Introduction to

Analysis of Variance

Chapter 11

Review t-tests

Single-sample t-test

Independent samples t-test

Related or paired-samples t-test

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Which test do I use?

What is the DV?

What is the IV?

What factor designates the groups?

What and how many levels make up the factor

Is it a between or within subject design?

How many groups of subjects?

How many measures for each subject?

Which t-test do you use?

What type of t-test? What is the DV? IV?

What design is used (between or within-Ss)?

A newspaper article reported that the typical American family spent an average of $81 on Halloween candy and costumes last year. You are interested in finding out if families in Spartanburg spend more or less than the national average.

A researcher would like to replicate a study that found that older adults that owned pets were less likely to go to the doctor after an upsetting event than those who did not own pets. She would like to compare a group of pet owners to a group of people without pets and measure the number of visits to the doctor per year.

Method: Compare multiple groups

A 2-level design may be too simple to answer the research question

Beins et al. (2007)

Present cartoons to Ss after told to expect something that was either

Not very funny

Very funny

No message about how funny to expect

Results: Ratings of jokes depended on what people were expecting

What are the benefits of the design?

ANOVA: Analysis of Variance

How can we compare 3 or more levels at same time?

Single-factor or One-way ANOVA

Between subject design

1 factor or IV– but 2+ levels

Example

DV = # colds

IV = drug type

3 levels bet-Ss: placebo, low dose, high dose of vitamin C

Null hypothesis: no difference between means

Alternative hypothesis: difference between means

Examine VARIANCE instead of sample mean difference

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Why use ANOVAs?

If = .05 for each t-test you compute

Each test, 5% chance of type I error

or 1 in 20 tests expect error

If need multiple t-tests – error

Calculate chance of making Type I error: 1-(1-α)c

Where c is number of comparisons

1-(1-.05)3 = 1 – (.95)3 = .14 = 14% chance!

Bonferroni adjustment: divide alpha by # tests

Instead perform all tests simultaneously in ANOVA

ANOVA test statistic

T-test ratio:

F-ratio (ANOVA):

chanceby expected difference

means samplebetween differencet

(error) chanceby expected difference variance

means samplebetween sdifference varianceF

Logic of ANOVA

Separates total variability in the DV into two parts

Between-treatments variability

Systematic variance = diff’s in means due to the IV

(treatment effect)

with error variance

Within-treatments variability

Error variance

Between-Treatment Variance

gro

up

rating0 5 10 15

1

2

3

1 2X

2 4X

3 12X

Includes systematic

variance plus error variance

Within-Treatment Variance

gro

up

rating0 5 10 15

1

2

3

1 2SS

2 18SS

3 2SS

1 2 3withinSS SS SS SS

2 18 2 22withinSS

Logic of ANOVA

Separates total variability in the DV into two parts

Between-treatments variability Systematic variance = diff’s in means due to the IV (treatment

effect)

with error variance

Within-treatments variability Error variance

Need to calculate “Sum of Squares” = “SS” Sum of squared deviations from each score from mean

∑(X – M)2

Note: if you then divide by N-1 you get variance (s2)! Right?!

Note: and then if you take square root you get standard deviation (s)! Right?!

Overall goal of ANOVA

Evaluate difference between groups

Distinguish if between-treatment differences due to:

Treatment effect

Chance or Error

Inter-Individual differences – between subject

Intra-Individual differences – within subject

ANOVA: Partitions the Variance

Total Variance

Between Treatment Variance

1. Treatment effects

2. Error

Within Treatment Variance

Error

Between variance ---------------------- Within variance

F =

The Analysis of Variance: The F-statistic

variation among sample means

variation among individuals in the same sample

Analysis of variance (ANOVA) looks at the ratio:

F=

Treatment Effect +differences due to chance

differences due to chanceF=

If there is no treatment effect, what would you expect

for the value of F?

0 +differences due to chance

differences due to chanceF= = 1 = no effect

The Analysis of Variance F-statistic

variation among sample means

variation among individuals in the same sample

Analysis of variance (ANOVA) looks at the ratio:

F=

Treatment Effect +differences due to chance

differences due to chanceF=

If there IS a treatment effect, what would you

expect for the value of F?

>0 +differences due to chance

differences due to chanceF= = >1 = an effect!

Error term

ANOVA formulas

MSB= MSW=

SS = sums of squares

Dogs, friends and stress

45 people were asked do a stressful task

and their HR was measured

15 were randomly assigned to one of three groups

Alone With a friend With their dog

Group 1 Group 2 Group 3

Results

Alone With a friend With their dog

Group 1 Group 2 Group 3

X

SS

85.5

1184

91.3 964

73.4 1372

What are hypotheses? 0 1 2 3:H

1 2 3: not all of , ,and are equalAH

.05

We cannot use the t-test, because there are more than 2

samples to be compared. So, use ANOVA

The ANOVA

Alone With a friend With their dog

Group 1 Group 2 Group 3

X

S

85.5

9.2

91.3 8.3

73.4 9.9

.05

F = 14.1

Between Group Variance = 1194

Within Group Variance = 84.9

Conclusion?

Is there evidence that the level of stress differs among

the experimental groups?

1. Determine correct df

2. Determine the critical F

3. Are the mean differences statistically significant?

ANOVA formulas: df

Total df = N – 1

Total = dfwithin + dfbetween

Within df = (n – 1) or = N – k

Total number of subjects – number of groups or levels

Between df = k – 1

Number of groups or levels - 1

ANOVA formulas: df

Total df = N – 1

Within df = (n – 1) or = N – k

Between df = k – 1 (where k = # grps or levels)

Total = dfwithin + dfbetween

Dog example: 45 people; 15 per condition

Total = 45 – 1 = 44

Within = (15-1)+(15-1)+(15-1) = 42

OR Within = 45 – 3 = 42

Between = 3 – 1 = 2

Total = 42 + 2 = 44

LOOK UP in table:

Between (numerator) and within (denominator) df

Critical F Table A.8 pp. 392-394

DF between

DF

wit

hin

t distribution for df = 18

F-ratios distribution for

df = 1, 18.

Notice that the critical

values for = .05 are t =

±2.101 and that F = 2.1012

= 4.41

Critical F Calculate between df and within df

Ex: between = 2, within = 42

Look up critical F

@ .05 = 3.23, @ .01 = 5.18

What is conclusion if: F (2, 42) = 14.1

Write-up

A one-way ANOVA was conducted to examine the effect of emotional support on heart rate. A significant difference in HR was found between the groups that dealt with a stressful event alone (M = 85.5, SD = 9.2), with a friend (M = 91.3, SD = 8.3), and with a dog (M = 73.4, SD = 9.9), F(2, 42) = 14.1, p < .01.

Anova: Definitional formulas

Between groups SS (sums of squares)

Sum of squared deviations from each group’s mean from grand mean multiplied by the number of Ss in group

Within groups SS

Sum of squared deviations of each score from group mean

Total SS

Sum of squared deviations of each score from the grand mean

])[( 2nMM Gg

2)( gMX

2)( GMX

ANOVA (theoretical) formulas: SS

Total SS =

= 46

Within SS =

= 16

Between SS =

= 30

SStotal = SSbetween + SSwithin

46 = 30 + 16

Temp Cond

1 2 3

0 4 1 X2=

1 3 2 106

3 6 2 G=30

1 3 0 N=15

0 4 0 k=3

T1=5 T2=20 T3=5 MG=2

SS1=6 SS2=6 SS3=4

n1=5 n2=5 n3=5

M1=1 M2=4 M3=1

])[( 2nMM Gg

2)( gMX

2)( GMX

ANOVA computational formulas:

Sum of Squares (SS)

Total SS: sum of squared deviations from grand mean

Within SS = SS = sum of squared deviations from grp mean

Between SS = sum of squared deviations of grp mean from

grand mean

SStotal = SSbetween + SSwithin

N

GXSSTOTAL

22

N

G

n

TSSbetween

22

Where G = grand (overall) sum of scores

Where N = total number of scores

Where T = sum of scores for group

Where n = number of scores in group

222 )()()(321 gggWithin MXMXMXSS

ANOVA formulas: SS

Total SS =

= 106 – 302/15 = 46

Within SS = SS

= 6 + 6 + 4 = 16

Between SS =

= 52/5 + 202/5 + 52/5 – 302/15

= 30

SStotal = SSbetween + SSwithin

46 = 30 + 16

N

GXSS

22

N

G

n

TSSbetween

22

Temp Cond

1 2 3

0 4 1 X2=

1 3 2 106

3 6 2 G=30

1 3 0 N=15

0 4 0 k=3

T1=5 T2=20 T3=5

SS1=6 SS2=6 SS3=4

n1=5 n2=5 n3=5

M1=1 M2=4 M3=1

ANOVA formulas

between

betweenbetween

df

SSMS

33.112

16withinMS

152

30betweenMS

within

withinwithin

df

SSMS

within

between

MS

MSF 28.11

33.1

15F

ANOVA summary table

Source df SS MS F

Between

Within

2

12

30

16

15

1.33

11.28*

Total 14 46

F (2, 12) = 11.28, p < .01

* Significant at .01 level

Effect size

How much of the variability in DV is

attributed to IV?

Effect size for ANOVA: eta-squared (η2)

Total

Between

SS

SS2

Post-hoc tests

If ANOVA result is significant…

How do you know which grps differ?

Need to do t-tests but reduce Type I error

Use Tukey’s HSD test

Pairwise comparisons while keeping same

alpha level