Thanks Benjamin Yvernaut and Brian Byod, for helping me figure out a lot of XNAT issues. Thank you Professor Hakmook Kang, for patiently explaining complex statistical theories and kindly helping me in my job search.
INTRODUCTION
Overview
To enable widespread use of multimodality imaging statistical studies, we introduced robust regression and nonparametric regression in the neuroimaging context for applying the general linear model (Yang, Beason-Held et al. 2011). An error term in a model is the part of the response variable that cannot be explained in the statistical equations.
General Linear Model in Brain Image
Although the GLM is only expressed as a simple linear model, it can also be used to express polynomial model if we include the polynomial term of the explanatory variables in the design matrix. The rate of type II error is denoted by β and related to the power of a test (which is equal to 1-β).
GLM in Structure Function Relationships Estimation
GLM in Resting-state fMRI Functional Connectivity
A widely used model is the first-order autoregressive (AR(1)) model (Friston, Glaser et al. 2002), whose correlated error has the time-domain form: 1.12) where is the white noise. The covariance matrix V can be estimated by maximizing the negative free energy using the restrictive maximum likelihood (ReML) method (Friston, Penny et al. 2002). This new model now conforms to sphericity assumptions and can be interpreted in the usual way at each voxel.
Assumptions in Ordinary Least Squares
Under the null hypothesis, the explanatory variables have no effect on the response variable, so any order of the explanatory data will result in the same answer. Then the corresponding corrected p-value for each voxel is the portion of the value in the maximum statistic vector that is greater than or equal to the voxel statistic.
Our Contributions The main contributions in the dissertation work are
Moreover, we evaluate the application of these methods by comparing with OLS in simulation and an empirical illustration in the context of multimodal image regression, demonstrating that random regressors are worth considering in images in image regression. We proposed models that can account for spatial correlations as well as temporal correlations in resting-state functional connectivity analysis within the general linear model framework.
Previous Publication
The ROI-based spatial-spectral mixed effects model for resting state fMRI analysis is proposed in (Kang, Yang et al. 2013). The voxel-wise spatio-temporal models and the multi-site large-scale data studies are under preparation.
BIOLOGICAL PARAMETRIC MAPPING WITH ROBUST AND NON-PARAMETRIC STATISTICS
Introduction
In the design of the M-estimator, we assume a conditional probability form of the data that allows for large deviations (Press 2007). The least squares approach is based on the parameterization of the error distribution as independent and Gaussian.
Implementation
Within the head of the caudate region, the true simulated intensity of the principal mode images is ―1.5* regressive images – constant‖. Second, to assess the relative performance of each method on empirical data, we tested nonrobust BPM, robust BPM, and BnPM (with 10,000 permutations) using a subset of the normal aging study of the BLSA neuroimaging project (Resnick, Goldszal et al 2000).
Results
If the user selects BnPM analysis, the subjobs are generated using a non-parametric regression method and sent to the supercomputer cluster. All the tools that have been implemented are listed in the table with the uptime compared to the SPM uptime.
Conclusions
The green arrow shows one voxel in the region where non-robust regression indicates non-significance, while robust regression indicates significance. The blue arrow shows one voxel in the region where non-robust regression indicates significance, while robust regression indicates non-significance.
BIOLOGICAL PARAMETRIC MAPPING ACCOUNTING FOR RANDOM REGRESSORS WITH REGRESSION CALIBRATION AND MODEL II REGRESSION
Regression analysis that accounts for errors in regressors will greatly improve the credibility of the BPM model by accounting for the randomness of the imaging modality in both the regressors and the regressors reasonably. Empirically successful statistical methods that account for random regressors have been developed, including regression calibration (Carroll, Ruppert et al. 2010) and model II regression (York 1966; Ludbrook 2010). Herein we focus on multi-modality inference; possible extensions to temporal modeling are discussed but left as future work.
Notation
Regression calibration and model II regression are included as regression method options in the BPM tool for SPM software using Matlab (Mathworks, Natick, MA). Meanwhile, regression calibration and Model II regression improve the true positive rate compared to OLS regression (100-FNR, Table 3.1). For the root mean square error when the true coefficient is zero ( ), the OLS method slightly outperforms the regression calibration and model II regression; when there is a relationship between and y ( ), regression calibration and model II regression are better than OLS.
EVALUATION OF STATISTICAL INFERENCE ON EMPIRICAL RESTING STATE FMRI
Therefore, artifacts are an increased problem with fMRI at 7T compared to the current 3T and 1.5T scanners (Triantafyllou, Hoge et al. 2005). In a recent pilot study, we investigated the distributional properties of resting-state fMRI (rs-fMRI) at 3T and 7T (Yang, Holmes et al. 2012). Data-driven predictive and reproducibility metrics enable such quantitative assessments through resampling/information theory (Strother, Anderson et al. 2002).
Theory
The correlation matrix is typically estimated using Restricted Maximum Likelihood (ReML) method with an AR(1) correlation matrix structure, and is estimated on the whitened data (i.e. the ―OLS‖ approach). The mean value of the original (unmined) voxel time course is approximately 905 (arbitrary intensity units), and the standard deviation is approximately 11. Note that both OLS and Huber were fit to the same subsample of the data (i.e., the random subsamples are paired).
Discussion and Conclusions
The similar performance of the robust method and the OLS method may be due to the fact that there are outliers, but the number is small. The blue circle shows that the OLS result is significant, while the robust result is not or less significant. The green circle indicates that the robust result is significant, while the OLS result is not or less significant.
WHOLE BRAIN FMRI CONNECTIVITY INFERENCE ACCOUNTING TEMPORAL AND SPATIAL CORRELATIONS
Kang, Ombao et al. 2012) proposed a spatial-spectral mixed-effects model to overcome the main barrier of including spatial correlations in fMRI data analysis. The authors demonstrated capturing the spatial and temporal correlation through simulation and empirical experiments, but the framework was limited to considering three ROIs. Herein, we proposed new functional connectivity analysis models that account for both the temporal correlation and the spatial correlation for ROI-based and voxel-wise analysis.
ROI-based Spatio-spectral Mixed Effects Model
Since we have taken spatial correlations and temporal correlations into account, the standard errors are not underestimated. The number of voxels and coordinates vary from ROI to ROI, but within a ROI the Euclidean correlation structures of the distances are the same (ie, the variogram function is the same). The average estimated seed connectivity coefficient ̂ and the average between ROI correlations ̂ among 25 subjects are shown in Figure 5.4.
Alternative Voxel-wise Spatio-temporal Model
In the voxel-wise spatiotemporal model, for each voxel, we consider the spatial correlations within its neighboring window. For functional connectivity analysis, instead of testing the estimated connectivity for each voxel, we can test the average connectivity in the voxel's neighbor window. Using equation (5.22), we can estimate the combined coefficients in the voxel's neighbor window for each voxel and its covariance.
Discussion
The left and right primary motor cortex and the premotor cortex show significance in the results, which is consistent with the motor network shown in (Newton, Rogers et al. 2012). The number of false positive voxels is much less in the winBeta and winCov results. However, there are many random significant voxels outside our expected region in the SPM results, while there are fewer in the winCov results and almost no false positive voxels in the winBeta results.
Appendix: Spatial Covariance Estimation with Exponential Variogram Function For each window
Therefore, considering spatial correlations, our proposed spatiotemporal model can better control the type I error and provide better estimates than the traditional first-level analysis model. Note that since we assume that the spatial covariance remains the same over time, the y here can be lumped across time series.
ASSESSMENT OF INTER MODALITY MULTI-SITE INFERENCE WITH THE 1000 FUNCTIONAL CONNECTOMES PROJECT
The results of the mega-analysis and the meta-analysis are shown in Figure 6.4 and Figure 6.5. The relationship results show that the correlation of the mega- and meta-analysis on the location difference is very high. For each regressor effect, the t-value from the mega- and meta-analysis is plotted (blue dots).
CONCLUSIONS AND FUTURE WORK
Summary
However, very few models can consider the temporal correlations and the spatial correlations simultaneously in estimation. We also applied the ideas of this model to the voxel-wise models within the statistical parametric mapping (SPM) framework to account for both the spatial and temporal correlations. The results show substantial and significant site difference across the whole brain and very strong effects observed with mega-analysis as opposed to meta-analysis.
Reliable Statistical Inference in Multi-modality Brain Image Analysis
We investigated possible statistical methods to increase the reliability of BPM analysis: robust regression and nonparametric mapping. We have evaluated the use of these methods in simulation and explained their use in empirical experiments that illustrate the applicability in the context of BPM and multimodal image regression. In a study of temporal lobe epilepsy, it was used to investigate the relationship between gray matter concentrations and functional connectivity at rest (hippocampus and thalamus).
Addressing Random Regressors in Multi-Modality Brain Image Analysis
The proposed statistical model treats the measurement error of the regressors separately, but does not directly include robust regression to exclude outliers. An accurate estimate of the measurement error of the random regressors will increase the accuracy of the estimate of the intermodal relationship. Obtaining repeated measures is a serious limitation in the neuroimaging community, but there may be specific study designs for which this is an appropriate solution.
Robust Statistics and Empirical Validation in Functional Connectivity Analysis We present a new technique to compare statistical methods using non-repeated finite samples of
Our early error investigation showed that artifacts in the 7T fMRI data set had a greater impact on statistical method than the 3T data sets --- such results are anecdotal, but consistent with expected scanner performance given known hardware stability issues. We proposed a new approach for the quantitative evaluation of statistical inference methods based on their resilience in the sense of decreasing data. Our proposed method is a promising but initial work to compare statistical inference methods on empirical data.
Spatial Temporal Models for Resting State fMRI Analysis
In the voxel analysis on an unsmoothed 7T dataset, significant results from traditional methods are too noisy while fewer false positive voxels appear using our spatiotemporal model with voxel test. Voxel-wise spatiotemporal models are interesting restatements of the estimation problem, but are relatively unexplored. A better window size should be able to balance the need to accurately estimate spatial correlations and significance size, and an adaptive window for each voxel may be more appropriate.
Multi-Site Brain Image Study
We explored mega-analysis and meta-analysis in statistical brain mapping using large-scale data. The difference between mega and meta analysis is due to fixed effects and random effects. Increased power with mega-over meta-analysis indicates encouraging full data sharing rather than summary statistics.
Overall Perspective
34;Asymptotics for the SIMEX estimator in nonlinear measurement error models." Journal of the American Statistical Association. Journal of the American Statistical Association. 34;Evaluating preprocessing choices in single-subject BOLD fMRI using NPAIRS performance metrics. " Neuroimaging.
