## 1 Introduction

Quantifying population-level differences in the brain that are attributable to neurological or psychiatric disorders is a major focus of neuroimaging research. Structural magnetic resonance imaging (MRI) is widely used to investigate changes in brain structure that may aid the diagnosis and monitoring of disease. A structural MRI of the brain consists of many voxels, where a voxel is the three dimensional analogue of a pixel. Each voxel has a corresponding intensity, and jointly the voxels encode information about the size and structure of the brain. Functional MRI (fMRI) also plays an important role in the understanding of disease mechanisms by revealing relationships between disease and brain function. In this work we focus on structural MRI data, but many of the concepts apply to fMRI studies.

One way to assess group-level differences in the brain is to take a “mass-univariate” approach, where statistical tests are applied separately at each voxel. This is the basic idea behind statistical parametric mapping (SPM) [1–3] and voxel-based morphometry (VBM) [4, 5]. Voxel-based methods are limited in the sense that they do not make use of information contained jointly among multiple voxels. Figure 1 illustrates this concept using toy data with two variables,

The goal of MVPA is often two-fold: (i) to understand underlying patterns in the brain that characterize a disease, and (ii) to develop sensitive and specific image-based biomarkers for disease diagnosis, the prediction of disease progression, or prediction of treatment response. Although the MVPA literature often uses terminology that suggests a causal interpretation of disease patterns in the brain, little has been done to formalize a causal framework for neuroimaging, with the notable exception of recent work by Weichwald et al. [62]. In this paper, we elucidate subtle differences between the two goals of MVPA and provide guidance for future implementation of MVPA in neuroimaging studies. We focus attention on the consequences of confounding on goal (i) and give a few remarks regarding goal (ii).

Confounding of the disease-image relationship by non-imaging variables such as age and gender can have undesirable effects on the output of MVPA. In particular, confounding may lead to identification of false disease patterns, undermining the usefulness and reproducibility of MVPA results. We discuss the implications of “regressing out” confounding effects using voxel-wise parametric models, a widely used approach for addressing confounding, and propose an alternative based on inverse probability weighting.

The structure of this paper is the following. Section 2 provides a brief overview of the use of MVPA in neuroimaging with focus on the use of the support vector machine (SVM) as a tool for MVPA. In Section 3, we address the issue of confounding by reviewing current practice in neuroimaging and proposing an alternative approach. In Section 4, we illustrate our method using simulated data, and Section 5 presents an application to data from an Alzheimer’s disease neuroimaging study. We conclude with a discussion in Section 6.

## 2 Multivariate pattern analysis in neuroimaging

Let

A popular MVPA tool used by the neuroimaging community is the support vector machine (SVM) [29, 30]. This choice is partly motivated by the fact that SVMs are known to work well for high dimension, low sample size data [31]. Often, the number of voxels in a single MRI can exceed one million depending on the resolution of the scanner and the protocol used to obtain the image. The SVM is trained to predict the group label from the vectorized set of voxels that comprise an image. Alternatives include penalized logistic regression [32] as well as functional principal components and functional partial least squares [33, 34].Henceforth, we focus on MVPA using the SVM.

The hard-margin linear SVM solves the contrained optimization problem

Where

where

In high-dimensional problems where the number of features is greater than the number of observations, the data are almost always separable by a linear hyperplane [36]. Thus, MVPA is often applied using the hard-margin linear SVM in (1). For example, this is the approach implemented by: Bendfeldt et al. [37] to classify subgroups of multiple sclerosis patients; Cuingnet et al. [7] and Davatzikos et al. [8] in Alzheimer’s disease applications; and Liu et al. [38], Gong et al. [39], and Costafreda et al. [40] for various classification tasks involving patients with depression. This is only a small subset of the relavant literature, which illustrates the widespread popularity of the approach.

## 3 Multivariate pattern analysis and confounding

### 3.1 Causal framework for descriptive aims

When the goal of MVPA is to understand patterns of change in the brain that are attributable to a disease, the ideal dataset would contain two images for each subject: one where the subject has the disease and another at the same point in time where the subject is healthy. Of course, this is the fundamental problem of causal inference, as it is impossible to observe both of these potential outcomes [41, 42]. In addition, confounding of the disease--image relationship presents challenges. Figure 2 depicts confounding of the

Let

The target parameter

We do not directly observe samples from

for all

Note that the expectation is over the marginal distribution of

where

To illustrate the effects of confounding on MVPA, consider a toy example with a single confounder

Note that model (3) has the property that

There is some variation in the definition of confounding in the imaging literature, making it unclear in some instances if, when, and why an adjustment is made. For example, some researchers recommend correcting images for age effects even after age-matching patients and contols [44]. In an age-matched study, age is not a confounder, and adjusting for its relationship with

*adjusted MVPA*.

A similar procedure is to fit model (4) using the control group only [44]. We refer to this approach as the *control-adjusted MVPA*. In applications where there is not a clear control group, i.e., comparing two disease subclasses, a single reference group is chosen. Let

A comparison of the adjusted and control-adjusted MVPA features is displayed in Figure 4. The first two plots of Figure 4 show the original feature

### 3.2 Inverse probability weighted classifiers

Having formally defined the problem of confounding in MVPA, we now propose a general solution based on inverse probability weighting (IPW) [45–48]. We have already shown that weighting observations by the inverse probability of

The inverse probability weights are often unknown and must be estimated from the data. One way to estimate the weights is by positing a model and obtaining fitted values for the probability that

IPW can be naturally incorporated into some classification models such as logistic regression. Subject-level weighting can be accomplished in the soft-margin linear SVM framework defined in expression (2) by weighting the slack variables. Suppose the true weights

However, in the approximately balanced pseudo-population, some of the

in (5) are equivalent to

In fact, assuming all observations in the original

The previous argument suggests one could use the true weights

The IPW-SVM algorithm only works when the data are not linearly separable. Otherwise, there are no slack variables in the optimization problem to weight. To provide intuition, suppose we are trying to separate two points in two-dimensional space. The optimization problem is then the hard-margin linear SVM formulation:

Adding copies of the data only adds redundant constraints that do not affect the optimization. This is a major issue in neuroimaging because the data often have more features than observations and are thus almost always linearly separable. When

## 4 Simulation study

In this section we evaluate the finite sample performance of the IPW-SVM relative to the regression methods discussed in Section 3.1. We simulate training data from the following generative model with

where

For each of

We compare the performance of the IPW-SVM (IPW) to an unadjusted SVM (Unadjusted), a SVM after “regressing out”

## 5 Application

The Alzheimer’s Disease Neuroimaging Initiative (ADNI) (http://www.adni.loni.usc.edu) is a $60 million study funded by public and private resources including the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, the Food and Drug Administration, private pharmaceutical companies, and non-profit organizations. The goals of the ADNI are to better understand progression of mild cognitive impairment (MCI) and early Alzheimer’s disease (AD) as well as to determine effective biomarkers for disease diagnosis, monitoring, and treatment development. MCI is characterized by cognitive decline that does not generally interfere with normal daily function and is distinct from Alzheimer’s disease [57]. However, individuals with MCI are considered to be at risk for progression to Alzheimer’s disease. Thus, studying the development of MCI and factors associated with progression to Alzheimer’s disease is of critical scientific importance. In this analysis, we study the effects of confounding on the identification of multivariate patterns of atrophy in the brain that are associated with MCI.

We apply the IPW-SVM to structural MRIs from the ADNI database. Before performing group-level analyses, each subject’s MRI is passed through a series of preprocessing steps that facilitate between-subject comparability. We implemented a multi-atlas segmentation pipeline [58] to estimate the volumes of

Although the ADNI study was approximately matched on age and gender, a logistic regression of disease group on age in our sample returns an estimated odds ratio of 1.06 with 95% confidence interval

In general, all four methods perform similarly and return patterns that closely resemble the pattern learned from the matched data. Table 1 gives the

Method | Distance |

IPW-SVM | 0.52 |

Unadjusted SVM | 0.76 |

Control-Adjusted SVM | 0.58 |

Adjusted SVM | 0.56 |

It should be noted that although there is a significant disease-age relationship in the observed data, it is unlikely representative of the true disease-age relationship in the population because the MCI cases are over-sampled. Thus, MVPA classifiers trained to study disease patterns in the brain may demonstrate suboptimal performance when classifying new subjects in the population. Dataset shift methods, or models that integrate imaging biomarkers with knowledge of the true disease-age relationship in the target population, may be applied to improve any MVPA imaging biomarkers derived from the ADNI data.

## 6 Discussion

We have proposed a framework for addressing confounding in MVPA that weights individual subjects by the conditional probability of observed class given confounders, i.e., inverse probability weighting (IPW). When the goal of MVPA is to estimate complex disease patterns in the brain, using IPW to address confounding is more principled that the current practice of “regressing out” confounder effects separately at each voxel without regard to the correlation structure of the data. When machine learning predictive models such as the SVM are used to perform MVPA, the IPW approach can recover underlying patterns in the brain associated with disease in the presence of measured confounding.

We believe there are several advantages to addressing confounding in MVPA using IPW. First, as demonstrated by simulation results, IPW better estimates the target parameter of interest, which is the disease pattern that would be present under no confounding. In cases where a matched study is too expensive or otherwise infeasible, IPW methods will enable researchers to perform MVPA and obtain correct, reproducible results. Finally, IPW is simple and intuitive, and the general idea is well-established in the causal inference and statistics communities. Thus, future research aiming to perform inference on the estimated disease patterns can rely on existing theory. We are currently working on extending existing inference methods for MVPA [14, 59] to account for confounding.

Further exploring the effects of confounding on high-dimensional classification models is imperative for neuroimaging research and may greatly impact current practice in the field. An interesting avenue for future research would be to develop dimension reduction techniques that could be applied before or concurrently with MVPA that account for possible confounding in the data. Developing sensitivity analysis methods for assessing the role of confounding in MVPA also merits attention in future work.

Although we have focused on the use of SVMs for binary classification problems, the idea of subject-level weighting to address confounding applies more generally to machine learning techniques for a variety of classification problems. In practice, incorporating subject-level weights into black box machine learning methods may not always be straightforward, and implementation of IPW might require specific tailoring to each problem. For example, generalizied versions of the propensity score exist for exposures with more than two groups and continuous exposures [60, 61]. Intuitively, it seems that applying generalized propensity score methods to multiclass classification problems or support vector regression for a continuous exposure is a natural extension of the methods proposed in this work. We believe these extensions are non-trivial and warrant focused attention in future research.

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