1. Marginal and Conditional distributions

2. Theorem: (Marginal distributions for the Multivariate Normal distribution) have p-variate Normal distribution with mean vector and Covariance matrix Then the marginal distribution of is q i -variate Normal distribution (q 1 = q, q 2 = p - q) with mean vector and Covariance matrix

3. Theorem: (Conditional distributions for the Multivariate Normal distribution) have p-variate Normal distribution with mean vector and Covariance matrix Then the conditional distribution of given is q i -variate Normal distribution with mean vector and Covariance matrix

4. is called the matrix of partial variances and covariances. is called the partial covariance (variance if i = j) between x i and x j given x 1, …, x q. is called the partial correlation between x i and x j given x 1, …, x q.

5. is called the matrix of regression coefficients for predicting x q+1, x q+2, …, x p from x 1, …, x q. Mean vector of x q+1, x q+2, …, x p given x 1, …, x q is:

6. Example: Suppose that Is 4-variate normal with

7. The marginal distribution of is bivariate normal with The marginal distribution of is trivariate normal with

8. Find the conditional distribution of given Now and

10. The matrix of regression coefficients for predicting x 3, x 4 from x 1, x 2.

12. Thus the conditional distribution of given is bivariate Normal with mean vector And partial covariance matrix

13. Using SPSS Note: The use of another statistical package such as Minitab is similar to using SPSS

14. The first step is to input the data. The data is usually contained in some type of file. 1.Text files 2.Excel files 3.Other types of files

15. After starting the SSPS program the following dialogue box appears:

16. If you select Opening an existing file and press OK the following dialogue box appears

17. Once you selected the file and its type

18. The following dialogue box appears:

19. If the variable names are in the file ask it to read the names. If you do not specify the Range the program will identify the Range: Once you “click OK”, two windows will appear

20. A window containing the output

21. The other containing the data:

22. To perform any statistical Analysis select the Analyze menu:

23. To compute correlations select Correlate then Bivariate To compute partial correlations select Correlate then Partial

24. for Bivariate correlation the following dialogue appears

25. the output for Bivariate correlation:

26. for partial correlation the following dialogue appears

27. - - - P A R T I A L C O R R E L A T I O N C O E F F I C I E N T S - - - Controlling for.. AGE HT WT CHL ALB CA UA CHL 1.0000.1299.2957.2338 ( 0) ( 178) ( 178) ( 178) P=. P=.082 P=.000 P=.002 ALB.1299 1.0000.4778.1226 ( 178) ( 0) ( 178) ( 178) P=.082 P=. P=.000 P=.101 CA.2957.4778 1.0000.1737 ( 178) ( 178) ( 0) ( 178) P=.000 P=.000 P=. P=.020 UA.2338.1226.1737 1.0000 ( 178) ( 178) ( 178) ( 0) P=.002 P=.101 P=.020 P=. (Coefficient / (D.F.) / 2-tailed Significance) ". " is printed if a coefficient cannot be computed the output for partial correlation:

28. Compare these with the bivariate correlation:

29. CHL ALB CA UA CHL 1.0000.1299.2957.2338 ALB.1299 1.0000.4778.1226 CA.2957.4778 1.0000.1737 UA.2338.1226.1737 1.0000 Partial Correlations Bivariate Correlations

30. In the last example the bivariate and partial correlations were roughly in agreement. This is not necessarily the case in all stuations An Example: The following data was collected on the following three variables: 1.Age 2.Calcium Intake in diet (CAI) 3.Bone Mass density (BMI)

31. The data

32. Bivariate correlations

33. Partial correlations

34. Scatter plot CAI vs BMI (r = -0.447)

35. 25 35 45 55 65 75

36. 3D Plot Age, CAI and BMI

40. Transformations Theorem Let x 1, x 2,…, x n denote random variables with joint probability density function f(x 1, x 2,…, x n ) Let u 1 = h 1 (x 1, x 2,…, x n ). u 2 = h 2 (x 1, x 2,…, x n ). u n = h n (x 1, x 2,…, x n ).  define an invertible transformation from the x’s to the u’s

41. Then the joint probability density function of u 1, u 2,…, u n is given by: where Jacobian of the transformation

42. Example Suppose that x 1, x 2 are independent with density functions f 1 (x 1 ) and f 2 (x 2 ) Find the distribution of u 1 = x 1 + x 2 u 2 = x 1 - x 2 Solving for x 1 and x 2 we get the inverse transformation

43. The Jacobian of the transformation

44. The joint density of x 1, x 2 is f(x 1, x 2 ) = f 1 (x 1 ) f 2 (x 2 ) Hence the joint density of u 1 and u 2 is:

45. Theorem Let x 1, x 2,…, x n denote random variables with joint probability density function f(x 1, x 2,…, x n ) Let u 1 = a 11 x 1 + a 12 x 2 +…+ a 1n x n + c 1 u 2 = a 21 x 1 + a 22 x 2 +…+ a 2n x n + c 2 u n = a n1 x 1 + a n2 x 2 +…+ a nn x n + c n  define an invertible linear transformation from the x’s to the u’s

46. Then the joint probability density function of u 1, u 2,…, u n is given by: where

47. Theorem Suppose that The random vector, [x 1, x 2, … x p ] has a p-variate normal distribution with mean vector and covariance matrix  then has a p-variate normal distribution with mean vector and covariance matrix

48. Theorem Suppose that The random vector, [x 1, x 2, … x p ] has a p-variate normal distribution with mean vector and covariance matrix  then has a p-variate normal distribution with mean vector and covariance matrix

49. Proof then

50. since Also and hence QED

51. Theorem Suppose that The random vector, has a p-variate normal distribution with mean vector and covariance matrix  with mean vector and covariance matrix then has a p-variate normal distribution Let A be a q  p matrix of rank q ≤ p

52. proof then is invertible. and covariance matrix Let B be a (p - q)  p matrix so that is p–variate normal with mean vector

53. Thus the marginal distribution of and covariance matrix is q–variate normal with mean vector