Multivariate semiparametric spatial methods for imaging data

SUMMARY

Univariate semiparametric methods are often used in modeling nonlinear age trajectories for imaging data, which may result in efficiency loss and lower power for identifying important age-related effects that exist in the data. As observed in multiple neuroimaging studies, age trajectories show similar nonlinear patterns for the left and right corresponding regions and for the different parts of a big organ such as the corpus callosum. To incorporate the spatial similarity information without assuming spatial smoothness, we propose a multivariate semiparametric regression model with a spatial similarity penalty, which constrains the variation of the age trajectories among similar regions. The proposed method is applicable to both cross-sectional and longitudinal region-level imaging data. We show the asymptotic rates for the bias and covariance functions of the proposed estimator and its asymptotic normality. Our simulation studies demonstrate that by borrowing information from similar regions, the proposed spatial similarity method improves the efficiency remarkably. We apply the proposed method to two neuroimaging data examples. The results reveal that accounting for the spatial similarity leads to more accurate estimators and better functional clustering results for visualizing brain atrophy pattern.

Functional clustering; Longitudinal magnetic resonance imaging (MRI); Penalized B-splines; Region of interest (ROI); Spatial penalty.

1. INTRODUCTION

As a noninvasive tool, magnetic resonance imaging (MRI) provides insightful information for understanding how brain changes due to a certain disorder or as people naturally grow. Structural MRIs are often preprocessed using FSL (http://fsl.fmrib.ox.ac.uk/fsl/fslwiki/) or Freesurfer (http://freesurfer.net) generating regions of interest (ROIs) data, which are based on well established neuroanatomic atlas. ROI parcellation results in a significant reduction in the number of brain ROIs latterly used in the statistical analysis. This reduction is desirable for clinical researchers who are interested in a priori ROIs typically built on popular neuroanatomic atlas. Accordingly, the goal of this work is to develop multivariate semiparametric methods for improving efficiency in ROI analysis. The complicated structures of imaging data drive the needs of efficient statistical methods that can account for the spatial and temporal structures. Along this line, for voxel-based analysis, Li and others (2011), Skup and others (2012), and Zhu and others (2014) proposed multiscale adaptive methods for neuroimaging data. Those methods combine spatial homogeneous neighboring voxels to improve efficiency. Aiming for prediction, Hyun and others (2014, 2016) developed functional principal component-based spatial and spatio-temporal Gaussian process models for longitudinal neuroimaging data. Goldsmith and others (2014) proposed spatial Bayesian methods for using neuroimaging data to predict clinical outcomes. For ROI analysis, George and Aban (2015) proposed a parametric spatial–temporal model for longitudinal imaging data.

Accurately and efficiently modeling neurodevelopmental/age trajectories plays an important role in studying neurodevelopmental disorders (Insel, 2014), neurodegenerative disorders, and aging. As a motivational example, Figure 1 shows the loess fitted smooth age trajectories of the normalized regional volumes (regional volume divided by the intracranial brain volume) for the healthy controls in the Predict-Huntington’s disease (HD) control sample (Paulsen and others, 2014). The corresponding left and right brain ROIs and different components of corpus callosum are similar in the mean age trajectories. In practice, univariate semiparametric approaches are often used in modeling the nonlinear age or time trajectory at each ROI. For instance, univariate semiparametric methods are applied in modeling the mean and quantile functions of neurodevelopmental trajectories in Reiss and others (2014) and Chen and others (2015), respectively. However, as they ignore the spatial information, univariate semiparametric methods may lead to low efficiency.

Fig. 1.

Predict-HD control sample: Loess fitted smooth age trajectories of the normalized regional volumes (regional volume divided by the intracranial volume). Top left panel: left/right cerebellum white matter; top right panel: left/right thalamus; bottom left panel: left/right cortical white matter; bottom right panel: anterior, middle-anterior, central, middle posterior, and posterior corpus callosum.

To improve estimation efficiency, we propose a spatial similarity penalty to account for similarity in nonlinear age or time trajectories among the ROIs. Specifically, we use penalized B-splines (de Boor, 1978; Eilers and Marx, 1996) to fit the trajectories of the ROIs, and add a spatial penalty to constrain the variation of the spline coefficients among those similar ROIs. Notably, our proposed methods are different from those temporal–spatial methods for voxel-based analysis, in the sense that we do not assume spatial smoothness for ROI data. Spatial smoothness is an appropriate assumption for voxel-based analysis in a homogeneous neighborhood. However, it is invalid for aggregated ROI data. Besides the spatial similarity, the proposed method can also account for the spatial correlation, and is applicable to both cross-sectional and longitudinal ROI-level imaging data. Furthermore, we conduct distance-based functional clustering analysis (Kaufman and Rousseeuw, 2009; Chen and others, 2014) to study the atrophy pattern of brain regional volumes and cortical thickness. Our simulation studies show that the proposed method improves the efficiency remarkably in estimating the mean and first-derivative functions. The two real data examples also reveal that incorporating spatial similarity leads to less variable estimates and improved cluster separation in functional clustering analysis.

For the asymptotic properties of penalized spline estimate, Claeskens and others (2009), Kauermann and others (2009), Li and Ruppert (2008), and Wang et al. (2011) derived asymptotic bias and variance rates of univariate estimate based on cross-sectional data. Chen and Wang (2011) and Chen and others (2013) studied the asymptotic properties of penalized spline estimate for longitudinal data. Yoshida and Naito (2012) provided asymptotic properties of penalized spline estimates under an additive model. In this work, we develop the asymptotic rate of the bias and covariance functions of the proposed spatial estimator and its asymptotic normality. Notably, because our outcomes are measured at multiple ROIs, the aforementioned asymptotic results for univariate outcomes do not carry over.

The rest of the paper is structured as follows. Section 2 reviews the univariate penalized B-spline regression. Section 3 presents the proposed multivariate semiparametric model and spatial method. Section 4 shows the asymptotic rates for the bias and variance function as well as the asymptotic normality for the proposed spatial estimator. Sections 5 and 6 present the simulation studies and two neuroimaging examples including both cross-sectional and longitudinal ROI-level imaging data.

2. PRELIMINARY: UNIVARIATE SEMIPARAMETRIC REGRESSION

We first introduce some notations. Let and be the number of subjects and the number of ROIs, respectively. For the th subject, is the imaging measure such as volume and cortical thickness at the th ROI, is a continuous variable of interest such as age or follow-up time, and are the covariates to be adjusted. Denote as the outcome at the th ROI, and denote as the imaging measure matrix. The following univariate semiparametric models are often used in modeling nonlinear neurodevelopmental trajectories (Reiss and others, 2014). Those models treat each ROI separately, and consequently fail to borrow information from other ROIs.

$$\begin{array}{ccc}& y_{ir}=f_r(t_i)+z_i^T{\beta }_{z,r}+{ϵ}_{ir},& \\ & {ϵ}_{ir}{\sim }^{\mathrm{i}.\mathrm{i}.\mathrm{d}.}N(0,{\sigma }_r^2),i=1,\cdots ,n;r=1,\cdots ,R,& \end{array}$$

where , and are the unspecified age or time trajectory, the coefficients of the covariates, and the variance of the measurement errors at the th ROI, respectively. Let be a vector of B-splines basis functions, which is defined recursively (de Boor, 1978). Assume , where is the spline coefficients. Denote as the matrix of basis functions, denote , denote as the stacked design matrix, and denote . We define the following penalized least squares function,

$$l_p({\theta }_r)=(Y_r-B{\beta }_r-Z{\beta }_{z,r}{)}^T(Y_r-B{\beta }_r-Z{\beta }_{z,r})+{\lambda }_r{\beta }_r^T{P}_2{\beta }_r.$$ (2.1)

Here is a matrix, and is a tuning parameter, which can be chosen by restricted maximum likelihood (REML, Reiss and Todd Ogden, 2009). The penalty term controls the smoothness of the estimated function and balances its bias and variance. Let be a block diagonal matrix, where is an matrix with all zero components. The estimate minimizing (2.1) is given by

$${\check{\theta }}_r=\arg \underset{{\theta }_r}{min}{l}_p({\theta }_r)=(X^{T}X+{\lambda }_r{P}_2^{\ast }{)}^{-1}{X}^T{Y}_r.$$

Hence, the estimated trajectory , where is the identity matrix. Besides the estimation procedure, the inference is often drawn separately for each without accounting for any spatial information either. To improve efficiency by borrowing information from similar ROIs, we extend the univariate method to multivariate methods in the next section, which account for the spatial similarity and correlation. The univariate semiparametric model-based estimates will be used as initial values for those multivariate methods.

3. MULTIVARIATE SEMIPARAMETRIC SPATIAL REGRESSION

For notation simplicity, we present the model and methods for the cross-sectional case in this section, and show an extension of the method to longitudinal imaging data under a spatial-independent structure in the supplementary material available at Biostatistics online.

Let be the age or time trajectories at ROIs , let , and let . Denote as the measurement error of the multivariate imaging outcome for the th subject, . We consider the following multivariate semiparametric model.

$$\genfrac{}{}{0ex}{}{Y}{n\times R}\genfrac{}{}{0ex}{}{=}{}\genfrac{}{}{0ex}{}{F}{n\times R}\genfrac{}{}{0ex}{}{+}{}\genfrac{}{}{0ex}{}{Z}{n\times m}\genfrac{}{}{0ex}{}{{\beta }_z}{m\times R}\genfrac{}{}{0ex}{}{+}{}\genfrac{}{}{0ex}{}{ϵ,}{n\times R}$$

where , and are independent and follow a multivariate normal distribution . Here is an covariance matrix of the ROIs, which measures the spatial correlations among them. Interestingly, the inverse covariance matrix is the concentration matrix, which measures the partial correlation between two ROIs conditional all the other ROIs (Yuan and Lin, 2007).

In the following part of this section, we adopt a multivariate semiparametric method to incorporate the spatial correlation first then propose a new spatial method accounting for both spatial correlation and similarity among the ROIs.

3.1 Multivariate semiparametric method accounting for spatial correlation

Denote and . The penalized log-likelihood function is given as following.

$$\begin{array}{c}{\tilde{l}}_p(\theta )=\sum _{i=1}^n[Y_{[i]}-(I_R\otimes X_i)\theta {]}^T{\mathrm{\Sigma }}^{-1}[Y_{[i]}-(I_R\otimes X_i)\theta ]+\sum _{r=1}^R\left\{{\lambda }_r{\theta }_r^T{P}_2^{\ast }{\theta }_r\right\},\end{array}$$

where is the identity matrix, and is the Kronecker product. Let be a diagonal matrix with diagonal components . The estimate minimizing (3.1) is given as following, while the derivation is provided in the supplementary material available at Biostatistics online.

$$\begin{array}{c}\tilde{\theta }={\left[{\mathrm{\Sigma }}^{-1}\otimes \left(\sum _{i=1}^n{X}_i^T{X}_i\right)+\mathrm{\Lambda }\otimes P_2^{\ast }\right]}^{-1}\sum _{i=1}^n(I_R\otimes X_i^T){\mathrm{\Sigma }}^{-1}{Y}_{[i]}.\end{array}$$

Under a parametric model, this method is similar to the seemingly unrelated regressions developed by Zellner (1962), which is implemented using the feasible generalized least squares method (Amemiya, 1985).

Since depends on the unknown , it is crucial to obtain an efficient estimate of or equivalently. {Practically, we can first obtain the residuals from the univariate models, and then use the residuals to estimate or . For a low-dimensional case, unstructured covariance or some parametric form can be assumed such as in George and Aban (2015). For a high-dimensional case, graphical lasso can be used to obtain a sparse and stable estimate of the inverse covariance matrix (Yuan and Lin, 2007; Zhao and others, 2012).

3.2. Multivariate semiparametric method accounting for spatial similarity

We propose to use a spatial penalty, when regional age trajectories show remarkable similarity. Specifically, we use the following penalty:

$$p(\beta )=\sum _{r=1}^R{\lambda }_r{\beta }_r^T{P}_2{\beta }_r+\gamma \sum _{1\le r,s\le R}{a}_{rs}({\beta }_r-{\beta }_s{)}^T{P}_1({\beta }_r-{\beta }_s),$$ (3.2)

where is a tuning parameter to control the spatial similarity penalty, and is a pre-determined or data-driven incidence matrix whose element is an indicating variable of similarity among the ROIs. The penalty matrix , thereby . In (3.2), the first penalty term is used to achieve smoothness in the estimated age or time trajectories, while the second penalty term is used to constrain variations of the trajectories among those similar ROIs defined by the incidence matrix .

First, assuming a spatial-independent structure for the ROIs, the penalized log-likelihood function is given as following.

$$\begin{array}{c}{l}_p(\theta )=\sum _{r=1}^R\left\{\frac{\Vert Y_r-X{\theta }_r{\Vert }^2}{{\sigma }_r^2}+{\lambda }_r{\theta }_r^T{P}_2^{\ast }{\theta }_r\right\}+\gamma \sum _{1\le r,s\le R}{a}_{rs}({\theta }_r-{\theta }_s{)}^T{P}_1^{\ast }({\theta }_r-{\theta }_s),\end{array}$$ (3.3)

where is a block diagonal matrix defined similarly as . Let, let with the row sum , , and let . The estimated coefficients are given by

$$\begin{array}{c}\hat{\theta }=M^{-1}(V^{-1}\otimes X{)}^T\mathrm{v}\mathrm{e}\mathrm{c}(Y),\end{array}$$ (3.4)

where is the vectorizing operator for a matrix. Consequently,

$$\begin{array}{ccc}\hfill \hat{F}(t)& \hspace{0.17em}=& \hspace{0.17em}[I_R\otimes \{B(t{)}^T,0_m^T\}]\hat{\theta },\text{and}\hfill \end{array}$$ (3.5)

$$\begin{array}{ccc}\hfill {\hat{f}}_r(t)& \hspace{0.17em}=& B(t{)}^T{\hat{\beta }}_r,\text{for}r=1,\cdots ,R.\hfill \end{array}$$ (3.6)

The derivation of the above formulas and their extension to longitudinal imaging data setting are given in the supplementary material available at Biostatistics online.

Second, given an estimated spatial covariance , the penalized log-likelihood function is

$$\begin{array}{ccc}\hfill l_p(\theta )& \hspace{0.17em}=& \hspace{0.17em}\{\mathrm{v}\mathrm{e}\mathrm{c}(Y)-(I_R\otimes X)\theta {\}}^T({\hat{\mathrm{\Sigma }}}^{-1}\otimes I_n)\{\mathrm{v}\mathrm{e}\mathrm{c}(Y)-(I_R\otimes X)\theta \}\\ & & \hspace{0.17em}+\sum _{r=1}^R{\lambda }_r{\theta }_r^T{P}_2^{\ast }{\theta }_r+\gamma \sum _{1\le r,s\le R}{a}_{rs}({\theta }_r-{\theta }_s{)}^T{P}_1^{\ast }({\theta }_r-{\theta }_s).\hfill \end{array}$$

By slight abuse of notation, we define . The estimated coefficients are given by

$$\begin{array}{c}\hat{\theta }=M^{-1}({\hat{\mathrm{\Sigma }}}^{-1}\otimes X{)}^T\mathrm{v}\mathrm{e}\mathrm{c}(Y).\end{array}$$ (3.7)

We can see in (3.4) and (3.7) that the spatial information is accounted by using all the relevant data in estimating the coefficients, i.e. borrowing information from similar ROIs. We use REML and 5-fold cross-validation to select tuning parameters and , respectively.

As in (3.5) is a linear smoother, its inference can be derived based on the distribution of . The bias and sandwich variance formula for are given as following.

$$\begin{array}{ccc}\hfill \mathrm{B}\mathrm{i}\mathrm{a}\mathrm{s}[\hat{F}(t)]& \hspace{0.17em}=& \hspace{0.17em}[I_R\otimes \{B(t{)}^T,0_m^T\}]M^{-1}(\gamma A\otimes P_1^{\ast }-\gamma A^{\ast }\otimes P_1^{\ast }-\mathrm{\Lambda }\otimes P_2^{\ast })\theta ,\hfill \end{array}$$ (3.8)

$$\begin{array}{ccc}\hfill \mathrm{V}\mathrm{a}\mathrm{r}[\hat{F}(t)]& \hspace{0.17em}=& \hspace{0.17em}[I_R\otimes \{B(t{)}^T,0_m^T\}]M^{-1}[V^{-1}\mathrm{\Sigma }{V}^{-1}\otimes X^{T}X]M^{-1}[I_R\otimes \{B(t{)}^T,0_m^T{\}}^T].\hfill \end{array}$$ (3.9)

The spatial penalty introduces bias which shows up in those terms involving in (3.8). Similarly, the smoothing penalty also introduces shrinkage bias showing in the last term with . However, the sandwich variance tends to be smaller due to the shrinkage in . We show that under certain conditions both biases go to zero as goes to infinity at certain rates. The variance of the estimated functions is inter-related, thus the proposed method could be more efficient by borrowing information from similar regions.

The spatial penalty depends on the incidence matrix . To obtain an matrix, we first compute the distance among those regional trajectories based on the initial estimates from the univariate semiparametric models in Section 2, then derive by thresholding . Scaling by the norm of each function or the column norm of is needed to account for the differences in scale across regions. Note that vertical shift effects can be removed by either centering the difference function or using the first-derivative function .

4. ASYMPTOTIC PROPERTIES

In this section, we show the asymptotic rates for the bias and covariance functions and asymptotic normality of the proposed estimates in (3.6), which is under an spatial-independent structure but assuming spatial similarity. For simplicity, we only present the results for the nonparametric term. Also notice that the parametric term converges at a faster rate. Let be the set of all the functions that are differentiable in the interval . Define , where are the spline knots, is the indicator function, and is the th Bernoulii polynomial. Zhou and others (1998) showed that if , then is the approximation bias of splines.

Theorem 4.1

Let for . Assume assumptions A1–A3 in the supplementary material available at Biostatistics online hold, , , , and . Then for ,

$$\begin{array}{ccc}& E\{{\hat{f}}_r(t)\}-f_r(t)=b_{r,a}(t)+b_{r,{\lambda }_r}(t)+b_{r,\gamma }(t)+o_p(K_n^{-(p+1)})+o_p(\frac{{\lambda }_r{K}_n^q}{n})+o_p(\frac{\gamma K_n}{n}),& \\ & \mathrm{c}\mathrm{o}\mathrm{v}\{{\hat{f}}_r(t),{\hat{f}}_s(t)\}=\frac{{\sigma }_{rs}}{n}B(t{)}^T{G}_r^{-1}G{G}_s^{-1}B(t)(1+o_p(1))+O_p(\frac{K_n}{n}),& \end{array}$$ (4.1)

where , , , and . Here is the design density function.

In (4.1), there are three bias terms due to splines approximation and shrinkage penalties. Specifically, is the approximation bias, and are the shrinkage bias due to the smoothing penalty and spatial penalty, respectively. The asymptotic rates for the bias and variance functions (when ) are the same as those in a univariate case shown in Claeskens and others (2009).

Remark: Due to the spatial penalty, there is an additional bias term in (4.1); however, the spatial shrinkage may lead to smaller asymptotic variances.

Theorem 4.2

Assume that there exists such that , and assumptions A1–A3 in the supplementary material available at Biostatistics online hold. Furthermore, assume , with , and . Then, for any fixed , as ,

$$\sqrt{\frac{n}{K_n}}\{{\hat{f}}_1(t)-E{\hat{f}}_1(t),\cdots ,{\hat{f}}_R(t)-E{\hat{f}}_R(t){\}}^T{⟶}^{d}N(0,\mathrm{\Omega }),$$

where with for .

When and , Theorem 4.2 is consistent with the univariate case in Kauermann and others (2009). Sketch of the proofs is provided in the supplementary material available at Biostatistics online.

5. SIMULATION STUDIES

In this section, we investigate the performance of the different methods in Section 3. Specifically, for the cross-sectional cases, we compare six different methods including the spatial similarity method with true (SSC-T), estimated covariances (SSC), and spatial independence (SS-IND), spatial correlation method with true (SC-T) and estimated spatial covariances (SC), and the massive univariate method (Univariate). {For the longitudinal cases, we compare the spatial similarity method with the massive univariate method. The findings are similar as the cross-sectional cases, which are shown in the supplementary material available at Biostatistics online.

The design point is generated from uniform distribution . The mean functions are derived from either the or sigmoid function . We generate outcomes for subjects with ROIs. The mean functions of those ROIs consecutively vary as following:

$$\begin{array}{c}{f}_r(t)=g(r)\ast f[t+{\delta }_1(r)]+{\delta }_2(r),r=1,\cdots ,R,\end{array}$$

where and are drift functions in the - and -direction, respectively, and is a function controlling the amplitudes of those mean functions. The derived mean functions are location and shape transformed functions of . We use , , for the sine functions and for the sigmoid functions. Note when the ROIs are close to each other, their mean functions have similar shapes. For example, when ROIs and are next to each other, the mean functions and are similar in the sense that the horizontal and vertical deviations between them are small.

The outcomes are generated from the following model:

$$\begin{array}{ccc}& y_{ir}=f_r(t_i)+{ϵ}_{ir},r=1,\cdots ,R,& \\ & {ϵ}_{[i]}=({ϵ}_{i1},\cdots ,{ϵ}_{iR}{)}^T{\sim }^{\mathrm{i}.\mathrm{i}.\mathrm{d}.}N(0,\mathrm{\Sigma }),i=1,\cdots ,n.& \end{array}$$

Both compound symmetry (CS) with and autoregressive model of order 1 (AR-1) with spatial correlation structure are considered. For simplicity, we assume the same variance for all the ROIs. for the sigmoid mean function cases and for the sine mean function cases.

We use cubic B-splines to fit the regional trajectories. We apply the six different methods: the spatial similarity method with true, estimated, and independent covariances, multivariate method with true and estimated spatial covariances, and the massive univariate method to the 500 simulated datasets. We use graphical lasso implemented in the huge package to estimate the inverse covariance matrix .The incidence matrix is chosen based on the initial estimates from the massive univariate nonparametric models. To evaluate the estimation accuracy, the average mean squared error (MSE) of the estimated functions is computed over the 500 simulations.

Figure 2 shows the MSE of the estimated mean (top four panels) and first-derivative functions (bottom four panels) with AR-1 (first two columns) or CS (last two columns) spatial correlation. The first and third columns are with the sigmoid mean functions, and the second and last columns are with the sine mean functions. Overall, we observe similar patterns of MSE in both the estimated mean and first-derivative functions across all the four cases. The proposed methods incorporating the spatial similarity (SSC and SS-IND) have the smallest MSEs compared to both the multivariate method accounting for spatial correlation (SC) and the massive univariate method (Univariate). By accounting for both spatial similarity and correlation, the reduction in MSE of SSC in estimating the mean functions is up to 33% compared to the massive univariate approach and is up to 31% compared to SC. Analogously, the reduction in MSE of SSC in estimating the first-derivative functions is up to 75% compared to the massive univariate approach and is up to 68% compared to SC. By accounting for the spatial correlation, SSC reduces the MSE up to 17% compared to SS-IND under the spatial independence. The graphical lasso provides comparable estimate to the corresponding ones assuming the true spatial correlation matrix. The spatial method with spatial-independent structure (SS-IND), estimated inverse spatial covariance (SSC), and the true spatial covariance (SSC-T) show increasing estimation accuracy. The best approach is the one incorporating both the spatial correlation and spatial similarity. The multivariate approach SC is more efficient than the massive univariate approach. The massive univariate approach is the least efficient one, since it ignores both the spatial correlation and the spatial similarity among the ROIs.

Fig. 2.

Simulation studies: MSE of the estimated mean (top four panels) and first derivative functions (bottom four panels) in the cross-sectional case with AR-1 (first two columns) or CS (last two columns) spatial correlation. The first and third columns are with the sigmoid mean functions, and the second and last columns are with the sine mean functions. Here, SSC-T, SSC, SS-IND, SC-T, SC, and Univariate are the spatial similarity method with true, estimated, and working independence spatial covariances, multivariate method with true and estimated spatial covariances, and the massive univariate method, respectively.

6. APPLICATIONS

In this section, we illustrate our methods using both cross-sectional and longitudinal neuroimaging datasets. Our goals are to investigate (i) how the brain regional structures change as people get older and (ii) how different ROIs grow/degenerate together or apart, and ultimately to understand and explain age-related physical and cognitive decline.

6.1. The control sample of the Predict-HD study

We consider the 248 controls in the Predict-HD study (Paulsen and others, 2014), whose age ranges from 19 to 67 years. Their baseline structural MRIs were preprocessed using Freesurfer 5.2, generating 42 global and subcortical regional brain volumes (see, Fischl and others, 2002; Paulsen and others 2014). To reduce the variation due to individual’s head size, we apply the following normalization procedure: (i) regional brain volumes are divided by the total intracranial volume; (ii) log-transformation is applied to handling the skewness of the outcomes; and (iii) centering and scaling are applied to reducing the heterogeneity among the ROIs, thus the regional volumes are transformed into z-scores.

For the first goal, we apply the massive univariate nonparametric method, spatial similarity methods with graphical lasso estimated inverse covariance (SSC) and with spatial-independent structure (SS-IND), and the spatial method with graphical lasso-estimated inverse covariance (SC) to the imaging data. We use cubic splines to fit the mean age trajectories. The SSC has similar results as the SS-IND method, while the SC method has similar results as the massive univariate method. Therefore, we only show the estimated mean age trajectories together with their first-derivative functions for the massive univariate and SS-IND methods in Figure 3. The first-derivative functions of these age trajectories measure the rate of brain structural changes. The differences between the univariate smoothing estimates and the SS-IND estimates are minor for the mean age trajectories. However, the differences in the first derivative of those age trajectories are non-ignorable especially in the two tails, that is before age of 30 years and after age of 60 years. For example, in estimating the rate of change at the Thalamus, Cerebellum Cortex, Ventral DC, Cortex gray matter, anterior and middle anterior parts of the corpus callosum, the SS-IND method leads to more stable estimates.

Fig. 3.

Predict-HD control sample: estimated mean age trajectories (first and third columns) and their first-derivative functions (second and fourth columns).

For the second goal, we utilize the distance-based functional clustering method via the pam function in the R package cluster. The interesting findings are summarized in the supplementary material available at Biostatistics online.

In summary, although the estimated mean age trajectories are similar for the univariate approach and multivariate spatial approaches, their first-derivative functions do show considerable deviations among the different approaches, and the spatial similarity approaches are more stable and lead to better clustering partition. We find that as people get older, the ventricular system shows remarkable expansion, while the cortical gray matter, white matter, corpus callosum, and subcortical ROIs show considerable atrophy, which may be associated with age-related physical and cognitive dysfunctions.

6.2. The HIV and aging study

Due to the effective antiretroviral therapy for human immunodeficiency virus (HIV), many infected people are able to live to an old age. Given the growing population, there are demanding needs to study how those HIV-infected people change in cognition and brain structure as they age. The HIV and aging study includes 106 HIV+ subjects and similar HIV controls with age ranging from 20 to 75 years (Seider and others, 2014). Among those HIV+ subjects, 63% are male, and 39% also have hepatic C virus (HCV). Their mean education is 12.4 (SD=2.2), and mean baseline HIV infection duration is 12.7 years (SD=6.9). Structural images were collected at baseline and yearly follow-up visits up to 3 years. The images were preprocessed using Freesurfer, leading to 148 regional cortical thickness measures. Specifically, 74 cortical ROIs listed in Destrieux and others (2010) are delineated in each hemisphere including 21 frontal ROIs, 8 insular ROIs, 8 limbic ROIs, 11 temporal ROIs, 11 parietal ROIs, and 15 occipital ROIs.

Analogous to the previous example, to investigate how their cortical ROIs change as HIV+ subjects age, we compute their mean age trajectories of cortical thickness, adjusting for infection duration, HCV status, education, and sex. We apply the SS-IND and the massive univariate semiparametric methods to this longitudinal example. Cubic splines are used to fit the mean age trajectories. Random intercepts are used to account for the temporal correlation for the massive univariate semiparametric approach. For SS-IND, we assume a CS structure for the temporal correlation and a spatial-independent structure.

The differences in the estimated mean age trajectories of cortical thickness are small between the massive univariate semiparametric method and the SS-IND method. However, the discrepancies in their first derivative functions are considerable. To efficiently visualize the age trajectories, we use functional clustering to group ROIs into homogeneous clusters according to their age trajectories. The centered age trajectories of cortical thickness in the nine clusters selected by silhouette information are displayed in Figure 4. The clustering partition in brain surface view is shown in Figure 5. Additional summary is provided in the supplementary material available at Biostatistics online. As shown in Figures 4 and 5, the left and right hemisphere show different atrophy patterns but no remarkable difference. All six areas including frontal, insular, limbic, temporal, parietal, and occipital areas are impacted by aging and HIV, {which may explain the HIV-associated neurocognitive disorder in older adults with long-standing HIV infection.} The insular, limbic, and temporal areas show more atrophy, as most of the ROIs in these areas are in clusters 3–6. The left frontal area has more atrophy compared to the right frontal area. Except for the frontal area, the other five areas show relatively symmetric atrophy pattern.

Fig. 4.

HIV and aging study: centered age trajectories of cortical thickness in the nine clusters with the cluster medoids in bold.

Fig. 5.

HIV and aging study: the nine-cluster partition cortical map based on the age trajectories of regional cortical thickness.

7. DISCUSSIONS

In this work, we propose multivariate semiparametric methods to account for spatial information in imaging data analysis. We show in both simulation studies and the two data examples that there is non-ignorable spatial similarity information, which is often ignored in practice. By accounting for the spatial similarity among ROIs, we are able to estimate the mean age trajectories and their first-derivative functions more accurately. Furthermore, incorporating the spatial information improves the functional clustering partitions, which is a desirable way to visualize brain growth or atrophy patterns.

The proposed methods can be extended to multimodal imaging data. For example, besides the structural MRI, the proposed methods can be applied to diffusion tensor imaging-derived white matter integrity measures such as fractional anisotropy and mean diffusivity, and magnetic resonance spectroscopy-derived brain metabolic measures including choline, N-acetyl-aspartate, and creatine. Between modality similarity can be considered in the same way as within modality similarity. As pointed out by a reviewer, the proposed methods could be applied to general multivariate longitudinal data as well.

Extension to more flexible models such as varying coefficient models and additive models is feasible. The estimation procedure can be directly carried over, while the inferences associated with those models are more complicated given the multiple comparison problem and worth further investigation. Additionally, missing data in longitudinal studies is not uncommon. However, due to the high-dimensionality and complex structure of neuroimaging data, missing data problem is usually not directly addressed in neuroimaging analysis. Our proposed method is valid under the assumption of missing completely at random. Further research along this line is needed to efficiently address the missing data issue for longitudinal neuroimaging data. Aiming to estimate the mean age trajectories of brain structural indices, we only considered marginal models. To improve the prediction performance especially by incorporating individual longitudinal trajectories, conditional models including subject-specific and region-specific random effects could be considered in the future.

SUPPLEMENTARY MATERIAL

Supplementary material is available at http://biostatistics.oxfordjournals.org.

FUNDING

Center for Cognitive Aging and Memory at the University of Florida, the McKnight Brain Research Foundation, and the Claude D. Pepper Center at the University of Florida (P30 AG028740) to H.C.; Simons Foundation (#354917) to G.C. National Institute of Mental Health (R01 MH074368) to R.C.