Matakuliah
Tahun
Versi
: I0214 / Statistika Multivariat
: 2005
: V1 / R1
Pertemuan 13
Analisis Ragam Peubah Ganda
(MANOVA I)
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan
mahasiswa akan mampu :
• Mahasiswa dapat menerangkan
konsep dasar analisis ragam peubah
ganda (manova)  C2
• Mahasiswa dapat menghitung
manova satu klasifikasi  C3
2
Outline Materi
• Konsep dasar analisis ragam peubah
ganda (manova)
• Analisis ragam peubah ganda satu
klasifikasi
3
<<ISI>>
MANOVA ~ the history
• Developed as a theoretical construct by
S.S. Wilks in 1932
• Published in Biometrika
• Wide availability of computers made
these methods practical for researchers
<<ISI>>
MANOVA ~ the definition
 Technique for assessing group
differences across multiple
metric dependent variables
(DV’s) simultaneously, based
on a set of categorical (nonmetric) variables acting as
independent variables (IV’s)
<<ISI>>
ANOVA vs MANOVA
• ANOVA ~ only 1 dependent variable
• MANOVA ~ 2 or more dependent variables
• Both are used with experimental designs in which
researchers manipulate or control one or more
independent variables to determine the effect on
one (ANOVA) or more (MANOVA) dependent
variables
<<ISI>>
Equations
• ANOVA
Y1
=
(metric DV)
X1 + X2 + X3 +...+ Xn
(non-metric IV’s)
• MANOVA
Y1 + Y2 + ... + Yn = X1 + X2 + X3 +...+ Xn
(metric DV’s)
(non-metric IV’s)
<<ISI>>
MANOVA and Regression
• Note the different terminology
• In multiple regression, univariate and
multivariate/multiple refer to the
number of IV’s
• In ANOVA and MANOVA
discussions, univariate and
multivariate refer to the number of
DV’s
<<ISI>>
Univariate Research Example
• Subjects shown
different advertising
messages
• Emotional or
Informational or ??
• Viewers rate appeal
of the message using
scores from 1 to 10
Ad appeal?
<<ISI>>
Univariate Review ~ t Test
 Two commercials shown
(emotional~informational)
 Single treatment/factor with two levels
 Use a t Test: One IV’s, one DV, two
treatment groups
M1 - M2
 The t statistic = ---------------------SEM1 M2
<<ISI>>
Univariate Review ~ ANOVA




Two or more commercials
(emotional~informational~funny~etc)
Single treatment/factor with two or more levels
Use ANOVA : Multiple IV’s, one DV, two or more
treatment groups
F statistic =
MSB
-----MSW
<<ISI>>
Univariate Hypothesis Testing
• Null Hypothesis (H0) ~ That there is no
difference between the DV means of the
treatment groups
• Alternate Hypothesis (HA) ~ That there is a
statistically significant difference between the
DV means of the treatment groups
<<ISI>>
Multivariate Research
Example
• Subjects shown
different advertising
messages
• Emotional or
Informational or ??
• Viewers rate appeal
of the message using
scores from 1 to 10
Ad appeal?
Will I buy?
<<ISI>>
Multivariate Procedures
 Hotelling’s T2


One IV, multiple DV’s, two groups
The k Group Case: MANOVA
Multiple IV’s, multiple DV’s, more than two
treatment groups
Null Hypothesis ~ that there is no difference
between vectors of means of multiple DV’s
across the treatment groups
<<ISI>>
Null Hypothesis Testing
MANOVA
ANOVA
• H0: M1 = M2 =...Mk
• H0: All the group
means are equal, that
is, they come from the
same populations
H0:
M11
M12 =...=
=
M21
M22
M1k
M2k
Mp1
Mpk
Mp2
All the group mean vectors are equal,
that is, they come from the same populations
<<ISI>>
Hotelling’s T2
• Direct extension of the t test, used when there
are only two groups, but multiple DV’s to be
measured
• Accounts for the fact that DV’s may be related to
one another (correlated)
• Provides a statistical test of the variate, formed
from the DV’s, that produces the greatest group
difference
<<ISI>>
Hotelling’s T2 ~ how it works
• Maximize group differences, using the equation
below:
C = W1Y1 + W2Y2 +...+ WnYn
where
C = composite or variate score for a respondent
Wi = weight for dependent variable i
Yi = dependent variable I
 Square the obtained t statistic to get T2 and
check statistical significance
<<ISI>>
MANOVA
• Extension of ANOVA
• Extension of Hotelling’s T2
 Establish dependent variable weights to produce
a variate for each respondent
 Adjust weights to maximize F statistic computed
on variate scores of all groups
<<ISI>>
MANOVA ~ how it works
• The first variate (called a discriminate function)
maximizes differences between groups and
therefore also the F value
• With maximum F, calculate greatest
characteristic root (grc) and check its
significance to reject null hypothesis (or not)
• Subsequent discriminant functions are
orthogonal and seek to explain remaining
variance
<<ISI>>
When to use MANOVA
• When you have multiple dependent variables
• Control of Experimentwide Error Rate
– Repeated univariate procedures can dramatically
increase Type I errors
– DV’s that are not highly correlated with one another
will cause the most trouble
• Differences among a Combination of Dependent
Variables
– Multiple univariate procedures do not equal a
multivariate procedure
– Multicollinearity is ignored
<<ISI>>
Discriminant Analysis
• MANOVA ~ sort of a mirror
image of discriminant analysis
• DV’s in MANOVA become IV’s
of DA
• DV of DA becomes IV of
MANOVA
<<ISI>>
Decision Process for
MANOVA
• Powerful analytic tool
suitable to a wide array of
research questions
• Six step process
• Logical progression
through all six will yield
best results
<<ISI>>
Step #1: Objectives of
MANOVA
•
Determine research question
 Multiple Univariate Questions ~ MANOVA used to
control experimentwide error rate before further
univariate analysis
 Structured Multivariate Questions ~ MANOVA used
to address multiple DV’s with known relationships
 Intrinsically Multivariate Questions ~ MANOVA used
with multiple DV’s where the principal concern is
how they differ/change as a whole...or how they
remain consistent across time
<<ISI>>
Step #1: continued
 Select Dependent Variables carefully
 There is a danger of including too many DV’s and a
tendency to do so...simply because you can
 One bad variable can skew all results
 Ordering of variables can also be important and can
lead to sequential effects
 MANOVA step-down analysis can help here
 Researcher responsibility to use tools properly
<<ISI>>
Stage #2: Research Design
• MANOVA requires greater sample sizes than
ANOVA ~ overall and by group (must exceed
specific thresholds in each cell)
• Factorial Designs ~ two or more IV’s or
treatments in the design
– Sometimes treatments are added post hoc
– Blocking factors (example: gender)
<<ISI>>
ANOVA Cereal Example
•
•
•
•
Three colors (red, blue, green)
Three shapes (stars, cubes, balls)
3 x 3 factorial design
With ANOVA, you would evaluate main effect for
color, main effect for shape, and interaction
effect of color and shape
• Each would be tested with an F statistic
<<ISI>>
Ordinal and Disordinal
• With MANOVA, we can establish the nature of
the interaction between two treatments
– No interaction
– Ordinal Interaction ~ effects of treatment are not
equal across all levels of another treatment...but
magnitude is in the same direction
– Disordinal Interaction ~ Effects of one treatment are
positive for some levels and negative for other levels
of the other treatment
<<ISI>>
MANOVA Interaction
• If significant interactions are ordinal, researcher
must interpret the interaction term carefully
• If significant interaction is disordinal however,
main effects of the treatments cannot be
interpreted and study must be redesigned
(treatments do not represent a consistent effect)
<<ISI>>
Covariates
• Metric independent variables, called covariates
can be used to eliminate systemic errors
• ANOVA becomes ANCOVA
• MANOVA becomes MANCOVA
• Procedures similar to linear regression are used
to remove variation in the DV associated with
covariates and then standard ANOVA and
MANOVA can be used
• Ideal covariate is highly correlated with DV and
not correlated with the IV
<<ISI>>
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<< CLOSING>>
• Sampai dengan saat ini Anda telah
mempelajari kosep dasar analisis ragam
peubah ganda, dan manova satu
klasifikasi
• Untuk dapat lebih memahami konsep
dasar analisis ragam peubah ganda dan
manova satu klasifikasi tersebut, cobalah
Anda pelajari materi penunjang,
website/internet dan mengerjakan latihan
33