One-Way Between Subjects ANOVA

Overview • Purpose • How is the Variance Analyzed?

• Assumptions • Effect Size

Purpose of the One-Way ANOVA • Compare the means of two or more groups • Usually used with three or more groups • Independent variable (factor) may or may not be manipulated; affects interpretation but not statistics

Why Not t-tests?

• Multiple t-tests inflate the experimentwise alpha level.

• ANOVA controls the experimentwise alpha level with an omnibus F-test.

Why is it One Way?

• Refers to the number of factors • How many WAYS are individuals grouped?

• NOT the number of groups (levels)

Why is it Called ANOVA?

• Analysis of Variance • Analyze variability of scores to determine whether differences between groups are big enough to reject the Null

HOW IS THE VARIANCE ANALYZED?

• Divide the variance into parts • Compare the parts of the variance

Dividing the Variance • Total variance: variance of all the scores in the study.

• Model variance: only differences between groups.

• Residual variance: only differences within groups.

Model Variance • Also called Between Groups variance • Influenced by: – effect of the IV (systematic) – individual differences (non-systematic) – measurement error (non-systematic)

Residual Variance • Also called Within Groups variance • Influenced by: – individual differences (non-systematic) – measurement error (non-systematic)

Sums of Squares • Recall that the SS is the sum of squared deviations from the mean • Numerator of the variance • Variance is analyzed by dividing the SS into parts: Model and Residual

Sums of Squares • SS Model = for each individual, compare the mean of the individual’s group to the overall mean • SS Residual = compare each individual’s score to the mean of that individual’s group

Mean Squares • Variance • Numerator is SS • Denominator is df – Model df = number of groups -1 – Residual df = Total df – Model df

Comparing the Variance MS Model F = MS Residual F = non - systematic + effect of i.v.

non - systematic

ASSUMPTIONS • Interval/ratio data • Independent observations • Normal distribution or large N • Homogeneity of variance – Robust with equal n’s

EFFECT SIZE FOR ANOVA • Eta-squared ( h 2 )indicates proportion of variance in the dependent variable explained by the independent variable h 2 = SS Model SS Total

Reporting F-test in APA Format A one-way between-subjects ANOVA indicated a significant difference among the three conditions,

F

(2,57) = 88.55,

p

h 2 = .76.

< .001,

Take-Home Points • ANOVA allows comparison of three or more conditions without increasing alpha.

• Any ANOVA divides the variance and then compares the parts of the variance.