January 7, 2009 - afternoon session

1

Multi-factor ANOVA and

Multiple Regression

January 5-9, 2008

Beth Ayers

January 7, 2009 - afternoon session

2

Thursday Session

• ANOVA‒ One-way ANOVA‒ Two-way ANOVA‒ ANCOVA‒ With-in subject‒ Between subject‒ Repeated measures‒ MANOVA‒ etc.

• Comparisons of different designs

January 7, 2009 - afternoon session

3

Some Terminology

• Between subjects design – each subject participates in one and only one group

• Within subjects design – the same group of subjects serves in more than one treatment‒ Subject is now a factor

• Mixed design – a study which has both between and within subject factors

• Repeated measures – general term for any study in which multiple measurements are measured on the same subject‒ Can be either multiple treatments or several

measurements over time

January 7, 2009 - afternoon session

4

With-in Subjects

• New methods are needed that do not make the assumption of no correlation (independence) of errors

• Since subjects are receiving more than one treatment in within-subjects designs, we expect outcomes to be correlated

January 7, 2009 - afternoon session

5

Why With-in Subjects Designs?

• We may want to study the change of an outcome over time

• Studying multiple outcomes for each subject allows each subject to be his or her own “control”

January 7, 2009 - afternoon session

6

Advantages

• All sources of variability between subjects is excluded from the experimental error

• Repeated measures economizes subjects, which is important when only a few subjects can be utilized for the experiment

• Increased power

January 7, 2009 - afternoon session

7

Disadvantages

• Interference/confounding‒ Order effect

‒ Connected to the position in the treatment order

‒ Carryover effect‒ Connected with the preceding treatment or

treatments

‒ Practice effect‒ Students get better with practice on preceding

treatment

• Various steps can be taken to minimize the dangers of interference

January 7, 2009 - afternoon session

8

Fixed vs. Random Factors

• Fixed factors – the levels are the same levels you would use if you repeated the experiment‒ Treatments are usually fixed factors

• Random factors – a different set of levels would be used if you repeated the experiment‒ Subjects are normally considered a random

factor

January 7, 2009 - afternoon session

9

Repeated Measures Analysis

• Repeated measures analysis is appropriate when one or more factors is a within-subjects factor

• Planned (main effect) contrasts are appropriate for both factors if there is no significant interaction

• Post-hoc comparisons can also be performed‒ Must take ® level into consideration if doing

post-hoc testing

January 7, 2009 - afternoon session

10

Assumptions of Repeated Measures

• Normal distribution of the outcome for each level of the with-in subjects factor

• Errors are assumed to be uncorrelated between subjects

• Within a subject, the multiple measurements are assumed to be correlated

• A technical condition called sphericity must be met‒ Population variances of repeated measures are

equal‒ Population correlations among all pairs of

measures are equal‒ Statistical packages can check this!

January 7, 2009 - afternoon session

11

Relation to Paired t-test

• If we have a treatment with two levels and each subject received both, a paired T-test gives the same results as a two-way ANOVA with subject and treatment as factors

January 7, 2009 - afternoon session

12

Keyboard Example

• Paired t-test results

• ANOVA results

January 7, 2009 - afternoon session

13

Example

• An experiment is conducted to compare energy requirements of three activities: running, walking, and biking

• 12 subjects are asked to run, walk, and bike a required distance and the number of kilocalories burned is measured

• The activities are done in a random order with recovery time in between

• Each subject does each activity once

January 7, 2009 - afternoon session

14

Example

• Why is random order used?

• Why can’t we used a paired t-test?

January 7, 2009 - afternoon session

15

Example

• Why is random order used?‒ Concerned about carry-over effect

• Why can’t we used a paired t-test?‒ There are three levels to the explanatory

variable

January 7, 2009 - afternoon session

16

Exploratory Analysis

January 7, 2009 - afternoon session

17

Exploratory Analysis

• Mean energy output for each activity

January 7, 2009 - afternoon session

18

Analysis

• Use Sphericity Assumed row, assuming that we’ve run the check and the assumption is met

January 7, 2009 - afternoon session

19

Contrasts

• Since there are k=3 levels of exercise, we can only do 2

• Level 1 = cycling, level 2 = walking, level 3 = running

• Can say that walking consumes more energy than cycling and that running consumes more than walking

January 7, 2009 - afternoon session

20

Comparisons

• Need to run comparisons to compare cycling to running

• The 1 vs. 3 shows us that there is a significant difference

January 7, 2009 - afternoon session

21

Mixed between/within ANOVA

• One factor is varied between subjects and the other is within subjects

• Need to check interaction

January 7, 2009 - afternoon session

22

Example

• Add gender to the previous within subjects exercise and energy consumption example

January 7, 2009 - afternoon session

23

Exploratory Analysis

January 7, 2009 - afternoon session

24

Exploratory Analysis

January 7, 2009 - afternoon session

25

Analysis

• Conclusions?

January 7, 2009 - afternoon session

26

Analysis

• Unfortunately SPSS doesn’t allow you to remove the interaction in repeated measures

• Options‒ Interpret main effects in presence of the non-

significant interaction‒ Use another statistical package

January 7, 2009 - afternoon session

27

Power

• A simple Google search for power repeated measures ANOVA turns up pages worth of online applets

• Pick one that you understand

January 7, 2009 - afternoon session

28

Name that Experimental Design

X1

X2

Level 1 Level 2

Level 1s1

s2

s3

s4

S5

s16

s17

s18

s19

s20

Level 2 s6

s7

s8

s9

s10

s21

s22

s23

s24

s25

Level 3s11

s12

s13

s14

s15

s26

s27

s28

s29

s30

X1

X2

Level 1 Level 2

Level 1s1

s2

s3

s4

s5

s1

s2

s3

s4

s5

Level 2 s1

s2

s3

s4

s5

s1

s2

s3

s4

s5

Level 3s1

s2

s3

s4

s5

s1

s2

s3

s4

s5

X1

X2

Level 1 Level 2

Level 1s1

s2

s3

s4

s5

s6

s7

s8

s9

s10

Level 2 s1

s2

s3

s4

s5

s6

s7

s8

s9

s10

Level 3s1

s2

s3

s4

s5

s6

s7

s8

s9

s10

321

January 7, 2009 - afternoon session

29

Notes on designs

• All three give interaction and main effects information, but vary in the number of subjects needed

• Two-factor repeated measures – provides good precision since all sources of variability between subjects is excluded

• Mixed design – reduce carryover effects since each subject is exposed to less treatments

• The mixed design is usually the design of choice when the researcher is studying learning and the process that influences the speed with which learning takes place

January 7, 2009 - afternoon session

30

MANOVA

• An extension of ANOVA where there is more than one dependent variable and the dependent variables can not be combined