Chapter Thirteen The One-Way Analysis of Variance.

Chapter Thirteen

The One-Way Analysis of Variance

Copyright © Houghton Mifflin Company. All rights reserved. Chapter 13 - 2

New Statistical Notation

1.Analysis of variance is abbreviated as ANOVA

2.An independent variable is called a factor

3.Each condition of the independent variable is also called a level or a treatment, and differences produced by the independent variable are a treatment effect

4.The symbol for the number of levels in a factor is k


An Overview of ANOVA


One-Way ANOVA

A one-way ANOVA is performed when

only one independent variable is tested in

the experiment


Between Subjects

• 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


Analysis of Variance

• The analysis of variance is the parametric procedure for determining whether significant differences occur in an experiment containing 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


Diagram of a Study Having ThreeLevels of One Factor


Experiment-Wise Error

• The overall probability of making a Type I error somewhere in an experiment is call 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 that is much larger than the one we have selected

• Using the ANOVA allows us to compare the means from all levels of the factor and keep the experiment-wise-error rate equal to


Assumptions of the ANOVA

1.The experiment has only one independent variable and all conditions contain independent samples

2.The dependent variable measures interval or ratio scores

3.The population represented by each condition forms a normal distribution

4.The variances of all populations represented are homogeneous


kH 210 :

equalaresallnot:a H

Statistical Hypotheses

•

•


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 means from a factor, one pair at a time, to determine which means differ significantly


Components of ANOVA


Sources of Variance

• There are two potential sources of variance

• Scores may differ from each other even when participants are in the same condition. This is called variance within groups

• Scores may differ from each other because they are from different conditions. This is called the variance between groups


Mean Squares

• The mean square within groups is an estimate of the variability in scores as measured by differences within the conditions of an experiment

• The mean square between groups is an estimate of the differences in scores that occurs between the levels in a factor


Performing the ANOVA


Sum of Squares

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

• Each of these terms is called the sum of squares and is symbolized by SS


Source Sum of df Mean FSquares Squares

Between SSbn dfbn MSbn Fobt

Within SSwn dfwn MSwn

Total SStot dftot

Summary Table of a One-way ANOVA


N

XXSS

2tot2

tottot

)(

Computing Fobt

1.Compute the total sum of squares (SStot)


N

X

nSS

2tot

2

bn

)(

columntheinscoresof

)columntheinscoresofsum(

Computing Fobt

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


Computing Fobt

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

SSwn = SStot - SSbn


Computing Fobt

4. Compute the degrees of freedom1. The degrees of freedom between groups

equals k - 1

2. The degrees of freedom within groups equals N - k

3. The degrees of freedom total equals N - 1


5.Compute the mean squares

bn

bnbn df

SSMS

wn

wnwn df

SSMS

Computing Fobt


wn

bnobt MS

MSF

Computing Fobt

6.Compute Fobt


The F-Distribution

The F-distribution is the sampling

distribution showing the various values of

F that occur when H0 is true and all

conditions represent one population


Sampling Distribution of F When H0 Is True


Critical F Value

• 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


Performing Post Hoc Comparisons


21wn

21obt

11nn

MS

XXt

Fisher’s Protected t-Test

• When the ns in the levels of the factor

are not equal, use Fisher’s protected

t-test


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

where qk is found using the appropriate

table

n

MSqHSD k

wn)(

Tukey’s HSD Test


Additional Procedures in the One-Way ANOVA


Xtn

MSXt

n

MS

)()( crit

wncrit

wn

Confidence Interval

• The computational formula for the confidence interval for a single is


A graph showing means from three conditions of an independent variable.

Graphing the Results in ANOVA


• Eta squared indicates the proportion

of variance in the dependent variable

that is accounted for by changing the

levels of a factor

2

tot

bn2

SS

SS

Proportion of Variance Accounted For


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

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


611.5518

2292969

)( 22tot2

tottot

N

XXSS

111.2218

229

6

82

6

67

6

80

)(

columntheinscoresof

)columntheinscoresofsum(

2222

2tot

2

bn

N

X

nSS

50.33111.22611.55bntotwn SSSSSS

Example


Example

• dfbn = k - 1 = 3 - 1 = 2

• dfwn = N - k = 18 - 3 = 15

• dftot = N - 1 = 18 - 1 = 17


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


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


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

• The mean of sample 3 is significantly different from the mean of sample 2

Chapter Thirteen The One-Way Analysis of Variance.

Documents

experiment slide

subjects anova slide

oneway anova slide

overview of anova slide

components of anova

homogeneous slide

ss slide

subjects factor