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Research Methodology: Tools

Applied Data Analysis (with SPSS)

Lecture 11: Repeated Measures ANOVA / ANCOVA

May 2011

Prof. Dr. Jrg Schwarz juerg.schwarz@hslu.ch

MSc Business Administration

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Contents

Aims ___________________________________________________________________________________________________ 5Introduction _____________________________________________________________________________________________ 6Outline ________________________________________________________________________________________________ 13Concepts of repeated measures ANOVA / ANCOVA ___________________________________________________________ 14ANOVA with SPSS: Two detailed examples __________________________________________________________________ 19

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Table of contents

Aims ___________________________________________________________________________________________________ 5Aims of the lecture .................................................................................................................................................................................................5

Introduction _____________________________________________________________________________________________ 6Example 1 ..............................................................................................................................................................................................................6Example 2 ..............................................................................................................................................................................................................9

Outline ________________________________________________________________________________________________ 13Concepts of repeated measures ANOVA / ANCOVA ___________________________________________________________ 14Repeated measures ANOVA................................................................................................................................................................................14ANCOVA..............................................................................................................................................................................................................15Homogeneity of variances....................................................................................................................................................................................16

Case of between-group ANOVA................................................................................................................................................................................................16Case of repeated measures ANOVA.........................................................................................................................................................................................17

Covariate..............................................................................................................................................................................................................18

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ANOVA with SPSS: Two detailed examples __________________________________________________________________ 19Repeated measures ANOVA (Dataset Diet1.sav) ................................................................................................................................................19

SPSS Elements: .......................................................................................................................19How to define repeated measures.............................................................................................................................................................................................20SPSS Output ANOVA (Diet1) Mauchly's Test of Sphericity ...................................................................................................................................................21SPSS Output ANOVA (Diet1) Tests of Within-Subjects Effects.............................................................................................................................................22SPSS Output ANOVA (Diet1) Tests of Between-Subjects Effects .........................................................................................................................................23SPSS Output ANOVA (Diet1) Contrasts.................................................................................................................................................................................24SPSS Output ANOVA (Diet1) Multivariate Tests ....................................................................................................................................................................25

ANCOVA (Dataset Diet2.sav)...............................................................................................................................................................................26SPSS Elements: ........................................................................................................................................26SPSS Output ANOVA (Diet2) Tests of Between-Subjects Effects .........................................................................................................................................27Homogeneity of regression........................................................................................................................................................................................................28Specify custom model................................................................................................................................................................................................................29Homogeneity of regression Output .........................................................................................................................................................................................30

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Aims

Aims of the lecture

You will understand the key steps in conducting a repeated measures ANOVA / ANCOVA.

You will understand the concept of repeated measures ANOVA / ANCOVA.

You will understand the concept of homogeneity of variances.

You will understand the concept of homogeneity of regression.

You can conduct a repeated measures ANOVA / ANCOVA with SPSS.

In particular, you will know how to

interpret the output

Mauchly's Test of sphericitycontrasts

check of homogeneity of regression

describe the output

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Introduction

Example 1 Repeated Measures ANOVA

Medical research: Effect on body weight of a diet over 5 weeks. (Dataset: Diet1.sav)

Typical questions

Is the diet effective?

Does it affect men and women differently?

15 women

15 men

all

Bodyweight[kg]

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Example 2 ANCOVA

Medical research: Effect of 3 different types of diet on body weight.

Each type of diet was tested on a subsample of 5 randomly allocated overweight men.

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Weightreduction[kg]

Type of diet

Dataset (Diet2.sav)

Sample of n = 15 overweight men

Variables for

type of diet (1,2,3) (diet)

weight reduction (kg) (reduction)

weight before diet (kg) (weight)

Typical questions

Is there an impact of the type of diet?

What is the influence of the initial weight?

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The Result Impact of type of diet

Significant ANOVA model ("Corrected Model" with p =.000).

Significant variable diet

There is a main effect of diet (types 1, 2, 3) on weight reduction (F(2, 12) = 20.812 p = .000).

The value of Partial Eta Squared = .776 shows that 78% of the variation can be explained by the

variable diet (all other variables fixed).

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What is the influence of weight before diet?

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It seems that "weight before diet" and

"weight reduction" are related.

The greater weight before diet, the more

weight reduction is possible.

This relation should also be modeled.

"Weight before diet" is a metric variable.

Enter the variable weightas a so called

covariate into the model.

Weightreduction[kg]

Weight before diet [kg]

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The Result Impact of type of diet plus weight before diet as covariate

There is a main effect of diet (types 1, 2, 3) on weight reduction (F(2, 11) = 25.017 p = .000).

The value of Partial Eta Squared = .820 shows that 82% of the variation can be explained by the

variable diet (all other variables fixed).

There is also a significant effect of the covariate weight (F(1, 11) = 6.659 p = .026).

The constant ("Intercept") is no more significant (p = .542).This is due to the covariate which "binds" the variation of the intercept.

The impact of diet is higher (2 = 82% vs. 2 = 78%) after controlling for the effects of weight

before diet. The model is more "selective" respecting the main effect.

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Outline

Basic situation for repeated measures ANOVA

Given: One dependent metric scaled variable which is measured multiple and many

independent variables with categorical scales.

Task: Find a relationship between the characteristics.

Analysis of repeated measures ANOVA Statistical method is the same as in analysis of variance.

The repeated measures model can test the main effects of within-subjects factors likesubsequent measurement times or different treatments.

Basic situation for ANCOVA

Given: One dependent variable with metric scale and many independent variables with categori-

cal scales. Included is a metric scaled independent variable.

Task: Find a relationship between the characteristics.

Analysis of Covariance (ANCOVA)

Controls for the effects of the metric scaled independent variable that covaries with thedependent variable.

The metric scaled independent variable is called "covariate" or "control variable".

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Concepts of repeated measures ANOVA / ANCOVA

Repeated measures ANOVA

Repeated measures

General term for a study in which multiple measurements are applied to the same subject

Can be several measurements over time or multiple treatments (e.g. Drug A, B, C)

Advantages

Source of variability between subjects is excluded from the experimental error

Less participants are neededThis will be important if only a few subjects are available for the experiment

Disadvantages

Carry-over effects (outcome is influenced by previous treatment in an unwanted way) Example: Subject changes his opinion on certain issues.

Order effects (outcome depends on the order of treatment)

Example: The order "treatment A > B" is different to "treatment B > A"

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ANCOVA

Design of experiment

At least one categorical variable (called factor) and one metric variable (called covariate)

Blending of regression and ANOVA

ANCOVA is used for different purposes

Experiment: to control for factors that cannot be randomized but that can bemeasured on an metric scale

Survey: to remove the effects of metric variables modifying the relationship of thecategorical independents to the dependent variable

In general, the covariate is considered to be a control variable

Advantages

Irrelevant variability can be reduced ("controlled")

Disadvantages If the covariate is correlated with the factors, then its inclusion will lead to an underestimation

of the effect of the treatment factors

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Homogeneity of variances

Case of between-group ANOVA

Back to the requirements of ANOVA in Lecture 10

3. Homogeneity of variances

Residuals (= Error) have constant variance

Correction => weight variances

Nurse example: Use homogeneity test

p < .050 => no constant variance

Weight with an appropriate variable

Nurse example of lecture 10

Homogeneity test "Options"

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Case of repeated measures ANOVA

Requirements of repeated measures ANOVA ("Compound Symmetry")

1. Residuals (= Error) have constant variance

=> the same as in lecture 10 "3. Homogeneity of variances"

2. Correlations between treatments are equal

Assumption of sphericity ("Sphrizitt")

Sphericity is a less restrictive form of compound symmetry

Technical implementation: A test whether variances of the differences between each pair of

treatment outcome are equal (=> at least three levels of treatment are needed)

Example (Diet1)

Calculate differences of scores between weeks 2 and 3,

and weeks 3 and 4. Variances of the differences of the

first pair must be the same as those of the second pair, etc.

Sphericity is measured using Mauchly's Test.

p > 0.05 => Sphericity is met

p < 0.05 => Sphericity is violated

If Mauchly's Test shows violation of sphericity, this may be compensated by an adjustment.

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Covariate

To control for metric variable x, introduce an additional (regression) term into the linear model.

(In example 2 the metric variable x is weight before diet)

gk g g gk gky y x= + + +

g

gk

y grand mean

effect of group g

random term

=

=

=

g

gk

Additional term

slope of covariate of group g

x covariate

=

=

Homogeneity of regression

ANCOVA demands that homogeneity of regression exists.

=> For each level of the covariate xg., the slope g must be equal.

The null hypothesis (H0) to verify is that all regression coefficients g have the same value

The alternative hypothesis (HA) is that this is not the case

H0: 1 = 2 = = G (= )

HA: at least two coefficients are not equal

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ANOVA with SPSS: Two detailed examples

Repeated measures ANOVA (Dataset Diet1.sav)

SPSS Elements:

Variables week1, week2, week5 are repeated measures of body weight

Define a name for a virtual Within-Subject-Factor

For example "week"

Enter the number of repetitions (called "levels")

This example: 5 for week1 to week5

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How to define repeated measures

Allocate each week variable to its level

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SPSS Output ANOVA (Diet1) Mauchly's Test of Sphericity

Mauchly's Test is significant with p = .010 => sphericity is not given

How to proceed if Mauchly's Test is significant: If Epsilon in Greenhouse-Geisser < .75 => correction as per Greenhouse-Geisser

If Epsilon in Greenhouse-Geisser > .75 => correction as per Huynh-Feldt

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SPSS Output ANOVA (Diet1) Tests of Within-Subjects Effects

This is the first of two main tables

The time elapsed (week 1, week 2, ) has a significant influence on body weight

(F(2.792, 78.175) = 178.349 p = .000, correction as per Greenhouse-Geisser).

The duration of the diet explains 86% of the variance.

There is also an interaction of time elapsed and sex on body weight

(F(2.792, 78.175) = 75.165 p = .000, correction as per Greenhouse-Geisser).

The interaction term explains 73% of the variance.

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SPSS Output ANOVA (Diet1) Multivariate Tests

This is an alternative to the test of "within-subjects effects" if sphericity is not given.

It is part of the MANOVA procedure, which makes no assumptions as regards sphericity.

The Multivariate Tests table provides F tests of the within-subjects factor week

and the interaction with the between-subjects grouping factor sex.

Wilks' lambda is the most commonly reported

Usually the same substantive conclusion emerges from any variant.

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ANCOVA (Dataset Diet2.sav)

SPSS Elements:

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SPSS Output ANOVA (Diet2) Tests of Between-Subjects Effects

There is a main effect of diet (types 1, 2, 3) on weight reduction (F(2, 11) = 25.017 p = .000).

The value of Partial Eta Squared = .820 shows that 82% of the variation can be explained by the

variable diet (all other variables fixed).

There is also a significant effect of the covariate weight (F(1, 11) = 6.659 p = .026).

The constant ("Intercept") is no more significant (p = .542).

The covariate "binds" the variation due to the intercept.

The impact of diet is higher (2 = 82% vs. 2 = 78% without covariate) after controlling for the ef-

fects of weight before diet. The model is more "selective" respecting the main effect.

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Homogeneity of regression

Are the slopes g between weight before diet and weight reduction equal for all levels of diet?

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Use a modification of the ANCOVA to quick-check whether the slopes g are equal.

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all diets

diet 1 diet 2 diet 3

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Specify custom model

Allocate main effects

Allocate interaction term

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Homogeneity of regression Output

If the interaction term of a main factor and a covariate is significant,

the assumption of homogeneity of regression is violated.

Here the interaction term is not significant=> homogeneity of regression is given

=> the covariate weight(weight before diet) may be introduced into the model.