This document provides an introduction to analysis of variance (ANOVA). It defines key terms like factors, levels, and independent/quasi-independent variables. It explains the advantages of ANOVA over t-tests in comparing more than two treatment conditions. Examples are given of one-way ANOVA to compare a single factor with repeated measures and independent measures designs. Two-way ANOVA is introduced for studying the interaction between two factors. Mauchly's test and the assumption of sphericity are also discussed.

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Analysis of Variance   
Chapter 13
INTRODUCTION TO
ANALYSIS OF VARIANCE

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Analysis of Variance   
INTRODUCTION
• Analysis of variance (ANOVA) is a
hypothesis testing procedure that is used to
evaluate mean differences between two or
more treatment
• ANOVA has a tremendous advantage over
t-test
• The major advantage is that it can be used
to compare two or more treatments

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Analysis of Variance   
TERMINOLOGY
• When a researcher manipulates a variable to
create treatment conditions, the variable is called
an independent variable
• When a researcher uses non-manipulated variable
to designate groups, the variable is called a quasi
independent variable
• An independent variable or a quasi independent
variable is called a factor
• The individual groups or treatment condition that
are used to make up a factor are called the levels
of the factor

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Analysis of Variance   
• Like the t test , ANOVA can be used with
either an independent measures or a repeated
measures design
• An independent-measures design means that
there is a separate sample for each of
treatments
• A repeated-measures design means that the
same sample is tested in all of the different
treatment condition
• ANOVA can be used to evaluate the results
from a research study that involves more than
one factor

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Analysis of Variance   
Two Factors Design
Temperature
Subjects
150
C 250
C 350
C
Ali
Bili

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Analysis of Variance   
STATISTICAL HYPOTHESES FOR ANOVA
• Suppose that a psychologist examined
learning performance under three temperature
conditions: 150
C, 250
C, and 350
C
• Three samples of subjects are selected, one
sample for each treatment condition
• The purpose of the study is to determine
whether room temperature affects learning
performance

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Analysis of Variance   
The HYPOTHESES
•H0 : µ1 = µ2 = µ3
In words, the null hypothesis states the
temperature has no effect on performance
•H1 : at least one condition mean is
different from another
In general, H1 states that the treatment
conditions are not all the same; that is, there is
a real treatment effect

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Analysis of Variance   
The TEST STATISTIC FOR ANOVA
F =
Variance (differences)
between samples means
Variance (differences)
expected by chance (error)
Note that the F-ratio is based on variances
instead of sample mean difference

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Analysis of Variance   
One-Way ANOVA
• The One-Way ANOVA procedure produces a
one-way analysis of variance for a
quantitative dependent variable by a single
factor (independent) variable.
• Analysis of variance is used to test the
hypothesis that several means are equal. This
technique is an extension of the two-sample
t-test.

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Analysis of Variance   
One-Way ANOVA
• Adakah pengaruh kelembapan terhadap
kecepatan mengetik?
• Bandingkan dengan t-test!
• Adakah perbedaan kecepatan mengetik
berdasarkan temperatur udara? Pada
temperatur berapakah kecepatan mengetik
yang paling cepat?

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Analysis of Variance   
One-Way ANOVA
Independent Variable
150
C 250
C 350
C
Mean 150
C Mean 250
C Mean 350
C
• Bersifat between subjects
• Contoh: Pengaruh temperatur udara terhadap
kecepatan mengetik
Analyze >> Compare Means >> One-Way Anova

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Analysis of Variance   
Two-Factor
Analysis of Variance
Independent Measures

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Analysis of Variance   
preview
Imagine that you are seated at your desk, ready
to take the final exam in statistics. Just before the
exam are handed out, a television crew appears
and set up a camera and lights aimed directly at
you. They explain they are filming students
during exams for a television special. You are
told to ignore the camera and go ahead with
your exam.
Would the presence of a TV camera affect
your performance on your exam?

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Analysis of Variance   
example
• Shrauger (1972) tested participants on a concept
formation task. Half the participants work alone
(no audience), and half with an audience of
people who claimed to be interested in observing
the experiment.
• Shrauger also divided the participants into two
groups on the basis of personality: those high in
self-esteem and those low in self-esteem
• The dependent variable for this experiment was
the numbers of errors on the concept formation
task

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Analysis of Variance   
result
10
8
6
4
2
Meannumberoferrors
Self-EsteemHIGH LOW
No
Audience
With
Audience
No
Audience
With
Audience

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Analysis of Variance   
result
• Notice that the audience had no effect on the
high-self-esteem participants
• However, the low-self-esteem participants
made nearly twice as many errors with an
audience as when working alone

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Analysis of Variance   
• Shrauger’s study have two independent
variables, which are:
–Audience (present or absent)
–Self-esteem (high or low)
• The result of this study indicate that the
effect of one variable depends on another
variable
• To determine whether two variables are
interdependent, it is necessary to examine
both variables together in single study

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Analysis of Variance   
• Most of us find it difficult to think clearly or to
work efficiently on hot days
• If you listen to people discussing this problem,
you will occasionally hear comments like, “It’s
not the heat; it’s the humidity”
• To evaluate this claim scientifically, you will
need to design a study in which both heat and
humidity are manipulated within the same
experiment and then observe behavior under a
variety of different heat and humidity
combinations

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Analysis of Variance   
The structure of a
two-factor
experiment
presented as matrix.
The factors are
humidity and
temperature
Temperature
150
C 250
C 350
C
Humidity
High
Low

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Analysis of Variance   
MAIN EFFECT
• The main differences among the level of one-
factor are referred to as the main effect of the
factor
• When the design of the research study is
represented as a matrix of one factor
determining the rows and the second factor
determining the columns, then the mean
differences among the row describe the main
effect of one factor, and the mean differences
among the column describe the main effect for
the second factor

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Analysis of Variance   
INTERACTION
An interaction between two factors occurs
whenever the mean differences between
individual treatment condition, or cells, are
different from what would be predicted from
the overall main effects of the factors

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Analysis of Variance   
Factorial ANOVA
• is used when we have two or more
independent variables (hence it called
factorial)
• Several types of factorial design:
– Unrelated factorial design
– Related factorial design
– Mixed design

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Analysis of Variance   
Several Types Factorial ANOVA
• Unrelated factorial design
This type of experiment is where there are several IV
and each has been measured using different subject
• Related factorial design
An experiment in which several IV have been
measures, but the same subjects have been used in
all conditions (repeated measures)
• Mixed design
A design in which several independent variables
have been measured; some have been measured
with different subject whereas other used the same
subject

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Analysis of Variance   
Factorial ANOVA
IV
Gol. Darah
A
Gol. Darah
B
Gol. Darah
AB
Gol. Darah
O
Laki-Laki Kel. 1 Kel. 2 Kel. 3 Kel. 4
Perempuan Kel. 5 Kel.6 Kel.7 Kel.8
• Bersifat between subject
• Contoh: Pengaruh golongan darah dan jenis
kelamin terhadap kemampuan meyelam
Analyze >> General Linear Model >> Univariat

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Analysis of Variance   
The GLM
Repeated Measures
ONE INDEPENDENT VARIABLE

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Analysis of Variance   
What is…
• ‘Repeated Measures’ is a term used when the
same subjects participate in all condition of an
experiment
• For example, you might test the effects of
alcohol on enjoyment of a party
• Some people can drink a lot of alcohol without
really feelings the consequences, whereas
other only have to sniff a pint of lager and
they fall to the floor and pretend to be a fish

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Analysis of Variance   
Repeated ANOVA
SUBJECT
Independent Variable
pagi siang malam
Subject-1
Subject-2
Subject-dst
• Bersifat within subjects
• Contoh: Pengaruh waktu (pagi/siang/malam)
terhadap kemampuan push-up
Analyze >> General Linear Model >> Repeated Measures

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Analysis of Variance   
Advantages…
• It reduces the unsystematic variability and so
provides greater power to detect effects
• More economical because fewer subjects are
required

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Analysis of Variance   
Disadvantages…
• In between-groups ANOVA, the accuracy of
the F-test depends upon the assumption that
scores in different conditions are independent.
When repeated measures are used this
assumption is violated: scores taken under
different experimental condition are related
because they come from the same subjects
• As such, the conventional F-test will lack
accuracy

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Analysis of Variance   
SPHERICITY
• The relationship between scores in different
treatment condition means that an additional
assumption has to be made and, put
simplistically, we assume that the relationship
between pairs of experimental condition is
similar
• This assumption is called the assumption of
sphericity

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Analysis of Variance   
What is SPHERICITY?
• Most of us are taught that is crucial to have
homogeneity of variance between conditions when
analyzing data from different subjects, but often we
are left to assume that this problem ‘goes away’ in
repeated measure design
• Sphericity refers to the equality of variances of the
differences between treatment level
• So, if you were to take each pair of treatment levels,
and calculate the difference between each pair of
scores, then it is necessary that differences have equal
variance

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Analysis of Variance   
How is sphericity measured?
variance A-B ≈ variance A-C ≈ variance B-C
Group
A
Group
B
Group
C
VARIANCE
A-B A-C B-C
10 12 8 -2 2 5
15 15 12 0 3 3
25 30 20 -5 5 10
35 30 28 5 7 2
30 27 20 3 10 7
15,7 10,3 10,7

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Analysis of Variance   
Assessing the severity of departures
from sphericity
• SPSS produces a test known as Mauchly’s,
which tests the hypothesis that the variances
of the differences between conditions are
equal
• Therefore, if Mauchly’s test statistic is
significant, we should conclude that there are
significant differences between the variance
differences, ergo the condition of sphericity is
not met

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Analysis of Variance   
Mixed ANOVA
• A design in which several independent
variables have been measured; some have
been measured with different subject whereas
other used the same subject
• Minimal ada 2 IV
• Bersifat between subject
Analyze >> General Linear Model >> Repeated Measures

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Analysis of Variance   
Mixed ANOVA
• Contoh: Pengaruh waktu (pagi/siang/malam) dan jenis
kelamin terhadap kemampuan push-up
• Semua subjek dilihat kemampuan push-up di pagi, siang,
dan malam. Tetapi ada dua kelompok yang sama sekali
berbeda, yaitu kelompok laki-laki dan perempuan
SUBJECT
Independent Variable
pagi siang malam
Laki-laki
Perempuan

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Analysis of Variance   
THE LOGIC OF ANALYSIS OF VARIANCE
150
C 250
C 350
C
0
1
3
1
0
M = 1
4
3
6
3
4
M = 4
1
2
2
0
0
M = 1
* Note that there are three separate samples, with
n = 5 in each sample. The dependent variable is
the number of problems solved correctly
One obvious characteristic of
the data is that the scores are
not all the same. Our goal is
to measure the amount of
variability and to explain
where it comes from