Smoothing Spline ANOVA Models: 297 (Springer Series in Statistics)

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Garrott Uncategorised lezen gratis online. Boek lezen gratis online lezen gratis online. The Boek is already in the wishlist! Launer Uncategorised lezen gratis online. Advances in Growth Curve Models: Topics from the Indian Statistical Institute: Progress in Partial Differential Equations: PET images for each subject were aligned and resliced [Woods et al. We propose the use of mixed effects regression spline models to characterize distributed patterns of localized brain responses to varying dose levels.

Corresponding to the ethanol study, we present our model for a PET study and later comment on modifications for fMRI studies. Define Y is v as the rCBF measurement for the i th subject at location voxel v at scan s , and x is is the level of the stimulus associated with scan s of subject i. We include the global rCBF term in the model to control for the variability in the global rCBF across scans that may confound the estimation of the effects of the stimulus. Inclusion of the subject random effect takes account of the between-subject variation in rCBF and also accounts for within-subject correlations among the serial rCBF measurements obtained from the same subject.

Regression splines are popular tools for performing nonparametric curve estimation. Essentially, a regression spline is a piecewise polynomial function. We divide the covariate space, here the dose range, into intervals and fit a local polynomial curve within each interval.

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The continuity of the overall curve across intervals is obtained by using a truncated power basis. From 2 , we see that the regression spline is a piecewise polynomial curve. A higher order power basis, i. In addition to the truncated power basis, other basis functions for regression splines also exist, such as those for B-splines and radial basis functions Green and Silverman, The truncated power basis is very popular due to its simplicity and generally good performance.

An important task in fitting a regression spline is to select the number and placement of knots. Knots are selected from the distinct values of the covariate being modeled. The number of knots depends on the characteristics of the data such as the observed covariate values and the complexity of the underlying response function Ruppert, The computations for these procedures may be time-consuming for imaging data if one considers all possible numbers of knots. In practice, regression splines generally require a small number of knots, with 3—10 knots typically sufficing Abrahamowicz et al.

Hence, one could fit regression splines at each voxel, with the number of knots ranging from 3—10, and then identify the optimum number of knots that has the minimum AIC for each voxel. Since the observed covariate values stimulus levels for our data are the same for all voxels, the optimum number of knots is usually very close across voxels.

To increase the comparability of the fitted regression splines across various brain areas, we propose to select a global optimal number of knots by minimizing the overall AIC statistic that is summarized across the brain.

In the special cases where the optimum number of knots is quite variable across voxels, one can use the voxel-specific number of knots determined by the AIC statistic, which allows the fitted regression splines to provide a more tailored fit for each voxel, especially for voxels with extreme response functions. However, varying number of knots reduces the comparability of the splines across voxels.

After deciding on the number of knots, we need to select the locations of the knots. A general guideline is that the placement of knots should appropriately capture the distribution of the associated covariate. For example, it is suggested that the best placement is to have about equal number of data points between knots Breiman, and hence one may place knots at equally spaced quantiles of the observed covariate values Ngo and Wand, Since the covariate values are the same across voxels, we specify the same placement of the knots for all brain areas.

The proposed mixed effects regression spline model in 1 not only allows us to flexibly model the dose-dependent effect of the stimulus but also provides a convenient framework for investigating the effects of other factors on brain function while controlling for the effects of the stimulus and subject variability. For example, subjects may be asked to perform certain tasks during the administration of the stimulus, and researchers may have an interest in the task-related effects on brain function after taking into account the current stimulus level.

The fitted regression spline model reveals a dynamic profile of regional brain activity associated with varying stimulus levels. From the fitted regression spline model, we can readily estimate the brain activity at any stimulus level within the observed range, which is a great advantage over correlation analysis and ANOVA.

The proposed mixed effects regression spline model also offers us a powerful tool to derive statistics for measuring and testing specific characteristics in brain response to the stimulus. For example, we can use the parameter estimates from the fitted regression spline model to construct a conventional t-statistic map to test the localized differences in brain activity between two levels of the stimulus; or we can evaluate the rate of change in brain activity in response to the increasing stimulus levels by examining the slope of the fitted spline at each voxel.

Another objective in dose-dependent neuroimaging studies involves identifying groups of brain areas with similar response patterns. To address this question, we can apply results from the mixed effects regression spline model to spatially group brain regions into clusters in such a way that voxels within the same cluster exhibit similar patterns of alterations in response to the stimulus. This approach treats all areas of the response profile equally in classifying the brain areas. In cases where certain areas in the response profile are of little interest or have too much noise to be estimated reliably, we can characterize the response pattern by subsampling several key points from the response profile.

Specifically, we can either choose stimulus levels that are associated with clinical interpretations or select several representative levels within the observed range of the stimulus. To control for the difference in the baseline activity across voxels, we evaluate the brain activity at selected stimulus levels a 1 , …, a q relative to a baseline level a 0.

The contrast statistic t j v has the same asymptotic distribution at all voxels under the null hypothesis and hence is comparable across different brain regions. This vector of contrasts depicts the relative activation of voxel v at selected stimulus levels.

A cluster analysis could be performed on this vector T v to identify groups of voxels that display similar response profiles. The proposed mixed effects regression spline model is convenient to fit in practice, but suffers from the limitation that it requires an analyst to select the number and placement of knots which is a time-consuming process for imaging data. As an alternative, we consider the use of a smoothing spline model to characterize the brain activity in response to a stimulus.

Smoothing spline is a spline method that avoids knot selection by treating every distinct covariate value as a knot and hence using the maximum set of knots. The complexity or smoothness of a smoothing spline is controlled by a single parameter. In the following, we first give a brief introduction of smoothing spline and then present a mixed effects smoothing spline model for modeling the dose-dependent effect of a stimulus on brain activity.

Estimation of the proposed mixed effects smoothing spline model can be performed under the linear mixed model framework through reparameterization. There are two major criteria in fitting a smoothing spline: The RSS reflects the two objectives in fitting a smoothing spline.

The first term measures the closeness of the curve to the data using a squared error loss function, and the second term penalizes curvature, or roughness, in the curve. The smoothing parameter can beusually chosen using the cross-validation approach Green and Silverman, Smoothing splines have a knot at every distinct value of the covariate the stimulus level x for our data and hence do not require the selection of the number and placement of knots, which is an advantage over regression splines.

Obviously, treating every covariate value as a knot results in over-parameterization. A natural cubic spline is a smooth curve that is piecewise cubic polynomial between adjacent knots and continuous up to its second derivative at each knot. Beyond the boundary knots, a natural cubic spline is forced to be linear to avoid wild behavior of cubic polynomials beyond boundaries.

Green and Silverman provide a detailed description of the natural cubic spline. To present our smoothing spline model, we use similar notations as in the regression spline model. Denote Y is v as the brain activity measurement for the i th subject at location voxel v at scan s , and x is is the level of the stimulus associated with scan s of subject i. To model the dose-dependent effect of the stimulus and to take into account of within-subject correlations among the repeated measurements, we propose the following mixed effects smoothing spline model,.

Inclusion of the subject random effect terms takes account of the between-subject variation in brain activity measurements and helps to account for within-subject correlations among the repeated measurements obtained from the same subject. As we recall from the previous section, the smoothing spline is estimated by minimizing the lack of fit of the curve and penalizing the roughness of overfitting. The penalized likelihood is composed of two terms which reflect the two objectives in fitting smoothing splines. The first term is the likelihood of the smoothing function and parameters based on the observed data, which boils down to the mean squared error that measures the lack of fit between the observed brain activity measurements and the smoothing spline model.

Estimation of the parameters in the mixed effects smoothing spline model 4 , although seemingly complicated, can be performed under the familiar framework of linear mixed models after appropriate reparameterization. In the following, we re-express model 4 in terms of a modified linear mixed model. Based on Green , the smoothing spline function f at each voxel can be written via a one-to-one linear transformation as. Through the linear transformation in 6 , we can re-write the mixed effects smoothing spline model in 4 as the following modified linear mixed model,.

In Appendix A , we provide details that establish the equivalence between the mixed effects smoothing spline model 4 and the linear mixed model 7.

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Through the above reparameterization, the mixed effects smoothing spline model 4 can then be estimated by fitting the linear mixed model 7 using standard linear mixed model procedures in existing statistical software such as SAS Pedan, ; Zhang et al.

It has been shown that the REML estimator of the smoothing parameter is identical to the generalized maximum likelihood GML estimator proposed by Wahba The REML estimation of the smoothing parameter is more computationally efficient than alternative estimation approaches such as cross-validation CV Rice and Silverman, and generalized cross-validation GCV Wahba, Another major advantage of the presented linear mixed model estimation approach is that it provides simultaneous estimation of the smoothing spline, other fixed effects such as the global normalization term and subject random effects.

Alternative estimation methods for smoothing splines do not provide easy accommodation of the other factors. The mixed effects spline models in 1 and 4 estimate the main effect of the dose level. For example, consider the ethanol data example in which scans are obtained across increasing BAC levels. If the scans were collected both during a resting state and during a task that engaged cognitive processes, it would be important to characterize the BAC-dependent profiles of measured brain activity separately for each task.

We provide the details of the extended mixed effects spline models and the hypothesis tests for the interaction effects in Appendix B. In this section we present the results from application of the mixed effects spline models to the PET study of ethanol. We also discuss the results from a simulation study that compared the performance of the proposed spline models against polynomial regression for voxels with different response patterns.

The primary interest in the study is to model the effect of ethanol on brain function and to identify brain regions with similar response patterns to ethanol. Secondly, researchers are also interested in whether the rate of ethanol consumption, measured by the rate of change in BAC, affects the brain. The shape of the brain response pattern to ethanol is unknown and varies considerably across different brain areas. Figure 1 presents the observed data smoothed by loess curves at four voxels, depicting different relationships between rCBF measurements and BAC levels.

The varying response patterns cannot be adequately accommodated by standard methods such as polynomial regression. Therefore, we apply the proposed mixed effects spline models for flexible estimation of the dose-dependent neural processing responses to ethanol. Descriptive plots showing varying dose-dependent response patterns at different voxels in the ethanol study. The observed data were smoothed using the loess curve. The varying response patterns cannot be accommodated by standard methods such as the polynomial regression models and prompt the need to develop methods that more effectively capture dose-dependent neural processing responses.

We first considered the mixed effects regression spline model. We modeled the effect of ethanol on brain function as follows,. One could also use a higher order polynomial basis such as the cubic basis to obtain a smoother spline curve. Inclusion of the subject random effects takes account of the between-subject variation in rCBF and helps to account for within-subject correlations among the serial rCBF measurements obtained from the same subject.

To determine the number of knots, we fit regression spline models at each voxel using the number of knots ranging from 3 to 10 and recorded the optimum number producing the minimum AIC statistic. The results across voxels indicated that the optimum number of knots at most voxels laid within the small range of 5—7 and the global optimum number of knots that minimized the overall AIC was 6. The locations of the 6 knots were placed at equally spaced quantiles of the empirical distribution of the observed BAC values so that there were approximately equal number of data points between knots Ngo and Wand, In our study, there was an interest in grouping voxels based on the similarity of their response pattern to ethanol.

Using estimates from the regression spline model 8 , we performed a cluster analysis on a vector of statistics that characterized the response profile of each voxel. This vector of contrasts depicted the relative brain activation of voxel v at selected BAC levels compared to the baseline. We then performed a cluster analysis on the vector T v. For the hierarchical clustering, we applied the Peudo-T 2 statistic as a stopping rule to determine the number of clusters, resulting in 12 clusters.

Figure 2 depicts the brain regions and within-cluster mean contrasts for two clusters of particular interest. Cluster A consisted of voxels in the anterior cingulate and medial frontal gyrus.

Maps of two brain region clusters identified by a cluster analysis for grouping brain locations based on their response patterns to ethanol. Each region is shown with coronal, saggital and axial slices. The figures on the right column plot the mean contrasts within each of the clusters. Cluster A exhibited increase in brain activity with rising BAC and includes anterior cingulate and medial frontal gyrus.

Cluster B demonstrated reduction in brain activity with increasing BAC and mainly includes cerebrum and occipital lobe. Another research objective in the study is to determine how the rate of change in BAC affects brain function. In our study, ethanol was administered in a continuous manner through the study session and BAC changed over time through the study. Hence ethanol-related brain function alterations at a particular time point were attributable to two sources: To accurately investigate how the rate of increase in BAC affects brain function, we need to control for the effect from the current level of BAC.

Hence, we considered the following model. Figure 3 identifies the brain areas that we found to be statistically significantly associated with the rate of change in BAC. Specifically, regions that are activated with the rapid increase in BAC include the left temporal gyrus and the right inferior temporal gyrus. Regions that are deactivated with rapid increase in BAC include the inferior frontal gyrus. Maps of brain regions that have significant association with the rate of increase in BAC, based on a mixed effects regression spline model. Axial slices are relative to the anterior commissural plane.

To avoid the need for knots specifications, we also applied the following mixed effects smoothing spline model to describe the dose-dependent effects of ethanol,. We estimated parameters in the mixed effects smoothing spline model under the linear mixed model framework through the reparameterization presented in 7.

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Smoothing spline ANOVA models

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