Introduction to Repeated Measures

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Introduction to Repeated Measures
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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

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MANOVA ? Repeated Measures ANOVA

• Put differently, we didnt have any specialness
  of an ordering among DVs
• Sometimes we take multiple measurements, and
  were 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

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Repeated Measures ANOVA

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

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When to use RM ANOVA

• Longitudinal Studies
• Experiments

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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

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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

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• Y
• Dependent variable
• Repeated measure

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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

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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

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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

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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 dont need to
  worry about sphericity

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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

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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

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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
  (Wilks, Pillais, 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

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Univariate and Multivariate Estimation

• It gets a little confusing here....because were
  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

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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

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RMANOVA Design Univariate Estimation
SST Total variance in the DV
SSBetween Total variance between subjects
SSWithin Total variance within subjects
SSM Effect of experiment
SSRES Within-subjects Error
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RMANOVA Design Multivariate
Lets 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

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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).

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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.

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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)

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Univariate or Multivariate?

• SPSS and SAS both give you the results of a
  RMANOVA using the
• Univariate approach
• Multivariate approach
• You dont have to do anything except decide which
  approach you want to use

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Effects

• RMANOVA gives you 2 different kinds of effects
• Within-Subjects effects
• Between-Subjects effects
• Interaction between the two

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Within-Subjects Effects

• This is the true repeated measures effect
• Is there a mean difference between measurement
  occasions within my participants?

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Between-Subjects Effects

• These are the effects on IVs 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

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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

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EXAMPLE

• Lets say Eric Kail does an intervention to
  improve the collegiality of his fellow IO
  students
• He uses a pretestinterventionposttest 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)

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EXAMPLE

• Within-Subjects effect
• Between-Subjects effect
• Mixed effect

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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 arent
  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

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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?

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You can name your within-subjects factor anything
you want.
3 reflects the number of occasions
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Put in your DVs for occasion 1, 2, 3
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We also get to do post-hoc comparisons
Just how was always do it!
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Total violation. What should we do?
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WHAT DOES THIS MEAN???
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These are the helmet contrasts. What are they
telling us?
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This is the previous 0.046 times 3 (for 3
comparisons)
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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 lt .05, ?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 F(1, 9)
  12.73, p lt .05, ?2.58. There was additional
  significant weight loss from month 2 to month 3
  F(1,9) 5.38, p lt .05, ?2.49.

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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

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The BS factors goes here!
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GLM spdcb1 spdcb2 BY intgrp /WSFACTOR
prepost 2 Repeated /MEASURE speed /METHOD
SSTYPE(3) /PLOT PROFILE( prepostintgrp )
/EMMEANS TABLES(intgrp) COMPARE
ADJ(BONFERRONI) /EMMEANS TABLES(prepost)
COMPARE ADJ(BONFERRONI) /EMMEANS
TABLES(intgrpprepost) COMPARE(prepost)
ADJ(BONFERRONI) /EMMEANS TABLES(intgrpprepost)
COMPARE(intgrp) ADJ(BONFERRONI) /PRINT
DESCRIPTIVE ETASQ HOMOGENEITY /CRITERIA
ALPHA(.05) /WSDESIGN prepost /DESIGN
intgrp .
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RMANOVA Data definition
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RMANOVA Assumption Check Sphericity test
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RMANOVA Multivariate estimation of
within-subjects effects
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RMANOVA Univariate estimation of
within-subjects effects
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RMANOVA Within subjects contrasts?
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RMANOVA Univariate estimation of
between-subjects effects
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This is the difference between the levels of the
IV collapsed across BOTH measures of speed (pre
and post)
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/EMMEANS TABLES(intgrpprepost)
COMPARE(intgrp) ADJ(BONFERRONI)
The only intgrp difference is speed versus all
others, and that is only at posttestexactly what
we would expect
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RMANOVA What does it look like?
I am missing something. What is it?
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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!