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
• x = mean for entire data set is
Group i has
• ni = # of individuals in group i
• xij = value for individual j in group i
• xi = mean for group i
• si = standard deviation for group i
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
( xi - x ) 2
• variation WITHIN groups
• for each data value we look at the difference between that
value and the mean of its group
( xij - xi ) 2
How ANOVA works
The ANOVA F-statistic is a ratio of the Between Group Variation divided
to the Within Group Variation:
F
Between
Within
MSG
MSE
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 Pvalue is greater than a, fail to reject H0.
How ANOVA works
 Step 1: The null hypothesis is
H0 :
1
2
3
• Step 2: The alternative hypothesis is
H a : not all of the
i
are equal
• 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:
F
Mean Square Group
MSG
or
Mean Square Error
MSE
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 = σ22 = σ32 = σi2 → Homogeneity of
variance
Ha: σ12 ≠ σ22 ≠ σ32 ≠ 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.
How to perform ANOVA in SPSS?
 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.
How to perform ANOVA in SPSS?
 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.
How to perform ANOVA in SPSS?
One IV or Factor
Is F-value significant?
Yes
No
Are there more than 2
groups?
Stop
Yes
No
Do Post Hoc
comparison
Stop
How to perform ANOVA in SPSS?
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”.
How to perform ANOVA in SPSS?
 Does management find evidence that
employees’ job satisfaction differs across
regions? Which branch differs from the
others?
How to perform ANOVA in SPSS?
This is how
the data set
is shown
How to perform ANOVA in SPSS?
How to perform ANOVA in SPSS?
Transfer Satisfaction
variable to
dependent variable
box and region
variable to Fixed
Factor(s) box. After
that click Options
How to perform ANOVA in SPSS?
Check the
homogeneity
check-box and
after that click
Continue
How to perform ANOVA in SPSS?
Click Post Hoc…
How to perform ANOVA in SPSS?
1. Transfer Location
variable from
factor(s) to Pos
Hoc Tests for:
2. Check the Tukey
Check-box
3. Click Continue
How to perform ANOVA in SPSS?
Click OK and wait a minute
How to perform ANOVA in SPSS?
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
How to perform ANOVA in SPSS?
Result of ANOVA
P-Value for ANOVA
Ho: μ1 = μ2 = μ3
Ha: At least one μ is
different than the others
Conclusion: There is a difference in
employees’ job satisfaction across
regions.
How to perform ANOVA in SPSS?
South region is significantly
different from others
How to perform ANOVA in SPSS?
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?
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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.
 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.