Chapter 14 – 1

Chapter 14: Analysis of Variance

• Understanding Analysis of Variance• The Structure of Hypothesis Testing with

ANOVA• Decomposition of SST• Assessing the Relationship Between

Variables• SPSS Applications• Reading the Research Literature

Chapter 14 – 2

Examples

• Using two sample t test, we could compare means between two groups, but how about more than two group?

• The differences on the mean GPA among racial groups.

• Is there any racial discrimination on annual income?

Chapter 14 – 3

ANOVA

• Analysis of Variance (ANOVA) - An inferential statistics technique designed to test for significant relationship between two variables in two or more samples.

• The logic is the same as in t-tests, just extended to independent variables with two or more samples.

Chapter 14 – 4

Understanding Analysis of Variance

• One-way ANOVA – An analysis of variance procedure using one dependent and one independent variable.

• ANOVAs examine the variation between samples, as well as the variation within a single sample.

Chapter 14 – 5

The Structure of Hypothesis Testing with ANOVA

Assumptions:• (1) Independent random samples are used.

Our choice of sample members from one population has no effect on the choice of members from subsequent populations.

• (2) The dependent variable is measured at the interval-ratio level. Some researchers, however, do apply ANOVA to ordinal level measurements.

Chapter 14 – 6

The Structure of Hypothesis Testing with ANOVA

Assumptions:

• (3) The population is normally distributed. Though we generally cannot confirm whether the populations are normal, we must assume that the population is normally distributed in order to continue with the analysis.

• (4) The population variances are equal.

Chapter 14 – 7

Stating the Research and Null Hypotheses

H1: At least one mean is different from the others.

H0: μ1 = μ2 = μ3 = μ4 =…

Chapter 14 – 8

The logic

• Why might sample means differ?

1. Group Membership

2. Differences not due to group membership

• The fundamental technique is a partitioning of the total sum of squares errors into between-group error and within-group error.

Chapter 14 – 9

The logic

Chapter 14 – 10

)()()(

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Chapter 14 – 11

Within-Group Sum of Squares

2

1 1

( )jnp

w ij jj i

SS x x

nj = the number of cases in a sample (j represents the number of different groups)

= each individual score in a sample = the mean of a groupjx

ijx

This tells us the variations within our groups; it also tells us the amount of unexplained variance.

Chapter 14 – 12

Between-Group Sum of Squares

2

1

( )p

B j jj

SS n x x

nj = the number of cases in a sample (j represents the number of different samples)

= the mean of a sample

= the overall mean

jx

x

This tells us the differences between the groups

Chapter 14 – 13

Decomposition of Variance

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Chapter 14 – 14

The Structure of Hypothesis Testing with ANOVATotal Sum of Squares

T B WSS SS SS

Chapter 14 – 15

The Structure of Hypothesis Testing with ANOVAMean Square Between

An estimate of the between-group variance obtained by dividing the between-group sum of squares by its degrees of freedom.

Mean square between (MSB) = SSB/dfb

where dfb = degrees of freedom between

dfb = j – 1 j = number of categories

Chapter 14 – 16

The Structure of Hypothesis Testing with ANOVAMean Square Within

An estimate of the within-group variance obtained by dividing the within-group sum of squares by its degrees of freedom.

Mean square within (MSW)= SSW/dfw

where dfw = degrees of freedom withindfw = N – j

N = total number of cases

j = number of categories

Chapter 14 – 17

The F test• It is simply the ratio of the two variance

estimates:

0

0

/

/

if then H

if then not H

where, , and

B B Bcalculated

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calculated critical

calculated critical

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SS df MS

F F

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SSSSMS MS

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DfB = j - 1

DfW = N - j

DfT = N - 1

Chapter 14 – 18

Definitions• F ratio (F statistic) – Used in an analysis of

variance, the F statistic represents the ratio of between-group variance to within-group variance

• F obtained – The test statistic computed by the ratio for between-group to within-group variance.

• F critical – The F score associated with a particular alpha level and degrees of freedom. This F score marks the beginning of the region of rejection for our null hypothesis.

Chapter 14 – 19

dfb

dfw

Chapter 14 – 20

Example: Obtained vs. Critical F

Since the obtained F is beyond the critical F value, we reject the Null hypothesis of no difference

Chapter 14 – 21

Example

• Research Question

Does the presence of others influence helping behavior?

• Hypotheses

  In Symbols In Words

H0 μ1= μ2= μ3The presence of others does not influence helping.

H1 Not H0

The presence of others does influence helping. Or at least one of means does not equal to others.

Chapter 14 – 22

The Data

  # of seconds it took for

folks to help

# people present

0 2 4

25 30 32

30 33 39

20 29 35

32 40 41

  36 44

4 5 5

26.8 33.6 38.2

Chapter 14 – 23

Chapter 14 – 24

Example• Assumptions

1) The subjects are sampled randomly.2) The groups are independent.3) The population variances are equal.4) The population distribution of the DV is normal in shape.

Chapter 14 – 25

critical

critical

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calculated

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Chapter 14 – 26

Example

• DecisionReject H0 and conclude that the presence of

others does influence helping. Or at least one of means does not equal to others.

Source SS df MS F p

Between 292.11 2 146.056 6.21 <.05

Within 258.75 11 23.520  

Total 550.86 13  

Chapter 14 – 27

SPSS Example:Bush’s Job Approval

Chapter 14 – 28

SPSS Example:Clinton’s Job Approval

Chapter 14 – 29

Exercise

• A manager wishes to determine whether the mean times required to complete a certain task differ for the three levels of employee training.

• He randomly selected 6 employees with each of the three levels of training (Beginner, Intermediate and Advanced).

Chapter 14 – 30

Datai adv med beg

1 31 23 31

2 24 24 23

3 20 24 34

4 22 22 34

5 25 27 30

6 17 24 30

Chapter 14 – 31

Reading the Research Literature

Chapter 14 – 32

Reading the Research Literature