introduction to repeated measures

Introduction to Repeated Measures

MANOVA Revisited

• MANOVA is a general purpose multivariate analytical tool which lets us look at treatment effects on a whole set of DVs

• As soon as we got a significant treatment effect, we tried to “unpack” the multivariate DV to see where the effect was

MANOVA Repeated Measures ANOVA

• Put differently, we didn’t have any specialness of an ordering among DVs

• Sometimes we take multiple measurements, and we’re interested in systematic variation from one measurement taken on a person to another

• Repeated measures is a multivariate procedure cause we have more than one DV

Repeated Measures ANOVA

• We are interested in how a DV changes or is different over a period of time in the same participants

When to use RM ANOVA

• Longitudinal Studies

• Experiments

Why are we talking about ANOVA?

• When our analysis focuses on a single measure assessed at different occasions it is a REPEATED MEASURE ANOVA

• When our analysis focuses on multiple measures assessed at different occasions it is a DOUBLY MULTIVARIATE REPEATED MEASURES ANALYSIS

Between- and Within-Subjects Factor

• Between-Subjects variable/factor– Your typical IV from MANOVA– Different participants in each level of the IV

• Within-Subjects variable/factor– This is a new IV – Each participant is represented/tested at each

level of the Within-Subject factor– TIME

Data are means and standard deviations

Y Dependent variable Repeated measure

exptal

control

Group Between-subjects factor Different subjects on each level

Period oftreatment

Within-subjects factor Same subjects on each level

y1 y2 y3Trial or Time

Between- and Within-Subjects Factor

• In Repeated Measures ANOVA we are interested in both BS and WS effects

• We are also keenly interested in the interaction between BS and WS– Give mah an example

RMANOVA

• Repeated measures ANOVA has powerful advantages– completely removes within-subjects variance, a

radical “blocking” approach – It allows us, in the case of temporal ordering, to

see performance trends, like the lasting residual effects of a treatment

– It requires far fewer subjects for equivalent statistical power

Repeated Measures ANOVA

• The assumptions of the repeated measures ANOVA are not that different from what we have already talked about– independence of observations– multivariate normality

• There are, however, new assumptions– sphericity

Sphericity

• The variances for all pairs of repeated measures must be equal– violations of this rule will positively bias the F statistic

• More precisely, the sphericity assumption is that variances in the differences between conditions is equal

• If your WS has 2 levels then you don’t need to worry about sphericity

Sphericity

• Example: Longitudinal study assessment 3 times every 30 days

variance of (Start – Month1) = variance of (Month1 – Month2) =

variance of (Start – Month 2) =

• Violations of sphericity will positively bias the F statistic

Univariate and Multivariate Estimation

• It turns out there are two ways to do effect estimation

• One is a classic ANOVA approach. This has benefits of fitting nicely into our conceptual understanding of ANOVA, but it also has these extra assumptions, like sphericity


• But if you take a close look at the Repeated Measures ANOVA, you suddenly realize it has multiple dependent variables. That helps us understand that the RMANOVA could be construed as a MANOVA, with multivariate effect estimation (Wilk’s, Pillai’s, etc.)

• The only difference from a MANOVA is that we are also interested in formal statistical differences between dependent variables, and how those differences interact with the IVs

• Assumptions are relaxed with the multivariate approach to RMANOVA


• It gets a little confusing here....because we’re not talking about univariate ESTIMATION versus multivariate ESTIMATION...this is a “behind the scenes” component that is not so relevant to how we actually run the analysis

Univariate Estimation

• Since each subject now contributes multiple observations, it is possible to quantify the variance in the DVs that is attributable to the subject.

• Remember, our goal is always to minimize residual (unaccounted for) variance in the DVs.

• Thus, by accounting for the subject-related variance we can substantially boost power of the design, by deflating the F-statistic denominator (MSerror) on the tests we care about

RMANOVA Design: Univariate Estimation

SST

Total variance in the DV

SSBetween

Total variance between subjectsSSWithin

Total variance within subjects

SSRES

Within-subjects Error

SSM

Effect of experiment

RMANOVA Design: MultivariateLet’s consider a simple design

Subject Time1 Time2 Time3 dt1-t2 dt1-t3 dt2-t3

1 7 10 12 3 5 2

2 5 4 7 -1 2 3

3 6 8 10 2 4 2

.......................................………………………………..

n 3 7 3 4 0 -3•In the multivariate case for repeated measures, the test statistic for k repeated measures is formed from the (k-1) [where k = # of occasions] difference variables and their variances and covariances

Univariate or Multivariate?• If your WS factor only has 2 levels the approaches give

the same answer!• If sphericity holds, then the univariate approach is more

powerful. When sphericity is violated, the situation is more complex

• Maxwell & Delaney (1990)• “All other things being equal, the multivariate test is

relatively less powerful than the univariate approach as n decreases...As a general rule, the multivariate approach should probably not be used if n is less than a + 10” (a=# levels of the repeated measures factor).

Univariate or Multivariate?

• If you can use the univariate output, you may have more power to reject the null hypothesis in favor of the alternative hypothesis.

• However, the univariate approach is appropriate only when the sphericity assumption is not violated.


• If the sphericity assumption is violated, then in most situations you are better off staying with the multivariate output. – Must then check homogeneity of V-C

• If sphercity is violated and your sample size is low then use an adjustment (Greenhouse-Geisser [conservative] or Huynh-Feldt [liberal])


• SPSS and SAS both give you the results of a RMANOVA using the – Univariate approach – Multivariate approach

• You don’t have to do anything except decide which approach you want to use

Effects

• RMANOVA gives you 2 different kinds of effects

• Within-Subjects effects

• Between-Subjects effects

• Interaction between the two

Within-Subjects Effects

• This is the “true” repeated measures effect

• Is there a mean difference between measurement occasions within my participants?

Between-Subjects Effects

• These are the effects on IV’s that examine differences between different kinds of participants

• All our effects from MANOVA are between-subjects effects

• The IV itself is called a between-subjects factor

Mixed Effects

• Mixed effects are another named for the interaction between a within-subjects factor and a between-subjects factor

• Does the within-subjects effect differ by some between-subjects factor

EXAMPLE • Lets say Eric Kail does an intervention to improve

the collegiality of his fellow IO students• He uses a pretest—intervention—posttest design• The DV is a subjective measure of collegiality• Eric had a hypothesis that this intervention might

work differently depending on the participants GPA (high and low)

EXAMPLE

• Within-Subjects effect =

• Between-Subjects effect =

• Mixed effect =

Within-Subjects RMANOVA• A within-subjects repeated measures ANOVA

is used to determine if there are mean differences among the different time points

• There is no between-subjects effect so we aren’t worried about anything BUT the WS effect

• The within-subjects effect is an OMNIBUS test

• We must do follow-up tests to determine which time points differ from one another

Example

• 10 participants enrolled in a weight loss program

• They got weighed when thy first enrolled and then each month for 2 months

• Did the participants experience significant weight loss? And if so when?

You can name your within-subjects factor anything you want.

“3” reflects the number of occasions

Put in your DV’s for occasion 1, 2, 3

Just how was always do it!

We also get to do post-hoc comparisons

Within-Subjects Factors

Measure: MEASURE_1

Start

Month1

Month2

occasion1

2

3

DependentVariable

Descriptive Statistics

171.9000 43.53657 10

162.0000 38.45632 10

148.5000 35.66900 10

Start

Month1

Month2

Mean Std. Deviation N

Mauchly's Test of Sphericityb

Measure: MEASURE_1

.454 6.311 2 .043 .647 .710 .500Within Subjects Effectoccasion

Mauchly's WApprox.

Chi-Square df Sig.Greenhouse-Geisser Huynh-Feldt Lower-bound

Epsilona

Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables isproportional to an identity matrix.

May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed inthe Tests of Within-Subjects Effects table.

a.

Design: Intercept Within Subjects Design: occasion

b.

Total violation. What should we do?

Tests of Within-Subjects Effects

Measure: MEASURE_1

2759.400 2 1379.700 8.769 .002 .494 17.539 .940

2759.400 1.294 2132.558 8.769 .009 .494 11.347 .833

2759.400 1.420 1943.811 8.769 .007 .494 12.449 .860

2759.400 1.000 2759.400 8.769 .016 .494 8.769 .750

2831.933 18 157.330

2831.933 11.645 243.179

2831.933 12.776 221.656

2831.933 9.000 314.659

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Sourceoccasion

Error(occasion)

Type III Sumof Squares df Mean Square F Sig.

Partial EtaSquared

Noncent.Parameter

ObservedPower

a

Computed using alpha = .05a.

Multivariate Tests c

.590 5.751 b 2.000 8.000 .028 .590 11.502 .704

.410 5.751 b 2.000 8.000 .028 .590 11.502 .704

1.438 5.751 b 2.000 8.000 .028 .590 11.502 .704

1.438 5.751 b 2.000 8.000 .028 .590 11.502 .704

Pillai's Trace

Wilks' Lambda

Hotelling's Trace

Roy's Largest Root

Effect

occasion

Value F Hypothesis df Error df Sig.

Partial Eta

Squared

Noncent.

Parameter

Observed

Powera


Exact statisticb.

Design: Intercept

Within Subjects Design: occasion

c. WHAT DOES THIS MEAN???

Tests of Within-Subjects Contrasts

Measure: MEASURE_1

2772.225 1 2772.225 12.729 .006 .586 12.729 .887

1822.500 1 1822.500 5.377 .046 .374 5.377 .543

1960.025 9 217.781

3050.500 9 338.944

occasionLevel 1 vs. Later

Level 2 vs. Level 3

Level 1 vs. Later

Level 2 vs. Level 3

Sourceoccasion

Error(occasion)


Partial EtaSquared

Noncent.Parameter

ObservedPower

a


These are the helmet contrasts. What are they telling us?

Estimates

Measure: MEASURE_1

171.900 13.767 140.756 203.044

162.000 12.161 134.490 189.510

148.500 11.280 122.984 174.016

occasion1

2

3

Mean Std. Error Lower Bound Upper Bound

95% Confidence Interval

Pairwise Comparisons

Measure: MEASURE_1

9.900* 3.199 .038 .517 19.283

23.400* 7.090 .028 2.602 44.198

-9.900* 3.199 .038 -19.283 -.517

13.500 5.822 .137 -3.578 30.578

-23.400* 7.090 .028 -44.198 -2.602

-13.500 5.822 .137 -30.578 3.578

(J) occasion2

3

1

3

1

2

(I) occasion1

2

3

MeanDifference

(I-J) Std. Error Sig.a

Lower Bound Upper Bound

95% Confidence Interval forDifference

a

Based on estimated marginal means

The mean difference is significant at the .05 level.*.

Adjustment for multiple comparisons: Bonferroni.a.

This is the previous 0.046 times 3(for 3 comparisons)

1 2 3

occasion

145

150

155

160

165

170

175

Est

imat

ed M

arg

inal

Mea

ns

Estimated Marginal Means of MEASURE_1

Write Up

• In order to determine if there was significant weight loss over the three occasions a repeated measures analysis of variance was conducted. Results indicated a significant within-subjects effect [F(1.29, 11.65) = 8.77, p < .05, η2=.49] indicating a significant mean difference in weight among the three occasions. As can be seen in Figure 1, the mean weight at month 2 and 3 was significantly lower relative to month 1 [F(1, 9) = 12.73, p < .05, η2=.58]. There was additional significant weight loss from month 2 to month 3 [F(1,9) = 5.38, p < .05, η2=.49.

Within and between-subject factors

• When you have both WS and BS factors then you are going to be interested in the interaction!

• IV = intgrp (4 levels)

• DV = speed at pretest and posttest

The BS factors goes here!

GLM spdcb1 spdcb2 BY intgrp /WSFACTOR = prepost 2 Repeated /MEASURE = speed /METHOD = SSTYPE(3) /PLOT = PROFILE( prepost*intgrp ) /EMMEANS = TABLES(intgrp) COMPARE ADJ(BONFERRONI) /EMMEANS = TABLES(prepost) COMPARE ADJ(BONFERRONI) /EMMEANS = TABLES(intgrp*prepost) COMPARE(prepost) ADJ(BONFERRONI)/EMMEANS = TABLES(intgrp*prepost) COMPARE(intgrp) ADJ(BONFERRONI) /PRINT = DESCRIPTIVE ETASQ HOMOGENEITY /CRITERIA = ALPHA(.05) /WSDESIGN = prepost /DESIGN = intgrp .

RMANOVA: Data definitionWithin-Subjects Factors

Measure: MEASURE_1

SPDCB1

SPDCB2

OCCASION1

2

DependentVariable

Between-Subjects Factors

Memory 629

Reasoning 614

Speed 639

Control 623

1

2

3

4

InterventionGroup

Value Label N

RMANOVA: Assumption Check: Sphericity test

RMANOVA: Multivariate estimation of within-subjects

effects

RMANOVA: Univariate estimation of within-subjects

effects

RMANOVA: Within subjects contrasts?

RMANOVA: Univariate estimation of between-subjects

effects

Tests of Between-Subjects Effects

Measure: speed

Transformed Variable: Average

2099.980 1 2099.980 349.858 .000 .123

1169.107 3 389.702 64.925 .000 .072

15011.948 2501 6.002

Source

Intercept

intgrp

Error


Partial EtaSquared

Pairwise Comparisons

Measure: speed

-.110 .139 1.000 -.477 .257

1.456* .138 .000 1.092 1.819

-.201 .138 .881 -.567 .165

.110 .139 1.000 -.257 .477

1.565* .138 .000 1.200 1.931

-.091 .139 1.000 -.459 .276

-1.456* .138 .000 -1.819 -1.092

-1.565* .138 .000 -1.931 -1.200

-1.656* .138 .000 -2.021 -1.292

.201 .138 .881 -.165 .567

.091 .139 1.000 -.276 .459

1.656* .138 .000 1.292 2.021

(J) Intervention groupReasoning

Speed

Control

Memory

Speed

Control

Memory

Reasoning

Control

Memory

Reasoning

Speed

(I) Intervention groupMemory

Reasoning

Speed

Control

MeanDifference

(I-J) Std. Error Sig.a Lower Bound Upper Bound

95% Confidence Interval forDifferencea

Based on estimated marginal means

The mean difference is significant at the .05 level.*.

Adjustment for multiple comparisons: Bonferroni.a.

This is the difference between the levels of the IV collapsed across BOTH measures of speed

(pre and post)

The only intgrp difference is speed versus all others, and that is only at posttest—exactly what we would expect

/EMMEANS = TABLES(intgrp*prepost) COMPARE(intgrp) ADJ(BONFERRONI)

RMANOVA: What does it look like?

I am missing something. What is it?

Practice

• IV = group ( 2 = training and 1 – control)

• DV = Letter series– Letser (pretest) and letser2 (posttest)

• Are the BS and WS effects

• More importantly is there an interaction?– If there is an interaction than you need to

decompose it!

introduction to repeated measures

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