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
• See also Miller and Chapman (2001). J Abnormal
  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

• ANCOVA addresses the same questions about IVs
  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