Analysis of Variance I
Analysis of Variance (ANOVA) is a statistical procedure for comparing means of two or more populations. As the name suggests, ANOVA is a method for studying differences in means by analysis of the variance components in the model. In earlier chapters we h
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Analysis of Variance I
8.1 Introduction Analysis of Variance (ANOVA) is a statistical procedure for comparing means of two or more populations. As the name suggests, ANOVA is a method for studying differences in means by analysis of the variance components in the model. In earlier chapters we have considered two sample location problems; for example, we compared the means of two groups using a two-sample t test. Let us now consider a generalization to the multi-sample location problem, where we wish to compare the location parameters of two or more groups. One-way ANOVA handles a special case of this problem, testing for equal group means. The following introductory example provides a context for reviewing terminology and statements about the models that follow. Example 8.1 (Eye color and flicker frequency). The data set “flicker.txt” measures ‘critical flicker frequency’ for 19 subjects with different eye colors.1 From the description of the data on OzDASL [38] “an individual’s critical flicker frequency is the highest frequency at which the flicker in a flickering light source can be detected. At frequencies above the critical frequency, the light source appears to be continuous even though it is actually flickering.” Is critical flicker frequency related to eye color? This data set has 19 observations and two variables, Colour (eye color) and Flicker (critical flicker frequency in cycles per second). The data is listed in Table 8.1. Here we have a quantitative variable (Flicker) and a group variable (Colour). To formulate a model and hypothesis to test, let us ask a more specific question: Does the mean critical flicker frequency differ by eye color? The dependent variable or response is Flicker. The explanatory variable is 1
See http://www.statsci.org/data/general/flicker.html, for the description and data; source [44].
J. Albert and M. Rizzo, R by Example, Use R, DOI 10.1007/978-1-4614-1365-3__8, © Springer Science+Business Media, LLC 2012
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8 Analysis of Variance I
Brown 26.8 27.9 23.7 25.0 26.3 24.8 25.7 24.5
(a) Green 26.4 24.2 28.0 26.9 29.1
Blue 25.7 27.2 29.9 28.5 29.4 28.3
Colour Brown Brown Brown Brown Brown Brown Brown Brown Green Green Green Green Green Blue Blue Blue Blue Blue Blue
(b) Flicker 26.8 27.9 23.7 25.0 26.3 24.8 25.7 24.5 26.4 24.2 28.0 26.9 29.1 25.7 27.2 29.9 28.5 29.4 28.3
Table 8.1 Two data layouts shown for the “Eye Color and Flicker Frequency” data. Table (a) on the left is in a spreadsheet-like layout (unstacked). Table (b) on the right is in stacked format. The response is Flicker (critical flicker frequency) and the factor is Colour (eye color). The factor Colour has three levels (Brown, Green, Blue).
the group variable, called the treatment or factor, and it has three levels in this example (Brown, Green, Blue). The null hypothesis is H0 : the mean response is equal for all groups. A more general null hypothesis is that the location parameter of each group is equal; for example, one could test whether the medians differ by group. Even more general is a null hypothesis that t
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