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## ANCOVA and MANCOVA

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### The relationship between ANCOVA and MANCOVA is the same as the relationship ... lecture: These techniques are often wrongly used in research with nonrandom ... – PowerPoint PPT presentation

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Title: ANCOVA and MANCOVA

1
ANCOVA and MANCOVA
• Covered in Tabachnick and Fidell (TF ANCOVA
chapter 8 MANCOVA chapter 9)
• The relationship between ANCOVA and MANCOVA is
the same as the relationship between ANOVA and
MANOVA
Psychology, 110, 40-48.
• Main message of lecture These techniques are
often wrongly used in research with nonrandom
assignment to groups

2
(M)ANCOVA
• 3 Uses (statistical operations are same)
• To increase power by reducing error term in
experimental work (with random assignment to
groups)
• To adjust for mismatch on nuisance variable in
nonexperimental work (N.B. this is the tricky
case)
• Stepdown analyses to follow-up MANOVA (as
discussed in lecture on MANOVA)

3
General Points
• ANCOVA can be used with all types of ANOVA
designs
• Can even have a changing covariate in repeated
measures designs (but not in SPSS)
• ANCOVA equivalent to multiple regression
• How does ANCOVA reduce the error term? (see Fig
8.1 in TF)

4
Types of Research Question
that ANOVA does (e.g. main and interaction
effects, specific comparisons and contrasts etc.)
• The effects of IVs are assessed holding
covariates constant (i.e., treating each subject
as if they scored at the mean for the covariate)
• Provides test of significance for the regression
of the covariate(s) on the DV ignoring group
effects

5
Theoretical Issues Choice of Covariates
• Ideal is small number of orthogonal covariates,
each correlated with the DV
• This gives maximum adjustment of the DV for
minimum reduction in df for the error term (each
covariate reduces error df by 1)

6
Theoretical Issues Random vs. Nonrandom
Assignment
• In random assignment (experimental) designs,
group differences in covariate will be due to
chance (as long as covariates measured before
assignment)
• With nonrandom assignment (common in psychology)
covariate differences may reflect meaningful
substantive differences related to group
membership

7
Why is ANCOVA invalid when groups differ on
covariate?
Grp(res) Grp with Cov removed
GRP
Cov
Cov
GRP
DV
DV
• Non-random assignment

Random assignment
8
Why is ANCOVA invalid when groups differ on
covariate?
• ANCOVA looks at relationship between DV and
Grp(res)
• Dont know what Grp(res) represents when Cov and
Grp are related
• ANCOVA may remove part of treatment effect or
produce a spurious effect
• Grp variable altered so that it may no longer
measure what it was intended to measure

9
ILLUSTRATIONS OF INVALID USE OF ANCOVA

10
Conceptual IllustrationsLords Example
• Do boys or girls (IVgender) end up weighing more
(DVfinal weight) when following a specific diet,
after correcting for initial weight (covariate)
differences between boys and girls?
• Problem of regression to the mean for matched
weight gender groups

11
Conceptual IllustrationsMiller Chapmans
Example
• Would six and eight year olds (IVage groups)
differ in weight (DV) if they did not differ in
height (covariate)?
• One cannot equate younger and older children in
height because height is an intrinsic part of the
age difference.

12
Typical Research Examples
• Comparing depressed participants vs. nondepressed
controls using trait anxiety score as a covariate
• Comparing schizophrenic participants vs. healthy
controls on memory performance using IQ as a
covariate

13
Can ANCOVA Ever be Valid with Group Differences
on Covariate?
• If group differences arose by chance (e.g. in
experiments with random assignment)
• Overall and Woodward (77) if group could NOT
have caused the covariate differences
• As a useful means of exploring the dataset and
clarifying the relationships between the variables

14
Alternatives to ANCOVA
• Incorporate the covariate as a substantive factor
into the analysis
• Rosenbaums propensity score method
• Extended regression equation in the comparison
group for the DV and the covariate analyse
residual scores for other group.

15
ANCOVA Practical Issues
• Absence of outliers (both univariate and
multivariate outliers among DVs and covariates)
• Eliminate highly correlated covariates
(multicollinearity and singularity)
• Homogeneity of variance for DV and covariates
• Relationships between DV and covariates, and
between covariates, should be linear

16
ANCOVA Extra Assumptions
• Homogeneity of Regression (see Fig 8.2 in TF)
• How to test this in SPSS?
• Reliability of covariates needs to be high in
nonexperimental research (gt0.8) in experimental
work unreliability just leads to conservative
reduction in error

17
Testing for Homogeneity of Regression
• Include covariate x IV interaction term(s) in the
model
• If these are significant then there is
heterogeneity of regression and ANCOVA is
inappropriate
• In SPSS, the Model button allows you to specify
the model
• Note a full factorial model (SPSS default)
does not include interactions between covariates
and IVs