Transcript PPT Lecture Notes

Part 1
Analysis of Covariance:
ANCOVA
Analysis of Covariance
• ANCOVA
• Like an analysis of variance in which one or
more variables (called covariates) have
been controlled for
• Analogous to a partial correlation
ANCOVA
1.
Why bother with ANCOVA? ANCOVA offers 2 benefits…
2.
First, ANCOVA can reduce the error term!
Recall that all of statistics in the “F Family” are based on
MSb / MSw.
If we can reduce the error term (MSw) by removing covariates we can
increase our sensitivity.
3.
Second, ANCOVA can eliminate confounding variables!
Confounding variables systematically co-vary with the independent
variable. (Emphasis on “systematically”.) If we can eliminate the
confounds, our inference can be stronger (…a better shot at drawing a
cause/effect relation).
What were the three necessary criteria for inferring causal relations?
ANCOVA
1.
Example 1: Viagra
D.V. = Libido
I.V. = Dosage of Viagra: 3 levels…
(Placebo, Low Dosage, High Dosage)
2.
An initial ANOVA indicated a non-significant difference in
libido (sex drive) across the 3 levels of Viagra.
3.
The researchers considered that a participant’s libido might
depend, too, on the partner’s libido. (It takes two to tango!)
So, the researchers used the data from the ANOVA but now
entered Partner’s Libido as a covariate, and ran an
ANCOVA…
ANCOVA
Initially ANOVA Was Run…no covariates.
There was a non-significant effect of the various Viagra dosages on libido.
ANCOVA
Subsequently, an ANCOVA Was Run: Covariate = Partner’s Libido.
The effect of Viagra-Dosage is significant now,
after ‘partialing out’ the (significant) effect of the partner’s libido!!!
(note the reduction in the Error Term, despite the same ‘total’ SS)
The Joy of Stats…
Stats
ANCOVA can
reduce an error term,
which can
render a non-significant effect
…significant!!
ANCOVA
1.
Example 2:
D.V. = Social adjustment in school-age boys.
I.V. = Parental Transitions…4 levels
(No transitions, loss of father, new stepfather 2 or more new step-fathers)
2.
An ANOVA indicated a significant difference in social
adjustment across the 4 levels of parental transition.
3.
To eliminate the possibility that the significant effect could be
explained by confounds with Parental SES and Per Capita
Income, those variables were made covariates in an
ANCOVA. The ANCOVA, too, was significant. So, parental
transitions alone are significantly correlated with the D.V..
Part 2
Introduction To
Multivariate Statistics
Independent vs.
Dependent Variables
• Independent variables
– Divide groups from each other
– Often based on random assignment
– Analogous to predictor variables in regression
• Dependent variables
– Represent the effect of the experimental
procedure
– Analogous to criterion variables in regression
Introduction To Multivariate Statistics
1. So far this semester, each of our analyses has
addressed just a single dependent variable at a time.
2. Univariate Analysis – Any statistical analysis that
focuses on a single dependent variable, regardless of
the number of I.V.s, or ‘predictor variables’.
3. We can now consider a more complicated case…
Multivariate Analyses
1. Multivariate Analysis – Any statistical
analysis that focuses two or more dependent
variable SIMULTANEOUSLY, regardless of
the number of I.V.s, or ‘predictor variables’.
2. There are many different multivariate
tests! We’ll begin with a MANOVA…
Part 3
MANOVA
And
Music Therapy For Chimpanzees
MANOVA
• More than one dependent variable
• Multivariate ANalysis Of VAriance
– MANOVA
– Like an analysis of variance with two or more
dependent variables
MANOVA
• Why bother with MANOVAs?
• To appreciate the motivation for MANOVAs, let’s
re-visit a question that we asked when began
factorial designs….
• Critical Thinking Question: Why bother with
factorial ANOVAs, when we can run a bunch of
one-way ANOVAs?
MANOVA
• Similarly, MANOVA offers a major advantage
over running ‘many little ANOVAs’ (i.e., one for
each D.V.)…
• MANOVA is sensitive to relationships among
dependent variables!!!!
• ANOVA is not, because it address only one D.V.
at a time.
MANOVA
• Example: Can experienced drivers (5+ years), new
drivers (1 year), and drunk drivers (legal
conviction) be distinguished from each other
based on a single DV –the number of pedestrians
they kill?
• ANOVA can tell us whether groups are
distinguishable from each other on the basis of a
single DV.
• In this example, the groups may be
indistinguishable -given this single D.V…
MANOVA
• However, these groups might become
readily distinguishable from each other if
you simultaneously analyze the combination
of….
• # of pedestrians killed, AND
• # of lamp posts hit, AND
• # of cars crashed into.
MANOVA
• Again, any of those D.V.s alone may have
produced a non-significant ANOVA…
• But a MANOVA is sensitive to the relations
among those variables and may be able to
achieve significance…
• In short, a MANOVA can be more sensitive
than ANOVA!!! That’s it’s first advantage.
MANOVA
• A second advantage of MANOVA is it (like other
multivariate tests) can evaluate “latent
variables”…
• Latent Variables – Are present implicitly, rather
than explicitly.
• A latent variable might only ‘potentially’ exist,
and is contingent on an operational definition that
synthesizes several explicitly defined D.V.s…
Agitated / Aggressive – From Article On Music Therapy for Chimps
MANOVA
Agitated / Aggressive
Operationally defined by the following explicit D.V.s
Aggression: Display-Charging: Display-Hunching: Threat: Pant Hoot
You could run separate ANOVAs on each explicit DV,
or run a MANOVA on “Agitated / Aggressive” (informally, an ‘uber’ variable)
Anxious/Fearful – From Article On Music Therapy for Chimps
MANOVA
Anxious/Fearful
Operationally defined by the following explicit D.V.s
Apprehension: Fear: Scratch: Yawning: Attachment: Locomotion: Vocalization
You could run separate ANOVAs on each explicit DV,
or run a MANOVA on “Anxious/Fearful ” (informally, an ‘uber’ variable)
Excited – From Article On Music Therapy for Chimps
MANOVA
Excited
Operationally defined by the following explicit D.V.s
Food Barks: Pant Hoot to Scream: Tandem Walk: Non-Directed Display
You could run separate ANOVAs on each explicit DV,
or run a MANOVA on “Excited ” (informally, an ‘uber’ variable)
Active/Explore – From Article On Music Therapy for Chimps
MANOVA
Active/Explore
Operationally defined by the following explicit D.V.s
Explore: Locomotion: Rough-And-Tumble Play
You could run separate ANOVAs on each explicit DV,
or run a MANOVA on “Active/Explore” (informally, an ‘uber’ variable)
Inactive / Relaxed – From Article On Music Therapy for Chimps
MANOVA
Inactive / Relaxed
Operationally defined by the following explicit D.V.s
Rest: Quiet Play: Groom: Foraging/Eating
You could run separate ANOVAs on each explicit DV,
or run a MANOVA on “Inactive / Relaxed ” (informally, an ‘uber’ variable)
MANOVA
MANOVA’s generate F statistics, just like ANOVAs.
The p-values (‘sig’) values are also typically evaluated
at the 0.05 level, just like ANOVAs.
(no big wup!)
MANOVA
The DF in this summary table is 2 (that’s 3 minus 1).
These are means, not p values!
There were three levels of the I.V.
Each D.V. (actually ‘uber’ variable) was measured
Before (pre), During (test) and After (post) the chimps heard music.
Now They’ve added Time-of-Day as an IV
MANOVA
Music Affected
Three Separate
Dependent Variables,
Each of which
was a latent variable
(‘uber variable’)
Agitated/Aggressive
Active/Explore
Inactive/Relaxed
The I.V.s
in this Analysis Were
Time of Day (AM/PM)
And
Music (pre, test, post)
So this was a
2x3 within Subjects
MANOVA!
Each chimp was evaluated
In each of the 2x3
Conditions
The df here refers
To the main effect of
Time-of-day…2 levels…df=1
Now They’ve added Social Group as an IV
MANOVA
Music Affected
Four Separate
Dependent Variables,
Each of which
was a latent variable
(‘uber variable’)
Agitated/Aggressive
Active/Explore – Solitary
Active/Explore – Social
Inactive/Relaxed
The I.V.s
in this Analysis Were
Social Group (M, F, Mixed)
And
Music (pre, test, post)
So this was a
3x3 Mixed
MANOVA!
Chimps were evaluated
In each of the 3x3
Conditions
The df here refers
To the main effect of
Social Group…3 levels…df=2
MANOVA
• We will NOT calculate MANOVAs by hand!
• Nor will we use SPSS to compute MANOVAs!
• But if you were to do MANOVAs for your senior research
here’s how you’d get started…
• Analyze GLM  Multivariate (not univariate)
The dependent variable box now allows you to slide in
multiple DVs (rather than just 1 in the univariate case)