and causes bias 11 / 17

12 NLSY data analysis Assume a quadratic mean and four clusters. Conditional Independence Exchangeable Correlation BMI (kg/m 2 ) BMI (kg/m 2 ) Age (years) Age (years) End up with a different story about the relationships! 12 / 17

13 NLSY data analysis Neither covariance assumption is correct. Component 1 Component 2 Autocorrelation Autocorrelation Time lag Time lag Component 3 Component 4 Autocorrelation Autocorrelation Time lag Time lag Autocorrelation estimate using residuals from conditional independence. 13 / 17

14 Concluding remarks If you have a heterogeneous population, consider using a mixture model. Estimates from mixture models are sensitive to covariance assumptions! If the clusters are well-separated, you are probably ok. Try a few covariance assumptions and compare. Always use robust standard errors. 14 / 17

15 Future work Determining a way to quantify the bias. Investigating other covariance structures that are closer to the true dependence but are simple enough to estimate within each group. Estimating the relationship between baseline covariates and group membership. Focusing on the pattern over time. 15 / 17

16 References Heggeseth, BC and Jewell, NP. The impact of covariance misspecification in multivariate Gaussian mixtures on estimation and inference an application to longitudinal modeling. Statistics in Medicine (in review). Liang K, Zeger S. Longitudinal data analysis using generalized linear models. Biometrika 1986; 73(1): McLachlan GJ, Peel D. Finite Mixture Models. Wiley Series in Probability and Statistics, Wiley: New York, Pickles A, Croudace T. Latent mixture models for multivariate and longitudinal outcomes. Statistical Methods in Medical Research 2010; 19(3): Available software for repeated measures: flemix in R (regression mean, conditional independence) mclust in R (mean vector, eigenvalue decomposition) Mplus (regression mean, mixed effects, robust SE) Proc Traj in SAS (regression mean, conditional independence) 16 / 17

17 Acknowledgements Thanks to: Nicholas Jewell Statistics in Epidemiology section for the Young Investigator Award ASA Bay Area Chapter for the travel award 17 / 17