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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 Hopkins Auckland University of Technology Auckland, NZ

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

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Repeated Measures: Data for Mixed Modeling One row per subject per trial: AthleteGroupTrialY Chrisexptalpre66 Chrisexptalmid68 Chrisexptalpost71 Samexptalpre74 Samexptalmid. Samexptalpost77 Jocontrolpre71 Jocontrolmid72 Missing value means loss of only one trial for the subject.

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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 means of Y for each level of Trial (e.g., Trial pre, Trial mid, Trial post ). Effect of the treatment = means of Y for Trial post – Trial pre. 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 [Trial post – Trial pre ] exptal – [Trial post – Trial pre ] control. AthleteGroupTrialY Chrisexptalpre66 Chrisexptalmid68 Chrisexptalpost71 Samexptalpre74

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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. Y Trial 1 Trial 2Trial 3 The between-subject SD represents typical variation in the true score between subjects. The within-subject SD represents typical variation in a subject's score between trials. 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 SD 2 + within SD 2 ).

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Repeated Measures: Data for ANOVA One row per subject: You have to define which columns represent your within-subjects factor. The fixed-effects models are effectively the same as 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. YpreYmidYpost 666871 74.77 7172 AthleteGroup Chrisexptal Samexptal Jocontrol Patcontrol 64 63 Measure = "Y" within-subjects factor = "Trial" Missing value means loss of subject.

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Repeated Measures: Data for T Test Calculate the most interesting change scores: AthleteGroupYpreYmidYpost Chrisexptal666871 Samexptal74.77 Jocontrol7172 Patcontrol64 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. 1 Ypost -Ypre 5 3

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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. Dependent variable pre midpost exptal control Mechanism variable pre midpost exptal control Trial 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: Mechanism variable = within-subjects covariate XpreXmidXpost 8.48.79.1 9.09.29.7 7.97.77.8 7.1 7.2 You can't use ANOVA, because it doesn't allow you to match up trials for the dependent and covariate. AthleteGroupYpreYmidYpost Chrisexptal666871 Samexptal747577 Jocontrol7172 Patcontrol64 63 Measure = "Y" within-subjects factor = "Trial"

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Mechanism Variables: Graphical Analysis - 1 Choose the most interesting change scores for the dependent and covariate: Then plot the change scores… AthleteGroupYpreYmidYpostXpreXmidXpost Chrisexptal6668718.48.79.1 Samexptal7475779.09.29.7 Jocontrol7172 7.97.77.8 Patcontrol64 637.1 7.2 1.5 0.7 -0.1 0.1 Xpost -Xpre Change score for covariate 5 3 1 Ypost -Ypre Change score for dependent

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

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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. Ypost - Ypre Xpost - Xpre 0 0 3. Little or no individual responses… …but mechanism variable tracks mean response.

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Relationship between change scores is often misinterpreted. Ypost - Ypre Xpost - Xpre 0 0 Mechanism Variables: Graphical Analysis - 5 0 "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." "The correlation between change scores for X and Y is trivial. Therefore X is not the 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 is one row per trial: AthleteGroupTrialY Chrisexptalpre66 Chrisexptalmid68 Chrisexptalpost71 Samexptalpre74 X 8.4 8.7 9.1 9.0 Mechanism variable = within-subjects covariate No problem with aligning trials for the dependent and covariate.

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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 means of Y for each level of Trial (e.g., Trial pre, Trial mid, Trial post ). Effect of the treatment = means of Y for Trial post – Trial pre. 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 [Trial post – Trial pre ] exptal – [Trial post – Trial pre ] control. AthleteGroupTrialY Chrisexptalpre66 Chrisexptalmid68 Chrisexptalpost71 Samexptalpre74

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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 ] 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.

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

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Subjects may differ in their response to a treatment… Individual Responses …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 Trial pre as a characteristic needs a special approach to avoid artifactual regression to the mean. See newstats.org. Y Trial pre midpost Data are values for individuals pre midpost boys girls

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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 GroupTrialY exptpre66 exptmid68 exptpost71 exptpre74 Athlete Chris Sam Athlete Chris Sam SexAge F23 F F M19 Subject characteristics = between-subject covariates

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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 (Trial post – Trial pre ) Sex female – (Trial post – Trial pre ) Sex male = 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 (Trial post – Trial pre ) Age 10 = 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: If no control group, use repeated-measures ANOVA (ANCOVA) and analyze the interaction Trial Covariate. With a control group, analyze Group Trial Covariate. GroupYpreYmidYpost exptal666871 exptal747577 control7172 control64 63 Within-subjects factor = "Trial" Athlete Chris Sam Jo Pat Athlete Chris Sam Jo Pat SexAge F23 M19 F M Covariates

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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. AthleteSexAgeGroupYpreYmidYpost ChrisF23exptal666871 SamM19exptal747577 JoF19control7172 PatM19control64 63 1 Ypost -Ypre 5 3

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= sets of several measurements for each trial, e.g. 4 bouts: Y pre midpost control Trial exptal Two Within-Subject Factors 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. Within Subject between Trials Within Subject within Trial Between Subjects within Bout Standard deviations: 1 2 3 4 Bout

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Two Within-Subject Factors: Mixed Modeling - 1 Data format is one row per bout per trial: AthleteSexAgeGroupTrialBout ChrisF23exptpre1 ChrisF23exptpre2 ChrisF23exptpre3 ChrisF23exptpre4 Y 68 67 65 64 ChrisF23exptmid1 ChrisF23exptmid2 ChrisF23exptmid3 ChrisF23exptmid4 72 70 68 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 (Trial post – Trial pre ) Bout 3. The change in Bout is 3 units. Overall increase in Y pre to post = Y for Trial post – Trial pre + (Trial post – Trial pre ) Bout 2.5. 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 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.

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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 SUMMARIZING DATA GENERALIZING TO A POPULATION Simple & Effect Statistics Precision of Measurement Precision of Measurement Confidence Limits Statistical Models Statistical Models Dimension Reduction Dimension Reduction Sample-Size Estimation Sample-Size Estimation newstats.org

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