Course Analysis of Variance Topic Repeated Measures Mixed Model 1 One Factor Repeated Measures The Multivariate Approach Previously we discussed the univariate or mixed model approach to analyzing withinsubject data In the next two lectures we will examine the multivariate approach to within subject data 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 twolevel 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 SCORE Let 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 which sneaker 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 Sneaker Subject B C 1 2 3 4 5 6 Marginal Means 18 14 19 13 19 15 16 33 10 8 14 10 15 11 11 33 Marginal Means 16 00 11 33 17 00 13 00 18 33 13 67 14 89 We 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 Subject 1 2 3 4 5 6 Marginal Means B 18 14 19 13 19 15 C 10 8 14 10 15 11 Difference C B 8 6 5 3 4 4 5 The 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 H 0 D 0 H 1 D 0 According to the null hypothesis the average population difference score equal zero i e there is no 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 Di D i Restricted Di 0 i Notice that error for the full model is simply the difference between a given persons score and the average difference score i Di D and error for the restricted model is simply a given persons score i Di 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 2 SSfull EF Di D 2 SSrestricted ER Di Course 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 ER EF F EF df R df F df F 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 for each participant Because each participant has only one difference score and the difference scores across participants are independent we no longer have a problem of independence Obviously not all within subjects variables are restricted to two levels When analyzing a within subject variable that has more than 2 levels the D score remains an integral aspect of the multivariate approach ANALYSIS OF A THREE LEVEL WITHIN SUBJECT FACTOR MULTIPLE DIFFERENCE SCORES Our discussion of a two level within subject factor introduced the difference score Dscore transformation Our discussion of a 3 level within subjects factor will provide all of the necessary concepts to the multivariate approach which generalize to n level factors To facilitate the discussion let s again return to our sneaker example
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