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UT Knoxville STAT 201 - 12) repeated_multivar_1factor

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SAS Code for Multivariate Repeated Measures ANOVA Using the Repeated StatementSPSS Code for Multivariate Repeated Measures ANOVA Using the Repeated StatementSAS OUTPUT FOR THE MULTIVARIATE REPEATED MEASURES ANOVA OF THE SHOE EFFECTSPSS OUTPUT FOR THE MULTIVARIATE REPEATED MEASURES ANOVA OF THE SHOE EFFECTSAS Code for Testing Contrasts with MANOVA StatementSPSS Code for Testing Contrasts with MANOVA StatementSAS OUTPUT FOR 2 CONTRASTS WITH SEPARATE ERROR TERMS USING THE MANOVA STATEMENTSPSS OUTPUT FOR 2 CONTRASTS WITH SEPARATE ERROR TERMS USING THE MANOVA STATEMENTCourse: Analysis of Variance Topic: Repeated-Measures: Mixed-Model 1 One-Factor Repeated Measures:The Multivariate ApproachPreviously we discussed the univariate or mixed-model approach to analyzing within-subject data. In the next two lectures we will examine the multivariate approach to within subjectdata. Recall that the problem posed by within-subject data is the lack of independence produced by taking repeated measurements on the same person (or unit of observation). The multivariate approach handles the dependence in the data by taking into account the degree of correlation (or overlap, or dependence) in the repeated measures and then removing the correlated (or overlapping, or dependent) information from the data. In essence, the multivariate approach calculates and analyzes the total amount of unique (i.e., independent) variability that exists across the repeated measures. In the current lecture we will examine a one-factor design and in the following lecture we will examine the multivariate approach to factorial designs. To understand the necessary concepts involved in the multivariate approach we will start with a two-level factor and then a three-level factor. The concepts underlying the analysis of n-level designs are a generalization of the 2 and 3-level designs.ANALYSIS OF A TWO-LEVEL WITHIN-SUBJECT FACTOR: THE DIFFERENCE SCORELet’s continue to use the sneaker example, which we developed previously, to examine the multivariate concept of a difference score. The sneaker example involves a track coach who is trying to determine which brand sneaker (A, B, or C) would be best for his/her team. For the current example, assume that the coach is deciding between only Sneaker B and C. To test whichsneaker is better, the coach uses a within-subjects design and records how quickly six persons run 100 yards with each sneaker. The following table contains the running time (in seconds) for each subject and sneaker.SneakerSubject B C MarginalMeans1 18 10 16.002 14 8 11.333 19 14 17.004 13 10 13.005 19 15 18.336 15 11 13.67Marginal Means 16.33 11.33 14.89We know that we cannot conduct a one-factor ANOVA (or an independent sample t-test) that simply compares the average running times for the two sneakers because the running times for sneakers B and C are not independent. (Keep in mind that the data within a particular subject is dependent not the data between the subjects. That is, it is assumed that the responses across participants are independent of one another) Recall that the univariate or (mixed-model) approach handles the dependence in the data by treating subject as a random factor in the design. An alternative approach, which serves as the basis of the multivariate approach, is to transform the repeated measures in such a way to remove the dependence.Course: Analysis of Variance Topic: Repeated-Measures: Mixed-Model 2 The transformation used by the multivariate approach is the difference score. That is, for each person we could subtract running time with shoe B from running time with shoe C. We would then be left with one score for each person that reflects the difference between the two sneakers. The following table demonstrates the difference score transformation.Sneaker DifferenceSubject B C (C-B)1 18 10 -82 14 8 -63 19 14 -54 13 10 -35 19 15 -46 15 11 -4Marginal Means -5The last column of the above table contains the difference score for each participant. This difference score reflects the difference in running times between sneakers C and B. Notice that negative scores indicate that a given person ran faster in sneaker C than B (e.g., the running time for subject 3 was 5 seconds faster with sneaker C than B). A negative score indicates that a given person ran faster in sneaker B than C. And, a score of 0 indicates that there was no difference in running time between the two sneakers. The average difference score for the 6 participants (M = –5) indicates that on average persons ran the 100 yards five seconds faster wearing sneaker C than sneaker B.The important aspect of the difference score is that we now have one score for each participant. Because the scores across participants are independent (i.e., running time for subject 1 is independent of subject 2), we can use the difference score as the dependent variable in a one factor ANOVA (or t-test). The null and alternative hypotheses for such an analysis are:0:0DH0:1DHAccording to the null hypothesis, the average population difference score equal zero (i.e., there isno difference between sneakers B and C). According to the alternative hypothesis, the average population difference score is not zero (i.e., there is a difference between sneakers B and C). The full and restricted models for the test of the null hypothesis are: Full:iDiDRestricted:iiD0Notice that error for the full model is simply the difference between a given persons score and the average difference score (DiiD) and error for the restricted model is simply a given persons score (iiD). We can calculate the total amount of error for each model by estimating the population difference score (D) with the average sample difference score (D) and then summing the squared errors (i.e., SS) for each person: SSfull = EF =2)( DDiSSrestricted = ER =2)(iDCourse: Analysis of Variance Topic: Repeated-Measures: Mixed-Model 3 We can then calculate the F-value by plugging the above values into our general formula for the F-ratio:FFFRFRdfEdfdfEEF)()(Because we need to estimate one parameter in the full model (i.e., D) we loose 1 degree of freedom, so dfF = n-1. Where as dfR = n. So, we were able to analyze a two-level within-subjects factor as a standard one-factor ANOVA by transforming the data with a difference score. This difference score transformation (D) produced one score


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UT Knoxville STAT 201 - 12) repeated_multivar_1factor

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