Analysis of variance (ANOVA)

Analysis of Variance (ANOVA)

By Tesfamichael Getu (PhD Student)

Bahir Dar University, Ethiopia

Outline

Introduction

Basic concepts

Types

One way Repeated Measures Two way

Introduction

♦Coined the phrase “analysis of variance” around

1920.♦defined it as “the

separation of variance ascribable to one group of causes from the variance ascribable to the other

groups.”

Sir Ronald Fisher (1890-1962)

Historical Background

Introduction ANOVA works by comparing the

differences between group means rather than the differences between group variances.

The procedure uses variances to decide whether the means are different.

ANOVA analysis uses the F-statistics.

Why ANOVA?In real life, things do not typically result in two groups being compared only.ANOVA might be used when we wish to:

compare means of more than two population

compare population each containing several levels of subgroups.

Why Not Use Lots of t-Tests?Computing multiple t-tests are problematic as it: is tedious when many groups are present increases the risk of a Type I error E.g. for three means computing three t tests

can increase the type 1 error rate from 5% to 14.3 %

At .05 level of significance, with 100 comparisons, 5 will show a difference when none exists (experiment wise error)

So the more t-tests you run, the greater the risk of a type I error (rejecting the null when there is no difference).

Assumptions in ANOVAThere are three basic assumptions in ANOVA.These are: normal distribution, homogeneity of variance and random sampling

What Does ANOVA Tell us? Null Hypothesis:

Like a t-test, ANOVA tests the null hypothesis that the means are the same.

Ho: 1= 2= 3 … Alternative Hypothesis:

The means differ. H1: 1 23 …

ANOVA is an Omnibus test It test for an overall difference between groups. It tells us that the group means are different. However, it doesn’t tell us exactly which means differ.

Basic Concepts

1

23

Variability Between

F ratio is the ratio of two variance.

F ratio

1

23

Variability Within

F ratioF ratio is central to ANOVA, so ANOVA is variability ratio.Variability Between

Variability Within

Distance from over all mean

Internal spread

Rejecting the hypothesis

=Reject Ho

=Fail to reject Ho

=fail to reject Ho

Variance between

Variance Within

Factorsare independent variables because of

which the groups might differ significantly in an ANOVA analysis, there might be just

one factor or more than one factor. A factor may have many levels. Eg. Gender has 2 levels: Male and FemaleFactors could be fixed or random.

Factors Cont.

Fixed V Random Factors A factor is fixed if it contains all the levels of

interest, and random if the levels under study are a random

sample of the possible levels that could be sampled and the goal is to make a statement regarding the larger population.

Experimental V Observational Experimental factors are “set” or “assigned” to

the experimental units; Observational factors are characteristics of the

experimental units that cannot be assigned.

Between, within and mixed Designs

An ANOVA design can be described by specifying three things: 1. Number of factors involved in the design. 2. Number of levels of each of the factors. 3. Whether the factor is a between- or within-

subjects factor.Between-groups design: the level of factor(s) varies between the subjects

of different groups subjects of each group will be exposed to only one

level of the factor(s). similar to the independent sample design

discussed in the case of t-tests.

Between, and within Designs

In within-groups design, the level of factor(s) varies within the subjects.

This is done by exposing the same subject to different levels of the factor(s) at different times.

It is also referred as repeated measures design.

This design is conceptually similar to the paired sample design discussed in the case of t-tests.

Main Effect and Interaction effect

Main effect is the direct effect of a single factor or

independent variable on the dependent variable.

It simply compares the mean of one level of a factor with the other level(s) of that factor.

Interaction effect is the combined effect of two or more

factors on the dependent variable.

Types of ANOVA

classified depending on the number of factorsOne-way ANOVA includes a single factor which gives rise to a single main effect. Two-way ANOVAtwo main effects of two factors and one two-way interaction effect between the twoThree-way ANOVA, there are 3 maineffects of 3 factors, 3 two-way interactions, and 1three-way interaction. If the three factors in three-way ANOVA are X, Y, and Z, then we will have 3 main effects of X, Y, and Z, three two-way interactions X*Y, X*Z and Y*Z, and one three-way interaction X*Y*Z.

Summary of classification

Types Independent Var.

Dependent Var.

Covariate

ANOVA One way 1 1Factorial >=2 1

Repeated measures ANOVA

One way 1 1Factorial >=2 1

MANOVA One Way 1 >=2Factorial >=2 >=2

ANCOVA 1 1 √MANCOVA 1 >=2 √

Exercise: Identify appropriate test for each research questions

1. Is there statistically significant difference in the sense of community among students based on type of course(online, residential and blended)?

2. Is there a change in sense of community score over three time periods(beginning, middle, end )of online program?

3. Is there statistically significant difference in the sense of community among students based on type of course(online, residential and blended) and gender(M, F)?

4. Is there statistically significant difference in the sense of social presence, cognitive presence, and teaching presence based on type of course(online, residential and blended)?

Ansewers1. One WAY Type of course(online, residential, blended):IV Sense of community: DV2. Repeated Measures Time(Time 1, Time 2, Time 3): IV Sense of community: DV3. Factorial Type of course(online, residential, blended):IV Gender : IV4. MANOVA Type of course(online, residential and blended): IV social presence: DV cognitive presence: DV teaching presence: DV

One Way ANOVA

How to perform One-way ANOVA in SPSS? Choose Analyze > General Linear Model >

Univariate Click the DV (only one click) to highlight it

and then transfer it to Dependent Variable box by clicking the corresponding arrow.

Doing a similar procedure for IV and transfer it to Fixed Factor(s) box by clicking the corresponding arrow.

After that, click the option button and check for homogeneity of Variance. Note: SPSS uses a Levene’s test of homogeneity of variance.

Back to the former box.

Post Hoc Test

The results from the ANOVA do not indicate which of the groups differ from one another.

To locate the source of this difference we use a post hoc test

commonly Tukey test and the more conservative one

Scheffé test are used in post hock. These tests assume equal variance.

Repeated Measures ANOVA

It tells the researcher if there is significant difference somewhere among a set of scores

Repeated Measures Designs are also called Dependent Measures Designs Within-Subjects Designs

Characteristics of repeated measures designs

Nature of the repeated measuresDuration between measures

Two Way ANOVA

In two-way analysis, we have two independent variables or factors and we are interested in knowing their effect on the same dependent variable. The following example illustrates how this analysis can be performed. Example Ho1: Gender will have no significant effect on

students Listening test score.Ho2: Age will have no significant effect on

students Listening test score.Ho3: Gender and age interaction will have no

significant effect on students Listening test score.

Exercise: Factorial ANOVA

A researcher seeks to understand whether level of self esteem and audience situation has effect on the number of errors people make . Then, 12 people (6 with high and 6 with low self esteem) performed in front of audience, and other 12 people (6 with high and 6 with low self esteem) alone. Ho1=Self esteem has no significant effect on the

number of errors people make.Ho2=Audience situation has no significant effect on

the number of errors people make.Ho3=Self-esteem and audience situation interaction

will have no significant effect on number of errors people make.

Factorial ANOVA output Table 1 Tests of Between-Subjects Effects

Source Type III Sum of Squares

df Mean Square F Sig.

Audience 96.000 1 96.000 22.326 .000

SelfEsteem 96.000 1 96.000 22.326 .000

Audience * SelfEsteem 24.000 1 24.000 5.581 .028

Error 86.000 20 4.300    

Total 1478.000 24      

A Report of Factorial ANOVA

A two way analysis of variance found that level of self-esteem did have a significant effect ( F(1,20)=22.326, p<.001, η2 =.527); high self-esteem performers averaged for fewer errors than low self-esteem performers (see Table 1). The presence of an audience also had a significant effect (F(1,20)=22.326, p<.001, η2 =.527) people who performed before an audience made about four more errors than who performed alone (see Table 1). The interaction of self-esteem and the audience situation was also significant (F(1, 20)=5.581, P=.028, η2 =.218). People with low self-esteem who performed in front of an audience made the most errors , while those with high-self esteem who performed alone made the fewest errors.

Thank you!