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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

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

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. ERP signal EEG trajectories EEG experiments Visual Stimulus

Geodesic Sensor Net The ERP data for Nicotine-Addicted Individuals • 180 smokers • 4 stimuli: Cigarettes, Neutral, Pleasant, Unpleasant. • 129 electrodes (channels). • Total number of ERP signals: 180X4X129 = 92880 — crossed design. • Each ERP curve: 225 time points range from -100 to 800 ms. • Goal: detect systematic differences among different visual stimuli.

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

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

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

Gaussian functional mixed models (Gfmm) • Guo (2002), Morris & Carroll(2006). Model Fitting Assume Gaussian Process: Disc screte Wave vele let Tran Transfor orm: Priors: separable structure

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

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

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

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

Posterior Inference – 2D smoothing across scalp

Posterior Inference Meyer et al. (2015).

Automated workflow

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

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

ERP data analysis – conclusion • 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 248-700ms in various cortical regions. • Within this period, the cigarette profile was more similar to the pleasant stimulus than unpleasant stimulus during the 248-512ms time period, and was similar to both pleasant and unpleasant stimuli from 516-700ms. • 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 • Griffin, 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) 1167-1179. Contact: Department of Statistics, Virginia Tech Hongxiao Zhu 406-A Hutcheson Hall hongxiao@vt.edu Blacksburg, VA 24061-0439 United States 1-540-231-0400

Appendix

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

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

Model Selection using Posterior Predictive Likelihood

ERP data analysis – model selection using posterior predictive likelihood • 140 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 Cigarette differ with emotional Stimuli. Various degree of similarities Between Cigarette and Emotional Stimuli. No evident similarities are found.

ERP data analysis – computational time