7/28/2019 6006499 ANOVA Introduction

1/14

bas statistics for Social Science slide 1

Analysis of Variance

(ANOVA)

7/28/2019 6006499 ANOVA Introduction

2/14

bas statistics for Social Science slide 2

Why ANOVA?

To compare differences between group means

Example: Is there any significant difference in satisfaction

between users of the following three computer operating

systems

The variables Type Scale

Satisfaction

Operating systems

Dependent

Independent

Interval

Nominal

7/28/2019 6006499 ANOVA Introduction

3/14

7/28/2019 6006499 ANOVA Introduction

4/14bas statistics for Social Science slide 4

Which of the following data sets indicate differences between thetwo groups?

Data 1:

Group

A B

Data 2:

Group

A B

30 21

14 28

26 14

15 31

27 16

13 1225 27

13 25

12 26

15 28

14 27

15 26

12 3017 29

Range 17 19

Mean 21.4 21.3

Range 5 5

Mean 14 27.3

7/28/2019 6006499 ANOVA Introduction

5/14

bas statistics for Social Science slide 5

Assumptions Underlying

One-way ANOVA

1. The dependent variable is normally distributed for each

of the populations as defined by the different levels of

the factor (independent variable)

2. The variances of the dependent variable are the same forall populations

3. The cases represent random samples from the

populations and the scores on the test variable areindependent of each other

7/28/2019 6006499 ANOVA Introduction

6/14

bas statistics for Social Science slide 6

What to expect?

Post hoc

comparison

Nosig.diffe

rence

z

Sig.diff

erence

Omega and

Eta squaresTesthypothesis

7/28/2019 6006499 ANOVA Introduction

7/14

bas statistics for Social Science slide 7

Steps in ANOVA

1. State the null and alternative hypotheses

HO: 1 = 2 = 3HA: Not all means are equal

2. Calculate the test statistic

F-ratio3. Determine critical value

4. Make your decision

5. Make conclusion

7/28/2019 6006499 ANOVA Introduction

8/14

bas statistics for Social Science slide 8

Step 1: State the hypotheses

HO: 1 = 2 = 3 = ..

HA: Not all means are equal

Depends on number of

groups

7/28/2019 6006499 ANOVA Introduction

9/14

bas statistics for Social Science slide 9

Step 2: Calculate test statistic

1. Calculate sum of squares

( )

=

22

. N

X

XSSTa

( )22

.N

X

n

TSBb

j =S

SSBSSTSSWc =.

2. Determine degrees of freedom

1. = kdfa B

kNdfb W =.

1. = Ndfc T

7/28/2019 6006499 ANOVA Introduction

10/14

bas statistics for Social Science slide 10

3. Calculate Mean Squares 4. Calculate F-ratio

Bdf

SSBMSBa =.

MSW

MSBF=

Wdf

SSWMSWb =.

Summary ANOVA table:

Source SS df MS F

Between-group SSB k-1 MSB F-ratio

Within-group SSW N-k MSW

Total SST N-1

7/28/2019 6006499 ANOVA Introduction

11/14

bas statistics for Social Science slide 11

Step 3: Determine critical valueCritical value is based on:

n Significance level ()

o Degrees of freedom1=

kkN

df

df FF bw

Step 4: Decision

Manual:Reject HO: Fcal > Fcritical

Fail to reject HO: Fcal Fcritical

SPSS:Reject HO: sig-F<

Fail to reject HO: sig-F

7/28/2019 6006499 ANOVA Introduction

12/14

bas statistics for Social Science slide 12

Step 5: Conclusion

Reject HO:

Significant difference between the groups

Fail to reject HO:No significant difference between the groups

7/28/2019 6006499 ANOVA Introduction

13/14

bas statistics for Social Science slide 13

Additional Analysis I:

Post-hoc Comparison

Tukeys HSD (Honestly Significant Difference)

n

MSWqHSD

Wdfk ,,=

Criteria to determine difference:

Sig difference: Mean difference > HSD

No sig. difference: Mean difference HSD

7/28/2019 6006499 ANOVA Introduction

14/14

bas statistics for Social Science slide 14

Additional Analysis II:

Omega SquareAmount of variance of the dependent variable attributed

to the independent variables

( )( ) NFdf

Fdf

B

B

+

=

1

1 2

Eta SquareProportion of variance of dependent variable that is

attributed to the independent variable

SST

SSB=

2