Lecture 10:One Way ANOVA

Between Subjects: Practice!

Laura McAvinue School of PsychologyTrinity College Dublin

Example 1: ANOVA by hand

• Our research interest is the treatment of social anxiety

• We would like to evaluate different therapies for social anxiety

• We took a sample of 15 people suffering from social anxiety & randomly assigned them to three groups– Placebo– Cognitive Behavioural Therapy (CBT)– Gestalt Therapy

• Did any of the treatments significantly improve social anxiety?

• Are any of the means significantly different?

StepsGroup A Group B Group C

Placebo CBT Gestalt

0 4 1

1 3 2

3 6 2

1 3 0

0 4 0

Mean 1 4 1

1. Sum of Squares

2. Degrees of Freedom

3. Mean Square

4. F Ratio

5. P ValueWhat is the grand mean of these observations?

2

SSTotal

(0-2)2 + (1-2) 2 + (3-2) 2 + (1-2) 2 + (0-2) 2

4 1 1 1 4+ (4-2) 2 + (3-2) 2 + (6-2) 2 + (3-2) 2 + (4-2) 2

4 1 16 1 4+ (1-2) 2 + (2-2) 2 + (2-2) 2 + (0-2) 2 + (0-2) 2

1 0 0 4 4

= 46

∑ (Xij – Grand mean)2

SSBetween

= 30

n∑ (Group mean – Grand mean)2

5 ∑ (1-2) 2 + (4-2) 2 + (1-2) 2

5 ( 1 + 4 + 1)5 (6)

SSwithin

= 16

∑ (Xij – Group meanj)2

SSplacebo (0-1) 2 + (1-1) 2 + (3-1) 2 + (1-1) 2 +(0-1) 2

1 0 4 0 1 = 6SSCBT (4-4) 2 + (3-4) 2 + (6-4) 2 + (3-4) 2 + (4-4) 2

0 1 4 1 0 = 6SSgestalt (1-1) 2 + (2-1) 2 + (2-1) 2 + (0-1) 2 + (0-1) 2

0 1 1 1 1 = 4

Degrees of Freedom

Dftotal N – 1

15 – 114

Dfbetween K – 1

3 – 12

Dfwithin K (n – 1)

3 (5 – 1)12

Mean Square

MSbetween SSbetween / dfbetween

30 / 215

MSwithin SSwithin / dfwithin

16 / 121.33

F Ratio

F Ratio MSbetween / MSwithin

15 / 1.3311.278

Is F > 1? Yes!

Is F big enough to reject Ho?Compare your F value to the F distribution

Dfnumerator = Dfbetween Dfdenominator = Dfwithin

= 2= 12

F Ratio

• What is the critical value of F when = .05?– 3.88

• What is the critical value of F when = .01?– 6.93

• Is your F value greater than the critical values?– Yes!

• Can you reject Ho? At what alpha level?– Yes! At P < .01

Example 2: ANOVA by computer

Group Social Anxiety

1 (Placebo)... 0

1 1

1 3

1 1

1 0

2 (CBT)... 4

2 3

2 6

2 3

Enter the data into SPSS...

Run the ANOVA

• Analyse, Compare Means, One Way ANOVA

– Dependent List: Social Anxiety– Factor: Group– Options: Descriptives

Homogeneity of variance testMeans plot

Examine the means plot

GestaltCBTplacebo

Therapy

4

3.5

3

2.5

2

1.5

1

Me

an

of

so

cia

lan

x

Are the means more or less the same or does one seem a little different?

Examine the test for Homogeneity of Variance

Is Levine’s statistic significant?

Can we assume homogeneity of variance among groups?

Test of Homogeneity of Variances

socialanx

.000 2 12 1.000

LeveneStatistic df1 df2 Sig.

Examine the ANOVA table

•Is it similar to the one you created?•What is the p value?•Is it statistically significant?•What can you conclude from this ANOVA?At least one mean is significantly different from the others

ANOVA

socialanx

30.000 2 15.000 11.250 .002

16.000 12 1.333

46.000 14

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

Multiple Comparisons

•According to the Bonferroni & Tukey posthoc tests, which means are significantly different from the others?

Multiple Comparisons

Dependent Variable: socialanx

-3.000* .730 .004 -4.95 -1.05

.000 .730 1.000 -1.95 1.95

3.000* .730 .004 1.05 4.95

3.000* .730 .004 1.05 4.95

.000 .730 1.000 -1.95 1.95

-3.000* .730 .004 -4.95 -1.05

-3.000* .730 .004 -5.03 -.97

.000 .730 1.000 -2.03 2.03

3.000* .730 .004 .97 5.03

3.000* .730 .004 .97 5.03

.000 .730 1.000 -2.03 2.03

-3.000* .730 .004 -5.03 -.97

(J) TherapyCBT

Gestalt

placebo

Gestalt

placebo

CBT

CBT

Gestalt

placebo

Gestalt

placebo

CBT

(I) Therapyplacebo

CBT

Gestalt

placebo

CBT

Gestalt

Tukey HSD

Bonferroni

MeanDifference

(I-J) Std. Error Sig. Lower Bound Upper Bound

95% Confidence Interval

The mean difference is significant at the .05 level.*.

Example 3: T tests v ANOVAs

Software/ Kevin Thomas/ ANOVA data setAnalyse the ‘age’ & ‘adherence’ variables using an independent samples t test & an ANOVA

Independent Samples Test

.000 1.000 -3.873 8 .005 -3.00000 .77460 -4.78622 -1.21378

-3.873 8.000 .005 -3.00000 .77460 -4.78622 -1.21378

Equal variancesassumed

Equal variancesnot assumed

adherenceF Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% ConfidenceInterval of the

Difference

t-test for Equality of Means

ANOVA

adherence

22.500 1 22.500 15.000 .005

12.000 8 1.500

34.500 9

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

Similar results?T2 = ?

• Example 4: Research example– Eysenck (1974) used three groups to investigate the impact of

levels of processing (Craik & Lockhart, 1972) on incidental learning - learning in the absence of expectation that the material will need to be recalled later

• Group 1 - count number of letters in each word - lowest level of processing

• Group 2 - think of an adjective that might be used to describe the word

• Group 3 - form a vivid image of the word

– What are Ho & H1?

Ho: There is no difference between the groupsLevel of processing has no effect on recall

H1 : At least one group is significantly differentAs level of processing increases, incidental memory increases

Run the ANOVA

ANOVA dataset: Group & Recall variables

ANOVA

recall

209.067 2 104.533 10.516 .000

268.400 27 9.941

477.467 29

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

•What is the F value?•Is F > 1?•Is it statistically significant?•What can we conclude?

Conclude:•At least one of the means is significantly different from the others•Level of processing does significantly affect incidental recall

Which means are different?

What does the means plot suggest?

imageryadjectivecounting

group

14.00

13.00

12.00

11.00

10.00

9.00

8.00

7.00

Mea

n o

f re

call

Which means are different?

What do the posthoc tests suggest?

Multiple Comparisons

Dependent Variable: recall

-4.00000* 1.41002 .022 -7.4960 -.5040

-6.40000* 1.41002 .000 -9.8960 -2.9040

4.00000* 1.41002 .022 .5040 7.4960

-2.40000 1.41002 .223 -5.8960 1.0960

6.40000* 1.41002 .000 2.9040 9.8960

2.40000 1.41002 .223 -1.0960 5.8960

-4.00000* 1.41002 .026 -7.5990 -.4010

-6.40000* 1.41002 .000 -9.9990 -2.8010

4.00000* 1.41002 .026 .4010 7.5990

-2.40000 1.41002 .301 -5.9990 1.1990

6.40000* 1.41002 .000 2.8010 9.9990

2.40000 1.41002 .301 -1.1990 5.9990

(J) groupadjective

imagery

counting

imagery

counting

adjective

adjective

imagery

counting

imagery

counting

adjective

(I) groupcounting

adjective

imagery

counting

adjective

imagery

Tukey HSD

Bonferroni

MeanDifference

(I-J) Std. Error Sig. Lower Bound Upper Bound

95% Confidence Interval

The mean difference is significant at the .05 level.*.

Effect Size

Calculate Eta squared Calculate Omega squared

2 SSbetweenSStotal

2 SSbetween (k 1)MSwithin

SStotal MSwithin209.09 (3 1)9.94

477.46 9.94189.18

487.4.39209.06 /

477.46

= .44

• Fully reporting the analysis– A one-way (or one-factor) analysis of variance

(ANOVA) compared the mean number of words recalled across three groups who processed the words differently: count, adjective or imagery. With an alpha level of .05, the analysis was statistically significant, F(2,27) = 10.516, p< .001. A Tukey HSD test indicated that the mean of 7 (SD =1.83) for the count group was significantly different to that of the adjective (M=11, SD=2.5) and imagery (M=13.4, SD=4.5) groups. The difference between the adjective and imagery groups was not significant (p > 10).