Revised July 1998 - One PDF per chapter
©1994 Andrew Peter Holmes
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Statistical Issues in functional Brain Mapping is a PhD thesis concentrating on the statistical analysis of functional mapping experiments, with particular attention to activation experiments using Positron Emission Tomography. The thesis is being made avaliable to the scientific community by Web HTTP and anonymous FTP, as a Adobe Acrobat files. This page gives an abstract, and a table of contents, and links to the appropriate files. I would be grateful if those uploading sections of this thesis could email me <andrew@fil.ion.ucl.ac.uk> to let me know they have it. Thanks. The work presented in this thesis was undertaken under the supervision of Professor Ian Ford in the Department of Statistics at the University of Glasgow, with the sponsorship of the Engineering and Physical Sciences Research Council. |
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Using Positron Emission Tomography (PET), it is possible to obtain quantitative images of regional cerebral blood flow, indicative of regional neuronal activity. This is used to examine the function of the brain through designed experiments. The simplest such functional mapping experiment is the simple activation study, which aims to locate the brain loci responsible for a certain function, by scanning under two conditions differing only in that function. These studies generate small numbers of observations of extremely high dimension, each data point being a three-dimensional image. The statistical analysis of such data sets requires new statistical techniques for significance testing, and it is this problem which this thesis addresses. Chapter 1 establishes the background to this work, giving a fairly comprehensive layman's description of PET and functional brain mapping. In the absence of apriori information regarding the location of a particular function, analysis of activation experiments proceeds at the voxel (pixel) level. For each voxel a model must be assumed for the data, and a null hypothesis of "no activation" expressed in terms of the model parameters. An added complication is the presence of global differences in cerebral blood flow between subjects. Computation of a statistic indicating evidence against the null hypothesis at each voxel, gives a statistic image. Model selection and the formation of statistic images is the subject of chapter 2. Particular attention is given to the two most popular models, namely Friston's AnCova and that of the t-statistic formed from subject difference images, with global changes removed by proportional scaling. Problems of simultaneous model fitting and the dangers of assuming homoscedascity are considered. The assessment of the statistic image presents a large multiple comparisons problem. Regions where the statistic image indicates evidence against the null hypothesis must be located, whilst maintaining strong control over familywise Type I error. The current methods for testing statistic images are discussed in detail in chapter 3, focusing on "random field" methods, their assumptions and properties. In the remaining chapters, three methods developed by the author for assessing simple activation studies are presented. The first of these is a two-stage approach, in which the group of subjects is split into a pilot group and a study group. A small number of regions of interest are identified from the pilot group data, and the study group data assessed over these regions of interest. A simulation study shows this approach to hold some promise. In chapter 5, the testing problem is reformulated as an image segmentation problem, to which empirical Bayesian techniques from statistical image processing are applied. A Markov random field is used to convey prior belief regarding the contiguous nature of activated areas. Here simulation results indicate that the incorporation of prior belief into a single threshold test results in a more conservative (and less powerful) test. The subject of the last chapter (Ch.6), is a non-parametric approach. A multiple comparisons randomisation test is developed for simple activation studies, which is shown to maintain strong control over familywise Type I error. A step- down procedure with strong control is introduced, and computationally feasible algorithms presented. The methods are illustrated on a real PET data set, with a pseudo t-statistic formed from subject difference images with a smoothed variance estimate. For the given data set the approach is found to outperform many of the parametric methods, particularly with the pseudo t-statistic. This, together with the flexibility and guaranteed validity of a non-parametric method, makes the approach very attractive, despite the computational burden imposed. The practicalities of the method are discussed, including extensions to other experimental paradigms, other test statistics, and permutation tests. This is an applied thesis, aimed at the statistically literate PET researcher. In addition to presenting the author's ideas, it is hoped that this document provides a useful summary and comprehensive critique of existing work, giving enough detail to serve as a useful reference. |
Table of Contents : Front | Introduction | Ch1 | Ch2 | Ch3 | Ch4 | Ch5 | Ch6 | Appendices | References
Title page
Dedication
Abstract
Table of Contents
List of Figures and Tables
Acknowledgements
Declaration
Table of Contents : Front | Introduction | Ch1 | Ch2 | Ch3 | Ch4 | Ch5 | Ch6 | Appendices | References
Pages 1-4
Table of Contents : Front | Introduction | Ch1 | Ch2 | Ch3 | Ch4 | Ch5 | Ch6 | Appendices | References
Pages 5-47
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1.1. Additional Reading...6
1.2. Data Acquisition...7
1.2.1. The Tomograph...7
1.2.2. Data acquisition...7
1.2.3. Sinograms...8
1.2.4. Additional data...10
1.3. 2D & 3D PET, SPECT...14
1.3.1. 2D & 3D PET...14
1.3.2. SPECT...15
1.4. Reconstruction...17
1.4.1. A statistical model for PET...17
1.4.2. Maximum Likelihood Reconstruction...19
1.4.3. Filtered Back-projection...22
1.4.3.1. Transmission tomography...22
1.4.3.2. Approximation of emission tomography problem to fit transmission tomography framework...23
1.4.3.3. Reconstruction from projections...24
1.4.3.4. Comments on filtered back-projection...27
1.4.4. Reconstructed images...28
1.5. Tracer and Modelling Issues...31
1.5.1. Radioactive isotopes and labelling...31
1.5.2. Short half lives and low activities of the isotopes...31
1.5.3. Cost of PET...32
1.5.4. Modelling...32
1.5.5. Imaging cerebral blood flow...32
1.6. Functional Mapping with PET...34
1.6.1. Imaging brain activity...34
1.6.1.1. Tracers for functional mapping...34
1.6.1.2. H215O infusion protocol...35
1.6.2. Types of study...36
1.6.2.1. Common factors...36
1.6.2.2. Activation studies...37
1.6.2.3. Correlation with an external variable ...37
1.6.2.4. Cross group comparison...38
1.6.2.5. Flow correlation, Functional connectivity...38
1.6.3. Region of interest methods...38
1.6.3.1. ROI methods...39
1.6.4. Voxel-by-voxel methods of analysis...39
1.6.4.1. Overview...39
1.6.4.2. Re-alignment...40
1.6.4.3. Anatomical normalisation...43
1.6.4.4. Primary smoothing...45
1.6.4.5. Adjusted images...46
Table of Contents : Front | Introduction | Ch1 | Ch2 | Ch3 | Ch4 | Ch5 | Ch6 | Appendices | References
Pages 49-94
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2.1. Global Changes...52
2.1.1. Computation of gCBF...53
2.1.2. Correcting for changes in gCBF: Normalised images...53
2.2. Single Subject Activation Experiments...54
2.2.1. Two sample t-statistic...54
2.2.2. Friston's model: AnCova...56
2.2.3. Model selection for single subject...58
2.2.3.1. Model selection for images...59
2.2.3.2. Model selection: Single subject activation studies...60
2.2.4. Conclusions: Single subject statistic images...63
2.3. Multiple Subject Activation Experiments...64
2.3.1. Proportional scaling approach...64
2.3.1.1. t-statistic on subject difference images...64
2.3.1.2. Discussion of t-test on subject difference images...65
2.3.2. AnCova models...67
2.3.2.1. Models...67
2.3.2.2. Model selection...68
2.3.2.3. Model selection for "V5" study...69
2.3.2.4. SPM and Friston's AnCova...70
2.3.3. Conclusions...72
2.3.3.1. t-statistic on subject difference images...72
2.3.3.2. Friston's AnCova...73
2.4. Additional Comments...74
2.5. A Multivariate Perspective...76
2.5.1. Sums of squares approaches...76
2.5.2. Multivariate regression formulation...77
2.5.3. Multivariate regression...78
2.5.4. Image regression...79
2.5.5. Multivariate regression revisited...80
2.6. Example-"V5" Study...83
2.6.1. Proportional scaling approach...83
2.6.1.1. Statistic images...83
2.6.1.2. A crude Bonferroni analysis...86
2.6.1.3. Empirical examination of assumptions...86
2.6.2. Friston's AnCova...89
2.7. Pooled Variance...91
2.7.1. Example-"V5" Study...92
2.7.2. Inappropriate use of pooled variance...93
Table of Contents : Front | Introduction | Ch1 | Ch2 | Ch3 | Ch4 | Ch5 | Ch6 | Appendices | References
Pages 95-141
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3.1. Preliminaries...96
3.1.1. Multiple comparisons...96
3.1.2. Random fields...97
3.1.3. Simulated Gaussian statistic images...99
3.1.4. Possible directions...101
3.2. MCP Tests...102
3.2.1. Bonferroni...102
3.2.2. Other MCP methods...103
3.3. Random Field Approaches...104
3.3.1. Worsley's Euler characteristic method...104
3.3.2. Friston's "Bonferroni" method...107
3.3.3. Transform functions...107
3.3.4. Comparison of Friston's and Worsley's methods...108
3.3.5. Estimation and specification of smoothness...109
3.3.6. Discussion of random field approaches...112
3.3.6.1. Assumptions...112
3.3.6.2. Normality...113
3.3.6.3. Model assumptions...113
3.3.6.4. Strict stationarity of statistic images...113
3.3.6.5. Low df, rough random fields, noisy statistic images...114
3.3.6.6. Smoothing Statistic images...114
3.4. Omnibus Tests...116
3.4.1. Friston's exceedence proportion test...116
3.4.2. Worsley's exceedence proportion test...118
3.4.3. Change distribution analysis...122
3.5. Suprathreshold Cluster Tests...123
3.5.1. Simulation approaches...123
3.5.2. Friston's Theoretical treatment...125
3.5.3. Comments on suprathreshold cluster size methods...128
3.6. Example-"V5" Study...131
3.6.1. Approaches on t-statistic images...131
3.6.2. Approaches on Gaussianised t-statistic image...134
3.6.3. Secondary smoothing...139
Table of Contents : Front | Introduction | Ch1 | Ch2 | Ch3 | Ch4 | Ch5 | Ch6 | Appendices | References
Pages 143-162
4.1. Two-Stage Testing...144
4.2. Simulation Methods...147
4.2.1. Image space...147
4.2.2. Simulating difference images...148
4.2.3. Two-Stage test implementation...151
4.2.4. Voxel-by-voxel tests...152
4.3. Results...155
4.3.1. Simulation parameters ...155
4.3.2. Size of tests...156
4.3.3. Power of two-stage test...157
4.4. Conclusions...159
4.4.1. Discussion...159
4.4.2. Further work...160
Table of Contents : Front | Introduction | Ch1 | Ch2 | Ch3 | Ch4 | Ch5 | Ch6 | Appendices | References
Pages 163-179
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5.1. Introduction and Motivation...164
5.2. MRFS and Gibbs Distributions...165
5.2.1. Markov Random Fields...165
5.2.2. Gibbs Random Fields...166
5.3. Image Segmentation...169
5.3.1. Segmentation...169
5.3.2. The Gibbs sampler...170
5.3.3. Example...170
5.4. A Bayesian Segmentation Test...173
5.5. Simulation Study...174
5.5.1.Simulation methods...174
5.5.2. Results...175
5.6. PET Example...176
5.7. Conclusions...178
Table of Contents : Front | Introduction | Ch1 | Ch2 | Ch3 | Ch4 | Ch5 | Ch6 | Appendices | References
Chapter 6: A Non-Parametric Approach
Pages 181-207
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6.1. Introduction and Motivation...182
6.2. Theory...183
6.2.1. Statistic images...183
6.2.2. Null hypothesis and labellings...185
6.2.3. Single threshold test...186
6.2.4. Multi-Step tests...188
6.2.4.1. Step-down test...189
6.2.4.2. Step-down in jumps variant...192
6.2.4.3. Direct computation of adjusted p-values...193
6.3. Exemplary Application...194
6.3.1. Raw t-statistic...194
6.3.2. Pseudo t-statistic image...197
6.4. Discussion...202
6.4.1. Permutation tests...202
6.4.2. Other applications...202
6.4.3. Other statistics...203
6.4.4. Number of possible labellings, size and power...204
6.4.5. Approximate tests...204
6.4.6. Step-down tests...205
6.4.7. Computational burden...206
6.5. Conclusions...207
Table of Contents : Front | Introduction | Ch1 | Ch2 | Ch3 | Ch4 | Ch5 | Ch6 | Appendices | References
Pages 209-227
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A: Co-ordinate Systems...210
A:1 Referring to voxels by position...210
A:2 Tri-linear interpolation...211
B: Smoothing Convolution...212
B:1 Smoothing convolution...212
B:2 Moving average filter...212
B:3 Edge effects & boundary truncation smoothing...212
B:4 Gaussian kernels...213
C: Some Results For Smoothing Convolution...214
C:1 Smoothing convolution: commutative for even kernels...214
C:2 Double smoothing convolution: Associative for even kernels...214
C:3 Double smoothing convolution: Order unimportant...215
C:4 Combining Gaussian kernels: Double smoothing...215
C:5 Covariance function of smoothed white noise processes...217
C:6 Smoothness of smoothed Gaussian white noise fields...218
C:7 Smoothness of (Gaussian) smoothed Gaussian white noise...218
C:8 Secondary smoothing...218
C:9 Effect of scaling on smoothness...219
D: Expected Euler Characteristics...220
D:1 The c2-field...220
D:2 The F-field...220
D:3 The t-field...221
E: "Transform" Functions...222
F: Ordering Theorem...223
G: Smoothness of t-Fields...224
H: Poline's Bivariate Approach...226
Table of Contents : Front | Introduction | Ch1 | Ch2 | Ch3 | Ch4 | Ch5 | Ch6 | Appendices | References
Pages 229-239
The thesis is avaliable as Adobe Acrobat (PDF) files, each named according to the chapter represented. These files are linked into the following table of contents. A UNIX tar file containing all the PDF files is available by anonyous FTP from ftp://ftp.fil.ion.ucl.ac.uk/spm/papers/APHthesis.tar (6Mb).
The files were prepared using Microsoft Word for Windows (v2.0a), and printed for A4 paper using Windows PostScript driver (MS & Aldus, V3.58). These PostScript files were then converted to PDFs using Acrobat Distiller (v2.0), then merged and optimised using Acrobat Exchange (v3.0)