Abstract
Many experiments examine the effects of multiple experimental conditions. When each measured response from a subject is a single-number, the data are usually analyzed with analysis of variance (ANOVA) .
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Notes
- 1.
For pedagogical simplicity, we wanted the number of subjects per group to be equal. This is not required for ANOVA; it merely makes things a bit easier to discuss. In the original data there were only 5 subjects in the 8-week control group. We therefore added the 12.35 value to the 8-week control group.
- 2.
Infants in the active-exercise group received stimulation of the walking and placing reflexes during four 3-minute sessions that were held each day from the beginning of the second week until the end of the eighth week. The infants in the passive-exercise group received equal amounts of gross motor and social stimulation as those who received active-exercise, but unlike the active-exercise group, these infants had neither the walking nor placing reflex exercised. Infants in the no-exercise group did not receive any special training, but were tested along with the active-exercise and passive-exercise subjects. The 8-week control group was tested only when they were 8 weeks of age; this group served as a control for the possible helpful effects of repeated examination.
- 3.
When \(\sigma \) is unknown the derivation is slightly different because \(\sigma \) must be included among the parameters in the loglikelihood function, so its MLE must be found and the likelihood ratio is different; but the end result is equivalent to the \(F\)-test.
- 4.
On the other hand, the paper by Zelazo et al. presented an additional measure where the results were more striking. On this subject, see Adolph (2002).
- 5.
It is also convenient to require the vectors to be orthogonal to one another, in which case they are called orthogonal contrasts. For orthogonal contrasts, each estimate is independent of the others. This is a topic discussed in many books on regression analysis and experimental design.
- 6.
The purpose of the study was to distinguish responses based on eye-centered coordinates, head-centered coordinates, and trunk-centered coordinates.
- 7.
ANOVA may also be applied, as a special case of regression, when one explanatory variable is quantitative and another variable is an ANOVA indicator variable. This is usually called analysis of covariance or ANCOVA. Its purpose is to adjust the ANOVA for effects of the quantitative variable. See p. 332.
- 8.
We are here simplifying by ignoring some aspects of the experimental design.
- 9.
A widely-cited source for many of these ideas is Hill (1971).
- 10.
We are here assuming that the reported regression is not being driven primarily by inclusion of lots of babies with zero percent breast milk, but rather holds among the non-zero percentage babies.
- 11.
We do not have the full results when percentage breast milk is used, so we don’t know whether these associations diminish or change sign in that case.
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Kass, R.E., Eden, U.T., Brown, E.N. (2014). Analysis of Variance. In: Analysis of Neural Data. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9602-1_13
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