BINF 702 SPRING Chapter 8 Hypothesis Testing: Two-Sample Inference. BINF702 SPRING 2014 Chapter 8 Hypothesis Testing: Two- Sample Inference 1

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1 BINF 702 SPRING 2014 Chapter 8 Hypothesis Testing: Two-Sample Inference Two- Sample Inference 1

2 A Poster Child for two-sample hypothesis testing Ex 8.1 Obstetrics In the birthweight data in Example 7.2, the underlying mean birthweight in one hospital was compared with the underlying mean birthweight in the United States, whose value was assumed known Def. 8.1 In a two-sample hypothesis testing problem, the underlying parameters of two different populations, neither of whose values is assumed known, are compared. Ex. 8.2 Cardiovascular Disease, Hypertension We might be interested in the relationship between the use of oral contraceptives (OC) and the level of blood pressure (BP) in women. Two-Sample Inference 2

3 Equation 8.1 Longitudinal Study 1. Identify a group of nonpregnant, premenenopausal women of childbearing age (16-49) who are not currently OC users, and measure their blood pressure (BP), which will be referred to as baseline blood pressure. 2. Rescreen these women 1 year later to ascertain a subgroup who have remained nonpregnant throughout the year and have become OC users. This subgroup will be the study population. 3. Measure the BP of the study population at the follow-up visit. 4. Compare the baseline and the follow-up BP of the women in the study population to determine the difference between the BP of women when they were using the pill at follow-up and when they were not using the pill at baseline. Two-Sample Inference 3

4 Equation 8.2 Cross-Sectional Study 1. Identify both a group of OC users and a group of non-oc users among non-pregnant, premenopausal women of childbearing age (16-49), and measure their BP. 2. Compare the BP of the OC users and nonusers. Two-Sample Inference 4

5 Paired-Sample Design Def Two samples are said to be paired when each data point of the first sample is matched and is related to a unique data point of the second sample. The longitudinal study described earlier represents a paired sample where each women is being used as her own control. Two-Sample Inference 5

6 Independent Samples The samples in the cross-sectional study are said to be independent because the data points in the first sample are not related to the data points in the second sample. Two-Sample Inference 6

7 Longitudinal or Cross-sectional? Twenty volunteers adopt a low-cholesterol diet for 3 months. The mean +- 1 sd of changes (baseline 3 months) in serum cholesterol over the 3-month period was (mg/dl). Two-Sample Inference 7

8 Longitudinal or Cross-sectional? Severe anxiety often occurs in patients who must undergo chronic hemodialysis. A set of progressive relaxation exercises was shown on a videotape to a group of 38 experimental subjects, while a set of neutral videotapes was shown to a control group of 23 patients who were also on chronic hemodialysis. A psychiatric questionnaire was administered to each patient. Two-Sample Inference 8

9 Longitudinal or Cross-sectional? A new sleeping pill was being tested on a number of volunteers. It was predicted that it would have a differential effect on men and women. There were six men and eight women who agreed to take part in the experiment. Over a two week period they took either a placebo or the sleeping pill. Participants were not aware of which pill they were taking each night. The number of extra hours slept during the seven pill night compared to the seven placebo nights was calculated. The men slept 4, 6, 5, 4, 5 and 6 extra hours and the women slept 3, 8, 7, 6, 7, 6, 7, and 6 extra hours. Is the prediction supported? Two-Sample Inference 9

10 Longitudinal or Cross-sectional? A teacher believed that the children in her class were better at their work in the morning than in the afternoon. She decided to test this out by using a mathematics test as this required the children to concentrate. If there was a post-lunch dip in performance the test should pick it up. She chose a random sample of 8 children from the class and gave them two tests matched on their difficulty. The samples were balanced on the two versions of the test, and what time they were tested first, to control carry-over effects. The tests gave a score out of 10, the higher the score the better the performance. Two-Sample Inference 10

11 Section 8.2 The Paired t Test Equation 8.4 (Paired t Test) Denote the test statistic by t, where s d is the sample standard deviation of the observed differences: s n n 2 i i1 i1 where n = number of matched pairs d 2 d di / n ( n 1) s d d / n If t > t n-1,1-a/2 or t < -t n-1,1-a/2 Then H 0 is rejected. If t t t n1,1 a/ 2 n1,1 a/ 2 then H 0 is accepted. Two-Sample Inference 11

12 Equation 8.5 Computation of the p-value for the Paired t Test If t < 0, If t>=0, d p 2 x[ the area to the left of t under a tn 1 distribution] s / n d d p 2 x[ the area to the right of t under a tn 1 distribution] s / n d Two-Sample Inference 12

13 t test in R Description: Performs one and two sample t-tests on vectors of data. Usage: t.test(x, y = NULL, alternative = c("two.sided", "less", "greater"),mu = 0, paired = FALSE, var.equal = FALSE,conf.level = 0.95,...) Two-Sample Inference 13

14 Paired t-test Example A teacher believed that the children in her class were better at their work in the morning than in the afternoon. She decided to test this out by using a mathematics test as this required the children to concentrate. If there was a post-lunch dip in performance the test should pick it up. She chose a random sample of 8 children from the class and gave them two tests matched on their difficulty. The samples were balanced on the two versions of the test, and what time they were tested first, to control carry-over effects. The tests gave a score out of 10, the higher the score the better the performance. Participant Morning Afternoon Two-Sample Inference 14

15 Paired t test Example (Continued) Two-Sample Inference 15

16 Paired t Test Ex. 8.5 Can you perform the paired t test for example 8.5 in R? Two-Sample Inference 16

17 Section 8.3 Interval Estimation for the Comparison of Means from Two Paired Samples Equation 8.6 (Confidence Interval for the True Difference (D) Between the Underlying Means of Two Paired Samples (Two-Sided)) A two-sided 100% x (1- a) confidence interval for the true mean difference (D) between two paired samples is given by ( d t s / n, d t s / n) n1,1 a/ 2 d n1,1 a/ 2 d Can you duplicate Example 8.7 in R? Two-Sample Inference 17

18 Section 8.4 Two-sample t Test for Independent Samples with Equal Variances Equation 8.11 (Two-Sample t Test for Independent Samples with Equal Variances) Suppose that we wish to test the hypothesis H 0 : m 1 = m 2 versus H 1 :m 1 not equal m 2 with a significance level of a for two normally distributed populations, where s 2 is assumed to be the same for each population. Compute the test statistic: t s s x x n n 1 2 where 1 1 n n 2 n s n s If t > t n1 + n 2-2, 1-a/2 or t < -t n1 + n 2-2, 1-a/2 then H 0 is rejected. If t n1 + n 2 2, BINF702 1 a/2 <= SPRING t <= 2014 t n1 Chapter + n 2-2, 81 Hypothesis a/2 then Testing: H 0 is accepted. Two-Sample Inference 18

19 Equation 8.12 Computation of the p-value for the Two-Sample t Test for Independent Samples with Equal Variances Compute the test statistic. x1 x2 t where 1 1 s n1 n2 n s n s s 1 1 n n If t <= 0, p = 2 x (area to the left of t under a t n1 + n 2 2 distribution) If t > 0, p = 2 x (area to the right of t under a t n1 + n 2 2 distribution) Two-Sample Inference 19

20 An Example of Computation of the p-value for the Two-Sample t Test for Independent Samples with Equal Variances A new sleeping pill was being tested on a number of volunteers. It was predicted that it would have a differential effect on men and women. There were six men and eight women who agreed to take part in the experiment. Over a two week period they took either a placebo or the sleeping pill. Participants were not aware of which pill they were taking each night. The number of extra hours slept during the seven pill night compared to the seven placebo nights was calculated. The men slept 4, 6, 5, 4, 5 and 6 extra hours and the women slept 3, 8, 7, 6, 7, 6, 7, and 6 extra hours. Is the prediction supported? Two-Sample Inference 20

21 Another Example of Computation of the p-value for the Two- Sample t Test for Independent Samples with Equal Variances 18 elastic bands were divided into 9 pairs, with bands of similar stretchiness placed in the same pair. One member of each pair was placed in ot water (60-65 C) for 4 minutes, while the other was left at ambient temperature. After a wait of abut ten minutes, the amounts of stretch, under a 1.35 kg weight, were recorded Heated Ambient Two-Sample Inference 21

22 Another Example of Computation of the p-value for the Two-Sample t Test for Independent Samples with Equal Variances Assuming equal variance test for the equality of the means in R. Two-Sample Inference 22

23 Section 8.5 Interval Estimation for the Comparison of Means from Two Independent Samples (Equal Variance Case) Equation 8.13 Confidence Interval for the Underlying Mean Difference (m 1 m 2 ) Between Two Groups (Two-Sided) (s 1 2 = s 22 ) A two-sided 100% x (1-a) confidence interval for the true mean difference m 1 - m 2 based on two independent samples is given by x x t, x x t 1 2 n1 n2 2,1 a/ n1 n2 2,1 a/ 2 n1 n2 n1 n2 Two-Sample Inference 23

24 Section 8.6 Testing for the Equality of Two Variances Equation 8.15 F-test for the Equability of two variances Suppose we wish to conduct a test of the hypothesis H 0 : s 1 2 = s 2 2 versus H 1 : s 1 2 not equal s 2 2 with significance level a. Compute the test statistic F = s 12 /s 22. If F F or F F n 1, n 1,1 a/ 2 n 1, n 1,1 a/ Then H 0 is rejected. If F F F n 1, n 1, a/ 2 n 1, n 1,1 a/ H 0 is accepted. Two-Sample Inference 24

25 Section 8.6 Testing for the Equality of Two Variances Equation 8.16 Computation of the p-value for the F test for the Equality of Two Variances Compute the test statistic F = s 12 /s 2 2 If F >= 1, then p 2*Pr Fn 1, n 1 F 1 2 If F < 1, then p 2*Pr Fn 1, n 1 F 1 2 Two-Sample Inference 25

26 Testing the Equality of Variances in R var.test In library STATS Description Performs an F test to compare the variances of two samples from normal populations. Usage var.test(x, y, ratio = 1, alternative = c("two.sided", "less", "greater"), conf.level = 0.95,...) var.test(formula, data, subset, na.action,...) Two-Sample Inference 26

27 A Pedagogical Example (Testing the Equality of Two Variances) Consider the following heart weight data Obs. # THW (disease) THW(normal) Two-Sample Inference Test for the equality of variances. 27

28 Our Pedagogical Example Continued Two-Sample Inference 28

29 Section 8.7 Two-Sample t Test for Independent Samples with Unequal Variances Equation 8.21 Two-Sample t Test for Independent Samples with Unequal Variances (Satterthwaite s Method) Compute the test statistic t x x s1 s2 ( ) n n 1 2 Compute the approximate degree of freedom d, where d ' s1 / n1 s2 / n2 2 2 s / n1 / n1 1 s2 / n2 / n2 1 Round d down to the nearest integer d 1. If t > y d,1-a/2 or t < -t d,1-a/2 then reject H If t d,1-a/2 <=t<= td,1-a/2 then accept H 0 Two-Sample Inference 29

30 Section 8.7 Two-Sample t Test for Independent Samples with Unequal Variances Equation 8.22 Computation of the p-value for the Two-Sample Test for Independent Samples with Unequal Variances (Satterthwaite Approximation) Compute the test statistic x1 x2 t 2 2 s1 s2 ( ) n n 1 2 If t <= 0, p = 2 * (area to the left of t under a t d distribution) If t > 0, then p = 2 * (area to the right of t under a t d distribution) Where d is as defined in equation Two-Sample Inference 30

31 Two-Sample Inference First column is patient id. Second column is mean hospital stay in days Third column is antibiotic code, 1 = yes, 2 = no. Compare the mean hospital stay between antibiotic patients and non-antibiotic patients 31

32 Pedagogical Example Continued (Setting up the data and testing for equality of variances) Two-Sample Inference 32

33 Pedagogical Example (cont.) Running the t Test Two-Sample Inference 33

34 A Simple Strategy for Testing the Equality of Means in Two Independent Normally Distributed Samples Significant Test for equality of variance with var.test Not significant Run t-test with t.test(x,y) Run t-test with t.test(x,y,var.equal=t ) Two-Sample Inference 34

35 Section 8.7 Two-Sample t Test for Independent Samples with Unequal Variances Equation 8.23 Two-Sided 100% x (1-a) Confidence Interval for m 1 m 2 (s 1 2 not equal s 22 ) 2 / 2 /, 2 / 2 / 1 2 d'',1 a/ d'',1 a/ x x t s n s n x x t s n s n Two-Sample Inference 35

36 Section 8.9 on the Treatment of Outliers Is Being Skipped Two- Sample Inference 36

37 Section 8.10 Estimation of Sample Size and Power for Comparing Two Means Equation 8.26 Sample Size Needed for Comparing the Means of Two Normally Distributed Samples of Equal Size Using a Two-Sided Test with Significance Level a and Power 1-b n 2 2 s 2 1 s2 z1 a/ 2 z1 b m m The means and variances of the two respective groups are (m 1, s 12 ) and (m 2, s 22 ). Two-Sample Inference 37

38 Equation 8.26 Example Blood flow is known to be very reduced in certain parts of the brain in people watching WWF television programs. Suppose we plan a n observational study to compare cerebral blood flow in stroke-prone people (with transient ischemic attacks, or TIAs, where neurological function is temporarily lost) vs stroke-age normal controls (age 50-70). We will use mean blood flow over the entire brain as an outcome variable. How many subjects do we need per group to have a 90% chance of finding a significant difference, if we plan to enroll an equal number of subjects per group? Write an R function to calculate this in general. Mean Sd Stoke prone Normal Two-Sample Inference 38

39 Our R function for Equation 8.26 Two-Sample Inference 39

40 Section 8.10 Estimation of Sample Size and Power for Comparing Two Means Equation 8.27 Sample Size Needed for Comparing the Means of Two Normally Distributed Samples of Unequal Size Using a Two-Sided Test with Significance Level a and Power 1-b 2 2 s1 s2 / k z1 a/ 2 z1 n 2 b 1 2 m m 2 1 k n n 2 1 n ks s2 z1 a/ 2 z1 b 2 2 m m 2 1 Two-Sample Inference 40

41 Section 8.10 Estimation of Sample Size and Power for Comparing Two Means Equation 8.28 Power for Comparing the Means of Two Normally Distributed Samples Using a Two-Sided Test with Significance Level a To test the hypothesis H 0 :m 1 = m 2 versus H 1 :m 1 not equal m 2 for the specific alternative m 1 m 2 = D with significance level a Power z 1 a / 2 D s / n s / n Where (m 1, s 12 ), (m 2, s 22 ) are the means and the variances of the two respective groups and n 1, n 2 are the two sample sizes. Two-Sample Inference 41

42 Write a function to calculate power using Eq Two-Sample Inference 42

43 Using Our Function to Plot a Power Curve Our Power Plot Two-Sample Inference 43

44 What if we have one-sided rather than two-sided tests? It is important to note that if we wish to compute sample sizes in the case of a one-sided test rather than a two-sided test one can merely replace a/2 by a in Equations 8.26 and It is important to note that to calculate power for a one-sided rather than a two-sided test, we can simply substitute a for a/2 in Equation 8.28 Two-Sample Inference 44

45 Section 8.11 Sample-Size Estimation for Longitudinal Studies Equation 8.30 Sample Size Needed for Longitudinal Studies Comparing Mean Change in Two Normally Distributed Samples with Two Time Points Suppose we are planning a longitudinal study with an equal number of subjects (n) in each of two groups. We wish to test the hypothesis H 0 :m 1 = m 2 versus H 1 :m 1 not equal m 2, where m 1 is the mean change over time t in group 1 and m 2 = mean change over time t in group 2. We will consider a two-sided test at level a and wish to have a power 1-b of detecting a significant difference m 1 m 2 = d under H 1. The require sample size per group is z 2 1 a/ 2 z1 b 2 2s d n 2 d s s s 2rs s d s 2 1 is the variance of baseline values within the treatment group, s 2 2 is the variance of follow-up values, and r is the correlation coefficient between baseline and follow-up values within a treatment group Two-Sample Inference 45

46 Section 8.11 Sample-Size Estimation for Longitudinal Studies Equation 8.31 Power of a Longitudinal Study Comparing Mean Change Between Two Normally Distributed Samples with Two Time Points To test the hypothesis H 0 :m 1 = m 2 versus H 1 :m 1 not equal m 2, where m 1 is the mean change over time t in group 1 and m 2 = mean change over time t in group 2 for the specific alternative m 1 m 2 = d under H 1 with a two-sided significance level a, and sample size n in each group, the power is given by nd Power z1 a / 2 s d 2 s s s 2rs s d s 1 2 is the variance of baseline values within the treatment group, s 2 2 is the variance of follow-up values, and r is the correlation coefficient between baseline and follow-up values within a treatment group Two-Sample Inference 46

47 Challenge Problems Two- Sample Inference 47

48 Challenge Problem #1 A study in Pittsburg looked at various cardiovascular risk factors in children, as measured at birth and during their first five years of life. In particular, heart rate was assessed at birth, 5 months, 15 months, 24 months, and annually thereafter until 5 years of age. Heart rate was related to age, sex, race, and socioeconomic status. The data in the table below presents heart rate to race among new-borns. Test for a statistically significant mean heart rate between white and black newborns. Mean Race (beats per minute) Sd N White Black 133 BINF SPRING Chapter 8 Hypothesis Testing: Two-Sample Inference 48

49 Challenge Problem # 1 Solution Two-Sample Inference 49

50 Homework HW 8.2, 8.3, 8.4, 8.5, 8.6, 8.19, 8.20, 8.21, 8.22, 8.25, 8.26, 8.27, 8.28, 8.29, 8.30 Two-Sample Inference 50

Two sample hypothesis testing

Statistics February 26, 2014 Debdeep Pati Two sample hypothesis testing 1. Suppose we want to study the relationship between use of oral contraceptives (OC) and level of blood pressure (BP) in women. 2.