Analysis of Covariance (ANCOVA)

Analysis of variance conducted after removing the relationship of some extraneous variable (covariate) with the dependent variable (Y)

What variability in DV can be explained by IV, AFTER removing variability

explained by the covariate?

Reasons for Using ANCOVA

1. reduce ‘error’ (MSW) by removing effects of extraneous variable

2. adjust DV scores, what would they be in the absence of the covariate

(estimated through regression)

Ideally

small number of covariates

each correlated with the DV

uncorrelated with each other

Analysis of Covariance (ANCOVA)

Example – even after ‘randomly assigning’ participants to levels of the IV, some differences still exist before IV is

introduced.

See example on next slide of ‘anxiety’ differences before any exposure to anxiety stimulus or any ‘treatment’ manipulation

Regression is used to adjust the DV scores

Adjusted scores reflect the removal of the variability in the DV explained by the Covariate

what would Anxiety at time 3 be, when Anxiety at time 1 is held constant?

The regression residuals reflect variability not explained by the covariate – it becomes the ‘error’ (MSW) term

and will be lower than the MSW would have been

Differences among the adjusted means (MSB) can be evaluated relative to the unexplained variance that remains

after removal of the relationship of the DV with the covariate

What is IV – DV relationship after removing the Covariate – DV relationship

The adjusted scores of the DV

are based on a regression equation

using the pooled regression coefficient

pooled across the separate regression

equations for each group in the design.

Thus, for each group the regression coefficient is the same, but the intercepts will vary (graph to follow)

1

11

1 1

1

2 2

22

22

3

3 3

33

3

DV (Y)

Covariate

Mean of Covariate

Original mean

Adjusted mean

Group means are adjusted to what they would be at mean of covariate

Slopes the same, intercepts differ

Assumptions for ANCOVA

Same as those for ANOVA, plus

Homogeneity of regression coefficientssince use ‘pooled’ estimate of regression coefficient

Linear relationship of DV with Covariatesince using linear regression to adjust scores on DV

What to report

Original Means and Standard Deviations

Adjusted Means

For multilevel variable use pairwise

comparisons option, not post hoc

Descriptive Statistics

Dependent Variable: Anxiety after treatment

32.6000 7.00049 30

35.3667 7.54519 30

39.6333 8.56812 30

35.8667 8.17945 90

Type of TreatmentHumor

Neutral Info

Wait

Total

Mean Std. Deviation N

Estimates

Dependent Variable: Anxiety after treatment

33.190a 1.273 30.659 35.721

34.835a 1.272 32.306 37.364

39.575a 1.267 37.056 42.094

Type of TreatmentHumor

Neutral Info

Wait

Mean Std. Error Lower Bound Upper Bound

95% Confidence Interval

Evaluated at covariates appeared in the model: Anxiety at baseline =37.2444.

a.

Tests of Between-Subjects Effects

Dependent Variable: Anxiety after treatment

1812.535a 3 604.178 12.545 .000 .304

2481.979 1 2481.979 51.535 .000 .375

1059.268 1 1059.268 21.994 .000 .204

658.206 2 329.103 6.833 .002 .137

4141.865 86 48.161

121732.000 90

5954.400 89

SourceCorrected Model

Intercept

ANXIETY1

TRTMNT

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

Partial EtaSquared

R Squared = .304 (Adjusted R Squared = .280)a.

Tests of Between-Subjects Effects

Dependent Variable: Anxiety after treatment

753.267a 2 376.633 6.300 .003 .127

115777.600 1 115777.600 1936.626 .000 .957

753.267 2 376.633 6.300 .003 .127

5201.133 87 59.783

121732.000 90

5954.400 89

SourceCorrected Model

Intercept

TRTMNT

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

Partial EtaSquared

R Squared = .127 (Adjusted R Squared = .106)a.

ANOVA Without Covariate

ANOVA With Covariate

MSW reduced due to covariate

MSB reduced due to covariate

Original Means and SDs Adjusted Means and SEs