Multivariate distributions

1. Multivariate distributions

2. The Normal distribution

3. Normal distribution with m = 50 and s =15 Normal distribution with m = 70 and s =20 1.The Normal distribution – parameters mands(or s2) Comment:If m = 0 and s = 1 the distribution is called the standard normal distribution

4. The probability density of the normal distribution If a random variable, X, has a normal distribution with mean mand variance s2 then we will write:

5. The multivariate Normal distribution

6. Let = a random vector Let = a vector of constants (the mean vector)

7. Let = a p ×p positive definite matrix

8. Definition The matrix Ais positive semi definite if Further the matrix Ais positive definite if

9. Suppose that the joint density of the random vector The random vector, [x1, x2, … xp]is said to have a p-variate normal distribution with mean vector and covariance matrix S We will write:

10. Example: the Bivariate Normal distribution with and

11. Now and

13. Hence where

14. Note: is constant when is constant. This is true when x1, x2 lie on an ellipse centered at m1, m2 .

16. Surface Plots of the bivariate Normal distribution

17. Contour Plots of the bivariate Normal distribution

18. Scatter Plots of data from the bivariate Normal distribution

19. Trivariate Normal distribution - Contour map x3 mean vector x2 x1

20. Trivariate Normal distribution x3 x2 x1

21. Trivariate Normal distribution x3 x2 x1

22. Trivariate Normal distribution x3 x2 x1

23. example In the following study data was collected for a sample of n = 183 females on the variables • Age, • Height (Ht), • Weight (Wt), • Birth control pill use (Bpl - 1=no pill, 2=pill) • and the following Blood Chemistry measurements • Cholesterol (Chl), • Albumin (Abl), • Calcium (Ca) and • Uric Acid (UA). The data are tabulated next page:

24. The data :

26. Alb, Chl, Bp

27. Marginal and Conditional distributions

28. 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 qi-variate Normal distribution (q1 = q, q2 = p - q) with mean vector and Covariance matrix

29. 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 qi-variate Normal distribution with mean vector and Covariance matrix

30. Proof: (of Previous two theorems) is The joint density of , and where

31. where , and

32. also and ,

33. ,

35. The marginal distribution of is

36. The conditional distribution of given is:

37. is called the matrix of partial variances and covariances. is called the partial covariance (variance if i = j) between xi and xj given x1, … , xq. is called the partial correlation between xi and xj given x1, … , xq.

38. is called the matrix of regression coefficients for predicting xq+1, xq+2,… , xpfrom x1, … , xq. Mean vector of xq+1, xq+2,… , xpgiven x1, … , xqis:

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

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

41. Find the conditional distribution of given Now and

43. The matrix of regression coefficients for predicting x3, x4from x1, x2.

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

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

47. The first step is to input the data. The data is usually contained in some type of file. • Text files • Excel files • Other types of files

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

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

50. Once you selected the file and its type