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.

Pc = N!

r!(N - r)!

p rqN -r

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

P(0 errors) = 6x5x4x3x2x1

(1)6x5x43x2x1

.050 .956

P(0 errors) = 1 1 .956 = .7351

So, the probability of making one or more errors is 1 - .7351 = .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

s 2 (x x)2

N 1

• 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:

Mean Square Sum of Squares

degrees of freedom

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 1 Group 2 Group 34 6 84 6 84 6 8

Example of Within Groups variance only:

Group 1 Group 2 Group 34 6 48 4 86 8 6

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

F = Between Groups Variance

Within Groups Variance

F = non - systematic + effect of i.v.

non - systematic

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 noise Low noise High noise

15 15 12

17 19 10

18 14 10

14 12 12

x=16 x=15 x=11

grand mean = 14

ANOVA Summary Table

Source SS df MS F p

Between

Within

Total

STEP 1: SS Total = S(x-xG)2 grand meanx x-x (x-x)2

15 1 117 3 918 4 1614 0 015 1 119 5 2514 0 012 -2 412 -2 410 -4 1610 -4 1612 -2 4 S = SS Total = 96

STEP 2: SS Between = S(xg-xG)2

group meanx x-x (x-x)2

16 2 416 2 416 2 416 2 415 1 115 1 115 1 115 1 111 -3 911 -3 911 -3 9 11 -3 9 S = SS Between = 56

STEP 3: SS Within = SS Total - SS Between

SS Within = 96 - 56 = 40

ANOVA Summary Table

Source SS df MS F p

Between 56

Within 40

Total 96

STEP 4: Calculate degrees of freedom.

df Total = N-1

df Total = 12-1 = 11

df Between = k-1 k=#groups

df Between = 3-1 = 2

df Within = N-k

df Within = 12-3 = 9

ANOVA Summary Table

Source SS df MS F p

Between 56 2

Within 40 9

Total 96 11

STEP 5: Calculate Mean Squares

28.00 2

56

Bet df

Bet SS =Between MS

4.44 9

40

Withindf

WithinSS = Within MS

ANOVA Summary Table

Source SS df MS F p

Between 56 2 28.00

Within 40 9 4.44

Total 96 11

STEP 6: Calculate F-ratio.

6.31 4.44

28.00

WithinMS

Between MS = F

STEP 7: Look up critical value of F.df numerator = df Betweendf 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

Source SS df MS F p

Between 56 2 28.00 6.31 <.05

Within 40 9 4.44

Total 96 11

Computing Effect Size

2 =SS Between

SS Total

2 =56

96 .58

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 noise Low noise High noise

17 19 9

18 16 8

14 12 12

16 11 8

13 10 7

STEP 1: Rank scores.

No noise Low noise High noise17 13 19 15 9 418 14 16 11.5 8 2.514 10 12 7.5 12 7.516 11.5 11 6 8 2.513 9 10 5 7 1

STEP 2: Sum ranks for each group.SR1 = 57.5 SR2 = 45 SR3 = 17.5

STEP 3: Compute H.

H =12

15(16)

57.52

5

452

5

17.52

5

3(16)

H = (.0500) 661.25 + 405 + 61.25 48

H = (.0500) 1127.50 48

H = 56.38 - 48 = 8.38

H =12

N(N+ 1)

R1 2

n1

R2 2

n2

...

3(N 1)

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