WAVELET-DOMAIN REGRESSION AND PREDICTIVE INFERENCE IN PSYCHIATRIC NEUROIMAGING

Abstract

An increasingly important goal of psychiatry is the use of brain imaging data to develop predictive models. Here we present two contributions to statistical methodology for this purpose. First, we propose and compare a set of wavelet-domain procedures for fitting generalized linear models with scalar responses and image predictors: sparse variants of principal component regression and of partial least squares, and the elastic net. Second, we consider assessing the contribution of image predictors over and above available scalar predictors, in particular via permutation tests and an extension of the idea of confounding to the case of functional or image predictors. Using the proposed methods, we assess whether maps of a spontaneous brain activity measure, derived from functional magnetic resonance imaging, can meaningfully predict presence or absence of attention deficit/hyperactivity disorder (ADHD). Our results shed light on the role of confounding in the surprising outcome of the recent ADHD-200 Global Competition, which challenged researchers to develop algorithms for automated image-based diagnosis of the disorder.

1. Introduction

A major goal of current psychiatric neuroimaging research is to predict clinical outcomes on the basis of quantitative image data. Many studies have focused on “predicting” current disease states from brain images (e.g., Craddock et al., 2009; Sun et al., 2009). While seemingly less difficult than accurate prediction of future outcomes, the goal of clinically useful imaging-based diagnosis has proved highly challenging (Kapur, Phillips and Insel, 2012; Honorio et al., 2012).

This paper addresses two important limitations of standard methods for using brain images to predict psychiatric outcomes:

• (i)Ordinarily, the voxels (volume units) of the brain are treated as interchangeable predictors or “features.” Accuracy might be improved by properly exploiting the spatial arrangement of the brain.
• (ii)In some cases brain images may prove successful for diagnostic classification, but only because the images are related to one or more scalar covariates that drive the association. This is a nonstandard form of confounding, and there seems to be no existing methodology for detecting it. In other words, little is known about how to assess whether image data o ers “added value” for prediction, beyond what is available from non-image data—which will typically be much simpler to acquire.

To address limitation (i), we approach the general problem as one of regressing scalar responses on image predictors, which are viewed, as in Reiss (2006) and Reiss and Ogden (2010), as a challenging special case of functional predictors (Ramsay and Silverman, 2005). The responses y₁, . . . , y_n are assumed to be generated independently by the model

$$y_i\sim EF({\mu }_i,\varphi ),$$ (1)

$$g\left({\mu }_i\right)={\mathit{t}}_i^T\delta +{\int }_{\mathcal{S}}{x}_i\left(\mathit{s}\right)\beta \left(\mathit{s}\right)d\mathit{s}.$$ (2)

Here EF (μ_i, ϕ) denotes an exponential family distribution with mean μ_i and scale parameter ϕ, along with a link function g; t_i is an m-dimensional vector of (scalar) covariates, of which the first is the constant 1; $x_i:\mathcal{S}\to \mathbb{R}$ is a functional predictor with domain $\mathcal{S}\subset {\mathbb{R}}^2$ or $\subset {\mathbb{R}}^3$; and the corresponding effect, the coefficient function or coefficient image $\beta :\mathcal{S}\to \mathbb{R}$, is the parameter of interest. The simplest special case is the linear model

$$y_i={\mathit{t}}_i^T\delta +{\int }_{\mathcal{S}}{x}_i\left(\mathit{s}\right)\beta \left(\mathit{s}\right)d\mathit{s}+{\epsilon }_i,$$ (3)

where the ε_i are independent and identically distributed errors with mean 0 and variance σ²(= ϕ). When t_i ≡ 1 (i.e., no scalar covariates), model (3) is the extension, from one-dimensional to multidimensional predictors, of the functional linear model that has been studied by Marx and Eilers (1999), Cardot, Ferraty and Sarda (1999), Müller and Stadtmüller (2005), Ramsay and Silverman (2005), Hall and Horowitz (2007), Reiss and Ogden (2007), Goldsmith et al. (2011), and many others.

For the case of one-dimensional functional predictors, a popular way to take spatial information into account is to restrict β(·) to the span of a spline basis (e.g., Marx and Eilers, 1999). Spline methods for two-dimensional predictors have been studied by Marx and Eilers (2005) and Guillas and Lai (2010), and by Reiss and Ogden (2010), whose work was motivated by neuroimaging applications.

Some more-recent work has considered neuroimaging applications with two- and three-dimensional predictors (Zhou, Li and Zhu, 2013; Goldsmith, Huang and Crainiceanu, 2013; Huang et al., 2013). In this paper, we propose a set of new approaches based on a wavelet representation of the coe cient image. The idea of transforming the images to the wavelet domain has previously appeared in the brain mapping literature, where it is customary to fit separate models at each voxel, with the image-derived quantity regressed on demographic or clinical variables of interest (e.g., Ruttimann et al., 1998; Van De Ville et al., 2007). But for our objective of using entire images in a single model to predict a scalar response, working in the wavelet domain has been mentioned as a natural idea (Grosenick et al., 2013) but rarely if ever pursued, at least until the very recent work of Wang et al. (2014). Unlike spline bases, wavelet bases are designed for sparse representation, and yield estimates that adapt to the features of the coefficient image.

Limitation (ii) was highlighted by the results of the recent ADHD-200 Global Competition for automated diagnosis of attention deficit/hyperactivity disorder (ADHD-200 Consortium, 2012). Teams were provided with functional magnetic resonance images from ADHD subjects and controls on which to train diagnostic algorithms, and then applied these algorithms to predict diagnosis in a separate set of images. A team of biostatisticians from Johns Hopkins University, whose methods are described by Eloyan et al. (2012), achieved the highest score for correct imaging-based classification, and were declared the winners. But a team from the University of Alberta, which discarded the images and used just four scalar predictors (age, sex, handedness, and IQ; see Brown et al., 2012), attained a slightly higher classification score (see Caffo et al., 2012, for related discussion).

To address limitation (ii), we test the e ect of image predictors via a permutation-based approach originally proposed in the neuroimaging literature (Golland and Fischl, 2003), which we extend to allow for scalar covariates. We also consider how to extend the traditional notion of confounding to settings with both scalar and image predictors. These ideas are illustrated using our wavelet methods, but are not specific to them; rather, they are applicable with other approaches to functional or high-dimensional regression.

Our contributions can be summarized as follows. (i) We propose novel wavelet-domain methodology for regression with image predictors. While Wang et al. (2014) and Zhao, Chen and Ogden (2014) have studied the wavelet-domain lasso for image predictors, we also propose and compare several other methods, and consider the generalized linear case and the role of scalar covariates. (ii) We extend predictive performance-based hypothesis testing (Golland and Fischl, 2003) to the case where scalar confounders are present, providing a new way to assess the usefulness of image-based prediction.

In §2 we introduce wavelet bases, and motivate and outline a general template for scalar-on-image regression in the wavelet domain. §3 describes three specific algorithms, which are evaluated in simulations in §4. §5 considers hypothesis testing and confounding with image predictors. In§ 6 the proposed methods are applied to a portion of the ADHD-200 data set, and the results point to a possible role of confounding in the competition's surprising result. §7 offers a concluding discussion.

2. Wavelets and their use in regression on images

2.1 A brief introduction to wavelet basis representations

Wavelet bases are a popular way to obtain a sparse representation for functional data, in particular when the degree of smoothness exhibits local variation (see Ogden, 1997; Vidakovic, 1999; Nason, 2008, for statistically-oriented treatments). A wavelet basis for $L^2\left(\mathbb{R}\right)$ is constructed from a scaling function (or “father wavelet”) ϕ and a wavelet function (“mother wavelet”) ψ (see Figure 1(a)–(b)), with the following properties:

• For each $j\in \mathbb{Z}$, the shifted and dilated functions $\{{\varphi }_{j,k}\left(x\right)=2^{j∕2}\varphi (2^{j}x-k):k\in \mathbb{Z}\}$ form an orthonormal basis for V_j, where $\dots \subset V_{-1}\subset V_0\subset V_1\subset \dots$ is a nested sequence of subspaces whose union is a dense subspace of $L^2\left(\mathbb{R}\right)$.

• For each $j\in \mathbb{Z}$, $\{{\psi }_{j,k}\left(x\right)=2^{j∕2}\psi (2^{j}x-k):k\in \mathbb{Z}\}$ form an orthonormal basis for a “detail space” Wj satisfying $V_{j+1}=V_j\oplus W_j$.

Hence, for any integer j₀ ≥ 0, $V_{j_0}\oplus W_{j_0}\otimes W_{j_0+1}\oplus \dots$ is a dense subspace of $L^2\left(\mathbb{R}\right)$.

Fig 1.

(a) Scaling function ϕ and (b) wavelet function for 1D Daubechies (1988) “least-asymmetric” wavelets with 10 vanishing moments. 2D basis functions are formed from tensor products such as (c) $(x,y)\mapsto \psi \left(x\right)\varphi \left(y\right)$ and (d) $(x,y)\mapsto \psi \left(x\right)\psi \left(y\right)$.

Given appropriate boundary handling, such as modifying the scaling and wavelet functions to be periodic, one can likewise construct orthonormal wavelet bases for L²[0, 1], of the form

$$\underset{\in V_{j_0}}{\underbrace{\{{\varphi }_{j_0,0},\dots ,{\varphi }_{j_0,2^{j_0}-1}\}}}\cup \underset{\in W_{j_0}}{\underbrace{\{{\psi }_{j_0,0},\dots ,{\psi }_{j_0,2^{j_0}-1}\}}}\cup \underset{\in W_{j_0+1}}{\underbrace{\{{\psi }_{j_0+1,0},\dots ,{\psi }_{j_0+1,2^{j_0+1}-1}\}}}\cup \dots .$$

—i.e., 2^j₀ scaling functions (corresponding to the large-scale features of the data), 2^j₀ wavelet functions at level j₀, 2^j₀+1 wavelet functions at level j₀+1, and so on, with higher wavelet levels capturing finer-scale details. This multiscale structure is what makes wavelet bases so useful for sparse representation of functions with varying degrees of smoothness.

The wavelet decomposition level j₀ acts as a tuning parameter. A small j₀ implies that a small number (2^j₀) of scaling functions are used to construct the macro features of the function, with most of the basis elements dedicated to providing detail at a variety of scales. A large j₀ allows for many more scaling functions, each at higher resolution, and thus fewer basis elements corresponding to detail.

In practice, a function f ∈ L²[0, 1] is observed at finitely many points, ordinarily taken to be the N = 2^J (for some positive integer J) equally spaced points 0, $\frac{1}{N},\dots ,\frac{N-1}{N}$. (When the function is observed at a number of points that is not a power of 2, one can insert zeroes before and after to attain the next-highest power of 2.) The observed values can then be interpolated by the N-dimensional truncated basis

$$\{{\varphi }_{j_0,0},\dots ,{\varphi }_{j_0,2^{j_0}-1}\}\cup \{{\psi }_{j_0,0},\dots ,{\psi }_{j_0,2^{j_0}-1}\}\cup \dots \cup \{{\psi }_{J-1,0},\dots ,{\psi }_{J-1,2^{J-1}-1}\}.$$

The discrete wavelet transform (DWT), implemented by the O(N) pyramid algorithm of Mallat (1989), expands f with respect to this basis. Given a judicious choice of ϕ and ψ, signals of varying smoothness can be well represented with a small number of coefficients. Throughout this paper we use the compactly supported Daubechies (1988) “least-asymmetric” wavelets with 10 vanishing moments, displayed in Figure 1.

Wavelet bases for two dimensions can be constructed by taking tensor products of the ϕ and ψ functions. The two-dimensional scaling function is ϕ(x)ϕ(y), and there are three types of 2D wavelets: ϕ(x)ψ(y), ψ(x)ϕ(y), roughly corresponding to “horizontal,” “vertical,” and “diagonal” detail respectively (see Figure 1(c)–(d)). These functions are dilated and translated just as their 1D counterparts are. Wavelet bases for 3D are constructed similarly. Morris et al. (2011) discuss alternative wavelet transforms for images that are not constructed as tensor products.

2.2 A meta-algorithm for scalar-on-image regression

Henceforth, the functional predictor x_i(·) of (2), (3) will be replaced by the ith discretized image observation x_i = (x₁, . . . , x_N)^T ≡ [x_i(s₁), . . . , x_i(s_N)]^T , where s₁, . . . , s_N ∈ $\mathcal{S}$ are distinct spatial locations at which the function x_i is measured. Often, in practice, each image is given as a matrix or 3D array; x_i is then obtained by converting this into a vector. From now until §3.5 we focus on the linear model (3), which can now be written in matrix form as

$$\mathit{y}=\mathit{T}\delta +\mathit{X}\beta +\epsilon .$$ (4)

Here y = (y₁, . . . , y_n)^T ε = (ε₁, . . . , ε_n)^T; T is the n × N matrix with ith row ${\mathit{t}}_i^T$; X is the n × N matrix with ith row ${\mathit{x}}_i^T$; and β = (β₁, . . ., β_N)^T is a similarly discretized version of the coefficient image β. More precisely, for j = 1, . . . , N, β_j = w_jβ(s_j), where the w_j's are quadrature weights such that ${\mathit{x}}_i^T\beta$ is a good approximation to the integral in (3); but for image data, s₁, . . . , s_N typically form an equally spaced grid, so these weights are taken as constant and hence ignored. With these definitions, (3) is just the ith of the n equations that make up the vector equation (4).

To simplify the notation, we shall use a single subscript and denote the wavelet basis functions for a given j₀ as {ψ₁, ψ₂, . . ., ψ_N}. The wavelet representation of the ith observed image x_i is $x_i\left(s\right)={\sum }_{k=1}^N{\tilde{x}}_{ik}{\psi }_k\left(s\right)$, in which the wavelet coefficients are given by ${\tilde{x}}_{ik}=\langle x_i,{\psi }_k\rangle$. The coefficient vector ${\tilde{\mathit{x}}}_i={({\tilde{x}}_{i1},\dots ,{\tilde{x}}_{iN})}^T$ can be written as ${\tilde{\mathit{x}}}_i=\mathcal{W}{\mathit{x}}_i$, where $\mathcal{W}$ is an N × N orthonormal matrix (which is not formed explicitly when ${\tilde{\mathit{x}}}_i$ is computed by the DWT). Similarly the discretized coefficient function β can be represented terms of its wavelet coefficients as $\tilde{\beta }=\mathcal{W}\beta$, leading to the wavelet-domain version of model (4):

$$\begin{array}{cc}\hfill \mathit{y}=& \mathit{T}\delta +\mathit{X}{\mathcal{W}}^T\mathcal{W}\beta +\epsilon \hfill \\ \hfill =& \mathit{T}\delta +\tilde{\mathit{X}}\tilde{\beta }+\epsilon ,\hfill \end{array}$$ (5)

where X̃ is the n × N matrix with ith row ${\tilde{\mathit{x}}}_i^T$.

The key point is that the wavelet-domain form (5) is better suited than the original form (4) for applying sparse techniques for high-dimensional regression—both because wavelet bases are designed for sparse representation of images (Mallat, 2009), and because the DWT approximately decorrelates or “whitens” data (Vidakovic, 1999). We can thus formulate a “meta-algorithm” for scalar-on-image regression in the wavelet domain:

1. Apply the DWT to the image predictors to transform model (4) into model (5).

2. Use some high-dimensional regression methodology to derive a sparse estimate $\widehat{\tilde{\beta }}$

3. Apply the inverse DWT to $\widehat{\tilde{\beta }}$ to obtain a coe cient image estimate $\widehat{\beta }$ for the original model (4).

Different choices for step 2 lead to specific algorithms, as described in the next section.

The above general scheme can be extended to multiple image predictors (cf. Zhu, Vannucci and Cox, 2010). We note that this meta-algorithm has been applied before for 1D functional predictors (Brown, Fearn and Vannucci, 2001; Wang, Ray and Mallick, 2007; Malloy et al., 2010; Zhao, Ogden and Reiss, 2012), and more for image predictors (Wang et al., 2014; Zhao, Chen and Ogden, 2014). Past work on wavelet-domain classification, as opposed to regression (e.g. Berlinet, Biau and Rouvière, 2008; Zhu, Brown and Morris, 2012; Chang, Chen and Ogden, 2014), may bear comparison to our proposed methods. Morris et al. (2011) develop wavelet-domain functional mixed models with images as responses.

3. Three wavelet-domain algorithms

3.1. Sparse wavelet-domain principal component regression

The functional linear model (3) is often fitted by assuming the coefficient function has a truncated functional principal component, or Karhunen-Loève, representation $\beta \left(s\right)={\sum }_{j=1}^m{c}_j{\rho }_j\left(s\right)$, where m is a positive integer and ρ₁, ρ₂, . . . , ρ_m are the first m eigenfunctions of the covariance operator associated with the predictor functions x_i (e.g., Cardot, Ferraty and Sarda, 1999; Müller and Stadtmüller, 2005; Cai and Hall, 2006). The eigenfunctions ρ₁, ρ₂ . . . , ρ_m can be estimated by viewing the functional predictors as (highly) multivariate data, and applying ordinary principal component analysis to the predictor matrix X.

Here and in §3.2, we assume that X has mean-centered columns, i.e., 1^TX = 0. The approach of the previous paragraph then amounts to assuming β = V_mγ for some $\gamma \in {\mathbb{R}}^m$, where UDV^T is the singular value decomposition of X, and V_m comprises the leading m columns of V. Hence estimation reduces to choosing δ, γ to minimize the principal component regression (PCR; Massy, 1965) criterion

$${\Vert \mathit{y}-\mathit{T}\delta -\mathit{X}{\mathit{V}}_m\gamma \Vert }^2.$$ (6)

(This is a slightly nonstandard PCR criterion, in that principal component reduction is applied only to X but not to T. A similar remark applies to the other criteria introduced below.)

As shown by Reiss and Ogden (2007), PCR can be implemented more e ectively by exploiting the functional character of the data. In the one-dimensional functional predictor case, this has usually meant forming smooth estimates of the eigenfunctions—as in the FPCR_C method of Reiss and Ogden (2007), which expands the eigenfunctions with respect to a B-spline basis (cf. Cardot, Ferraty and Sarda, 2003). But for image predictors, local adaptivity—the ability to capture sharp features in some areas, vs. a high degree of smoothness elsewhere—becomes particularly important. This motivates using a wavelet basis, rather than a spline basis, to represent the eigenfunctions; or in other words, developing a wavelet-domain version of PCR as an instance of the meta-algorithm of §2.2.

A non-sparse wavelet-domain PCR estimate would minimize

$${\Vert \mathit{y}-\mathit{T}\delta -\tilde{\mathit{X}}{\tilde{\mathit{V}}}_m\gamma \Vert }^2,$$ (7)

which is analogous to (6) but based on the SVD of X̃ rather than of X. However, the advantage of working in the wavelet domain is to obtain a sparse coe cient estimate, by replacing the PC weights Ṽ with weights from a sparse version of PCA. Several penalty-based methods have been proposed for sparse PCA (e.g., Zou, Hastie and Tibshirani, 2006; Shen and Huang, 2008; Witten, Tibshirani and Hastie, 2009); but we opted for the approach of Johnstone and Lu (2009), which is simpler than the penalized methods and, unlike them, was developed with a view toward sparse wavelet representations of signals. Johnstone and Lu (2009) propose to select the features or coordinates with highest variance, and apply PCA only to these. The resulting sparse PCR criterion is

$${\Vert \mathit{y}-\mathit{T}\delta -{\tilde{\mathit{X}}}^{\ast }{\tilde{\mathit{V}}}_m^{\ast }\gamma \Vert }^2;$$ (8)

here X̃* consists of the c columns of X̃ having highest variance, and ${\tilde{V}}_m^{\ast }$ consists of the leading m columns of Ṽ*, where Ũ* D̃* Ṽ*^T is the SVD of X̃*. The minimizer $(\widehat{\delta },\widehat{\gamma })$ of (8) can be obtained by simple least squares. The vector of wavelet coefficient estimates is then $\widehat{\tilde{\beta }}={\tilde{V}}_m^{\ast }\widehat{\gamma }$, and the coefficient image estimate $\widehat{\beta }={\mathcal{W}}^T\widehat{\tilde{\beta }}$ is derived by the inverse DWT.

3.2 Sparse wavelet-domain partial least squares

Whereas PCR reduces dimension by regressing on the leading PCs of the predictors, partial least squares (PLS; Wold, 1966) works by regressing on a set of components that are relevant to predicting the responses. A (non-sparse) wavelet-domain PLS estimate (cf. Nadler and Coifman, 2005) is derived by minimizing

$${\Vert \mathit{y}-\mathit{T}\delta -\tilde{\mathit{X}}{\tilde{\mathit{R}}}_m\gamma \Vert }^2$$ (9)

(cf. (7)), where the columns of R̃_m are defined iteratively as follows (Stone and Brooks, 1990):

• r̃₁ = arg min_∥r∥=1 Cov(y, X̃r);

• for j = 2, . . ., c,

  $${\tilde{\mathit{r}}}_j=\underset{\Vert r\Vert =1,r^T{\tilde{X}}^T\tilde{X}{r}_m=0\forall m=1,\dots ,j-1}{\arg \min }\mathrm{Cov}(\mathit{y},\tilde{\mathit{X}}\mathit{r}).$$

Once again, however, the point of working in the wavelet domain is to obtain a sparse estimate. To define sparse wavelet-domain PLS, as with PCR, we could have used penalization to derive sparse PLS components (Chun and Keleş, 2010); but we instead opted to build on the aforementioned approach of Johnstone and Lu (2009) to sparse PCA. A natural PLS analogue of that approach is to select those features x̃_j whose covariance with y has the greatest magnitude. This results in the sparse PLS criterion

$${\Vert \mathit{y}-\mathit{T}\delta -{\tilde{\mathit{X}}}^{†}{\tilde{\mathit{R}}}_m^{†}\gamma \Vert }^2;$$ (10)

here X̃^† consists of the c columns of X̃ having highest covariance with y, and the columns of ${\tilde{\mathit{R}}}_m^{†}$ are defined analogously to those of R̃_m in (9). As for PCR, the least-squares minimizer $(\widehat{\delta },\widehat{\gamma })$ of (10) leads directly to estimates of the wavelet coefficients $\tilde{\beta }$ and of the resulting coefficient image β.

Our PLS algorithm is a wavelet-domain counterpart of the spline-based functional PLS procedure denoted by FPLS_C in Reiss and Ogden (2007). We note that Preda and Saporta (2005) and Delaigle and Hall (2012b) have proposed more explicitly functional formulations of PLS, based on covariance operators on function spaces.

3.3 Wavelet-domain elastic net

Since wavelet bases are well suited for sparse representation of functions, recent work has considered combining them with sparsity-inducing penalties, both for semiparametric regression (Wand and Ormerod, 2011) and for regression with functional or image predictors (Zhao, Ogden and Reiss, 2012; Wang et al., 2014; Zhao, Chen and Ogden, 2014). The latter papers focused on ${\ell }_1$ penalization, also known as the lasso (Tibshirani, 1996), in the wavelet domain. Alternatives to the lasso include the SCAD penalty (Fan and Li, 2001) and the adaptive lasso (Zou, 2006). Here we consider the elastic net (EN) estimator for wavelet-domain model (5), which minimizes

$${\Vert \mathit{y}-\mathit{T}\delta -\tilde{\mathit{X}}\tilde{\beta }\Vert }^2+\lambda \left[\alpha {\Vert \tilde{\beta }\Vert }_1+(1-\alpha ){\Vert \tilde{\beta }\Vert }_2^2\right]$$ (11)

over $(\delta ,\tilde{\beta })$ for a regularization parameter λ > 0 and a mixing parameter α ∈ [0, 1] which controls the relative strength of the ${\ell }_1$ and ${\ell }_2$ penalties on the coe cients (Zou and Hastie, 2005).

In the original nomenclature of Zou and Hastie (2005), the minimizer of (11) is the “naïve” EN, whereas EN is a rescaled version. Since we shall make use of the generalized linear extension of EN as implemented by Friedman, Hastie and Tibshirani (2010), we follow these authors in omitting the rescaling step. When α > 0, the ${\ell }_1$ penalty shrinks small coefficients to zero, leading to a sparse wavelet representation. The wavelet-domain lasso is obtained when α = 1. As explained by Zou and Hastie (2005), given a group of important features that are highly correlated, the lasso tends to select just one whereas EN selects the entire group, which is often preferable—even in the wavelet domain, notwithstanding the “whitening” property of the discrete wavelet transform.

3.4 Summary: Alternative routes to sparsity

All three of the above methods seek to represent the coefficient image β (·) sparsely, as a linear combination of a subset of the wavelet basis functions; but they deploy very di erent strategies to choose that subset. The ${\ell }_1$ penalty in the elastic net criterion (11) has the effiect of shrinking small coefficients to zero. This can be interpreted as imposing a prior that favors a sparse estimate. The PCR criterion (8) eliminates basis elements before performing regression, based on an implicit assumption that those basis elements with low variance in the data have little to contribute to the coe cient image. This assumption is broadly consistent, on the one hand, with the assumption of Johnstone and Lu (2009) that such basis elements are merely capturing noise; and on the other hand, with the underlying assumption of PCR, namely that the highest-variance principal components are most relevant in regression (see Cook, 2007, for some relevant discussion). The PLS criterion (10) likewise lets the data determine which basis elements to include; but here, instead of considering only the wavelet-transformed image data X̃ as in PCR, we define relevant components by iteratively maximizing covariance with the responses y.

3.5 Extension to the generalized linear case

The above three wavelet-domain algorithms can be straightforwardly extended from linear to generalized linear models (GLMs) of the form

$$g\left[E\left(\mathit{y}\right)\right]=\mathit{T}\delta +\tilde{\mathit{X}}\tilde{\beta },$$ (12)

for a link function g, generalizing (5). For PCR, one simply fits a GLM, as opposed to a linear model, to the sparse PCs. For the elastic net, the glmnet algorithm of Friedman, Hastie and Tibshirani (2010) is available for the generalized linear case.

PLS is sometimes performed in an iteratively reweighted manner for GLMs (Marx, 1996); but in high-dimensional settings, such algorithms may require nontrivial modification (e.g. Ding and Gentleman, 2005) to avoid convergence problems. Here we view PLS as a generic approach to constructing relevant components, which may be employed beyond the linear regression setting (e.g. Nguyen and Rocke, 2002; Delaigle and Hall, 2012a). Thus we construct PLS components exactly as we would for a linear model, but then use these components to fit a GLM.

3.6 Tuning parameter selection

For wavelet-domain PCR and PLS, three tuning parameters must be selected: the resolution-level parameter j₀; the number c of wavelet coefficients to retain (i.e., the number of columns of X̃* in (8), or of X̃^† in (10)); and the number m of PCs or PLS components. We generally fix j₀ = 4, since we have found that resolution level to be generally either optimal or near-optimal as measured by cross-validation (CV). For wavelet-domain elastic net, one must choose j₀ and the two penalty parameters α and λ in (11), but we again prefer to fix j₀ = 4.

These tuning parameters are chosen by repeated K-fold CV. In the rth of R repetitions we divide the data points (y_i, t_i, x_i) (i = 1, . . . , n) into K equal-sized validation sets indexed by I_r,1, . . . , I_r,K. We can then choose the tuning parameters to minimize the CV score

$$\frac{1}{RK}\sum _{r=1}^R\sum _{k=1}^K\sum _{i\in I_{r,k}}L(y_i;{\widehat{\delta }}_{-r,k},{\widehat{\tilde{\beta }}}_{-r,k}),$$ (13)

where ${\widehat{\delta }}_{-r,k},{\widehat{\tilde{\beta }}}_{-r,k}$ are the estimates that result when model (12) is fitted (by PCR, PLS, or EN) with the observations indexed by I_r,k excluded, and L is an appropriate loss function. For linear regression the standard loss function is the squared error $L({\mathit{y}}_i;\delta ,\tilde{\beta })={(y_i-{\mathit{t}}_i^T\delta -{\tilde{\mathit{x}}}_i^T\tilde{\beta })}^2$. For the generalized linear case, following Zhu and Hastie (2004), we use the deviance $D(y_i;\delta ,\tilde{\beta })$ as the loss function. Specifically for logistic regression, unusually large summands can dominate criterion (13). Therefore, similarly to Chi and Scott (2013), we instead choose the tuning parameters by a robust CV score that takes the median rather than the mean over each set of K validation sets:

$$\frac{1}{R}\sum _{r=1}^R{\text{median}}_{k\in \{1,\dots ,K\}}\sum _{i\in I_{r,k}}D(y_i;{\widehat{\delta }}_{-r,k},{\widehat{\tilde{\beta }}}_{-r,k}).$$ (14)

4. Comparative simulation study

To test the performance of our methods with realistic image predictors, we created a data set based on the positron emission tomography (PET) data previously studied by Reiss and Ogden (2010). That data set included axial slices from 33 amyloid beta maps, from which we extracted a square region of 64 × 64 voxels. To generate a larger sample of n = 500 images, we applied a procedure similar to that of Goldsmith, Huang and Crainiceanu (2013):

1. We estimated the (vectorized) principal components (eigenimages)

   $${\widehat{\rho }}_1,\dots ,{\widehat{\rho }}_{32}\in {\mathbb{R}}^{{64}^2},$$

   with corresponding eigenvalues λ₁, . . . , λ₃₂.

2. Fir i = 1, . . ., 500, we generated the i^th simulated predictor image as ${\mathit{x}}_i={\sum }_{j=1}^{32}{c}_{ij}{\widehat{\rho }}_j$ with the c_ij's simulated independently from the N(0, λ_j) distribution.

In step 1 above we used the sparse PCA method of Johnstone and Lu (2009), including the 492 wavelet coe cients having the highest variance. This number of wavelet coefficients was su cient to capture 99.5% of the “excess” variance, in the sense of section 4.2 of Johnstone and Lu (2009).

We used two different true coefficient images $\beta \in {\mathbb{R}}^{{64}^2}$, which are shown in Figure 2. The first image β⁽¹⁾ is similar to that used by Goldsmith, Huang and Crainiceanu (2013). Taking its domain to be [1, 64]², this coefficient image is given by β⁽¹⁾ = g₁ – g₂ where g₁, g₂ are the densities of the bivariate normal distributions

$$N\left[\left(\begin{array}{c}\hfill 30\hfill \\ \hfill 20\hfill \end{array}\right),10{\mathit{I}}_2\right]\text{and}N\left[\left(\begin{array}{c}\hfill 20\hfill \\ \hfill 55\hfill \end{array}\right),10{\mathit{I}}_2\right],$$

respectively. The second image β⁽²⁾ is a two-dimensional analogue of the “bumps” function used by Donoho and Johnstone (1994), and many subsequent authors, to illustrate the properties of wavelets.

Fig 2.

Coefficient images β⁽¹⁾ (left) and β⁽²⁾ (right) used in the simulation study.

We then simulated continuous or binary outcomes y₁, . . . , y_n with specified approximate values of the coefficient of determination R², in the sense detailed in Supplementary Appendix A.1 (Reiss et al., 2015). We generated 100 sets of n = 500 continuous outcomes, and 100 sets of 500 binary outcomes, for each of the R² values 0.1, 0.5.

We compared the performance of the three wavelet-domain methods described in §3 with three analogous “voxel-domain” methods, i.e., sparse PCR, sparse PLS and elastic net without transformation to the wavelet domain. The wavelet- and voxel-domain methods are denoted by WPCR, WPLS and WNet and by VPCR, VPLS and VNet, respectively. We also included the B-spline-based functional PCR method (“FPCR_R”, or simply FPCR) of Reiss and Ogden (2007, 2010). Tuning parameter selection was as described in Supplementary Appendix A.1 (Reiss et al., 2015).

Performance was evaluated in terms of estimation error and prediction error. Estimation error is defined by the scaled mean squared error (MSE) ${\Vert \widehat{\beta }-\beta \Vert }^2∕{\Vert \beta \Vert }^2$, where β, $\widehat{\beta }$ are the true and estimated coefficient images. Prediction error is defined using a separate set of outcomes $y_1^{\ast },\dots ,y_n^{\ast }$, generated from the same conditional distribution as y₁, . . .,y_n. We use the scaled mean squared prediction error $\frac{1}{n{\sigma }^2}{\sum }_{i=1}^n{(y_i^{\ast }-{\widehat{y}}_i)}^2$ as our criterion for linear regression, and the mean of the deviances of $y_1^{\ast },\dots ,y_n^{\ast }$ for logistic regression.

Figure 3 presents boxplots of the results. In general all seven methods differ only slightly in prediction error. Much greater di erences are seen for estimation error. Compared with the corresponding voxel-domain methods, the estimation MSE for wavelet methods is either roughly equal or clearly lower on average, and the variability of the MSE is often much lower. The wavelet methods also markedly outperform B-spline-based FPCR. Somewhat contrary to expectation, the superior performance of wavelet methods is not clearly more pronounced for β⁽²⁾ than for β⁽¹⁾.

Fig 3.

Estimation error, displayed as log(scaled MSE) (left subfigure), and prediction error (right subfigure) in the simulation study.

While the wavelet-domain methods do not clearly attain lower estimation error than voxel-domain methods for logistic regression with R² = 0.5, they do appear superior for the R² = 0.1 setting (which seems more realistic) and for linear regression. Moreover, qualitatively, wavelet-domain modeling helps to capture the main features of the coefficient image. Figure 4 displays an example of the training-set estimates derived by wavelet-domain lasso versus ordinary lasso. The wavelet-domain estimates are clearly more similar to each other and to the true coefficient image than are the ordinary lasso estimates.

Fig 4.

True coe cient function β⁽¹⁾ from the comparative simulation study (top left) compared with five training-set coefficient function estimates (for data simulated under R² = 1 setting) based on wavelet-domain lasso (other top panels) and voxel-domain lasso (bottom panels). The wavelet-based estimates are reasonably accurate, while each of the voxel-domain estimates has about 20–25 scattered voxels with nonzero values. Note the unequal scales.

The wavelet-domain EN appears to have a slight edge overall compared with PCR and PLS. For this reason, and because wavelet EN (or at least its special case, the lasso) are now somewhat established in the literature (Zhao, Ogden and Reiss, 2012; Wang et al., 2014; Zhao, Chen and Ogden, 2014), the simulations and real-data analyses in the next two sections consider only wavelet-domain EN.

5. Inferential issues

We now turn to what the Introduction referred to as limitation (ii) of predictive analyses in neuroimaging: the need for methodology to assess the predictive value of image data, in particular when scalar covariates are present.

5.1 Permutation testing

Consider testing the null hypothesis β(s) ≡ 0 in the the general model (1), (2), i.e., testing the null parametric model $g\left({\mu }_i\right)={\mathit{t}}_i^T\delta$ versus the alternative (2). Informally, we are asking whether the images have predictive value beyond the information contained in the scalar predictors. We propose a permutation test procedure in which the CV criterion (13) or (14) serves as the test statistic. If the true-data CV falls in the left tail of the distribution of permuted-data CV values, significance is declared. Permutation techniques of this kind have previously appeared in the neuroimaging and machine learning literature (Golland and Fischl, 2003; Ojala and Garriga, 2010).

The way the permutation distribution is constructed depends on the null model under consideration. When t_i ≡ 1 in (2) (no scalar covariates), one can simply permute the responses: i.e., we repeatedly reorder the responses as y_π(1), . . ., y_π(n) for some permutation π, refit the model, and record the CV value. For the linear model (3) with scalar covariates, a common approach is to permute the residuals from the null parametric model: i.e., model (3) is refitted repeatedly with ith response of the form ${\widehat{y}}_i+{\widehat{\epsilon }}_{\pi \left(i\right)}$, where the hats refer to fitted values and residuals from the model $y_i={\mathit{t}}_i^T\delta +{\epsilon }_i$. For some GLMs, however, such pseudo-responses based on permuted residuals are not of the correct form (e.g., for logistic regression, they are not binary). One can instead form pseudo-predictors, by regressing the predictor of interest on the nuisance covariates and permuting the residuals from this fit. In other words, we replace the design matrix (T|X) with

$$[\mathit{T}\mid {\mathit{P}}_T\mathit{X}+\Pi (\mathit{I}-{\mathit{P}}_T)\mathit{X}],$$ (15)

where P_T = T(T^TT)⁻¹T^T and Π is a permutation matrix. Although a similar idea was proposed by Potter (2005) for (ordinary) logistic regression, we have adopted it as our preferred permutation approach even for the linear case; see Supplementary Appendix B (Reiss et al., 2015) for further discussion.

We conducted a simulation study, using the ADHD-200 image data analyzed in §6, to assess the type-I error rate and power of the permutation test procedure. Here we focus on logistic regression (see Supplementary Appendix C Reiss et al., 2015, for linear regression results) and the wavelet-domain lasso. We first considered the case without scalar covariates, and generated binary responses y_i ~ Bernoulli(p_i), i = 1, . . . , n = 333, where

$$\log \frac{p_i}{1-p_i}={\delta }_0+{\mathit{x}}_i^T\beta$$ (16)

where δ₀ is a constant used to adjust the base rate (probability of event); ${\mathit{x}}_i\in {\mathbb{R}}^{{64}^2}$ is the ith image (expressed as a mean-zero vector); β is the true coefficient image shown in Figure 5(a) (similarly vectorized), multiplied by an appropriate constant to attain a specified value of R² (see Supplementary Appendix A Reiss et al., 2015, regarding the definition of R²). For each of the base rates .25, .5, .75 and each of the R² values .0, .07, .1, .15, .2, .25, .3, we simulated 200 response vectors to assess power to reject H₀ : β = 0 at the p = .05 level, as well as 1000 response vectors with β = 0 (R² = 0) to assess the type-I error rate. Next we considered testing the same null hypothesis for the model

$$\log \frac{p_i}{1-p_i}={\delta }_0+t_i{\delta }_1+{\mathit{x}}_i^T\beta ,$$ (17)

with a scalar covariate t_i such that R² for the submodel E(y_i|t_i) = t_iδ is approximately 0.2. We generated the same number of response vectors as above for each of the above R² values, but here R² refers to the partial R² adjusting for t_i (see Supplementary Appendix A.2 Reiss et al., 2015).

Fig 5.

(a) True coefficient image β used in the power study: gray denotes 0, black denotes 1. (b) Estimated probability of rejecting the null hypothesis β = 0 as a function of R², with 95% confidence intervals, for model (16). (c) Same, for model (17).

The results, displayed in Figure 5(b) and (c), indicate that the nominal type-I error rate is approximately attained for both models. For a given R² > 0, the power is somewhat higher for model (16) than for model (17), and highest for either model when the base rate is .5. Evidently, for base rates closer to 0 or 1, the CV deviance under the null hypothesis tends to be lower, and thus a stronger signal is needed to reject the null.

Basing a test of the hypothesis β(·) ≡ 0 on the prediction performance of an estimation algorithm, rather than on an estimate of β, is admittedly somewhat unconventional. In neuroimaging specifically, inference typically proceeds by fitting separate models at each voxel, and then applying some form of multiple testing correction (Nichols, 2012). In the present setting of a single model that uses the entire image to predict a scalar response, it might be possible to assign p-values to individual voxels as in Meinshausen, Meier and Bühlmann (2009). In practice, however, predictive algorithms tend to produce rather unstable estimates, as a number of authors have acknowledged (e.g., Craddock et al., 2009; Honorio et al., 2012; Sabuncu, Van Leemput and the Alzheimer Disease Neuroimaging Initiative, 2012). Our hypothesis testing approach thus sets the more modest inferential goal of verifying that the coefficient image as a whole yields better-than-chance prediction.

5.2. Confounding

For ordinary, as opposed to functional, regression, confounding is said to occur when (i) x appears predictive of y, but this relationship can be attributed to a third variable t such that (ii) t is predictive of y and (iii) t is correlated with x. For example, birth order (x) is associated with the occurrence of Down syndrome (y), but this is due to the effect of the confounding variable maternal age (t) (Rothman, 2012).

To extend the above definition to the case of a functional predictor x(·), suppose that (i) x(·) is ostensibly related to y, in the sense that β(·) is not identically zero when model (2) includes no scalar covariates; but (ii) the scalar variable t is also predictive of y. A functional-predictor analogue of point (iii) is to suppose that t is correlated with $\int x\left(s\right)\widehat{\beta }\left(s\right)ds$, where $\widehat{\beta }(\cdot )$ is an estimate obtained with t excluded from model (2). Aside from this “global” analogue of (iii), it may be useful to consider a “local” analogue which holds if t is correlated with x(s), specifically for s such that β(s) ≠ 0; but this is somewhat less straightforward to assess.

6. Application: fALFF and ADHD

6.1 ADHD-200 data set and candidate models

We now apply the wavelet-domain elastic net to “predicting” ADHD diagnosis using maps of fractional amplitude of low-frequency fluctuations (fALFF) (Zou et al., 2008) from a portion of the ADHD-200 sample referred to in the Introduction (http://fcon_1000.projects.nitrc.org/indi/adhd200/). fALFF is defined as the ratio of BOLD signal power spectrum within the 0.01–0.08 Hz range to total over the entire range. Yang et al. (2011) reported altered levels of fALFF in a sample of children with ADHD relative to controls, specifically in frontal regions. That study relied on the traditional analytic approach in neuroimaging, which regresses the imaged quantity (in this case fALFF) on diagnostic group, separately at each voxel. Here we employed scalar-on-image logistic regression, which reverses the roles of response and predictor, to regress diagnostic group on fALFF images. Our sample consisted of 333 individuals: 257 typically developing controls and 76 with combined-type ADHD. The sample included 198 males and 135 females, with age range 7–20 (see Supplementary Appendix D Reiss et al., 2015, for further details). We chose the 2D slice for which the mean across voxels of the SD of fALFF was highest. This was the axial slice located at z = 26 (just dorsal to the corpus callosum) in the coordinate space of the Montreal Neurological Institute's MNI152 template (4mm resolution). We fitted two models. The first was

$$\text{logit Pr}\left(i\text{th subject has ADHD}\right)=\delta +{\int }_{\mathcal{S}}{x}_i\left(\mathit{s}\right)\beta \left(\mathit{s}\right)d\mathit{s},$$ (18)

where x_i(s) denotes the ith subject's fALFF image. The second model was

$$\text{logit Pr}\left(i\text{th subject has ADHD}\right)={\mathit{t}}_i^T\delta +{\int }_{\mathcal{S}}{x}_i\left(\mathit{s}\right)\beta \left(\mathit{s}\right)d\mathit{s},$$ (19)

where the vector t_i includes the ith subject's age, sex, IQ and mean FD, as well as a leading 1 for the intercept.

Figure 6 shows the coefficient images attained for model (18) with each value of the mixing parameter α. As expected, increasing values of lead to more-sparse estimates in the wavelet domain and hence in the voxel domain. Figure 7 shows the CV deviance as a function of λ for α = 0.1, which had the lowest CV deviance overall, as well as for α = 1.

Fig 6.

Coefficient image estimates for model (18) applied to the ADHD-200 data, using wavelet-domain elastic net with four diffierent values of the mixing parameter α.

Fig 7.

Cross-validated deviance +/− one approximate standard error, for the wavelet-domain elastic net models with α = 0.1, 1.

The left subfigure of Figure 8 shows that the CV deviance lies in the left tail of the permutation distribution for model (18), indicating a significant e ect of the fALFF image predictors (p = 0.015). However, with the scalar covariate adjustment of model (19), this e ect disappears. The next subsection examines more closely how the scalar covariates may be acting as confounders.

Fig 8.

Permutation test results. For the full sample (left), a significant e ect of the fALFF images is seen in model (18), but not in model (19), which adjusts for scalar covariates. When only younger individuals are included (right), neither model shows a significant fALFF effect.

Our test of model (18) entailed 999 permuted-data fits with four candidate values of α and 100 of λ, requiring 14.25 hours on an Intel Xeon E5-2670 processor running at 2.6 GHz. In practice we recommend parallelizing the permutations via cluster computing to make the computation time more manageable. In addition, truncated sequential probability ratio tests (Fay, Kim and Hachey, 2007) could in some cases reduce computation time via early stopping. We also explored fitting model (18) with the full 3D fALFF images as predictors; see Supplementary Appendix E (Reiss et al., 2015).

6.2 Assessing and remedying confounding

As discussed in §5.2, the notion of confounding entails three elements (see Figure 9). Point (i), an apparent e ect of the image predictor fALFF on diagnosis, was established by the above permutation test result for model (18). To check point (ii) of the definition for each of the four scalar covariates under consideration, we performed an ordinary logistic regression with diagnosis (1=ADHD, 0=control) as response and the above four scalar predictors. In Table 1 (at left), sex, age and IQ are all seen to be significantly related to diagnosis. See also Figure 10, which compares the fitted probabilities from this ordinary logistic regression with those resulting from models (18) and (19). The scalar-covariates model is seen to separate the two groups (black vs. gray dots) quite well; the image predictors increase the spread of the predicted probabilities without clearly improving the two groups’ separation. Based on these results, each of these three variables may be acting as a confounder.

Fig 9.

Relationships among a putative predictor X, outcome Y and confounder T (see § 5.2), illustrated with respect to the ADHD-200 data.

Table 1.

To examine element (ii) of confounding, an ordinary logistic regression was fitted with the four scalar predictors and with ADHD diagnosis as response; the resulting estimates are shown with 95% confidence intervals. For element (iii), we display the correlations of each predictor with the logit probabilities estimated by fitting model (18).

	(ii)		(iii)
	Log odds ratio	p-value	Correlation	p-value
Intercept	3.90 (1.11, 6.78)	0.007
Sex (M - F)	1.26 (0.65, 1.91)	0.00008	0.14 (0.03,0.24)	0.011
Age	−0.20 (−0.32, −0.09)	0.0005	−0.35 (−0.44,−0.25)	6 × 10–11
IQ	−0.03 (−0.05, −0.01)	0.003	−0.09 (−0.19,0.02)	0.10
Mean FD	−2.51 (−8.80, 3.56)	0.42	−0.04 (−0.15,0.07)	0.47

Fig 10.

Predicted probabilities of ADHD diagnosis, according to the images-only model (18); an ordinary logistic regression with the four scalar covariates; and model (19), which includes both. Also shown are the R² values, as defined in Supplementary Appendix A.1, for the three models.

Next we consider point (iii), i.e., the correlations of each scalar covariate with ${\int }_{\mathcal{S}}{x}_i\left(\mathit{s}\right)\widehat{\beta }\left(\mathit{s}\right)d\mathit{s}$ where $\widehat{\beta }$ is the coefficient image estimate from the fALFF-only model (18), or equivalently with the predicted logit probability of ADHD from that model. The results, shown at right in Table 1, point to age and sex as the principal confounders. (Here sex was treated as a binary variable, with 1 for male and 0 for female; a t-test and a Mann-Whitney test yielded similar results.) “Local” examination in the sense of §5.2 reveals that the fALFF x(s) tends to be higher in males and in younger individuals for many voxels s; and such regions overlap considerably with those in which $\widehat{\beta }\left(\mathit{s}\right)>0$. In other words, the ostensible association between fALFF and ADHD likely reflects the dependence of fALFF on age and sex, which in turn are related to ADHD in our sample.

Further inspection revealed that, of the 67 individuals with age above 14.0, only 8 had ADHD, with maximum age 17.43—whereas the controls had ages as high as 20.45. This led us to suspect that these older individuals might be driving the confounding with age that results in a spurious effect of fALFF on diagnosis. To investigate this possibility, we repeated the analysis using only the 266 individuals of age 14.0 or lower. Figure 8 shows that in this subsample, the fALFF effect is no longer significant, even without adjusting for the scalar covariates. Moreover, given how far the test statistic is from the left tail of the permutation distribution, it seems unlikely that the loss of significance is due merely to the lower sample size.

In general, absent careful matching at the design stage, it would be advisable to match the two diagnostic groups optimally on a complete set of clearly relevant variables, via algorithms such as those described in Rosen-baum (2010). Our aim here, however, was to show how a straightforward new notion of confounding for functional predictors can be used to identify a principal scalar confounder, whose impact can be removed by the crude device of simply truncating the age range.

7. Discussion

Our analysis in §6 included only one imaging modality and only a subset of the individuals from the ADHD-200 Global Competition database. At any rate, our essentially negative result is consistent with the finding (Brown et al., 2012) that diagnostic accuracy was optimized by basing prediction on scalar predictors, while ignoring the image data. In a blog comment on that outcome, cited both by ADHD-200 Consortium (2012) and by Brown et al. (2012), the neuroscientist Russ Poldrack suggested that “any successful imaging-based decoding could have been relying upon correlates of those variables rather than truly decoding a correlate of the disease.” Stated a bit di erently, the competing teams’ successes in using the image data to predict diagnosis may have been brought about by confounding. But there appear to have been few attempts, if any, to study systematically how confounding may give rise to spurious relationships between quantitative image data and clinical variables. Similarly, analyses of the ADHD-200 data, and related work on brain “decoding,” have devoted little attention to formally testing the contribution of imaging data to prediction of scalar responses.

As we have shown, these two interrelated issues—testing the e ect of image predictors, and investigating possible confounders—can be handled straightforwardly within our scalar-on-image regression framework. The permutation test procedure of §5.1 found a statistically significant relationship between fALFF images and ADHD diagnosis, but this disappeared when four scalar covariates were adjusted for. Further examination, in light of our extension of the notion of confounding to functional/image predictors in §5.2, pointed to age and sex as the key confounders.

The ADHD-200 project is one of a number of recent initiatives to make large samples of neuroimaging data publicly available (Milham, 2012). These initiatives have been a boon for statistical methodology development, but it must be borne in mind that even as neuroimaging sample sizes increase rapidly, they remain much smaller than the data dimension. No approach to scalar-on-image regression can completely escape the ensuing non-identifiability of the coe cient image. We can, however, (i) put forth assumptions, likely to hold approximately in practice, that reduce the e ective dimension of the coefficient image; and (ii) employ multiple methods in the hope that these will converge upon similar coe cient image estimates, at least when the signal is sufficiently strong.

With these considerations in mind, we have introduced three methods for scalar-on-image regression, each relying on a different set of assumptions to achieve dimension reduction in the wavelet domain. Implementations of these three methods, for 2D and 3D image data, are provided in the refund.wave package (Huo, Reiss and Zhao, 2014) for R (R Development Core Team, 2012), available at http://cran.r-project.org/web/packages/refund.wave. This new package, a spino of the refund package (Crainiceanu et al., 2014), relies on the wavethresh package (Nason, 2013) for wavelet decomposition and reconstruction.

As discussed in §2.2, the three methods described here are merely three instances of a meta-algorithm for scalar-on-image regression. The refund.wave package allows for straightforward incorporation of alternative penalties, and other extensions may allow for more-refined wavelet-domain algorithms, which may improve the stability and reproducibility of the coefficient image estimates (Rasmussen et al., 2012). For instance, in wavelet-based nonpara-metric regression, thresholding is often performed in a level-specific manner. Analogously, it might be appropriate to modify criterion (11) so as to di erentially penalize coe cients at di erent levels. One might also employ resampling techniques (cf. Meinshausen and Bühlmann, 2010) to select those wavelet basis elements that are consistently predictive of the outcome. Finally, wavelets whose domain is anatomically customized, such as the wavelets defined on the cortex by Özkaya and Van De Ville (2011), offer a promising new way to confine the analysis to relevant portions of the brain.