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Covariates in Repeated-Measures Analyses

Covariates in Repeated-Measures Analyses. 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?

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Covariates in Repeated-Measures Analyses

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  1. Covariates in Repeated-Measures Analyses • 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)? Will G HopkinsAuckland University of TechnologyAuckland, NZ

  2. Period oftreatment exptal Data are means and standard deviations control pre mid post Group • Between-subjects factor • Different subjects on each level Trial • Within-subjects factor • Same subjects on each level Repeated Measures • Two or more measurements (trials) per subject • longitudinal or monitoring studies • interventions or experiments: Y • Dependent variable • Repeated measure

  3. Missing valuemeans loss ofonly one trial for the subject. 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

  4. 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 – Trialpre]exptal – [Trialpost – Trialpre]control. Athlete Group Trial Y Chris exptal pre 66 Chris exptal mid 68 Chris exptal post 71 Sam exptal pre 74

  5. The between-subject SD represents typical variation in the true score between subjects. Y • The within-subject SD represents typical variation in a subject's score between trials. Trial1 Trial2 Trial3 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).

  6. Athlete Group Chris exptal Sam exptal Jo control Pat control Missing value means loss of subject. Repeated Measures: Data for ANOVA Measure = "Y"within-subjects factor = "Trial" • One row per subject: • 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

  7. Ypost-Ypre 5 3 1 -1 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.

  8. Mechanism variable Dependent variable exptal exptal control control pre mid post pre mid post Trial 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.

  9. Measure = "Y"within-subjects factor = "Trial" Mechanism variable =within-subjects covariate Athlete Group Ypre Ymid Ypost Xpre Xmid Xpost Chris exptal 66 68 71 8.4 8.7 9.1 Sam exptal 74 75 77 9.0 9.2 9.7 Jo control 71 72 72 7.9 7.7 7.8 Pat control 64 64 63 7.1 7.1 7.2 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.

  10. Change scorefor dependent Ypost-Ypre Xpost-Xpre Athlete Group Ypre Ymid Ypost Xpre Xmid Xpost Chris exptal 66 68 71 8.4 8.7 9.1 5 1.5 Sam exptal 74 75 77 9.0 9.2 9.7 3 0.7 Jo control 71 72 72 7.9 7.7 7.8 1 -0.1 Pat control 64 64 63 7.1 7.1 7.2 -1 0.1 Change scorefor covariate Mechanism Variables: Graphical Analysis - 1 • Choose the most interesting change scoresfor the dependent and covariate: • Then plot the change scores…

  11. 1. Large individual responses… …tracked by mechanism variable… …even in the control group. exptal Ypost - Ypre 0 control 0 Xpost - Xpre Mechanism Variables: Graphical Analysis - 2 • Three possible outcomes with a real mechanism variable: • The covariate is an excellent candidate for a mechanism variable.

  12. Ypost - Ypre 0 0 Xpost - Xpre 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.

  13. 3. Little or no individual responses… …but mechanism variable tracks mean response. Ypost - Ypre 0 0 Xpost - Xpre Mechanism Variables: Graphical Analysis - 4 • Three possible outcomes with a real mechanism variable: • The covariate is a good candidate for a mechanism variable.

  14. Ypost - Ypre 0 0 0 Xpost - Xpre 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." 

  15. Mechanism variable =within-subjects covariate Athlete Group Trial Y X Chris exptal pre 66 8.4 Chris exptal mid 68 8.7 Chris exptal post 71 9.1 Sam exptal pre 74 9.0 • No problem with aligning trials for the dependent and covariate. 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:

  16. 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 forTrialpost – 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 – Trialpre]exptal – [Trialpost – Trialpre]control. Athlete Group Trial Y Chris exptal pre 66 Chris exptal mid 68 Chris exptal post 71 Sam exptal pre 74

  17. 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]Trial:4.6 units (95% likely limits, 2.1 to 7.1 units). • Effect of treatment from model Y [Group]Trial X:2.5 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.

  18. 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 >2 trials to estimate this SD. • All with confidence limits, which you interpret, of course.

  19. boys girls Y Data arevalues forindividuals pre mid post pre mid post Trial 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.

  20. Subject characteristics = between-subject covariates Athlete Athlete Sex Age Group Trial Y Chris Chris F 23 expt pre 66 Chris Chris F 23 expt mid 68 Chris Chris F 23 expt post 71 Sam Sam M 19 expt pre 74 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

  21. 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.

  22. Covariates Athlete Sex Age Chris F 23 Sam M 19 Jo F 19 Pat M 19 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Covariate.

  23. Ypost-Ypre Athlete Sex Age Group Ypre Ymid Ypost Chris F 23 exptal 66 68 71 5 Sam M 19 exptal 74 75 77 3 Jo F 19 control 71 72 72 1 Pat M 19 control 64 64 63 -1 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.

  24. 1 2 Bout 3 exptal 4 Y Standard deviations: Between Subjects within Bout control Within Subject between Trials pre mid post Within Subject within Trial Trial 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.

  25. 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

  26. 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.

  27. 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Trial;implies 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.

  28. 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!

  29. 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

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