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 How can you estimate population variation from variance between samples? 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 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

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