One-Way Analysis of One-Way Analysis of Covariance (ANCOVA)Covariance (ANCOVA)

Extension of Analysis of Variance Extension of Analysis of Variance (ANOVA)(ANOVA) One categorical independent (grouping) One categorical independent (grouping)

variablevariable One continuous dependent variableOne continuous dependent variable Add additional continuous covariate(s)Add additional continuous covariate(s)

Covariates hypothesized to have Covariates hypothesized to have potential effect on outcome of interestpotential effect on outcome of interest

ANCOVA allows statistical adjustment ANCOVA allows statistical adjustment in group analysis, increases likelihood in group analysis, increases likelihood that can detect differences between that can detect differences between groupsgroups

Uses of ANCOVA Uses of ANCOVA

Used when have two-group pre/post-test Used when have two-group pre/post-test design (comparing impact on two different design (comparing impact on two different interventions, taking before and after interventions, taking before and after measures for each group) measures for each group)

Research Question: Do males and females Research Question: Do males and females differ in their reading abilities (measured differ in their reading abilities (measured by reading post-test), following an by reading post-test), following an intervention, controlling for their initial intervention, controlling for their initial differences in reading (measured by differences in reading (measured by reading pre-test)?reading pre-test)?

Uses for ANCOVA Uses for ANCOVA Control for pre-existing differences Control for pre-existing differences

between groupsbetween groups Control for variables that vary by group Control for variables that vary by group

and also affect dependent variable (PCV – and also affect dependent variable (PCV – potentially confounding variables)potentially confounding variables)

Have small samples sizesHave small samples sizes Small to medium effect sizesSmall to medium effect sizes Quasi-experimental studies where cannot Quasi-experimental studies where cannot

randomly assign study participants to randomly assign study participants to groups groups

Choosing CovariatesChoosing Covariates Based on theory and previous research Based on theory and previous research

literature guiding your researchliterature guiding your research Ideally choose 2-3 covariates to reduce error Ideally choose 2-3 covariates to reduce error

variance and to increase chance of detecting variance and to increase chance of detecting significant differences between groupssignificant differences between groups

Need to be continuous variablesNeed to be continuous variables Correlate significantly with dependent variableCorrelate significantly with dependent variable Moderately (not highly) correlated with each Moderately (not highly) correlated with each

otherother Covariate measured before Covariate measured before

treatment/intervention so not affected by treatment/intervention so not affected by treatmenttreatment

Examples - ANCOVAExamples - ANCOVA Is there a significant difference in the Fear Is there a significant difference in the Fear

of Statistics test scores (FOST) for of Statistics test scores (FOST) for participants in the math skills group and the participants in the math skills group and the confident building group, while controlling confident building group, while controlling for their scores on this test at Time 1?for their scores on this test at Time 1?

Is there a difference in self-efficacy levels Is there a difference in self-efficacy levels for low/medium/high performing students, for low/medium/high performing students, controlling for their parents’ level of controlling for their parents’ level of education (number of years of formal education (number of years of formal education completed)?education completed)?

Assumptions of ANCOVAAssumptions of ANCOVA

NormalityNormality Homogeneity of variancesHomogeneity of variances Influence of treatment on covariate Influence of treatment on covariate

measurement measurement Reliability of covariatesReliability of covariates MulticollinearityMulticollinearity LinearityLinearity Homogeneity of regressionHomogeneity of regression Unequal sample sizes (unbalanced design)Unequal sample sizes (unbalanced design) OutliersOutliers

Breathe….Breathe….

Take deep breathsTake deep breaths Inhale slowlyInhale slowly Hold for 5 secondsHold for 5 seconds Exhale slowlyExhale slowly Repeat many timesRepeat many times

AssumptionsAssumptions

Influence of treatment on covariate Influence of treatment on covariate measurement measurement Ensure covariate measured Ensure covariate measured beforebefore the the

treatment or interventiontreatment or intervention If violated, covariate may be correlated If violated, covariate may be correlated

with dependent variable, thus removing with dependent variable, thus removing some of treatment effect some of treatment effect

Assumptions (cont.)Assumptions (cont.)

Reliability of covariatesReliability of covariates ANCOVA assumes covariates measured ANCOVA assumes covariates measured

without error (hard to attain)without error (hard to attain) To minimize violation, need to improve To minimize violation, need to improve

reliability of measurement instrumentsreliability of measurement instruments Use good, well-validated scales & Use good, well-validated scales &

questionnaires (make sure they measure questionnaires (make sure they measure what you think they measure and are suited what you think they measure and are suited for your sample)for your sample)

Check internal consistency (form of Check internal consistency (form of reliability) – Cronbach’s alpha > .7 reliability) – Cronbach’s alpha > .7 8 8 preferred)preferred)

Assumptions (cont.)Assumptions (cont.) Reliability of covariates (cont.)Reliability of covariates (cont.)

To minimize violation, need to improve reliability To minimize violation, need to improve reliability of measurement instrumentsof measurement instruments

If design own instruments, make sure questions If design own instruments, make sure questions clear, appropriate, unambiguous. Pilot-test clear, appropriate, unambiguous. Pilot-test questions before official data collection!questions before official data collection!

If using equipment/measuring instrumentation, If using equipment/measuring instrumentation, makes sure it is functioning properly, is makes sure it is functioning properly, is calibrated, and that person operating calibrated, and that person operating equipment is trained and competent to use.equipment is trained and competent to use.

If study involves other people to observe/rate If study involves other people to observe/rate behavior, make sure they are trained and behavior, make sure they are trained and calibrated to use same criteria. Preliminary calibrated to use same criteria. Preliminary pilot-testing to check inter-rater consistency pilot-testing to check inter-rater consistency (reliability) is essential. (reliability) is essential.

Assumptions (cont.)Assumptions (cont.) Multicollinearity (a.k.a. correlations among Multicollinearity (a.k.a. correlations among

covariates)covariates) To minimize violation, avoid covariates that are highly To minimize violation, avoid covariates that are highly

correlated (strongly related) (r= .8 or above)correlated (strongly related) (r= .8 or above) Examine scatter plots, run preliminary correlation Examine scatter plots, run preliminary correlation

analyses to examine strength of relationship among analyses to examine strength of relationship among proposed covariatesproposed covariates

Linearity (a.k.a. linear relationship Linearity (a.k.a. linear relationship between dependent variable and covariate)between dependent variable and covariate) Use scatter plots to check linearity by Use scatter plots to check linearity by

subgroupsubgroup If curvilinear, eliminate covariate or transformIf curvilinear, eliminate covariate or transform

Add scatter plot example to Add scatter plot example to demonstrate correlation and demonstrate correlation and linearitylinearity

Assumptions (cont.)Assumptions (cont.)

Homogeneity of regression slopesHomogeneity of regression slopes Equal “slopes” between covariate and Equal “slopes” between covariate and

dependent variabledependent variable Interaction between covariate and Interaction between covariate and

dependent variable is problematicdependent variable is problematic Unequal sample sizes (unbalanced Unequal sample sizes (unbalanced

design)design) Outliers Outliers

Check on case-by-case basis Check on case-by-case basis

ProceduresProcedures AnalyzeAnalyze

General Linear Model, then UnivariateGeneral Linear Model, then Univariate

Enter Dependent variables, Enter Dependent variables, Independent/grouping variable (Fixed factor), Independent/grouping variable (Fixed factor), covariatescovariates

Click on Model, Specify Full FactorialClick on Model, Specify Full Factorial

Options: Options:

Estimated Marginal Means Estimated Marginal Means grouping variablegrouping variable Move into ‘Display Move into ‘Display Means for’Means for’

Options – descriptives, effect size, Options – descriptives, effect size, homogeneityhomogeneity

Click OKClick OK

An exampleAn example

1.1. Dataset: experim3ED.sav (Pallant)Dataset: experim3ED.sav (Pallant)2.2. Use ANCOVA to assess whether there are Use ANCOVA to assess whether there are

significant differences between students’ fear of significant differences between students’ fear of statistics (FOST) following the math skills class statistics (FOST) following the math skills class (Group 1) or the confidence building class (Group 1) or the confidence building class (Group 2), while controlling for their pre-test.(Group 2), while controlling for their pre-test.

3.3. The grouping variable will be: Time.The grouping variable will be: Time.

4.4. Check data (Ns for each group, missing data? Check data (Ns for each group, missing data? Coding?)Coding?)

5.5. Check assumptions (e.g., equal variances, Check assumptions (e.g., equal variances, linearity)linearity)

Experim3EDExperim3ED example example (cont.)(cont.)

6.6. Determine overall significance Determine overall significance (p<.05)(p<.05)

7.7. Compare adjusted means– which is Compare adjusted means– which is higher? T1 or T2?higher? T1 or T2?

8.8. Calculate effect sizeCalculate effect size

9.9. Present resultsPresent results

Your Turn!Your Turn!

Based on your research interests, Based on your research interests, what research questions would what research questions would require an ANCOVA analysis? require an ANCOVA analysis?

Try it out with Omnibus datasetTry it out with Omnibus dataset