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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. - PowerPoint PPT Presentation
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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
Univariate and Multivariate Estimation
• 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
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 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.
Univariate or Multivariate?
• 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])
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
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
Computed using alpha = .05a.
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)
Type III Sumof Squares df Mean Square F Sig.
Partial EtaSquared
Noncent.Parameter
ObservedPower
a
Computed using alpha = .05a.
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
Type III Sumof Squares df Mean Square F Sig.
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!