1 / 41

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.

hhunt
Download Presentation

Profile Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related