Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Chapter 10 Introduction to the Analysis of Variance

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Analysis of Variance Abbreviated as “ANOVA” Used to compare the means of more than two samples –Null hypothesis is that all populations being studied have the same mean –Reject null if at least one population has a mean that differs from the others Actually works by analyzing variances

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Two Different Ways of Estimating Population Variance Estimate population variance from variation within each sample –Is not affected by whether or not null hypothesis is true Estimate population variance from variation between each sample –Is affected by whether or not null hypothesis is true

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Two Important Questions 1.How can you estimate population variation from variance between samples? 2.How is that estimate affected by whether or not the null is true?

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Estimate population variance from variation between means of samples First, variation among means of samples is related directly to the amount of variation within each population from which samples are taken –The more variation within each population, the more variation in means of samples taken from those populations –Note that populations on the right produce means that are more scattered

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Estimate population variance from variation between means of samples And second, when null is false there is an additional source of variation When null hypothesis is true (left), variation among means of samples caused by –Variation within the populations When null hypothesis is false (right), variation among means of samples caused by –Variation within the populations –And also by variation among the population means

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Basic Logic of ANOVA ANOVA entails a comparison between two estimates of population variance –Ratio of between-groups estimate to within-groups estimate called an F ratio –Compare obtained F value to an F distribution Analogy of signal-to-noise ratio…

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Carrying Out an ANOVA Step 1: Estimate population variance from variation of scores within each group –Pool variance estimates from each sample

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Carrying Out an ANOVA Step 2: Estimate population variance from variation of scores within each group –Estimate the variance of distribution of means –Figure the estimated variance of the population

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Carrying Out an ANOVA Step 3: Compute F ratio –Divide between-groups estimate of the population variance by the within- groups estimate of the population variance

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Carrying Out an ANOVA Step 4: Compare obtained F to an F distribution –Compute degrees of freedom for both between- groups estimate and within- groups estimate –Consult an F table

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Assumptions of an ANOVA Populations follow a normal curve Populations have equal variances As for t tests, ANOVAs often work fairly well even when those assumptions are violated

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Rejecting the Null Hypothesis A significant F tells you that at least one of the means differs from the others –Does not indicate how many differ –Does not indicate which one(s) differ For more specific conclusions, a researcher must conduct follow-up t tests Problem: Lots of t tests increases the chances of finding a significant result just by chance (i.e., increases chances beyond p =.05)

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Effect Size To determine effect size of an ANOVA, divide square root of F by square root of n, the number of scores in each group –Small =.10 –Medium =.25 –Large =.40

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Factorial ANOVA Procedure that allows one to examine two or more variables in the same study –Efficient –Allows for examination of interaction effects An ANOVA with only one variable is a one-way ANOVA, an ANOVA with two variables is a two-way ANOVA, and so on

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Main Effects vs. Interactions A main effect refers to the effect of one variable, averaging across the other(s) An interaction effect refers to a case in which the effect of one variable depends on the level of another variable

Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall In a two-way ANOVA, there can be two main effects and one interaction Any combination of significant and non-significant results is possible Main Effects vs. Interactions