Introduction to Repeated Measures

Transcript 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

  Y Dependent variable Repeated measure exptal Period of treatment control Data are means and standard deviations   Group Between-subjects factor y1 Trial or Time Different subjects on each level   y2 y3 Within-subjects factor Same subjects on each level

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

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

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

Univariate and Multivariate Estimation

• But if you take a close look at the Repeated Measures ANOVA, you suddenly realize it has

multiple

• The only dependent variables. That helps us understand that the RMANOVA could be construed as a MANOVA, with multivariate effect estimation (Wilk’s, Pillai’s, etc.)

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

Univariate and Multivariate Estimation

• 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

-statistic denominator (MS error ) on the tests we care about

RMANOVA Design: Univariate Estimation

SS T Total variance in the DV SS Within Total variance within subjects SS Between Total variance between subjects SS M Effect of experiment SS RES Within-subjects Error

RMANOVA Design: Multivariate

Let’s consider a simple design Subject Time1 Time2 Time3 d t1-t2 d t1-t3 d t2-t3 1 2 3 7 5 10 4 12 7 3 5 2 -1 2 3 6 8 10 2 4 2 .......................................………………………………..

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

decreases...As a general rule, the multivariate approach should probably not be used if

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.

Univariate or Multivariate?

• If the sphericity assumption

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

]

)

Univariate or Multivariate?

• 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

Meas ure: MEASURE_1 occas ion 1 2 3 Dependent Variable Start Month1 Month2 Start Month1 Month2

Descriptive Statistics

Mean 171.9000

162.0000

148.5000

Std. Deviation 43.53657

38.45632

35.66900

N 10 10 10

Mauchly's Test of Sphericity b

Measure: MEASURE_1 Eps ilon a Within Subjects Effect occas ion Mauchly's W .454

Approx.

Chi-Square 6.311

df 2 Sig.

.043

Greenhous e-Geis s er .647

Huynh-Feldt .710

Lower-bound .500

Tes ts the null hypothes is that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix.

a. May be us ed to adjus t the degrees of freedom for the averaged tes ts of s ignificance. Corrected tests are displayed in the Tests of Within-Subjects Effects table.

b. Des ign: Intercept Within Subjects Design: occas ion Total violation. What should we do?

Multivariate Tests c

Effect occasion Pillai's Trace Wilks' Lambda Hotelling's Trace Roy's Largest Root Value .590

.410

1.438

F 5.751

b 5.751

b Hypothesis df 2.000

2.000

Error df 8.000

8.000

a. Computed using alpha = .05

b. Exact statistic

WHAT DOES THIS MEAN???

c. Design: Intercept Within Subjects Design: occasion Sig.

.028

Partial Eta Squared .590

.590

Noncent.

Parameter 11.502

11.502

Observed a Power .704

.704

Tests of Within-Subjects Effects

Meas ure: MEASURE_1 Source occas ion Error(occasion) Sphericity As sumed Greenhouse-Geiss er Huynh-Feldt Lower-bound Sphericity As sumed Greenhouse-Geiss er Huynh-Feldt Lower-bound a. Computed using alpha = .05

Type III Sum of Squares 2759.400

2759.400

2831.933

df 2 1.294

1.420

1.000

18 11.645

12.776

9.000

Mean Square 1379.700

2132.558

1943.811

2759.400

157.330

243.179

221.656

314.659

F 8.769

8.769

Sig.

.002

.009

.007

.016

Partial Eta Squared .494

.494

Noncent.

Parameter 17.539

11.347

12.449

8.769

Obs erved Power a .940

.833

.860

.750

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

Measure: MEASURE_1 Source occas ion Error(occasion) occas ion Level 1 vs. Later Level 2 vs. Level 3 Level 1 vs. Later Level 2 vs. Level 3 a. Computed using alpha = .05

Type III Sum of Squares 2772.225

1822.500

1960.025

3050.500

Tests of Within-Subjects Contrasts

df 1 1 9 9 Mean Square 2772.225

1822.500

217.781

338.944

F 12.729

5.377

Sig.

.006

.046

Partial Eta Squared .586

.374

Noncent.

Parameter 12.729

5.377

Obs erved Power a .887

.543

Estimates

Meas ure: MEASURE_1 occas ion 1 2 3 Mean 171.900

162.000

148.500

Std. Error 13.767

12.161

11.280

95% Confidence Interval Lower Bound 140.756

Upper Bound 203.044

134.490

122.984

189.510

174.016

Pairwise Comparisons

Meas ure: MEASURE_1 (I) occas ion 1 2 3 (J) occasion 2 3 1 3 1 2 Mean Difference (I-J) 9.900* 23.400* -9.900* 13.500

-23.400* -13.500

Std. Error 3.199

7.090

3.199

5.822

7.090

5.822

Bas ed on es timated marginal means *. The mean difference is significant at the .05 level.

a. Adjus tment for multiple comparisons : Bonferroni.

Sig.

a .038

.028

.038

.137

.028

.137

95% Confidence Interval for Difference a Lower Bound .517

2.602

Upper Bound 19.283

44.198

-19.283

-3.578

-44.198

-30.578

-.517

30.578

-2.602

3.578

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

Estimated Marginal Means of MEASURE_1

160 155 150 145 175 170 165 1 2

occasion

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 [

(1.29, 11.65) = 8.77,

< .05,

η 2

=.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

(1, 9) = 12.73,

< .05,

η 2

=.58]. There was additional significant weight loss from month 2 to month 3 [

(1,9) = 5.38,

< .05,

η 2

=.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 definition

Within-Subjects Factors

Meas ure: MEASURE_1 OCCASION 1 2 Dependent Variable SPDCB1 SPDCB2

Between-Subjects Factors

Intervention Group 1 2 3 4 Value Label Memory Reas oning Speed Control N 629 614 639 623

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 Source Intercept intgrp Error Type III Sum of Squares 2099.980

1169.107

15011.948

df 1 3 2501 Mean Square 2099.980

389.702

6.002

F 349.858

64.925

Sig.

.000

Partial Eta Squared .123

.072

This is the difference between the levels of the IV collapsed across BOTH measures of speed (pre and post)

Pairwise Comparisons

Measure: speed (I) Intervention group Memory Reasoning Speed Control (J) Intervention group Reasoning Speed Control Memory Speed Control Memory Reasoning Control Memory Reasoning Speed Based on estimated marginal means *. The mean difference is significant at the .05 level.

Mean Difference (I-J) -.110

1.456* -.201

.110

1.565* -.091

-1.456* -1.565* -1.656* .201

.091

1.656* a. Adjustment for multiple comparisons: Bonferroni.

Std. Error .139

.138

.139

.138

.139

.138

.139

.138

Sig.

a 1.000

.000

.881

1.000

.000

1.000

.000

.881

1.000

.000

95% Confidence Interval for Difference a Lower Bound Upper Bound -.477

1.092

-.567

.257

1.819

.165

-.257

1.200

-.459

-1.819

-1.931

-2.021

-.165

-.276

1.292

.477

1.931

.276

-1.092

-1.200

-1.292

.567

.459

2.021

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

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

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

Transcript Introduction to Repeated Measures