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© aSup-2007 1
Analysis of Variance   
Chapter 13
INTRODUCTION TO
ANALYSIS OF VARIANCE
© aSup-2007 2
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
© aSup-2007 3
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
© aSup-2007 4
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
© aSup-2007 5
Analysis of Variance   
Two Factors Design
Temperature
Subjects
150
C 250
C 350
C
Ali
Bili
© aSup-2007 6
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
© aSup-2007 7
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
© aSup-2007 8
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
© aSup-2007 9
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.
© aSup-2007 10
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?
© aSup-2007 11
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
© aSup-2007 12
Analysis of Variance   
Two-Factor
Analysis of Variance
Independent Measures
© aSup-2007 13
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?
© aSup-2007 14
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
© aSup-2007 15
Analysis of Variance   
result
10
8
6
4
2
Meannumberoferrors
Self-EsteemHIGH LOW
No
Audience
With
Audience
No
Audience
With
Audience
© aSup-2007 16
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
© aSup-2007 17
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
© aSup-2007 18
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
© aSup-2007 19
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
© aSup-2007 20
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
© aSup-2007 21
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
© aSup-2007 22
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
© aSup-2007 23
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
© aSup-2007 24
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
© aSup-2007 25
Analysis of Variance   
The GLM
Repeated Measures
ONE INDEPENDENT VARIABLE
© aSup-2007 26
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
© aSup-2007 27
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
© aSup-2007 28
Analysis of Variance   
Advantages…
• It reduces the unsystematic variability and so
provides greater power to detect effects
• More economical because fewer subjects are
required
© aSup-2007 29
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
© aSup-2007 30
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
© aSup-2007 31
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
© aSup-2007 32
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
© aSup-2007 33
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
© aSup-2007 34
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
© aSup-2007 35
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
© aSup-2007 36
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

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Analysis of variance

  • 1. © aSup-2007 1 Analysis of Variance    Chapter 13 INTRODUCTION TO ANALYSIS OF VARIANCE
  • 2. © aSup-2007 2 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
  • 3. © aSup-2007 3 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
  • 4. © aSup-2007 4 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
  • 5. © aSup-2007 5 Analysis of Variance    Two Factors Design Temperature Subjects 150 C 250 C 350 C Ali Bili
  • 6. © aSup-2007 6 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
  • 7. © aSup-2007 7 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
  • 8. © aSup-2007 8 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
  • 9. © aSup-2007 9 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.
  • 10. © aSup-2007 10 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?
  • 11. © aSup-2007 11 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
  • 12. © aSup-2007 12 Analysis of Variance    Two-Factor Analysis of Variance Independent Measures
  • 13. © aSup-2007 13 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?
  • 14. © aSup-2007 14 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
  • 15. © aSup-2007 15 Analysis of Variance    result 10 8 6 4 2 Meannumberoferrors Self-EsteemHIGH LOW No Audience With Audience No Audience With Audience
  • 16. © aSup-2007 16 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
  • 17. © aSup-2007 17 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
  • 18. © aSup-2007 18 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
  • 19. © aSup-2007 19 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
  • 20. © aSup-2007 20 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
  • 21. © aSup-2007 21 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
  • 22. © aSup-2007 22 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
  • 23. © aSup-2007 23 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
  • 24. © aSup-2007 24 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
  • 25. © aSup-2007 25 Analysis of Variance    The GLM Repeated Measures ONE INDEPENDENT VARIABLE
  • 26. © aSup-2007 26 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
  • 27. © aSup-2007 27 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
  • 28. © aSup-2007 28 Analysis of Variance    Advantages… • It reduces the unsystematic variability and so provides greater power to detect effects • More economical because fewer subjects are required
  • 29. © aSup-2007 29 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
  • 30. © aSup-2007 30 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
  • 31. © aSup-2007 31 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
  • 32. © aSup-2007 32 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
  • 33. © aSup-2007 33 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
  • 34. © aSup-2007 34 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
  • 35. © aSup-2007 35 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
  • 36. © aSup-2007 36 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