Analysis of Variance II
Analysis of Variance (ANOVA) is a statistical procedure for comparing means of two or more populations. In Chap. 8 we considered one-way ANOVA models, which help to analyze differences in the mean response corresponding to the levels of a single group var
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Analysis of Variance II
9.1 Introduction Analysis of Variance (ANOVA) is a statistical procedure for comparing means of two or more populations. In Chapter 8 we considered one-way ANOVA models, which help analyze differences in the mean response corresponding to the levels of a single group variable or factor. In this chapter we consider randomized block designs and two-way ANOVA models. Randomized block designs model the effects of a single group variable or factor while controlling for another source of variation using blocks. Two-way ANOVA models explain differences in the mean response corresponding to the levels of two group variables (factors) and their possible interaction.
9.2 Randomized Block Designs In this section we focus on randomized block designs, which model the effect of a treatment or factor where the samples are not independent, but matched by a variable called a block variable. In experiments some sources of variation are known and controllable. In Example 9.1 the experimental units are students who have taken a series of exams. The objective is to compare the student performance on different types of exams. There is of course, inherent variability among students, but the effect of this variability on the statistical analysis of exam scores can be controlled by introducing a variable to identify the student (the block variable). Other sources of variation may be unknown and uncontrollable. Randomization is a design technique that guards against this type of variation.
J. Albert and M. Rizzo, R by Example, Use R, DOI 10.1007/978-1-4614-1365-3__9, © Springer Science+Business Media, LLC 2012
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9 Analysis of Variance II
Example 9.1 (Exam scores). The scor data is provided in the bootstrap package [31]. It contains exam scores in five subjects for 88 students. The five exams are Name Description Type mec mechanics closed book vec vectors closed book alg algebra open book ana analysis open book sta statistics open book The layout of the scor data is shown in Table 9.1. In this example, a oneway analysis is not appropriate. A one-way ANOVA analyzes a completely randomized design, where the samples are assumed to be independent. If the exam topics (mechanics, vectors, algebra, analysis, statistics) are considered to be the treatments, the exam scores for each topic are not independent samples because the same 88 students took each exam. Table 9.1 Layout of the scor data in Example 9.1. There are 88 rows corresponding to 88 students, and five columns corresponding to five different exam scores for each student. mec 77 63 75 55 63 53 .. . 0
vec 82 78 73 72 63 61
alg 67 80 71 63 65 72
40 21
ana 67 70 66 70 70 64
sta 81 81 81 68 63 73
9 14
To use this data set, install the bootstrap package (see the example in Chapter 1, page 33). Once the bootstrap package is installed, load it using library. Then the data frame scor is available, and we can print the first part of the data frame using the head function. > library(bootstrap) > head(scor)
1 2 3 4 5 6
mec vec alg ana sta 77 82 67 67 81 63 78 80 70
Data Loading...
