Covariates in Repeated-Measures Analyses

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Covariates in Repeated-Measures Analyses
Will G HopkinsAuckland University of
TechnologyAuckland, NZ

• Repeated Measures
• What change has occurred (in response to a
  treatment)?
• Mechanism Variables
• How much of the change was due to a change in
  whatever?
• Individual Responses to a Treatment
• What's the effect of subject characteristics on
  the change?
• Two Within-Subject Factors
• What's the effect of the treatment on a pattern
  of responses in sets of trials (e.g., the effect
  on fatigue)?

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Repeated Measures

• Two or more measurements (trials) per subject
• longitudinal or monitoring studies
• interventions or experiments

• Y
• Dependent variable
• Repeated measure

3
Repeated Measures Data for Mixed Modeling

• One row per subject per trial

Athlete
Group
Trial
Y
Chris
exptal
pre
66
Chris
exptal
mid
68
Chris
exptal
post
71
Sam
exptal
pre
74
Sam
exptal
mid
.
Sam
exptal
post
77
Jo
control
pre
71
Jo
control
mid
72
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Repeated Measures Fixed Effects in Mixed Modeling

• Fixed effects means.
• Fixed effects model without control group
  Y ? Trial
• This model estimates meansof Y for each level of
  Trial(e.g., Trialpre, Trialmid, Trialpost).
• Effect of the treatment means of Y for
  Trialpost Trialpre.
• Fixed-effects model with a control or other
  groups Y ? Group?Trial
• This model estimates means for each level of
  Group and Trial.
• Effect of treatment means of Y for Trialpost
  Trialpreexptal Trialpost Trialprecontrol.

Athlete
Group
Trial
Y
Chris
exptal
pre
66
Chris
exptal
mid
68
Chris
exptal
post
71
Sam
exptal
pre
74
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Repeated Measures Random Effects in Mixed
Modeling

• Random effects standard deviations (SDs).
• In the simplest repeated-measures model, there is
  one between-subject SD and one within-subject SD.

• In SAS, random Athlete specifies and estimates
  the pure between-subject SD and a residual error
  representing the within-subject SD (the same SD
  for each trial).
• Observed SD in any trial ?(between SD2 within
  SD2).

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Repeated Measures Data for ANOVA

• One row per subject

Measure "Y"within-subjects factor "Trial"

• You have to define which columns represent your
  within-subjects factor.
• The fixed-effects models are effectively the
  sameas for mixed models, but...
• You have less control over the random effects.
• If there is no control group, use a 1-way
  repeated-measures ANOVA (1 way Trial) or a
  paired t test.
• With a control group, use a 2-way (Trial, Group)
  repeated-measures ANOVA and analyze the
  interaction Group?Trial.

Ypre
Ymid
Ypost
66
68
71
74
.
77
71
72
72
64
64
63
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Repeated Measures Data for T Test

• Calculate the most interesting change scores

Athlete
Group
Ypre
Ymid
Ypost
Chris
exptal
66
68
71
Sam
exptal
74
.
77
Jo
control
71
72
72
Pat
control
64
64
63

• Use an unpaired t test to analyze the difference
  in the change score between exptal and control
  groups.
• Usually the post-pre change score, but
• can be for any parameter ("within-subject
  modelling").
• Missing values not a problem.
• More robust but less powerful than more complex
  analyses.

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Mechanism Variables

• Mechanism variable something in the causal path
  between the treatment and the dependent variable.
• Necessary but not sufficient that it "tracks" the
  dependent.

• Important for PhD projects or to publish in
  high-impact journals.
• It can put limits on a placebo effect, if it's
  not placebo affected.
• Can't use ANOVA can use graphs and mixed
  modeling.

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Mechanism Variables ANOVA?

• For ANOVA, data have to be one row per subject

• You can't use ANOVA, because it doesn't allow you
  to match up trials for the dependent and
  covariate.

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Mechanism Variables Graphical Analysis - 1

• Choose the most interesting change scoresfor the
  dependent and covariate

• Then plot the change scores

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Mechanism Variables Graphical Analysis - 2

• Three possible outcomes with a real mechanism
  variable

• The covariate is an excellent candidate for a
  mechanism variable.

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Mechanism Variables Graphical Analysis - 3

• Three possible outcomes with a real mechanism
  variable

2. Apparently poor tracking of individual
responses
could be due to noise in either variable.

• The covariate could still be a mechanism variable.

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Mechanism Variables Graphical Analysis - 4

• Three possible outcomes with a real mechanism
  variable

• The covariate is a good candidate for a mechanism
  variable.

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Mechanism Variables Graphical Analysis - 5

• Relationship between change scores is often
  misinterpreted.

?

• "The correlation between change scores for X and
  Y is trivial.
• Therefore X is not the mechanism."

• "Overall, changes in X track changes in Y well,
  but
• Noise may have obscured tracking of any
  individual responses.
• Therefore X could be a mechanism."

?
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Mechanism Variables Mixed Modeling Overview

• Need to quantify the role of the mechanism
  variable, with confidence limits.
• Mixed modeling with restricted maximum likelihood
  estimation does the job.
• Data format isone row per trial

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Mechanism Variables Mixed Modeling WITHOUT
Covariate

• To remind you
• Fixed effects ( means) model without control
  group Y ? Trial
• This model estimates meansof Y for each level of
  Trial(e.g., Trialpre, Trialmid, Trialpost).
• Effect of the treatment means of Y for
  Trialpost Trialpre.
• Fixed-effects model with a control or other
  groups Y ? Group?Trial
• This model estimates means for each level of
  Group and Trial.
• Effect of treatment means of Y for Trialpost
  Trialpreexptal Trialpost Trialprecontrol.

Athlete
Group
Trial
Y
Chris
exptal
pre
66
Chris
exptal
mid
68
Chris
exptal
post
71
Sam
exptal
pre
74
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Mechanism Variables Mixed Modeling WITH
Covariate

• Fixed-effects model Y ? Group?Trial X
• Estimates means for each level of Trial with X
  held constant.
• So, contrasts of interest derived from Trial
  represent effects of treatment not explained by
  putative mechanism variable.
• Example
• Effect of treatment from usual model Y ?
  Group?Trial4.6 units (95 likely limits, 2.1
  to 7.1 units).
• Effect of treatment from model Y ?
  Group?Trial X2.5 units (95 likely limits,
  -1.0 to 7.0 units).
• So, a little more than half the effect (2.5
  units) is not explained by X, but we need a
  larger sample or more reliable Y and/or X to
  reduce the uncertainty (-1.0 to 7.0 units).
• If changes in X can't be due to any placebo
  effect, the placebo effect is ?2.5 units.

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Mechanism Variables Random Effects in the Mixed
Model

• Simple random-effects ( standard deviations)
  model random Athlete
• In the Statistical Analysis System, this model
  specifies and estimates a pure between-subject SD
  and a residual error representing within-subject
  SD (the same SD for each trial).
• More complex model random Athlete
  Athlete?X
• This model implies X has a different effect for
  each subject.
• The coefficient of X in the fixed-effects model
  represents the mechanism effect averaged over all
  subjects.
• The SD from Athlete?X is the typical variation in
  this mechanism effect between subjects. Need gt2
  trials to estimate this SD.
• All with confidence limits, which you interpret,
  of course.

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Individual Responses

• Subjects may differ in their response to a
  treatment

• due to subject characteristics interacting
  with the treatment.
• It's important to measure and analyze their
  effect on the treatment.
• Use mixed modeling, ANOVA, or "within-subject
  modeling".
• Using Y for Trialpre as a characteristic needs a
  special approach to avoid artifactual regression
  to the mean. See newstats.org.

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Individual Responses Mixed Modeling - 1

• Data format is one row per trial

• Fixed-effects model
• without a control group Y ? Trial
  Trial?Covariate
• with a control group or other groups Y ?
  Group?Trial Group?Trial?Covariate

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Individual Responses Mixed Modeling - 2

• If Covariate is nominal (e.g., Sex),
  Group?Trial?Covariate represents different
  means for each level of Sex and Trial.
• Y for (Trialpost Trialpre)?Sexfemale
  (Trialpost Trialpre)?Sexmale difference
  between effect of treatment on females and males.
• Overall effects of treatment from Group?Trial
  represent effects for equal numbers of females
  and males, even if unequal in the study.
• If Covariate is numeric (e.g., Age),
  Group?Trial?Covariate represents different
  slopes for each level of Trial.
• Y for (Trialpost Trialpre)?Age10 increase in
  the effect per decade of age, e.g. 2.1
  units.10y-1.
• Overall effects of treatment from Group?Trial
  represent effects for subjects on the mean age.
• Random-effects model include special term to
  quantify individual responses before and after
  adding covariate. See newstats.org.

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Individual Responses Repeated-Measures ANOVA

• Data format is one row per subject

Within-subjectsfactor "Trial"
Group
Ypre
Ymid
Ypost
Athlete
exptal
66
68
71
Chris
exptal
74
75
77
Sam
control
71
72
72
Jo
control
64
64
63
Pat

• If no control group, use repeated-measures ANOVA
  (ANCOVA) and analyze the interaction
  Trial?Covariate.
• With a control group, analyze Group?Trial?Covariat
  e.

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Individual Responses Within-Subject Modeling

• Calculate the most interesting change scores or
  other within-subject parameters

• If no control group, analyze effect of Covariate
  on change score with unpaired t test, linear
  regression, or simple ANOVAs.
• With a control group, analyze effect of
  Group?Covariate on the change score with a simple
  ANOVA.
• Less powerful, more robust than mixed modeling or
  ANOVA.

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Two Within-Subject Factors

• sets of several measurements for each trial,
  e.g. 4 bouts

• We want to estimate the overall increase in Y in
  the exptal group in the mid and post trials, and
• the greater decline in Y in the exptal group
  within the mid and post trials (representing, for
  example, increased fatigue).
• Use mixed modeling, ANOVA, or within-subject
  modeling.

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Two Within-Subject Factors Mixed Modeling - 1

• Data format is one row per bout per trial

Athlete
Sex
Age
Group
Trial
Bout
Y
Chris
F
23
expt
pre
1
68
Chris
F
23
expt
pre
2
67
Chris
F
23
expt
pre
3
65
Chris
F
23
expt
pre
4
64
Chris
F
23
expt
mid
1
72
Chris
F
23
expt
mid
2
70
Chris
F
23
expt
mid
3
68
Chris
F
23
expt
mid
4
66
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Two Within-Subject Factors Mixed Modeling - 2

• Fixed-effects model
• Bout can be nominal (like Sex) or numeric (like
  Age).
• In the example, Bout is best modeled as a linear
  numeric effect. Polynomials are also possible.
• Model (without control group) Y ? Trial
  Trial?Bout
• Increase in the linear fatigue effect between
  Bouts 1 and 4 from pre to post Y for (Trialpost
  Trialpre)?Bout3.
• The change in Bout is 3 units.
• Overall increase in Y pre to post Y for
  Trialpost Trialpre (Trialpost
  Trialpre)?Bout2.5.
• The middle of each trial corresponds to Bout
  2.5.
• Add Trial?Covariate and Trial?Bout?Covariate to
  the model to explore individual responses.

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Two Within-Subject Factors Mixed Modeling - 3

• Random-effects models
• It takes time to get used to random-effects
  models!
• Simplest is random Athlete Athlete?Trial
• Athlete gives the pure between-subjects
  variation.
• The residual is the within-trial (between-bouts)
  error.
• Athlete?Trial gives the pure within-subject
  variation between trials. Add the residual
  variance to get observed between-bouts
  between-trials variation.
• If Bout is numeric, random Athlete
  Athlete?Bout Athlete?Trialimplies Bout has a
  different slope for each subject.
• The SD from Athlete?Bout is the typical variation
  in the slope between subjects.
• Other random effects are as above, sort of.

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Two Within-Subject Factors ANOVA Within-Subject
Modeling

• With sufficiently powerful ANOVA, you can specify
  two nominal within-subject effects and take into
  account various within-subject errors (using
  adjustments for asphericity).
• Specifying a linear or polynomial fatigue effect
  is possible but difficult (for me, anyway).
• Within-subject modeling is much easier.
• In the example, derive the Bout slope (or any
  other parameter) within each trial for each
  subject.
• Derive the change in the slope between pre and
  post for each subject.
• Do an unpaired t test for the difference in the
  changes between the exptal and control groups.
• Simple, robust, highly recommended!

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This presentation was downloaded from
A New View of Statistics
newstats.org
SUMMARIZING DATA
GENERALIZING TO A POPULATION
Simple Effect Statistics
Precision of Measurement
Confidence Limits
Statistical Models
Dimension Reduction
Sample-Size Estimation