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Multivariate Analysis: Can use ANCOVA as a follow-up to a significant MANOVA ... Conduct follow-up tests between groups. Ancova Example ... – PowerPoint PPT presentation

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What is Analysis of Covariance?
  • When you think of Ancova, you should think of
    sequential regression, because really thats all
    it is
  • Covariates enter in step 1, grouping variable(s)
    in step 2
  • In this sense we want to assess how much variance
    is accounted for in the DV after controlling for
    (partialing out) the effects of one or more
    continuous predictors (covariates)
  • ANCOVA always has at least 2 predictors (i.e., 1
    or more categorical/ grouping predictors, and 1
    or more continuous covariates).
  • Covariate
  • Want high r with DV low with other covariates
  • It is statistically controlled in an adjusted DV
  • If covariate correlates with categorical
    predictor heterogeneity of regression
    (violation of assumption)

Similarities with ANCOVA
  • Extends ANOVA Assess significant group
    differences on 1 DV
  • While controlling for the effect of one or more
  • Is Multiple Regression with 1 continuous
    predictor (covariates) and 1 dummy coded
  • So incorporates both ANOVA and Multiple
  • Allows for greater sensitivity than with ANOVA
    under most conditions

When to Use ANCOVA?
  • Experimental Design Best use, but often
    difficult to implement
  • Manipulation of IV (2 group double blind drug
  • Random Selection of Subjects
  • Random Assignment to Groups
  • Multivariate Analysis Can use ANCOVA as a
    follow-up to a significant MANOVA
  • Conduct on each DV from a significant MANOVA with
    remaining DVs used as Covariates in each
    follow-up ANCOVA
  • Question to ask yourself however is, why are you
    interested in uni output if you are doing a
    multivariate analysis?
  • Quasi-Experimental Design (controversial use)
  • Intact groups (students taking stats courses
    across state)
  • May not be able to identify all covariates
    effecting outcome

ANCOVA Background Themes
  • Sample Data Random selection assignment
  • Measures Group predictor, Continuous Covariate,
    Continuous DV
  • Methods Inferential with experimental design
  • Assumptions Normality, Linearity,
    Homoscedasticity, and Homogeneity of Regression
  • HoR Slopes between covariate and DV are similar
    across groups
  • Indicates no interaction between IV and covariate
  • If slopes differ, covariate behaves differently
    depending on which group (i.e., heterogeneity of
  • When slopes are similar, Y is adjusted similarly
    across groups

Testing for Homogeneity of Regression
  • First run the Ancova model with a Treatment X
    Covariate interaction term included
  • If the interaction is significant, assumption
  • Again, the interaction means the same thing it
    always has. Here we are talking about changes in
    the covariate/DV correlation depending on the
    levels of the grouping factor
  • If not sig, rerun without interaction term

  • Y GMy ? Bi(Ci Mij) E
  • Y is a continuous DV (adjusted score)
  • GMy is grand mean of DV
  • ? is treatment effect
  • Bi is regression coefficient for ith covariate,
  • M is the mean of ith covariate
  • E is error
  • ANCOVA is an ANOVA on Y scores in which the
    relationships between the covariates and the DV
    are partialed out of the DV.
  • An analysis looking for adjusted mean differences

Central Themes for ANCOVA
  • Variance in DV
  • As usual, we are interested in accounting for the
    (adjusted) DV variance, here with categorical
  • Covariance between DV Covariate(s)
  • in ANCOVA we can examine the proportion of shared
    variance between the adjusted Y score and the
    covariate(s) and categorical predictor
  • Ratio Between Groups/ Within Groups
  • Just as with ANOVA, in ANCOVA we are very
    interested in the ratio of between-groups
    variance over within-groups variance.

ANCOVA Macro-Assessment
  • F-test
  • The significance test in ANCOVA is the F-test as
    in ANOVA
  • If significant, 2 or more means statistically
    differ after controlling for the effect of 1
  • Effect Size ES ?2
  • ?2 SSEFFECT / SSTOTAL, after adjusting for
  • Partial ?2 SSEFFECT /(SSEFFECT SSerror)

ANCOVA Micro-Assessment
  • Test of Means, Group Comparisons
  • Follow-up planned comparisons (e.g., FDR)
  • d-family effect size
  • While one might use adjusted means, if
    experimental design (i.e. no correlation b/t
    covariate and grouping variable) the difference
    should be pretty much the same as original means
  • However, current thinking is that the
    standardizer should come from the original
    metric, so run just the Anova and use the sqrt of
    the mean square error from that analysis
  • Graphs of (adjusted) means for each group also
    provide a qualitative examination of specific
    differences between groups.

Steps for ANCOVA
  • Consider the following
  • All variables reliable?
  • Are means significantly different, i.e., high BG
  • Do groups differ after controlling for covariate?
  • Are groups sufficiently homogeneous, i.e., low WG
  • Low to no correlation between grouping variable
  • Correlation between DV covariates?
  • Can the design support causal inference (e.g.,
    random assignment to manipulated IV, control

  • Descriptive Statistics
  • Means, standard deviations, skewness and kurtosis
  • Correlations
  • Across time for test-retest reliability
  • Across variables to assess appropriateness of
  • Test of Homogeneity of Regression
  • ANOVA (conduct as a comparison)
  • ANCOVA (ANOVA, controlling for covariates)
  • Conduct follow-up tests between groups

Ancova Example
  • This example will show the nature of Ancova as
  • Ancova is essentially performing a regression
    and, after seeing the difference in output
    between the stages in a hierarchical regression,
    we can see what Ancova is doing

Ancova Example
  • Consider this simple prepost setup from our mixed
    design notes
  • We want to control for differences at pre to see
    if there is a posttest difference b/t treatment
    and control groups

Pre Post treatment 20 70 treatment 10 50 treatme
nt 60 90 treatment 20 60 treatment 10 50 control
50 20 control 10 10 control 40 30 control 20 50
control 10 10
Ancova Example
  • Dummy code grouping variable
  • Though technically it should probably already be
    coded as such anyway for convenience

Ancova Example
  • To get the compare the SS output from regression,
    well go about it in hierarchical fashion
  • First enter the covariate as Block 1 of our
    regression, then run Block 2 with the grouping
    variable added

Ancova Example
  • Ancova output from GLM/Univariate in SPSS
  • Regression output
  • 4842.105-642.623 4199.482

Ancova Example
  • Compared to regression output with both
    predictors in
  • Note how partial eta-squared is just the squared
    partial correlations from regression
  • The coefficient for the dummy-coded variable is
    the difference between marginal means in the

  • So again, with Ancova we are simply controlling
    for (taking out, partialing) the effects due to
    the covariate and seeing if there are differences
    in our groups
  • The only difference between it and regular MR is
    the language used to describe the results (mean
    differences vs. coefficients etc.) and some
    different options for analysis