anova: analysis of variance - Åbo...

One-way ANOVA

Introduction

Two variables: 1 Categorical variable (factor/IV), 1 Quantitative variable (response/DV)

Main Question: Do (the means of) the quantitative variables depend on which group (given by categorical variable) the individual is in?

ANOVA looks at differences between groups.

Note: We usually refer to the sub-populations or the same population but with different treatments as “groups” when doing ANOVA.

Introduction

At its simplest ANOVA tests the following

hypotheses:

H0: The means of all the groups are equal

μ1 = μ2 = μ3 = μi

Ha: Not all the means are equal

Introduction

Usefulness:

– Similar to t-test

– More versatile than t-test

– Compare one parameter (response variable) between two or more groups

Introduction

Why Not Just Use t-tests?

– Tedious when many groups are present

– Using all data increases stability

– Large number of comparisons some may appear significant by chance

Introduction

Examples:

– ”An organization has three different branches.

Turnover level differs across the three branches and

management wants to know whether this may be

explained by the extent to which employees are

satisfied with their working environment across the

branches. Fifty employees are randomly selected at

each branch and given a questionnaire measuring

how satisfied they currently are with the working

environment”.

Introduction

– Researchers investigate the effects of control type

on firm performance. The research question is

whether a real difference exists in performance

between owner- and manager-controlled firms

(McKean and Kania, 1978).

– Investigators want to investigate whether

demographic factors (e.g. age groups, races,

education level, annual income level, and

employment status) and investment experience

(novice, intermediate, advance) have influence on

retirement planning intention.

Introduction

– Researchers investigate the behavior of noise traders

and their impact on the market. There are three

groups in the experiment (accordingly with

researchers’ treatments): informed traders (who

possess fundamental information), liquidity traders

(who have to trade for exogenous reasons), and

noise traders (who do not possess fundamental

information and have no exogenous reasons to

trade); (Bloomfield, O’Hara, and Saar, 2007).

– Researchers investigate the impact of moods (i.e.

Negative, positive, and neutral) on ethical judgment

of auditors (Cianci and Bierstaket, 2009).

Introduction

The researchers investigate the effects of

advertising models’ eye color (blue, green,

and brown) in ad viewers responses to the

ad (Simpson, Sturgen, and Tanguma)

Introduction

What can we conclude from the examples?

ANOVA Assumptions

There are Three basic assumptions used in

ANOVA:

The populations from which the samples

were taken are normally distributed.

Homogeneity of variance

Random sampling.

Notation for ANOVA

• n = number of individuals all together

• i = number of groups

• = mean for entire data set is

Group i has

• ni = # of individuals in group i

• xij = value for individual j in group i

• = mean for group i

• si = standard deviation for group i

ix

x

How ANOVA works

ANOVA measures two sources of variation in the data and

compares their relative sizes

• variation BETWEEN groups

• for each data value look at the difference between its group

mean and the overall mean

• variation WITHIN groups

• for each data value we look at the difference between that

value and the mean of its group

2)-( xxi

2)-( iij xx

The ANOVA F-statistic is a ratio of the Between Group Variation divided

to the Within Group Variation:

MSE

MSG

Within

BetweenF

This compares the variation between groups (group means to overall mean)

to the variation within groups (individual values to group means). This is

what gives it the name “Analysis of Variance.”

A large F is evidence against H0, since it indicates that there is more

difference between groups than within groups.

Note: it is easier to look at the P-value to indicate whether the H0 is

rejected or not If the P-value is less than or equal to a, reject H0. If the P-

value is greater than a, fail to reject H0.

How ANOVA works

How ANOVA works

Step 1: The null hypothesis is

3210 :H

• Step 2: The alternative hypothesis is

equal are theof allnot :H ia

• Step 3: The significance level is = ?

(usually is set to one of the values {0.01, 0.05, 0.1}

How ANOVA works

Step 4: Calculate the F-statistic:

MSE

MSGor

Error SquareMean

Group SquareMean F

MSG, MSE and the F-statistic are found in the

ANOVA table when the analysis is run on the SPSS

How ANOVA works

Step 5: Find the P-value

Step 6. Reject or fail to reject H0 based on the

P-value.

Step 7. State your conclusion.

How ANOVA works

Levene’s test:

H0: σ12 = σ2

2 = σ32 = σi

2 → Homogeneity of

variance Ha: σ1

2 ≠ σ22 ≠ σ3

2 ≠ σi2

– Homogeneity fulfilled → Equal variances assumed.

– Homogeneity rejected → Equal variances not assumed.

Note: •ANOVA is still robust even when the homogeneity assumption is not fulfilled, as long as the sample sizes are roughly equal or the deviation is only of a moderate level. As a rule of thumb, if the largest std.dev < (2 x the smallest std.dev) then we need not to be concerned about this assumption. •Equal variance assumed or not assumed will affect to Post Hoc test methods (p.20)

How to perform ANOVA in SPSS?

One-way ANOVA – 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 three 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 is

Scheffé test; equal variance is assumed in these tests).

– Click Post Hoc and check Tukey box, click Continue button.

– Last, click OK button and wait a moment while SPSS analyzes the

data.

Note:

• Tukey performs all of the pairwise comparisons between groups.

• Scheffe performs simultaneous joint pairwise comparisons for all

possible pairwise combinations of means. Can be used to examine all

possible linear combinations of group means, not just pairwise

comparisons.


If equal variance is not assumed, some post hoc tests could be used:

– Tamhane's T2. Conservative pairwise comparisons test based on a t-test.

– Dunnett's T3. Pairwise comparison test based on the Studentized maximum modulus.

– Games-Howell. Pairwise comparison test that is sometimes liberal.

– Dunnett's C. Pairwise comparison test based on the Studentized range.


One IV or Factor

Is F-value significant?

Yes

Are there more than 2

groups?

Yes

Do Post Hoc

comparison

No

Stop

No

Stop


Exercise 1:

Open job satisfaction.sav

An organization has three branches in three different region. Management wishes to know whether employees are satisfied with their job differs across regions. A total of 218 employees are randomly selected at the regions and given a questionnaire measuring how satisfied they currently are with their job”.


Does management find evidence that

employees’ job satisfaction differs across

regions? Which branch differs from the

others?


This is how

the data set

is shown


Transfer Satisfaction

variable to

dependent variable

box and region

variable to Fixed

Factor(s) box. After

that click Options


Check the

homogeneity

check-box and

after that click

Continue


Click Post Hoc…


1. Transfer Location

variable from

factor(s) to Pos

Hoc Tests for:

2. Check the Tukey

Check-box

3. Click Continue


Click OK and wait a minute


The number of sample

in each region

Homogeneity test’s

result

P-value for Levene’s Test

Ho: σ1 = σ2 = σ3

Ha: At least one σ is different than

the others


Result of ANOVA

Conclusion: There is a difference in

employees’ job satisfaction across

regions.

P-Value for ANOVA

Ho: μ1 = μ2 = μ3

Ha: At least one μ is

different than the others


South region is significantly

different from others


Exercise 2:

Open Training.xlsx file.

Read the instruction in the Training.xlsx file and the raw data.

Open Training.sav file.

Observe how we handle the raw data and convert it into three treatments in order to analysis it using ANOVA.

Perform the ANOVA test using file Training.sav.

Answer the questions.

Report this exercise 2 in written form by the end of this course week.

Test yourself

What is ANOVA?

Why do we use ANOVA?

What are ANOVA assumptions?

How to test ANOVA assumptions?

What do we do when the equal variance is not fulfilled?

What does it mean when the F value in ANOVA result is statistically significant?

What does the post hoc test answer?

References

Agresti, A. (2007) Ch 12: Comparing group: Analysis of Variance (ANOVA) method, Retrieved on 26/04/2012, from http://www.stat.ufl.edu/~aa/sta6127/ch12.pdf.

Bloomfield, R., O’Hara M., and Saar G. (2007) ”How Noise Trading Affects Markets: An Experimental Analysis”, Available at SSRN: http://ssrn.com/abstract=994379 or http://dx.doi.org/10.2139/ssrn.994379.

Cianci, A. and Bierstaker, J. 2009. "The Effect of Client Importance and Performance Feedback on Auditors' Technical and Ethical Judgments." Managerial Auditing Journal, Vol. 24 Iss: 5, pp.455 – 474.

Ghozali, I. (2005) Multivariate analysis application with SPSS, Diponegoro University Publishing, Semarang.

Ho, R. (2006) Handbook of univariate and multivariate data analysis and interpretation with SPSS, Taylor & Francis Group, Boca Raton, FL.

http://www.stat.ufl.edu/~aa/sta6127/ch12.pdf

http://dx.doi.org/10.2139/ssrn.994379

McKean, J. R., and Kania, J. J. (1978) “An Industry approach to

owner-manager control and profit performance”, Journal of

Business, Vol. 51 No. 2, pp. 327-342.

Murray, J. (2010) Analysis of Variance – Homework and Exam,

Retrieved on 27/04/2012, from

http://www.murraylax.org/bus735/fall2010.

Pruim, R. (nd) ANOVA: Analysis of Variance, Retrieved on

30/04/2012, from

http://www.calvin.edu/~rpruim/courses/m243/F03.

http://www.calvin.edu/~rpruim/courses/m243/F03

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