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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
  • 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
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