One-Way ANOVA

Introduction to Analysis of Variance (ANOVA)

What is ANOVA?

ANOVA is short for ANalysis Of VAriance Used with 3 or more groups to test for MEAN

DIFFS. E.g., caffeine study with 3 groups:

No caffeine Mild dose Jolt group

Level is value, kind or amount of IV Treatment Group is people who get specific

treatment or level of IV Treatment Effect is size of difference in means

Rationale for ANOVA (1)

We have at least 3 means to test, e.g., H0: 1 = 2 = 3.

Could take them 2 at a time, but really want to test all 3 (or more) at once.

Instead of using a mean difference, we can use the variance of the group means about the grand mean over all groups.

Logic is just the same as for the t-test. Compare the observed variance among means (observed difference in means in the t-test) to what we would expect to get by chance.

Rationale for ANOVA (2)

Suppose we drew 3 samples from the same population. Our results might look like this:

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4

3

2

1

0

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Raw Scores (X)

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Three Samples from the Same Population

Mean 1

Mean 2

Mean 3

Standard Dev Group 3

Note that the means from the 3 groups are not exactly the same, but they are close, so the variance among means will be small.

Rationale for ANOVA (3)

Suppose we sample people from 3 different populations. Our results might look like this:

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4

3

2

1

0

Three Samples from 3 Diffferent Populations

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Three Samples from 3 Diffferent Populations

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Three Samples from 3 Diffferent Populations

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Raw Scores (X)

Three Samples from 3 Diffferent Populations

Mean 1

Mean 2Mean 3

SD Group 1

Note that the sample means are far away from one another, so the variance among means will be large.

Rationale for ANOVA (4)Suppose we complete a study and find the following results (either graph). How would we know or decide whether there is a real effect or not?

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4

3

2

1

0

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Raw Scores (X)

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Three Samples from the Same Population

Mean 1

Mean 2

Mean 3

Standard Dev Group 3

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4

3

2

1

0

Three Samples from 3 Diffferent Populations

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Three Samples from 3 Diffferent Populations

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Three Samples from 3 Diffferent Populations

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Raw Scores (X)

Three Samples from 3 Diffferent Populations

Mean 1

Mean 2Mean 3

SD Group 1

To decide, we can compare our observed variance in means to what we would expect to get on the basis of chance given no true difference in means.

Review

When would we use a t-test versus 1-way ANOVA?

In ANOVA, what happens to the variance in means (between cells) if the treatment effect is large?

Rationale for ANOVA We can break the total variance in a study into meaningful pieces that correspond to treatment effects and error. That’s why we call this Analysis of Variance.

Definitions of Terms Used in ANOVA:

GX The Grand Mean, taken over all observations.

AX

1AX

The mean of any level of a treatment.

The mean of a specific level (1 in this case) of a treatment.

iX The observation or raw data for the ith person.

The ANOVA ModelA treatment effect is the difference between the overall, grand mean, and the mean of a cell (treatment level).

GA XXEffectIV

Error is the difference between a score and a cell (treatment level) mean.

Ai XXError

The ANOVA Model:

)()( AiGAGi XXXXXX

An individual’s score is The grand

mean+

A treatment or IV effect + Error

The ANOVA Model

)()( AiGAGi XXXXXX The grand

meanA treatment or IV effect

Error

40

30

20

10

0

Fre

quen

cy

ANOVA Data by Treatment LevelANOVA Data by Treatment Level

ANOVA Data by Treatment Level

Grand Mean

Treatment Mean

Error

IV Effect

The graph shows the terms in the equation. There are three cells or levels in this study. The IV effect and error for the highest scoring cell is shown.

ANOVA CalculationsSums of squares (squared deviations from the mean) tell the story of variance. The simple ANOVA designs have 3 sums of squares.

2)( Gitot XXSSThe total sum of squares comes from the distance of all the scores from the grand mean. This is the total; it’s all you have.

2)( AiW XXSS The within-group or within-cell sum of squares comes from the distance of the observations to the cell means. This indicates error.

2)( GAAB XXNSS The between-cells or between-groups sum of squares tells of the distance of the cell means from the grand mean. This indicates IV effects.WBTOT SSSSSS

Computational Example: Caffeine on Test Scores

G1: Control G2: Mild G3: Jolt

Test Scores

75=79-4 80=84-4 70=74-4

77=79-2 82=84-2 72=74-2

79=79+0 84=84+0 74=74+0

81=79+2 86=84+2 76=74+2

83=79+4 88=84+4 78=74+4

Means

79 84 74

SDs (N-1)

3.16 3.16 3.16

G1 75 79 16

Control 77 79 4

M=79 79 79 0

SD=3.16 81 79 4

83 79 16

G2 80 79 1

M=84 82 79 9

SD=3.16 84 79 25

86 79 49

88 79 81

G3 70 79 81

M=74 72 79 49

SD=3.16 74 79 25

76 79 9

78 79 1

Sum 370

GXiX 2)( Gi XX

Total Sum of Squares

2)( Gitot XXSS

In the total sum of squares, we are finding the squared distance from the Grand Mean. If we took the average, we would have a variance.

2)( Gitot XXSS

Scores on the Dependent Variable by Group

0.5

0.4

0.3

0.1

0.0

Rel

ativ

e F

requ

ency

Low High

Grand Mean

G1 75 79 16

Control 77 79 4

M=79 79 79 0

SD=3.16 81 79 4

83 79 16

G2 80 84 16

M=84 82 84 4

SD=3.16 84 84 0

86 84 4

88 84 16

G3 70 74 16

M=74 72 74 4

SD=3.16 74 74 0

76 74 4

78 74 16

Sum 120

Within Sum of Squares

iXAX 2)( Ai XX

2)( AiW XXSS

Within sum of squares refers to the variance within cells. That is, the difference between scores and their cell means. SSW estimates error.

Scores on the Dependent Variable by Group

0.5

0.4

0.3

0.1

0.0

Rel

ativ

e F

requ

ency

Low High

Cell or Treatment Mean

2)( AiW XXSS

G1 79 79 0

Control 79 79 0

M=79 79 79 0

SD=3.16 79 79 0

79 79 0

G2 84 79 25

M=84 84 79 25

SD=3.16 84 79 25

84 79 25

84 79 25

G3 74 79 25

M=74 74 79 25

SD=3.16 74 79 25

74 79 25

74 79 25

Sum 250

Between Sum of Squares

2)( GAAB XXNSS

AX GX 2)( GA XX

The between sum of squares relates the Cell Means to the Grand Mean. This is related to the variance of the means.

Scores on the Dependent Variable by Group

0.5

0.4

0.3

0.1

0.0

Rel

ativ

e F

requ

ency

Low High

Cell Mean

Grand Mean

Cell MeanCell Mean

2)( GAAB XXNSS

ANOVA Source Table (1)

Source SS df MS F

Between Groups

250 k-1=2 SS/df

250/2=

125 =MSB

F = MSB/MSW = 125/10

=12.5

Within Groups

120 N-k=

15-3=12

120/12 = 10 =

MSW

Total 370 N-1=14

ANOVA Source Table (2)

df – Degrees of freedom. Divide the sum of squares by degrees of freedom to get

MS, Mean Squares, which are population variance estimates.

F is the ratio of two mean squares. F is another distribution like z and t. There are tables of F used for significance testing.

The F Distribution

F Table – Critical ValuesNumerator df: dfB

dfW 1 2 3 4 5

5 5%

1%

6.61

16.3

5.79

13.3

5.41

12.1

5.19

11.4

5.05

11.0

10 5%

1%

4.96

10.0

4.10

7.56

3.71

6.55

3.48

5.99

3.33

5.64

12 5%

1%

4.75

9.33

3.89

6.94

3.49

5.95

3.26

5.41

3.11

5.06

14 5%

1%

4.60

8.86

3.74

6.51

3.34

5.56

3.11

5.04

2.96

4.70

Review

What are critical values of a statistics (e.g., critical values of F)?

What are degrees of freedom? What are mean squares? What does MSW tell us?

Review 6 Steps

1. Set alpha (.05).

2. State Null & Alternative

H0:

H1: not all are =.

3. Calculate test statistic: F=12.5

4. Determine critical value F.05(2,12) = 3.89

5. Decision rule: If test statistic > critical value, reject H0.

6. Decision: Test is significant (12.5>3.89). Means in population are different.

321

Post Hoc Tests

If the t-test is significant, you have a difference in population means.

If the F-test is significant, you have a difference in population means. But you don’t know where.

With 3 means, could be A=B>C or A>B>C or A>B=C.

We need a test to tell which means are different. Lots available, we will use 1.

Tukey HSD (1)

HSD means honestly significant difference.

A

W

N

MSqHSD

is the Type I error rate (.05).

q Is a value from a table of the studentized range statistic based on alpha, dfW (12 in our example) and k, the number of groups (3 in our example).

WMS Is the mean square within groups (10).

AN Is the number of people in each group (5).

33.55

1077.305. HSD

From table

MSW

ANResult for our example.

Use with equal sample size per cell.

Tukey HSD (2)

To see which means are significantly different, we compare the observed differences among our means to the critical value of the Tukey test.

The differences are:1-2 is 79-84 = -5 (say 5 to be positive).1-3 is 79-74 = 52-3 is 84-74 = 10. Because 10 is larger than 5.33, this result is significant (2 is different than 3). The other differences are not significant. Review 6 steps.

Review

What is a post hoc test? What is its use? Describe the HSD test. What does HSD

stand for?

Test

Another name for mean square is _________.

1. standard deviation

2. sum of squares

3. treatment level

4. variance

Test

When do we use post hoc tests? a. after a significant overall F test b. after a nonsignificant overall F test c. in place of an overall F test d. when we want to determine the impact of

different factors