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Chapter 15 Repeated Measures Designs 1 Analysis of Correlated Measures Repeated Measures Section 15.1 pp 492-495 data a; do Treatment = "Amiodarone","Vehicle","Saline"; do Rabbit = 1,2,3,4,5; do Time = 0,30,60,90; input EarTempDiff @@; output; end; end; end; cards; -.3 -.2 1.2 3.1 -.5 2.2 3.3 3.7 -1.1 2.4 2.2 2.7 1.0 1.7 2.1 2.5 -.3 .8 .6 .9 -1.1 -2.2 .2 .3 -1.4 -.2 -.5 -.1 -.1 -.1 -.5 -.3 -.2 .1 -.2 .4 -.1 -.2 .7 -.3 -1.8 .2 .1 .6 -.5 0 1 .5 -1 -.3 -2.1 .6 .4 .4 -.7 -.3 -.5 .9 -.4 -.3 ;;;; A profile plot is a good start for this. Proc sort; by Treatment Time; Proc means data=a noprint mean; var EarTempDiff; output out=junk mean=mean; by Treatment Time; run; Proc Plot data=junk hpercent=25 vpercent=25 ; Plot Mean*Time = "*" /vaxis = -1 to 3 by 1 ; by Treatment; run;

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Chapter 15 Repeated Measures Designs 2 Early Detection of Phlebitis in Amiodarone Therapy Treatment=Amiodarone Plot of mean*Time='*'. mean 3 ˆ * 2 ˆ * * 1 ˆ 0 ˆ * -1 ˆ Šˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ 0 30 60 90 Time Treatment=Saline 2 ˆ * * * * Time Treatment=Vehicle 0 ˆ * * * * Time Experimental units are subjects Treatments are between subject factors Time is the within subject factor
Chapter 15 Repeated Measures Designs 3 Relationships Among Repeated Measurements Section 15.2 pp 495-498 Dividing covariance by the product of the standard deviations puts the correlation coefficient between ±1. The covariance is a measure of how two variables will vary together. The variance is a particular type of covariance.

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Chapter 15 Repeated Measures Designs 4 The theoretical variances and covariances for repeated measures taken 12 3 4 successively as y ,y , y and y can be shown as this 4 X 4 Ó matrix. SAS calls this the unstructured structure and it says that all of the variances and covariances are unique. 1234 yyyy 1 1 12 13 14 y FFFF 2 2 21 2 23 24 y 2 3 31 32 3 34 y 2 4 41 42 43 4 y 2 There are three types of covariance structure (i.e. sets of assumptions) that will allow for a univariate ANOVA for repeated measures. Independent, Compound Symetric, Huynh-Feldt condition.
Chapter 15 Repeated Measures Designs 5 Independent (normal assumptions for ANOVA) 1234 yyyy 1 y F 000 2 2 y0 F 00 2 3 y00 F 0 2 4 y000 F 2 Compound Symmetry (assumption for split-plot). This structure specifies that measures at all times have the same variance and that all pairs of measures on the same animal have the same correlation. 1 y F DF DF DF 2 222 2 y DF F DF DF 22 2 2 3 y DF DF F DF 222 2 4 y DF DF DF F 2222

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Chapter 15 Repeated Measures Designs 6 Huynh and Feldt showed that the compound symmetric structure was not essential for repeated measures, but rather only that there is the same variance of the difference for all pairs of observations taken at different time periods ij say y and y Huynh-Feldt or Type H Matrix 1234 yyyy 1 y 2 y 3 y 4 y
Chapter 15 Repeated Measures Designs 7 proc print data=a; run; Obs Treatment Rabbit Time EarTempDiff 1 Amiodarone 1 0 -0.3 2 Amiodarone 2 0 -0.5 3 Amiodarone 3 0 -1.1 . . . . .

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