General linear model

Abstract: + Abstract: Statistical parametric maps are spatially extended statistical processes that are used to test hypotheses about regionally specific effects in neuroimaging data. The most established sorts of statistical parametric maps (e.g., Friston et al. (1991): J Cereb Blood Flow Metab 11:690-699; Worsley et al. 119921: J Cereb Blood Flow Metab 12:YOO-918) are based on linear models, for example ANCOVA, correlation coefficients and t tests. In the sense that these examples are all special cases of the general linear model it should be possible to implement them (and many others) within a unified framework. We present here a general approach that accommodates most forms of experimental layout and ensuing analysis (designed experiments with fixed effects for factors, covariates and interaction of factors). This approach brings together two well established bodies of theory (the general linear model and the theory of Gaussian fields) to provide a complete and simple framework for the analysis of imaging data. The importance of this framework is twofold: (i) Conceptual and mathematical simplicity, in that the same small number of operational equations is used irrespective of the complexity of the experiment or nature of the statistical model and (ii) the generality of the framework provides for great latitude in experimental design and analysis.

Abstract: LINEAR MODELS A simple linear model Linear models in general The theory of linear models The geometry of linear modelling Practical linear models Practical modelling with factors General linear model specification in R Further linear modelling theory Exercises GENERALIZED LINEAR MODELS The theory of GLMs Geometry of GLMs GLMs with R Likelihood Exercises INTRODUCING GAMS Introduction Univariate smooth functions Additive models Generalized additive models Summary Exercises SOME GAM THEORY Smoothing bases Setting up GAMs as penalized GLMs Justifying P-IRLS Degrees of freedom and residual variance estimation Smoothing Parameter Estimation Criteria Numerical GCV/UBRE: performance iteration Numerical GCV/UBRE optimization by outer iteration Distributional results Confidence interval performance Further GAM theory Other approaches to GAMs Exercises GAMs IN PRACTICE: mgcv Cherry trees again Brain imaging example Air pollution in Chicago example Mackerel egg survey example Portuguese larks example Other packages Exercises MIXED MODELS and GAMMs Mixed models for balanced data Linear mixed models in general Linear mixed models in R Generalized linear mixed models GLMMs with R Generalized additive mixed models GAMMs with R Exercises APPENDICES A Some matrix algebra B Solutions to exercises Bibliography Index

Abstract: I. INTRODUCTION. 1. Experimental Design. II. SINGLE FACTOR EXPERIMENTS. 2. Sources of Variability and Sums of Squares. 3. Variance Estimates and F Ratio. 4. Analytical Comparisons Among Means. 5. Analysis of Trend. 6. Simultaneous Comparisons. 7. The Linear Model and Its Assumptions. 8. Effect Size and Power. 9. Using Statistical Software. III. FACTORIAL EXPERIMENTS WITH TWO FACTORS. 10. Introduction to the Factorial Design. 11. The Principal Two-Factor Effects. 12. Main Effects and Simple Effects. 13. The Analysis of Interaction Components. IV. NONORTHOGONALITY AND THE GENERAL LINEAR MODEL. 14. General Linear Model. 15. The Analysis of Covariance. V. WITHIN-SUBJECT DESIGNS. 16. The Single-Factor Within-Subject Design. 17. Further Within-Subject Topics. 18. The Two-Factor Within-Subject Design. 19. The Mixed Design: Overall Analysis. 20. The Mixed Design: Analytical Analyses. VI. HIGHER FACTORIAL DESIGNS AND OTHER EXTENSIONS. 21. The Overall Three-Factor Design. 22. The Three-Way Analytical Analysis. 23. Within-Subject and Mixed Designs. 24. Random Factors and Generalization. 25. Nested Factors. 26. Higher-Order Designs. Appendix A: Statistical Tables.