12-1 One-Way ANOVA

• The analysis of variance (ANOVA) methods of this chapter require the F distribution

• The F distribution has the following properties:

  • The F distribution is NOT symmetric. It is skewed right.

  • Values of the F distribution cannot be negative

  • The exact shape of the F distribution depends on the 2 different degrees of freedom

• One way analysis of variance (ANOVA) is a method of testing the equality of three or more population means by analyzing sample variances. One way analysis of variance is used with data categorized with one factor (or treatment), so there is one characteristic used to separate the sample data into different categories

• Understand that a small p-value (<= 0.05) leads to rejection of the null hypothesis of equal means. With a large p-value (>0.05), fail to reject the null hypothesis of equal means

• When we conclude that there is sufficient evidence to reject the claim of equal population means, we cannot conclude from ANOVA that any particular mean is different from the others

• Test statistic for One-way ANOVA: F = variance between samples / variance within samples

• When testing for equality of 3 or more populations, use analysis of variance. Using multiple hypothesis tests with 2 samples at a time could adversely affect the significance level

• How to calculate the test statistic F with equal sample sizes n:

  • 1. Find the variance between samples

  • 2. Find the variance within samples

  • 3. Calculate the test statistic

• Degrees of freedom (using k = number of samples and n = sample size), numerator degrees of freedom = k -1 and denominator degrees of freedom = k(n-1)

• The F test statistic is very sensitive to sample means, even though it is obtained through 2 different estimates of the common population variance

• One way to reduce the effect of the extraneous factors in an experiment is to use a completely randomized design (each sample value is given the same chance of belonging to the different factor groups) or rigorously controlled design (sample values are carefully chosen so that all other factors have no variability)

• Two informal methods for comparing means:

  • 1. Construct boxplots of the different samples and examine any overlap to see if 1 or more of the boxplots is very different from the others

  • 2. Construct CI estimates of the means for each of the different samples, then compare those CI to see if 1 or more of them does not overlap with the others

• Range tests allow us to identify subsets of means that are not significantly different from each other

• Multiple comparison tests use pairs of means and make adjustments to overcome the problem of having a significance level that increases as the number of individual tests increases. Examples include the Bonferroni Multiple Comparison test.

12-2 Two-Way ANOVA

• There is an interaction between 2 factors if the effect of one of the factors changes for different categories of the other factor (examples like peanut butter and jelly)

• A two-way analysis of variance is used to conduct the following 3 tests: test for an effect from an interaction between the row factor and the column factor, test for an effect from the row factor, test for an effect from the column factor

• Requirements to conduct the test:

  • Normality, variation, sampling, independence, two-way, balanced design