Check back soon! We are working on some diagrams that Wil presented at a recent Brain Imaging workshop that the audience found quite helpful.

Preamble

There are several objectives of this brief document.

• A brief review of some of the basic principles of analysis of variance models is presented, with specific emphasis on both the modeling error terms for between- and within-subject designs and modeling error estimates in terms of Fixed or Random components.
• The general mechanics of effecting such analyses are discussed, with specific reference to SPM analyses. (As an aside, many or all of the Random Effects analysis discussed here can be performed by other programs such as AFNI.)
• References to additional original sources are provided which discuss these topics from a variety of perspectives

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Basics of ANOVA

Independent variables are either fixed or random.

• A fixed-effect independent variable (the most common) is specifically chosen and is of interest. For example, in a study on the effect of memory, there may be a low, medium & high load on working memory. The "load" is the item of interest, and the possible values for it are fixed at low, medium, and high denominations.
• A random-effect independent variable is randomly selected (as a sample) from a larger population. The results of a given sample is of no specific interest, rather the objective is to infer population parameters based on sample statistics. This type of conclusion is typically, and incorrectly, drawn from fixed-effect models.

The power of ANOVA is ultimately determined by the appropriate estimate of an error term (and, of course, the available degrees of freedom).

• Error estimates in fixed-effect models ignore possible subject-within-condition interactions. That is, when marginal means are compared, the presence of a few subjects with high scores will yield an overall significant effects, even when the effect is actually only present in, for example, 2 subjects. One can think of such a result as typical, if not qualitative, in that it speaks to the particular sample rather than to the general phenomenon.
• Error estimates in random-effect models explicitly utilize the variability of subject-within-condition variance. Such an analysis yields an average result in the sense that a significant result means that a given effect is present in every subject (and, in the case of neuroimaging data, at similar orders of magnitude and in similar spatial locations).

Fixed- and Random-Effects Analysis in SPM

• In general, single-subject analyses are fixed-effect analyses and yield some parametric output which represents the effect of some manipulation in a given subject.
• In general, within-subject, between-group and mixed ?factorial? designs are random-effect analyses based on the single-subject, fixed-effect analyses. The output is some general estimate of the effect a given manipulation would have on the population.

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References

• Online
  • Eric Zarahn's .pdf discussion.
  • Daren Gitelman's web-page.
  • Matthew Brett's web-page.
• Journal Articles
  • Friston, K. J., Holmes, A. P. and Worsley, K. J. (1999). How many subjects constitute a study? Neuroimage, 10, 1-5.
  • Frisson, L. & Pocock, S. J. (1992). Repeated measures in clinical trials: Analysis using mean summary statistics and its implications for design. Statistics in Medicine, 11, 1685-1704.
  • Keselman, H. J., Algina, J., Kowalchuk, R. K. and Wolfinger, R. D. (1999). A comparison of recent approaches to the analysis of repeated measures. British Journal of Mathematical & Statistical Psychology, 52, 63-78.
  • Ogenstad, S. (1997). Analysis and design of repeated measures in clinical trials using summary statistics. Journal of Biopharmaceutical Statistics, 7, 593-604.
  • Senn, S., Stevens, L., & Chaturvedi, N. (2000). Repeated measures in clinical trials: simple strategies for analysis using summary measures. Statistics in Medicine, 19, 861- 877.
• Book Chapters
  • Keppel, G. (1991). Design and analysis. New Jersey: Prentice Hall, pp. 495-487, 561-576.
  • Snedecor, G. W. & Cochran, W. G. (1967). Statistical Methods (3rd Edition). Ames, Iowa: The Iowa State University Press, pp. 275-296 (chapter on one-way ANOVA), 364-369 (sections on expected values of mean squares). Note, this volume is currently in its 8th edition so page numbers will vary.

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