A Model for Interpretable High Dimensional Interactions Bhatnagar SR1,2, Yang Y3, Blanchette M4, Greenwood CMT1,2 1Department of Epidemiology, Biostatistics and Occupational Health, McGill University 2Lady Davis Institute, Jewish General Hospital, Montreal, QC 3Department of Mathematics and Statistics, McGill University, QC 4School of Computer Science, McGill University Summary • Environmental exposures may induce subtle system-wide changes in high- dimensional genomic data such as gene expression or epigenetic measures – Can such situations be exploited to improve prediction models? • Large system-wide changes are observed in many environments and hence this assumption can possibly be exploited to aid analysis of high dimensional data • We develop and implement a multivariate penalization procedure for predict- ing a continuous or binary disease outcome while detecting interactions between high dimensional data (p >> n) and an environmental factor. R software: http://sahirbhatnagar.com/eclust/ – Dimension reduction is achieved through leveraging the environmental-class- conditional correlations – Also, we develop and implement a strong heredity framework within the penal- ized model Motivation Figure 1: Microarray study of COPD. Top: Heatmap of Pearson correlations. Bottom: Heatmap of gene expression data (2,900 genes) rows are genes and columns are subjects. There are 7 subjects in each group, matched on COPD case status, gender and age. Methods Figure 3: Method overview. First step involves measuring gene similarity in both exposure groups. We then cluster these and create a cluster representation. The last step involves entering these terms in a penalization model that follows the strong heredity principle [1] Figure 4: Clustering based on Fisher’s Z transformation of exposure dependent correla- tions. Let ρijk be the correlation between genes i and j in class k. Transform the corre- lations into z values: zijk = 0.5log|(1 + ρijk)/(1 −ρijk)|. The Z-test statistic is given by |zij0 −zij1|/ p 1/(n0 −3) + 1/(n1 −3) ∼N(0, 1) • Model: g(µ) = β0 + β1X1 + · · · + βpXp + βEE | {z } main eﬀects + α1E(X1E) + · · · + αpE(XpE) | {z } interactions • Strong Hierarchy Principle [1]: ˆαjE ̸= 0 ⇒ ˆβj ̸= 0 and ˆβE ̸= 0 • Reparametrization [2]: αjE = γjEβjβE. • Variable Selection: arg minβ0,β,γ 1 2 ∥Y −g(µ)∥2 + λβ w1β1 + · · · + wqβq + wEβE  + λγ w1Eγ1E + · · · + wqEγqE  • Adaptive weights: wj = 1 ˆβj , wjE = ˆβj ˆβE ˆαjE • Why strong heredity? – Statistical Power: large main eﬀects are more likely to lead to detectable interactions than small ones – Interpretability: Assuming a model with interaction only is generally not biologically plausible – Practical Sparsity: X1, E, X1 · E (2 variables to measure) vs. X1, E, X2 · E (3 variables to measure). Simulation Study Results Figure 5: Stability of results: Average Jaccard distance from 10-fold cross validation. A Jaccard distance of 1 indicates perfect agreement between two sets while no agreement will result in a distance of 0. Figure 6: Prediction accuracy: Test set mean squared error This work was supported by the Ludmer Centre for Neuroinformatics and Mental Health. Software available at http://sahirbhatnagar.com/eclust/. References [1] Hugh Chipman. Bayesian variable selection with related predictors. Canadian Journal of Statis- tics, 24(1):17–36, 1996. [2] Nam Hee Choi, William Li, and Ji Zhu. Variable selection with the strong heredity constraint and its oracle property. Journal of the American Statistical Association, 105(489):354–364, 2010.