Inference on the Significance of Modalities in Multimodal Generalized Linear Models

TL;DR

This work addresses the challenge of testing the significance of an individual modality dentro high-dimensional multimodal GLMs by introducing the Expected Relative Entropy (ERE) $H_m = 2 \mathbb{E}_{\boldsymbol{x}}[D_{KL}(p(y|\boldsymbol{x}) \parallel p(y|\boldsymbol{x}_{-m}))]$ as a rigorous information-based metric. A two-step estimator combining Sure Independence Screening (SIS) with a partially penalized maximum likelihood yields $\widehat{H}_m = \dfrac{2}{n}[\log L_n(\widehat{\boldsymbol{\beta}}) - \log L_n(\widehat{\boldsymbol{\beta}}_0)]$, enabling inference on modality significance. Theoretical results show that $\widehat{H}_m$ is consistent with rate $O_P\Big( \max\{ \tfrac{s}{n}\log n, \tfrac{1}{\sqrt{n}} \} \Big)$ and that $n\widehat{H}_m$ converges to a non-central $\chi^2$ distribution with $\widetilde{s}_m$ degrees of freedom and non-centrality $\Gamma_n$, allowing confidence intervals and p-values without requiring variable-selection consistency. Simulations and an ADNI multimodal neuroimaging analysis demonstrate good coverage properties and that FDG-PET can be more informative than Amyloid-PET for certain outcomes, illustrating the practical impact of modality-level inference in high-dimensional settings. Overall, the paper provides a principled, scalable framework for modality significance testing in multimodal regression, with potential extensions to other models such as Cox processes.

Abstract

Despite the popular of multimodal statistical models, there lacks rigorous statistical inference tools for inferring the significance of a single modality within a multimodal model, especially in high-dimensional models. For high-dimensional multimodal generalized linear models, we propose a novel entropy-based metric, called the expected relative entropy, to quantify the information gain of one modality in addition to all other modalities in the model. We propose a deviance-based statistic to estimate the expected relative entropy, prove that it is consistent and its asymptotic distribution can be approximated by a non-central chi-squared distribution. That enables the calculation of confidence intervals and p-values to assess the significance of the expected relative entropy for a given modality. We numerically evaluate the empirical performance of our proposed inference tool by simulations and apply it to a multimodal neuroimaging dataset to demonstrate its good performance on various high-dimensional multimodal generalized linear models.

Figures (4)

• Figure 1: Empirical coverage probabilities and variable selection performance (Sensitivity and Specificity) by the three methods for Model 1. The oracle method (blue), SIS + SCAD penalized method (red), SIS + refit method (gray).
• Figure 2: Empirical coverage probabilities and variable selection performance (Sensitivity and Specificity) by the three methods for Model 2. The oracle method (blue), SIS + SCAD penalized method (red), SIS + refit method (gray).
• Figure 3: Empirical coverage probabilities and variable selection performance (Sensitivity and Specificity) by the three methods for Model 3. The oracle method (blue), SIS + SCAD penalized method (red), SIS + refit method (gray).
• Figure 4: Selected regions of interest for MEM, EF and DX by two modalities. MEM, the memory score; EF, the executive function score; DX, the AD diagnostic label.

Theorems & Definitions (12)

• Proposition 1
• Example 1: Linear regression
• Example 2: Logistic regression
• Example 3: Exponential Regression
• Example 4: Poisson Regression
• Theorem 1
• Theorem 2
• Proposition 2
• Lemma 1
• proof