Chapter 10: Analysis of variance (ANOVA)

Transcription

1 Chapter 10: Analysis of variance (ANOVA) ANOVA (Analysis of variance) is a collection of techniques for dealing with more general experiments than the previous one-sample or two-sample tests. We first present the most basic method, which is single-factor, single-classification ANOVA, also called one-way ANOVA. This method assumes data sampled from a set of (more than two) numerical populations. The characteristic, which identifies each population is called a factor and different values of the factor are called levels. Typical questions answered by ANOVA: Does the brand of gasoline pumped into a vehicle affect its mpg? Does the number of days of study before the test affect the test score? (Assuming the total amount of hours studied is constant per student.) Does the textbook used in a Statistics class affect student performance? Single-factor ANOVA The mathematical problem setting of a Single-factor ANOVA is as follows: I is the number of populations compared µ 1, µ 2,..., µ I are the true population means (unobservable) the null hypothesis H 0 : µ 1 = µ 2 = = µ I. the alternative hypothesis H a : at least two of the µ i are different We first focus on the case, where the same number of observations J is available from each population. Assumptions: we first assume that all I populations are normal with the same variance σ 2, so E[X i,j ] = µ i and V [X i,j ] = σ 2. Test procedure 1. Identify the parameter of interest. 2. State the null and alternative hypothesis. 3. Identify the distribution of the test statistic and the rejection region based on the significance level α. 4. Compute the test statistic. 5. Reject or do not reject the null hypothesis

2 Notation and quantitites We will use X i,j as the random variable denoting the j-th measurement of the i-th population and x i,j as its observed value. The sample means from each population are denoted X i,, X i, = The sample variance of each population is S 2 i = j=1 X i,j J. (X i,j X i, ) 2. J 1 i=1 The average over all measured values is called the grand mean and computed as Test statistic X, = Let the mean square for treatments be given by and the mean square for errors be MSTr = J I 1 X i,j IJ. ( X i, X, ) 2 MSE = S2 1 + S S2 I I Then the test statistic for a single-factor ANOVA test is i=1 F = MSTr MSE. Under the null hypothesis, the sample means X i, should be close to each other and MSTr should be small. On the other hand, whether or not the null hypothesis is true does not affect the value of MSE, because it only depends on the relationship of samples within the populations. Thus the value of F should be small under H 0 but large if H 0 is false. Test statistic distribution and the rejection region The statistic F is F -distributed (hence the name). The F -distribution arises as a ratio of appropriately scaled χ 2 -distributed random variables. A critical value in the F -distribution is thus denoted as F α,ν1,ν 2. For the single-factor ANOVA test when H 0 is true, the test statistic F is F -distributed with ν 1 = I 1 and ν 2 = I(J 1). We conduct a right-tailed hypothesis test at a given level of α and reject the null hypothesis when f F α,i 1,I(J 1), where f is the computed value of our test statistic F..

3 Example Example 10.3 on p What is the parameter of interest? What is the null hypothesis and the alternative hypothesis? What is the test statistic and what is its distribution? Compute the value of the test statistic. Based on the alternative hypothesis, the statistic distribution and the level of significance α, find the critical value and the rejection region for the test. Decide to reject or not to reject and summarize your result in the context of the problem.

4 Alternative view of ANOVA The sum of squares statistics help us get another view of ANOVA and its rationale. We define, SST = (X i,j X, ) 2, SSTr = ( X i, X, ) 2, SSE = (X i,j X i, ) 2. where SST is the total sum of squares, SSTr is the treatment sum of squares and SSE is the error sum or squares. The important identity holds SST = SSTr + SSE. Also MSTr = SSTr I 1, MSE = SSE I(J 1) The SSE is a measure of variation in the data that is independent of the truth value of H 0, so it is the variation that unexplained. SSTr is the part of the variation that can be explained by the possible differences between µ i (if H 0 happens to be false). ANOVA idea: if the variance explained by the alternative hypothesis that the means differ is sufficiently large, than we reject H 0. Single-factor ANOVA for unequal sample sizes The main difference with respect to the case with equal sample sizes J i is the computation of the degrees of freedom, which impacts the distribution of the computed statistic F. We define n = I i=1 J i and formulate the statistics using sums of squares, which are slightly modified: SST = J i (X i,j X, ) 2 = J i Xi,j 2 1 n X2, SSTr = J i ( X i, X, ) 2 = Xi, 2 J i=1 i 1 n X2, SSE = (X i,j X i, ) 2 = SST SSTr MSTr = SSTr SSE, MSE = I 1 n I. As above, the test statistic is F = MSTr MSE but now its distribution under the null hypothesis is F α,i 1,n I.

5 Example In previous research, the proportion of the human brain engaged in viewing TV programs was investigated for several types of TV shows. The proportion was estimated by counting the number of engaged voxels in an fmri experiment out of a total of voxels. The effect of 3 TV programs were shown to the subjects with the following results. type J i x i, x 2 i, reality show newscast action movie Are the assumptions for an ANOVA satisfied? What is the parameter of interest? What is the null hypothesis and the alternative hypothesis? What is the test statistic and what is its distribution? Compute the value of the test statistic. Continued on next page...

6 Based on the alternative hypothesis, the statistic distribution and the level of significance α, find the critical value and the rejection region for the test. Decide to reject or not to reject and summarize your result in the context of the problem.