Analysis of CovarianceAnalysis of CovarianceANCOVAANCOVA

Chapter 11Chapter 11

ANOVA TerminologyANOVA Terminology

The purpose of this experiment was to The purpose of this experiment was to compare the effects of the dose of ginseng compare the effects of the dose of ginseng 人蔘 (placebo, low, high) on the number of (placebo, low, high) on the number of bench press repetitions, bench press repetitions, while controlling for while controlling for the energy level of the strength coachthe energy level of the strength coach..

The independent variable The independent variable dose of ginseng dose of ginseng is is called a called a FACTORFACTOR. .

The FACTOR has The FACTOR has 3 LEVELS3 LEVELS (placebo, low, high)(placebo, low, high) The dependent variable in this experiment is The dependent variable in this experiment is

the number of repetitions performed.the number of repetitions performed. Strength coach energy level is a covariate.Strength coach energy level is a covariate.

Assumptions of ANOVAAssumptions of ANOVA

Dependent variable is interval or Dependent variable is interval or ratio.ratio.

The distributions within groups are The distributions within groups are normally distributed.normally distributed.

The variances between groups are The variances between groups are equal.equal.

The effects of ginseng on repetitionsThe effects of ginseng on repetitions

The effects of ginseng on repetitionsThe effects of ginseng on repetitions

The purpose of this experiment was to compare the effects of the dose of ginseng 人蔘 (placebo, low, high) on the number of bench press repetitions, while controlling for the energy level of the strength coach. 15 subjects were randomly assigned to one of the following conditions (placebo, low dose, high dose). Strength coach energy (enthusiasm) was coded as 0 = boring thru 7 = very high enthusiasm.

StrCoach_Energy as CovariateStrCoach_Energy as Covariate

Check homogeneity of variance if you have a between subjects factor.

Choose the Sidak post hoc test.

Plot number of repetitions by ginseng dose

Check homogeneity of variance if you have a between subjects factor. The null hypothesis is that the groups have equal variance. In this case you retain the null. You don’t want this to be significant, if it is significant you are violating an assumption of ANVOA: homogeneity of variance.

The groups DO NOT have equal variance, Levine’s test F(2,27) = 4.618, p = .019

See page 405 of Field for an additional test to check for homogeneity of variance.

The Str_Coach energy level was a sig. covariate F(1,26) = 4.959, p = .035.

After controlling for the effects of strength coach energy, there the dose of ginseng significantly effected the number of repetitions performed F(2,26) = 4.142, p = .027, power = .68.Look at posthoc tests to find which groups are different.

ANCOVA ResultsANCOVA Results

Post Hoc Post Hoc ResultsResults

Placebo is different from High Dose.

Low Dose is NOT different from High Dose

Effect SizeEffect Size

The dose of ginseng explained a bigger proportion of the variance not attributed to other variables than strength coach energy. (see p 415-516).

Assumptions of ANCOVAAssumptions of ANCOVA

The covariate should be independent from the The covariate should be independent from the experimental factor. experimental factor. Run a single factor ANOVA with the covariate Run a single factor ANOVA with the covariate

as the dependent variable and the as the dependent variable and the experimental variable as a fixed factor.experimental variable as a fixed factor.

Homogeneity of regression slopesHomogeneity of regression slopes Do a scatter plot of each experimental Do a scatter plot of each experimental

condition with the covariate on X axis and the condition with the covariate on X axis and the dependent variable on the Y axis.dependent variable on the Y axis.

Run the ANCOVA with a custom model see p Run the ANCOVA with a custom model see p 413 – 415.413 – 415.

Is Covariate Independent of Experimental Factor?Is Covariate Independent of Experimental Factor?

Is there a difference in the covariate (Strength Coach) by experimental factor (Dose)

Is Covariate Independent of Experimental Factor?Is Covariate Independent of Experimental Factor?

No problem. The covariate is independent from the experimental factor (Dose), F(2,27) = 1.98, p = .16

Homogeneity of Regression Slopes?Homogeneity of Regression Slopes?

Click the Model Button1.Highlight Dose, StrCoach then click Main Effects.2.Highlight Dose, StrCoach then click Interaction.

Homogeneity of Regression Slopes?Homogeneity of Regression Slopes?

Look at the interaction term Dose * StrCoach. You don’t want this to be significant.

In this example the effect is significant, therefore we are violating the assumption that the regression slopes are equal.

HomeworkHomework

Analyze the Task 1 and Task 2 data sets from the book, see page 419.

Do a Sidak post hoc test instead of the planned contrast suggested in the book.

Compute the effect size using:

Use the Sample Methods and Results section as a guide to write a methods and results section for your homework.

Sidak Post Hoc Lacks Power!Sidak Post Hoc Lacks Power!

According to ANOVA F(2,11) = 4.32, p = .041At least 1 pair should be different, but Sidak lacks power. You can either report this p = 0.052, or run a planned contrast, see the next page for a planned contrast example.

Planned Contrast 1-3, 1-2 for HangoverCure Planned Contrast 1-3, 1-2 for HangoverCure Data SetData Set

Click Contrasts, Choose Simple, Chose First, Click Change, then Continue. This will run a simple planned contrast of:1 – 3 and 1 – 2, it does not do 2 – 3.

Planned Contrast 1-3, 1-2 Results for Planned Contrast 1-3, 1-2 Results for HangoverCure Data SetHangoverCure Data Set

The planned contrast finds the difference between 1-2 at p = 0.018.

Where Sidak reported this difference as p = 0.052.

This shows that a planned contrast is more powerful than a post hoc test.