one-way anova anova = analysis of variance this is a technique used to analyze the results of an...

One-Way ANOVA

• ANOVA = Analysis of Variance

• This is a technique used to analyze the results of an experiment when you have more than two groups

Example• You measure the number of days 7

psychology majors, 7 sociology majors, and 7 biology majors are absent from class

• You wonder if the average number of days each of these three groups was absent is significantly different from one another

Results

Sociology Psychology Biology4 1 13 2 12 0 23 2 03 4 24 3 02 2 1

X = 3.00 X = 2.00 X = 1.00

Hypothesis

• Alternative hypothesis (H1)

• H1: The three population means are not all equal

Hypothesis

• Null hypothesis (H0)

psych = socio = bio

Between and Within Group Variability

• Two types of variability

• Between– the differences between the mean scores of the

three groups– The more different these means are, the more

variability!

Results


X = 3.00 X = 2.00 X = 1.00

Between Variability


X = 3.00 X = 2.00 X = 1.00

S2 = .66

Between Group Variability

• What causes this variability to increase?

• 1) Effect of the variable (college major)

• 2) Sampling error


• Two types of variability

• Within– the variability of the scores within each group

Results


X = 3.00 X = 2.00 X = 1.00

Within Variability


X = 3.00 X = 2.00 X = 1.00

S2 =.57 S2 =1.43 S2 =.57

Within Group Variability

• What causes this variability to increase?

• 1) Sampling error


Between-group variability

Within-group variability


sampling error + effect of variable

sampling error



sampling error

Thus, if null hypothesis was true this would result in a value of 1.00



sampling error

Thus, if null hypothesis was not true this value would be greater than 1.00

Calculating this Variance Ratio

Degrees of Freedom

• dfbetween

• dfwithin

• dftotal

• dftotal = dfbetween + dfwithin

Degrees of Freedom

• dfbetween = k - 1 (k = number of groups)

• dfwithin = N - k (N = total number of observations)

• dftotal = N - 1

• dftotal = dfbetween + dfwithin

Degrees of Freedom

• dfbetween = k - 1 3 - 1 = 2

• dfwithin = N - k 21 - 3 = 18

• dftotal = N - 1 21 - 1 = 20

• 20 = 2 + 18

Sum of Squares

• SSBetween

• SSWithin

• SStotal

• SStotal = SSBetween + SSWithin

Sum of Squares

• SStotal

Sum of Squares

• SSWithin

Sum of Squares

• SSBetween

Sum of Squares

• Ingredients:

XX2

Tj2

• N

• n

To Calculate the SSSocio Psych Bio

4 1 1

3 2 1

2 0 2

3 2 0

3 4 2

4 3 0

2 2 1

XSocio Psych Bio

4 1 1

3 2 1

2 0 2

3 2 0

3 4 2

4 3 0

2 2 1

Xs = 21 Xp = 14 XB = 7

XSocio Psych Bio

4 1 1

3 2 1

2 0 2

3 2 0

3 4 2

4 3 0

2 2 1

Xs = 21 Xp = 14 XB = 7

X = 42

X2

Socio X2 Psych X2 Bio X2

4 16 1 1 1 1

3 9 2 4 1 1

2 4 0 0 2 4

3 9 2 4 0 0

3 9 4 16 2 4

4 16 3 9 0 0

2 4 2 4 1 1

X2s = 67 X2

P = 38 X2B = 11

Xs = 21 Xp = 14 XB = 7

X = 42

X2


4 16 1 1 1 1

3 9 2 4 1 1

2 4 0 0 2 4

3 9 2 4 0 0

3 9 4 16 2 4

4 16 3 9 0 0

2 4 2 4 1 1

X2s = 67 X2

P = 38 X2B = 11

Xs = 21 Xp = 14 XB = 7

X = 42

X2 = 116

T2 = (X)2 for each groupSocio X2 Psych X2 Bio X2

4 16 1 1 1 1

3 9 2 4 1 1

2 4 0 0 2 4

3 9 2 4 0 0

3 9 4 16 2 4

4 16 3 9 0 0

2 4 2 4 1 1

X2s = 67 X2

P = 38 X2B = 11

Xs = 21 Xp = 14 XB = 7

T2s = 441 T2

P = 196 T2B = 49

X = 42

X2 = 116

Tj2


4 16 1 1 1 1

3 9 2 4 1 1

2 4 0 0 2 4

3 9 2 4 0 0

3 9 4 16 2 4

4 16 3 9 0 0

2 4 2 4 1 1

X2s = 67 X2

P = 38 X2B = 11

Xs = 21 Xp = 14 XB = 7

T2s = 441 T2

P = 196 T2B = 49

X = 42

X2 = 116

Tj2

= 686

NSocio X2 Psych X2 Bio X2

4 16 1 1 1 1

3 9 2 4 1 1

2 4 0 0 2 4

3 9 2 4 0 0

3 9 4 16 2 4

4 16 3 9 0 0

2 4 2 4 1 1

X2s = 67 X2

P = 38 X2B = 11

Xs = 21 Xp = 14 XB = 7

T2s = 441 T2

P = 196 T2B = 49

X = 42

X2 = 116

Tj2

= 686

N = 21

nSocio X2 Psych X2 Bio X2

4 16 1 1 1 1

3 9 2 4 1 1

2 4 0 0 2 4

3 9 2 4 0 0

3 9 4 16 2 4

4 16 3 9 0 0

2 4 2 4 1 1

X2s = 67 X2

P = 38 X2B = 11

Xs = 21 Xp = 14 XB = 7

T2s = 441 T2

P = 196 T2B = 49

X = 42

X2 = 116

Tj2

= 686

N = 21

n = 7

IngredientsX = 42

X2 = 116

Tj2

= 686

N = 21

n = 7

Calculate SSX = 42

X2 = 116

Tj2

= 686

N = 21

n = 7• SStotal

Calculate SSX = 42

X2 = 116

Tj2

= 686

N = 21

n = 7• SStotal

11642

2132

Calculate SS

• SSWithin

X = 42

X2 = 116

Tj2

= 686

N = 21

n = 7

Calculate SS

• SSWithin

X = 42

X2 = 116

Tj2

= 686

N = 21

n = 7

116686

718

Calculate SS

• SSBetween

X = 42

X2 = 116

Tj2

= 686

N = 21

n = 7

Calculate SS

• SSBetween

X = 42

X2 = 116

Tj2

= 686

N = 21

n = 7

686

7

42

21

14

Sum of Squares

• SSBetween

• SSWithin

• SStotal

• SStotal = SSBetween + SSWithin

Sum of Squares

• SSBetween = 14

• SSWithin = 18

• SStotal = 32

• 32 = 14 + 18

Calculating the F value


14

27


7


7

18

181


7

17

How to write it out

Source SS df MS F

Between 14 2 7 7

Within 18 18 1

Total 32 20

Significance

• Is an F value of 7.0 significant at the .05 level?

• To find out you need to know both df

Degrees of Freedom

• Dfbetween = k - 1 (k = number of groups)

• dfwithin = N - k (N = total number of observations)

Degrees of Freedom

• Dfbetween = k - 1 3 - 1 = 2

• dfwithin = N - k 21 - 3 = 18

• Page 390 Table F

• Dfbetween are in the numerator

• Dfwithin are in the denominator

• Write this in the table

Critical F Value

• F(2,18) = 3.55

• The nice thing about the F distribution is that everything is a one-tailed test

Decision

• Thus, if F value > than F critical– Reject H0, and accept H1

• If F value < or = to F critical– Fail to reject H0

Current Example

• F value = 7.00

• F critical = 3.55

• Thus, reject H0, and accept H1

• Alternative hypothesis (H1)

H1: The three population means are not all equal– In other words, psychology, sociology, and

biology majors do not have equal class attendence

– Notice: It does not say where this difference is at!!

How to write it out

Source SS df MS F

Between 14 2 7 7*

Within 18 18 1

Total 32 20

one-way anova anova = analysis of variance this is a technique used to analyze the results of an...

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