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## ANCOVA

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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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Title: ANCOVA

1
ANCOVA
2
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)

3
Similarities with ANCOVA
• Extends ANOVA Assess significant group
differences on 1 DV
• While controlling for the effect of one or more
covariate
• Is Multiple Regression with 1 continuous
predictor (covariates) and 1 dummy coded
predictor
• So incorporates both ANOVA and Multiple
Regression
• Allows for greater sensitivity than with ANOVA
under most conditions

4
When to Use ANCOVA?
• Experimental Design Best use, but often
difficult to implement
• Manipulation of IV (2 group double blind drug
trial)
• 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

5
ANCOVA Background Themes
• Sample Data Random selection assignment
• Measures Group predictor, Continuous Covariate,
Continuous DV
• Methods Inferential with experimental design
assumptions
• 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
regression)
• When slopes are similar, Y is adjusted similarly
across groups

6
Testing for Homogeneity of Regression
• First run the Ancova model with a Treatment X
Covariate interaction term included
• If the interaction is significant, assumption
violated
• 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

7
ANCOVA Model
• 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,
Ci
• 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

8
Central Themes for ANCOVA
• Variance in DV
• As usual, we are interested in accounting for the
(adjusted) DV variance, here with categorical
predictor(s)
• 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.

9
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
covariates
• Effect Size ES ?2
• ?2 SSEFFECT / SSTOTAL, after adjusting for
covariates
• Partial ?2 SSEFFECT /(SSEFFECT SSerror)

10
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.

11
Steps for ANCOVA
• Consider the following
• All variables reliable?
• Are means significantly different, i.e., high BG
variance?
• Do groups differ after controlling for covariate?
• Are groups sufficiently homogeneous, i.e., low WG
variance?
• Low to no correlation between grouping variable
covariates?
• Correlation between DV covariates?
• Can the design support causal inference (e.g.,
random assignment to manipulated IV, control
confounds)?

12
ANCOVA Steps
• Descriptive Statistics
• Means, standard deviations, skewness and kurtosis
• Correlations
• Across time for test-retest reliability
• Across variables to assess appropriateness of
ANCOVA
• Test of Homogeneity of Regression
• ANOVA (conduct as a comparison)
• ANCOVA (ANOVA, controlling for covariates)
• Conduct follow-up tests between groups

13
Ancova Example
• This example will show the nature of Ancova as
regression
• 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

14
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
15
Ancova Example
• Dummy code grouping variable
• Though technically it should probably already be
coded as such anyway for convenience

16
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

17
Ancova Example
• Ancova output from GLM/Univariate in SPSS
• Regression output
• 4842.105-642.623 4199.482

18
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
Ancova

19
Ancova
• 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