chapters 11&12 factorial and mixed factor anova and ancova
TRANSCRIPT
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