ONE-WAY BETWEEN SUBJECTS ANOVA Also called One-Way Randomized ANOVA Purpose: Determine whether there is a difference among two or more groups – used mostly with three or more groups – does not show which groups differ (unless there are only two)

Design and Assumptions Design: – One way means one independent variable – Between subjects means different people in each group. Assumptions: same as for independent samples t-test

Why not t-tests? Multiple t-tests inflate the experimentwise alpha level. experimentwise alpha level is the total probability of Type I error for all tests of significance in the study. ANOVA controls the experimentwise alpha level.

If I am doing six t-tests, each with a.05 alpha level, what is the experimentwise alpha?

So, the probability of making one or more errors is =.2649.

Concept of ANOVA Analysis of Variance Variance is a measure of variability Two step process: – divides the variance into parts – compares the parts

About Variance Numerator is the Sum of Squares Denominator is the Degrees of Freedom

Mean Square Variance is also called Mean Square Formula for variance in ANOVA terms:

Part I: Dividing the Variance Total Variance is divided into two parts: – Between Groups Variance - only differences between groups. – Within Groups Variance - only differences within groups. Between Groups + Within Groups = Total

Example of Between Groups variance only: Group 1Group 2Group 3 468

Example of Within Groups variance only: Group 1Group 2Group

What Influences Between Groups Variance? effect of the i.v. (systematic) individual differences (non-systematic) measurement error (non-systematic)

What Influences Within Groups Variance? individual differences (non-systematic) measurement error (non-systematic)

Part II: Comparing the Variance

About the F-ratio Larger with a bigger effect of the IV Expected to be 1.0 if Ho is true Never significant below 1.0 Can’t be negative

Sampling Distribution of F 1.0

Computation of One-Way BS ANOVA EXAMPLE: Twelve participants were randomly assigned to one of three noise conditions and given a spelling test. Determine whether noise had a significant effect on spelling performance. (See data on next page)

No noiseLow noiseHigh noise x=16x=15x=11 grand mean = 14

ANOVA Summary Table SourceSSdfMSFp Between Within Total

STEP 1: SS Total =  (x-x G ) 2 grand mean xx-x(x-x)  = SS Total = 96

STEP 2: SS Between =  (x g -x G ) 2 group mean xx-x(x-x)  = SS Between = 56

STEP 3: SS Within = SS Total - SS Between SS Within = = 40

ANOVA Summary Table SourceSSdfMSFp Between56 Within40 Total96

STEP 4: Calculate degrees of freedom. df Total = N-1 df Total = 12-1 = 11 df Between = k-1k=#groups df Between = 3-1 = 2 df Within = N-k df Within = 12-3 = 9

ANOVA Summary Table SourceSSdfMSFp Between562 Within409 Total9611

STEP 5: Calculate Mean Squares

ANOVA Summary Table SourceSSdfMSFp Between Within Total9611

STEP 6: Calculate F-ratio.

STEP 7: Look up critical value of F. df numerator = df Between df denominator = df Within F-crit (2,9) = 4.26

APA Format Sentence A One-Way Between Subjects ANOVA showed a significant effect of noise, F (2,9) = 6.31, p <.05.

ANOVA Summary Table SourceSSdfMSF p Between <.05 Within Total9611

Computing Effect Size Eta-squared is the proportion of variance in the DV that can be explained by the IV.

KRUSKAL-WALLIS ANOVA Non-parametric replacement for One-Way BS ANOVA Assumptions: – independent observations – at least ordinal level data – minimum 5 scores per group

EXAMPLE: Fifteen participants were randomly assigned to one of three noise conditions and given a spelling test. Determine whether noise had a significant effect on spelling performance. (See data on next page) Calculating the Kruskal-Wallis ANOVA

No noiseLow noiseHigh noise

STEP 1: Rank scores. No noiseLow noiseHigh noise STEP 2: Sum ranks for each group.  R 1 = 57.5  R 2 = 45  R 3 = 17.5

STEP 3: Compute H.

STEP 4: Compare to critical value from  2 table. df = 2, critical value = 5.99 A Kruskal-Wallis ANOVA showed a significant difference among the three noise conditions, H(2) =8.38, p <.05.

ANOVA for Within Subjects Designs When the IV is within subjects (or matched groups), a Repeated Measures ANOVA should be used The logic of the ANOVA is the same Calculation differs to take advantage of the design

ANOVA for Within Subjects Designs The Friedman ANOVA is the non- parametric replacement for One-Way Repeated Measures ANOVA