Robust Functional Mixed Models for Spatially Correlated Functional Regression -- with Application to Event-Related Potentials for Nicotine-Addicted Individuals Hongxiao Zhu Virginia Tech June 15, 2015 ICSA/Graybill 2015 Collaborated with Francesco Versace, Paul Cinciripini, Jeffrey S. Morris 1

2 ERP data in a smoking cessation study. Review existing methods. Proposed approach. ERP analysis results. Outline

The Event-related potentials (ERPs) data ERPs : electrophysiological signals recorded from human’s scalp surface. Obtained by averaging over electroencephalogram (EEG) measurements. Visual stimuli is often presented during experiment. 3 EEG trajectories ERP signal EEG experiments Visual Stimulus

The ERP data for Nicotine-Addicted Individuals 180 smokers 4 stimuli: Cigarettes, Neutral, Pleasant, Unpleasant. 129 electrodes (channels). Total number of ERP signals: 180X4X129 = — crossed design. Each ERP curve: 225 time points range from -100 to 800 ms. Goal: detect systematic differences among different visual stimuli. 4 Geodesic Sensor Net

Characteristics of the ERP data set ERP curves are functional data. Effects of stimuli vary across cortical regions. Spatial correlation between electrodes. 5

Commonly used methods on ERP data analysis Typically focus on features: amplitude of the peak, mean voltages in a time window, are often extracted. 6 ERP components: waveforms with + or - deflections.

Main idea: the functional mixed model (FMM) framework 7 Fixed effect Random effect To be estimated

Gaussian functional mixed models (Gfmm) Guo (2002), Morris & Carroll(2006). 8 Priors: Model Fitting Discrete Wavelet Transform: Assume Gaussian Process: separable structure

Robust functional mixed models (Rfmm) Zhu, Brown and Morris (2011, JASA) 9 Column (j,k) Key idea: scale mixture of normal Priors: Equivalent Bayesian Lasso Allow functions with outliers and outlying regions. Fast scalable computation using Gibbs. i.i.d.

Generalization 1 – correlated channel-specific fixed effects. 10 stimulus 1 stimulus 2 stimulus 3 stimulus 4 Assume correlated shrinkage prior for (Griffin & Brown, 2012)

Generalization 2 – Model the correlation in E(t) in Gfmm 11

Generalization 3 – Model the correlation in E(t) in Rfmm 12

Posterior Inference – 2D smoothing across scalp 13

Posterior Inference 14 Meyer et al. (2015).

Automated workflow 15

ERP data analysis -- regions flagged using BFDR (delta = 0.7) 16

ERP data analysis – summary plots using BFDR (delta=0.7) 17

ERP data analysis – conclusion 18 Our analysis reveals significant neurological effects induced by different types of visual stimuli. There are extensive differences between neutral images and emotion-evoking images of cigarette, pleasant, and unpleasant types during the time period ms in various cortical regions. Within this period, the cigarette profile was more similar to the pleasant stimulus than unpleasant stimulus during the ms time period, and was similar to both pleasant and unpleasant stimuli from ms. Compared with the initial results of Versace et al. (2011), our FMMc models provide more refined information about the timing and locations in which the stimuli effects differ, and are able to produce these results in a semi-automated fashion without having to preselect temporal or cortical regions to consider.

Reference Gri ffi n, J. E., and Brown, P. J. (2012), “Structuring shrinkage: some correlated priors for regression,” Biometrika, 99(2), 481–487. Meyer, M. J., Coull, B. A., Versace, F., Cinciripini, P., and Morris, J. S. (2015), “Bayesian function-on-function regression for multilevel functional data,” Biometrics. Morris, J. S. and Carroll R. J. (2006). Wavelet-based functional mixed models. J R Stat Soc, Ser B. 68,179–199. Ruppert, D., Wand, M. P., and Carroll, R. J. (2003), Semiparametric Regression, Cambridge Series in Statistical and Probabilistic Mathematics, UK: Cambridge University Press. Versace, F., Minnix, J. A., Robinson, J. D., Lam, C. Y., Brown, V. L., and Cinciripini, P. M. (2011), Brain reactivity to emotional, neutral and cigarette-related stimuli in smokers, Addiction Biology, 16, 296–307. Zhu, H., Brown, P. J. and Morris, J. S. (2011) Robust, adaptive functional regression in functional mixed model framework, J Am Stat Assoc 106 (495) Contact: Hongxiao Zhu Department of Statistics, Virginia Tech 406-A Hutcheson Hall Blacksburg, VA United States

Appendi x

Use SimBaS to flag regions on which |C(s,t)|>0 21

Use BFDR to flag regions on which |C(s,t)|>delta 22

Model Selection using Posterior Predictive Likelihood 23

ERP data analysis – model selection using posterior predictive likelihood subjects in the training set. Applied 6 models to each regions. Calculate the LPPL for each model on each region. Select the model with the highest LPPL.

ERP data analysis -- summary of time intervals with pronounced flagging patterns 25 Various degree of similarities Between Cigarette and Emotional Stimuli. Cigarette differ with emotional Stimuli. No evident similarities are found.

ERP data analysis – computational time 26