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

ANCOVA1. 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?

ANCOVA1. Example 1: Viagra

D.V. = LibidoI.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)

Stats

The Joy of Stats…

ANCOVA canreduce an error term,

which canrender a non-significant effect

…significant!!

ANCOVA1. Example 2: D.V. = Social adjustment in school-age boys.

I.V. = Parental Transitions…4 levels (No transitions, loss of father, new step- father 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 ToMultivariate 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 Statistics1. 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

MANOVAAnd

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…

MANOVAAgitated / Aggressive – From Article On Music Therapy for Chimps

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)

MANOVAAnxious/Fearful – From Article On Music Therapy for Chimps

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)

MANOVAExcited – From Article On Music Therapy for Chimps

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)

MANOVAActive/Explore – From Article On Music Therapy for Chimps

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)

MANOVAInactive / Relaxed – From Article On Music Therapy for Chimps

Inactive / RelaxedOperationally 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 evaluatedat the 0.05 level, just like ANOVAs.

(no big wup!)

MANOVAThe DF in this summary table is 2 (that’s 3 minus 1).

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.

These are means, not p values!

MANOVAMusic AffectedThree Separate

Dependent Variables,Each of which

was a latent variable(‘uber variable’)

Agitated/AggressiveActive/Explore

Inactive/Relaxed

The I.V.sin this Analysis Were

Time of Day (AM/PM)And

Music (pre, test, post)

So this was a 2x3 within Subjects

MANOVA!

Each chimp was evaluatedIn each of the 2x3

Conditions

The df here refersTo the main effect of

Time-of-day…2 levels…df=1

Now They’ve added Time-of-Day as an IV

MANOVAMusic AffectedFour Separate

Dependent Variables,Each of which

was a latent variable(‘uber variable’)

Agitated/AggressiveActive/Explore – SolitaryActive/Explore – Social

Inactive/Relaxed

The I.V.sin this Analysis Were

Social Group (M, F, Mixed)And

Music (pre, test, post)

So this was a 3x3 MixedMANOVA!

Chimps were evaluatedIn each of the 3x3

Conditions

The df here refersTo the main effect of

Social Group…3 levels…df=2

Now They’ve added Social Group as an IV

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)