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Profile Analysis

Profile Analysis. Definition Let X 1 , X 2 , … , X p denote p jointly distributed variables under study Let m 1 , m 2 , … , m p denote the means of these variables s denote the means these variables The profile of these variables is a plot of m i vs i. m i. i.

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Profile Analysis

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  1. Profile Analysis

  2. Definition • Let X1, X2, … , Xp denote p jointly distributed variables under study • Let m1, m2, … , mpdenote the means of these variables s denote the means these variables • The profile of these variables is a plot of mi vs i. • mi • i

  3. The multivariate Test Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix S. Let denote a sample of m from the p-variate normal distribution with mean vector and covariance matrix S. Suppose we want to test

  4. Hotelling’s T2 statisticfor the two sample problem if H0 is true than has an F distribution with n1= p and n2= n +m – p - 1

  5. Profile Comparison X Group A Group B p … 1 2 3 variables

  6. Hotelling’s T2 test, tests against

  7. Profile Analysis

  8. Parallelism

  9. Variables not interacting with groups(parallelism) X groups … p 1 2 3 variables

  10. Variables interacting with groups(lack of parallelism) X groups p … 1 2 3 variables

  11. Parallelism • Group differences are constant across variables Lack of Parallelism • Group differences are variable dependent • The differences between groups is not the same for each variable

  12. Test for parallelism

  13. Let denote a sample of n from the p-variate normal distribution with mean vector and covariance matrix S. Let denote a sample of m from the p-variate normal distribution with mean vector and covariance matrix S.

  14. Let Then

  15. The test for parallelism is Consider the data This is a sample of n from the (p -1) -variate normal distribution with mean vector and covariance matrix . Also is a sample of m from the (p -1) -variate normal distribution with mean vector and covariance matrix .

  16. Hotelling’s T2 test for parallelism if H0 is true than has an F distribution with n1= p – 1 and n2= n +m – p Thus we reject H0 if F > Fawith n1= p – 1 and n2= n +m – p

  17. To perform the test for parallelism, compute differences of successive variables for each case in each group and perform the two-sample Hotelling’s T2 test.

  18. Test for Equality of Groups (Parallelism assumed)

  19. Groups equal X groups … p 1 2 3 variables

  20. If parallelism is proven: It is appropriate to test for equality of profiles i.e.

  21. The t test Thus we reject H0 if |t|> ta/2with df = n= n +m - 2 To perform this test, average all the variables for each case in each group and perform the two-sample t-test.

  22. Test for equality of variables (Parallelism Assumed)

  23. Variables equal X groups i … 1 2 3 variables

  24. Let Then

  25. The test for equality of variables for the first group is: Consider the data This is a sample of n from the p-variate normal distribution with mean vector and covariance matrix .

  26. Hotelling’s T2 test for equality of variables if H0 is true than has an F distribution with n1= p – 1 and n2= n - p + 1 Thus we reject H0 if F > Fawith n1= p – 1 and n2= n – p + 1

  27. To perform the test, compute differences of successive variables for each case in the group and perform the one-sample Hotelling’s T2 test for a zero mean vector A similar test can be performed for the second sample. Both of these tests do not assume parllelism.

  28. If parallelism is assumed then Then This is a sample of n + m from the p-variate normal distribution with mean vector and covariance matrix . The test for equality of variables is:

  29. Hotelling’s T2 test for equality of variables if H0 is true than has an F distribution with n1= p – 1 and n2= n +m - p Thus we reject H0 if F > Fawith n1= p – 1 and n2= n + m – p

  30. To perform this test for parallelism, • Compute differences of successive variables for each case in each group • Combine the two samples into a single sample of n + m and • Perform the single-sample Hotelling’s T2 test for a zero mean vector.

  31. Example • Two groups of Elderly males • Groups • Males identified with no senile factor • Males identified with a senile factor • Variables – Scores on WAIS (intelligence) test • Information • Similarities • Arithmetic • Picture completion

  32. Summary Statistics

  33. Hotellings T2 test (2 sample) H0 :equal means, is rejected

  34. Profile Analysis

  35. Hotelling’s T2 test for parallelism Decision: Accept H0 :parallelism

  36. The t test for equality of groups assuming parallelism Thus we reject H0 if t > tawith df = n= n +m - 2 = 47

  37. Hotelling’s T2 test for equality of variables Thus we reject H0 if F > Fawith n1= p – 1= 3 and n2= n + m – p = 45 F0.05= 6.50 if n1 = 3 and n2= 45

  38. Example 2: Profile Analysis for Manova In the following study, n = 15 first year university students from three different School regions (A, B and C) who were each taking the following four courses (Math, biology, English and Sociology) were observed: The marks on these courses is tabulated on the following slide:

  39. The data

  40. Summary Statistics

  41. Repeated Measures

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