Chapter 11

HYPOTHESIS TESTING USING THEONE-WAY ANALYSIS OF VARIANCE

Moving Forward

Your goals in this chapter are to learn:• The terminology of analysis of variance• When and how to compute • Why should equal 1 if H0 is true, and why

it is greater than 1 if H0 is false

• When and how to compute Tukey’s HSD• How eta squared describes effect size

obtF

obtF

Analysis of Variance

• The analysis of variance is the parametric procedure for determining whether significant differences occur in an experiment with two or more sample means

• In an experiment involving only two conditions of the independent variable, you may use either a t-test or the ANOVA

An Overview of ANOVA

One-Way ANOVA

• Analysis of variance is abbreviated as ANOVA• An independent variable is also called a factor• Each condition of the independent variable is

called a level or treatment• Differences produced by the independent

variable are a treatment effect

Between-Subjects

• A one-way ANOVA is performed when one independent variable is tested in the experiment

• When an independent variable is studied using independent samples in all conditions, it is called a between-subjects factor

• A between-subjects factor involves using the formulas for a between-subjects ANOVA

Within-Subjects Factor

• When a factor is studied using related (dependent) samples in all levels, it is called a within-subjects factor

• This involves a set of formulas called a within-subjects ANOVA

Diagram of a Study Having ThreeLevels of One Factor

Assumptions of the ANOVA

1. All conditions contain independent samples

2. The dependent scores are normally distributed, interval or ratio scores

3. The variances of the populations are homogeneous

Experiment-Wise Error

• The probability of making a Type I error somewhere among the comparisons in an experiment is called the experiment-wise error rate

• When we use a t-test to compare only two means in an experiment, the experiment-wise error rate equals

Comparing Means

• When there are more than two means in an experiment, the multiple t-tests result in an experiment-wise error rate much larger than the we have selected

• Using the ANOVA allows us to make all our decisions and keep the experiment-wise error rate equal to

Statistical Hypotheses

kH 210 :

equalaresallnot:a H

The F-Test

• The statistic for the ANOVA is F

• When Fobt is significant, it indicates only that somewhere among the means at least two of them differ significantly

• It does NOT indicate which specific means differ significantly

• When the F-test is significant, we perform post hoc comparisons

Post Hoc Comparisons

• Post hoc comparisons are like t-tests

• We compare all possible pairs of level means from a factor, one pair at a time to determine which means differ significantly from each other

Components of the ANOVA

Mean Squares

• The mean square within groups describes the variability in scores within the conditions of an experiment. It is symbolized by MSwn.

• The mean square between groups describes the differences between the means of the conditions in a factor. It is symbolized by MSbn.

The F-Ratio

• The F-ratio equals the mean square between groups divided by the mean square within groups

• When H0 is true, Fobt should equal 1

• When H0 is false, Fobt should be greater than 1

wn

bnobt MS

MSF

Performing the ANOVA

Sum of Squares

• The computations for the ANOVA require the use of several sums of squared deviations

• The sum of squares is the sum of the squared deviations of a set of scores around the mean of those scores

• It is symbolized by SS

Summary Table of a One-way ANOVA

Computing Fobt

1. Compute the sums and means• • •

for each level. Add the from all levels to get . Add together the from all levels to get . Add the ns together to get N.

X2X

X

X

totX 2X2totX

Computing Fobt

2. Compute the total sum of squares (SStot)

N

XXSS

2tot2

tottot

)(

Computing Fobt

3. Compute the sum of squares between groups (SSbn)

N

X

n

XSS

2tot

2

bn

)(

columnin

)columnin(

Computing Fobt

4. Compute the sum of squares within groups (SSwn)

bntotwn SSSSSS

Computing Fobt

Compute the degrees of freedom• The degrees of freedom between groups

equals k – 1 where k is the number of levels in the factor

• The degrees of freedom within groups equals N – k

• The degrees of freedom total equals N – 1

Computing Fobt

Compute the mean squares

•

•

bn

bnbn df

SSMS

wn

wnwn df

SSMS

Computing Fobt

Compute Fobt

wn

bnobt MS

MSF

Sampling Distribution of FWhen H0 Is True

Degrees of Freedom

The critical value of F (Fcrit) depends on

• The degrees of freedom (both the dfbn = k – 1 and the dfwn = N – k)

• The selected

• The F-test is always a one-tailed test

Tukey’s HSD Test

When the ns in all levels of the factor are equal, use the Tukey HSD multiple comparisons test

where qk is found using Table 5 in Appendix B

n

MSqHSD k

wn)(

Tukey’s HSD Test

• Determine the difference between each pair of means

• Compare each difference between the means to the HSD

• If the absolute difference between two means is greater than the HSD, then these means differ significantly

Effect Size and Eta2

Proportion of Variance Accounted For

Eta squared indicates the proportion of variance in the dependent variable scores that is accounted for by changing the levels of a factor

)( 2

tot

bn2

SS

SS

Example

Using the following data set, conduct a one-way ANOVA. Use = 0.05.

Group 1 Group 2 Group 3

14 14 10 13 11 15

13 10 12 11 14 13

14 15 11 10 14 15

Example

611.5518

2292969

)(

2

2tot2

tottot

N

XXSS

Example

111.2218

229

6

82

6

67

6

80

)(

columnin

)columnin(

2222

2tot

2

bn

N

X

n

XSS

Example

50.33

111.22611.55bntotwn

SSSSSS

Example

• dfbn = k – 1 = 3 – 1 = 2

• dfwn = N – k = 18 – 3 = 15

• dftot = N – 1 = 18 – 1 = 17

Example

055.112

111.22

bn

bnbn

df

SSMS

233.215

50.33

wn

wnwn

df

SSMS

951.4233.2

055.11

wn

bnobt

MS

MSF

Example

• Fcrit for 2 and 15 degrees of freedom and = 0.05 is 3.68

• Since Fobt = 4.951, the ANOVA is significant

• A post hoc test must now be performed

Example

242.26

233.2675.3)( wn

n

MSqHSD k

334.0333.13667.13

500.2167.11667.13

166.2167.11333.13

13

23

21

XX

XX

XX

Example

Because 2.50 > 2.242 (HSD), the mean of sample 3 is significantly different from the mean of sample 2.