January 7, 2009 - afternoon session 1 Multi-factor ANOVA and Multiple Regression January 5-9, 2008...

January 7, 2009 - afternoon session

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Multi-factor ANOVA and

Multiple Regression

January 5-9, 2008

Beth Ayers

January 7, 2009 - afternoon session

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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

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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

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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

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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”

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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

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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

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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

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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

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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!

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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

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Keyboard Example

• Paired t-test results

• ANOVA results

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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

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Example

• Why is random order used?

• Why can’t we used a paired t-test?

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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

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Exploratory Analysis

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Exploratory Analysis

• Mean energy output for each activity

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Analysis

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

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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

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Comparisons

• Need to run comparisons to compare cycling to running

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

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Mixed between/within ANOVA

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

• Need to check interaction

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Example

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

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Exploratory Analysis

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Exploratory Analysis

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Analysis

• Conclusions?

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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

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Power

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

• Pick one that you understand

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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

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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

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MANOVA

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