1 Introduction to Biostatistics (PUBHLTH 540) Multiple Random Variables

2 Multiple Random Variables Linear Combinations of Random Variables Linear Combinations of Random Variables –Expected Value –Variance Stochastic Models Stochastic Models Covariance of two Random Variables Covariance of two Random Variables Independence Independence Correlation Correlation

3 SPH&HS, UMASS Amherst 3 An Example Choose a Simple Random Sample with Replacement of size n=2 from a Population of N=3 Choose a Simple Random Sample with Replacement of size n=2 from a Population of N=3 Observe: Observe: –1 Response (i.e. Age) on each Subject in the Sample Question: Question: –What is the average age of subjects in the population? Use the sample mean to estimate the Population Average Age Use the sample mean to estimate the Population Average Age DaisyLilyRose Introducing….

4 SPH&HS, UMASS Amherst 4 Population

5 Population of N=3 Note: Population mean Variance. ID (s) Subject Response (Age) 1Daisy25 2Lily32 3Rose33

6 Pick SRS with Replacement of n=2 a random variable representing the 1 st selection ID (s) SubjectResponse 1Daisy25 2Lily32 3Rose33 i=1,…,n=2 a random variable representing the 2nd selection

7 Use as an Estimator: Sample Mean A Linear Estimator- a sum of random variables When n=2,

8 Linear Combination of Random Variables Example: Sample Mean

9 Models for Response Models for Response 3 (=N) 2 1 ID (s) Lily Rose DaisyResponseSubject Non-Stochastic model (Deterministic) Stochastic model

10 SPH&HS, UMASS Amherst 10 Finite Population Pick a SRS with replacement of size n=2 Stochastic model

11 SPH&HS, UMASS Amherst 11 Finite Population with replacement Stochastic model

12 SPH&HS, UMASS Amherst 12 Finite Population with replacement Stochastic model

13 SPH&HS, UMASS Amherst 13 Sampling- n=2 with replacement Random Variables Linear Combination of Random Variables Stochastic model

14 SPH&HS, UMASS Amherst 14 Sampling- n=2 with replacement Realized Values

15 SPH&HS, UMASS Amherst 15 Other Possible Samples with replacement

16 SPH&HS, UMASS Amherst 16 Other Possible Samples with replacement

17 Sample (t)Probability 11/925 21/ / / /932 61/ / /93332 All Possible Samples

18 Sample (t)Probability 11/ / / / / / / / / Expected Values

19 Sample (t)Probability 11/ / / / / / / / /

20 Sample (t)Probability 11/ / / / / / / / /

21 Covariance of Two Random Variables

22 Sample (t)Probability 11/ / / / / / / / / Based on simple random sampling with replacement

23 Variance Matrix When n=2, and SRS with replacement: When n=2, and SRS with replacement: Identity Matrix

24 Variance Matrix for n Random Variables

25 Covariance of Random Variables When SRS without Replacment (n=2) Sample (t)Probability 11/ / / / / /

26 Covariance of two random variables when sampling without replacement

27 Estimating the Covariance Estimate the variance: assuming srs assuming srs Estimate the covariance: assuming srs assuming srs

28 Independence Two random variables, Y and Z are independent if Two random variables, Y and Z are independent if P(Y=y|Z=z)=P(Y=y) P(Y=y|Z=z)=P(Y=y) P(Y=y|Z=z) means the probability that Y has a value of y, given Z has a value of z (see Text, sections 6.1 and 6.2)

29 Example: SRS with rep n=2 Areandindependent? Does? ID (s) SubjectResponse1Daisy25 2Lily32 3Rose33

30 SPH&HS, UMASS Amherst 30 Sampling n=2 (with rep) Areandindependent? Yes

31 SPH&HS, UMASS Amherst 31 Sampling n=2 (with rep) Areandindependent? Yes

32 SPH&HS, UMASS Amherst 32 Sampling n=2 (with rep) Areandindependent? Yes

33 Example: SRS without rep n=2 Areandindependent? Does? ID (s) SubjectResponse1Daisy25 2Lily32 3Rose33

34 SPH&HS, UMASS Amherst 34 Sampling n=2 (without replacement) Areandindependent? No

35 SPH&HS, UMASS Amherst 35 Areandindependent? No Sampling n=2 (without replacement)

36 SPH&HS, UMASS Amherst 36 Sampling n=2 (without replacement) Areandindependent? No

37 Relationship between Independence and Covariance If two random variables are independent, then their covariance is 0. If two random variables are independent, then their covariance is 0. If the covariance of two random variables is zero, the two may (or may not) be independent If the covariance of two random variables is zero, the two may (or may not) be independent

38 Expected Value of a Linear Combination of Random Variables Write linear combinations using vector notation. Write linear combinations using vector notation. Constants Random variables

39 where Example: SRS of size n:

40 Example 2: Suppose two independent SRS w/o replacement are selected from populations of boy and girl babies, and the weight recorded. Let us represent the boy weight by Y and the girl weight by X. Suppose sample results are given as follows: Boysn=25Girlsn=40 Sample Mean Variance An estimate is wanted of the average birth weight in Europe, where for every 1000 births, 485 are girls, while 515 are boys. Write a linear combination that can be used to construct an estimator.

41 Variance of a Linear Combination of Random Variables ConstantsRandom variables Example: Sample mean, n=2 srs with replacement

42 Matrix Multiplication Hence

43 Practice: Variance of a Linear Combination of Random Variables Constants Random variables Example: Sample mean, n=2 srs withOUT replacement from a population of N

44 Correlation (see 17.1, 17.2 in text) The correlation between two random variables is defined as The correlation between two random variables is defined as Based on a simple random sample, we estimate the correlation by Based on a simple random sample, we estimate the correlation by