Chapters 11&12Chapters 11&12

Factorial and Mixed Factor ANOVA and ANCOVA

ANOVA ReviewANOVA Review

Compare 2+ mean scores One way (1 factor or IV) Repeated measures (multiple factors)

Main effects Interactions F-ratio P-value Post hoc tests and corrections

Within and between

Multiple Factor ANOVAMultiple Factor ANOVA

aka Factorial ANOVA; incorporates more than one IV (factor).

Only one DV Factor = IV Levels are the “groups” within each factor.

In the reaction time example, there was one factor (“drug”) with three levels (beta blocker, caffeine, and placebo).

Mixed factor is both within and between in the same analysis.

Factorial ANOVA ExampleFactorial ANOVA Example

Studies are explained by their levels 2 x 3 or 3 x 3 x 4

The effect of three conditions of muscle glycogen at two different exercise intensities on blood lactate. There are 2 IV (factors: glycogen and exercise intensity) and 1 DV (blood lactate). 3 levels of muscle glycogen: depleted, loaded, normal. 2 levels of exercise intensity: 40% and 70% VO2max. 2 x 3 ANOVA, two-way ANOVA. 60 subjects randomized to the 6 cells (n = 10 per cell).

Between subjects.

Factorial ANOVA ExampleFactorial ANOVA Example

Each subject, after appropriate glycogen manipulation, performs 30 minute cycle ergometer ride at either low intensity (40%) or high intensity (70%).

Blood is sampled following ride for lactate level.

3 F ratios in 2-way ANOVA3 F ratios in 2-way ANOVA

2 “Main Effects” – a ‘main effect” looks at the effect of one IV while ignoring the other IV(s), i.e., “collapsed across” the other IV(s). Based on the “marginal means” (collapsed). Main effect for Intensity –

based on “row” marginal means (collapsed across glycogen state).

If significant, look at mean values to see which one is larger (since there are only 2 means).

3 F ratios in 2-way ANOVA3 F ratios in 2-way ANOVA Main effect for glycogen state

Compare column marginal means. If significant, perform follow-up procedures on the 3

means (collapsed across intensity).

Main effects are easily followed up if the “interaction” (see below) is not significant.

Each main effect is treated as a single factor ANOVA while ignoring the other factor.

If the interaction is significant, focus on the interaction even if the main effects are significant. Ignore the main effects

Depleted Loaded Normal

40% D40 L40 N40 MM40%

70% D70 L70 N70 MM70%

MMD MML MMN

ExerciseIntensity

Glycogen Condition

Glycogen Marginal Means

Exercise IntensityMarginal Means

Main Effect For ExerciseIntensity

40% 70%0

5

10

15

Exercise Intensity

Blo

od

Lac

tate

(m

M)

Main Effect for Intensity

Main Effect for Glycogen State

Depleted Loaded Normal0

5

10

15

Glycogen State

Blo

od

Lac

tate

(m

M)

Main Effect for Glycogen

3 F ratios in 2-way ANOVA3 F ratios in 2-way ANOVA Interaction – does the effect of one IV (factor)

change across levels of the other factor(s). Significant interaction indicates that the effects of muscle

glycogen on blood [lactate] differs across levels of exercise intensity.

Or equivalently, a significant interaction indicates that the effects of exercise intensity on blood [lactate] differs across different levels of muscle glycogen.

Interactions tell you that the slopes of lines of the plotted data are not parallel.

In other words the groups did not react the same way.

InteractionsInteractions The first F ratio to consider is the highest

order (most complicated) interaction. In this example, there is only one interaction.

If the interaction is significant, then ignore the main effects and analyze the interaction. When a significant interaction occurs, the main

effects can be misleading.

Interaction Example

Depleted Loaded Normal0

10

2070%40%

Glycogen State

Blo

od

Lac

tate

(m

M)

Interaction of Intensity and Glycogen

Main Effect for Glycogen State

Depleted Loaded Normal0

5

10

15

Glycogen State

Blo

od

Lac

tate

(m

M)

Main Effect For ExerciseIntensity

40% 70%0

5

10

15

Exercise Intensity

Blo

od

Lac

tate

(m

M)

InteractionsInteractions

Options for Follow Up Procedures Perform multiple pairwise comparisons; need to control

familywise Type I error rate. (Bonferroni) Tests of Simple Main Effects

Compare cell means within the levels of each factor. Examples:

1. Perform two 1x3 ANOVAs; one for each level of exercise intensity.

2. Perform three 1x2 ANOVAs (single df comparisons, t tests) for each level of glycogen state.

3. Perform both 1 and 2 above.

Depleted Loaded Normal

40% D40 L40 N40

70% D70 L70 N70

D40 L40 N40and

D70 L70 N70

D40

D70

L40

L70

N40

N70

Or

InteractionsInteractions

Options for Follow Up Procedures (continued) Analysis of interaction comparisons – transform the

factorial into a set of smaller factorials. Plot interaction and describe.

The choice of follow-up procedure depends on the research question(s); one may be better in one situation vs. another.

Main Effect for Intensity Main Effect for Glycogen

Interaction of Intensity and Glycogen

ANCOVAANCOVA Analysis of Covariance

Combined use of ANOVA and Regression Adjust for covariate by regressing covariate on the DV,

then doing an ANOVA on the adjusted DV. Can remove pre-treatment variations (as measured by the

covariate) from the post-treatment means prior to testing groups for differences in the DV.

Example – compare strength in subjects who did Swiss Ball exercise vs. controls.

Randomization may not equate groups on body weight. Covary for body weight prior to comparing groups.

ANCOVAANCOVA Issues with ANCOVA

Covariate should be highly correlated with DV. Covariate should not be correlated with IV.

Homogeneity of Regression Slopes of regression lines between covariate and DV must be

equal across levels of the IV. Violation implies an interaction between the covariate and IV.

Groups may differ on other variables that are not adjusted.

Abuse – arguably inappropriate to correct for pre-existing group differences if those groups were not formed by randomization.

ANCOVAANCOVA Advantages

More Power – due to decreased variance that must be explained by the IV (smaller error term in the F ratio).

Covariate “accounts for” some of the variance in the DV variance that must be explained by IV to reach significance.

Some suggest use of covariate solely to increase power.

Adjusts for pre-treatment differences between groups. If pre-treatment differences exist because groups were not

randomly formed, then ANCOVA will not magically eliminate the bias that may exist with non-random assignment.

Next ClassNext Class

Tonight: factorial ANOVA and ANCOVA in lab and stat practice

Research paper due and stat practice Final exam next week