aseline ded to DBP evel of of the tudy. ding pabil- at the DBP ges 0- Cur ed ng ce is 7. this question. *7.35 State the hypotheses that we can use to consider $7.36 Carry out the hypothesis test in Problem 7.35 using the critical-value method with an a level of .05, and sum- marize your findings. 7.36? *7.37 What is the p-value for the test conducted in Problem The standard deviation of daily iron intake in the larger pop- ulation of 9- to 11-year-old boys was 5.56 mg. We want to test whether the standard deviation from the low-income group is comparable to that of the general population. question. *7.38 State the hypotheses that we can use to answer this *7.39 Carry out the test in Problem 7.38 using the critical- findings. value method with an a level of .05, and summarize your *7.40 What is the p-value for the test conducted in Problem 7.39? *7.41 Compute a 95% CI for the underlying variance of daily iron intake in the low-income group. What can you infer from this CI? 7.42 Compare the inferences you made from the proce- dures in Problems 7.39, 7.40, and 7.41. General Suppose a new test is proposed to identify patients with a specific condition. One specimen is obtained from each of 220 subjects in the population to whom the test will be applied. Fifty-one (51) subjects have the condition and 169 do not, according to a gold standard used to assess the presence of the condition (referred to as a reference standard below). The following results are obtained: TABLE 7.5 Comparison of a new test with a reference standard New test + 1 Total Reference standard Condition present 44 7 51 Condition absent 1 168 169 7.43 What is the estimated sensitivity of the test? 7.44 Provide a 95% CI for the sensitivity. Total 45 175 220 7.45 What is the estimated specificity of the test? 7.46 Provide a 95% CI for the specificity (Hint: Use a com- puter program.) Suppose a quality score is constructed for the test. A true positive identified by the test is given a score of +20. A true negative identified by the test is given a score of +10. A false positive identified by the test is given a score of of -20. -5. A false negative identified by the test is given a score 7.47 The FDA will approve the new test if the lower bound of a 90% Cl for the mean score is 2+5. Should the FDA approve the test? A yes/no answer is not sufficient. Hint: Assume that the score distribution is approximately normal or that the central limit theorem is applicable. mately normal. Hint: Assume that a t-distribution with > 200 df is approxi- x = Σfix₁/Σf₁ i=1 data, use the formulas Hint: To compute the mean and variance from grouped i=1 Problems Σfix? 271 (Σ ₁ fix₂)² i=1 N N-1 where x, is the ith score value, f, is the frequency of the ith score value, k = number of score values, and N = Σ 1 f = 220. Occupational Health The mortality experience of 8146 male employees of a research, engineering, and metal-fabrication plant in Tonawa- nda, New York, was studied from 1946 to 1981 [2]. Potential workplace exposures included welding fumes, cutting oils, asbestos, organic solvents, and environmental ionizing ra- diation, as a result of waste disposal during the Manhattan Project of World War II. Comparisons were made for specific causes of death between mortality rates in workers and U.S. white-male mortality rates from 1950 to 1978. Suppose that 17 deaths from cirrhosis of the liver were observed among workers who were hired prior to 1946 and who had worked in the plant for 10 or more years, whereas 6.3 were expected based on U.S. white-male mortality rates. 7.48 What is the SMR for this group? 7.49 Perform a significance test to assess whether there is an association between long duration of employment and mortality from cirrhosis of the liver in the group hired prior to 1946. Report a p-value. 7.50 A similar analysis was performed among workers who were hired after 1945 and who were employed for 10 or more years. It found 4 deaths from cirrhosis of the liver, whereas only 3.4 were expected. What is the SMR for this group? 7.51 Perform a significance test to assess whether there is an association between mortality from cirrhosis of the liver and duration of employment in the group hired after 1945. Report a p-value. 272 CHAPTER 7 Hypothesis Testing: One-Sample Inference Ophthalmology Researchers have reported that the incidence rate of cata- racts may be elevated among people with excessive expo- sure to sunlight. To confirm this, a pilot study is conducted among 200 people ages 65-69 who report an excessive tendency to burn on exposure to sunlight. Of the 200 peo- ple, 4 develop cataracts over a 1-year period. Suppose the expected incidence rate of cataracts among 65- to 69-year- olds is 1% over a 1-year period. *7.52 What test procedure can be used to compare the 1-year rate of cataracts in this population with that in the general population? *7.53 Implement the test procedure in Problem 7.52, and report a p-value (two-sided). The researchers decide to extend the study to a 5-year pe- riod and find that 20 of the 200 people develop a cataract over a 5-year period. Suppose the expected incidence of cataracts among 65- to 69-year-olds in the general popula- tion is 5% over a 5-year period. *7.54 Test the hypothesis that the 5-year incidence rate of cataracts is different in the excessive-sunlight-exposure group compared with the general population, and report a p-value (two-sided). *7.55 Construct a 95% Cl for the 5-year true rate of cata- racts among the excessive-sunlight-exposure group. Cardiovascular Disease, Pediatrics Left ventricular mass (LVM) is an important risk factor for subsequent cardiovascular disease. A study is proposed to assess the relationship between childhood blood pressure levels and LVM in children as determined from echocar- diograms. The goal is to stratify children into a normal bp group (< 80th percentile for their age, gender, and height) and an elevated bp group (≥ 90th percentile for their age, gender, and height) and compare change in LVM between the 2 groups. Before this can be done, one needs to dem- onstrate that LVM actually changes in children over a 4-year period. Total cholesterol (mg/dL) n 20 Baseline TABLE 7.6 Pilot data on left ventricular mass (LVM) in children from the Bogalusa Heart Study High fiber ID 1 2 3 4 5 6 7 8 9 10 Mean sd *4-year LVM minus baseline LVM Baseline LVM (g) To help plan the main study, a pilot study is conducted where echocardiograms are obtained from 10 random chil- dren from the Bogalusa Heart Study at baseline and after 4 years of follow-up. The data are given in Table 7.6. 7.56 What test can be used to assess if there is a change in mean LVM over 4 years? TABLE 7.7 Serum-cholesterol levels before and during high-fiber and low-fiber supplemention 139 134 86 98 78 90 102 73 93 162 105.5 29.4 Low fiber 186 ± 31 172 ± 28 172 ± 25 Note: Plus-minus (±) values are mean ± sd. Values in parentheses are 95% confidence limits. 4-year LVM (g) Difference (high fiber low fiber) 163 126 142 96 111 108 167 82 77 172 124.4 35.2 (-8, +7) 7.57 Implement the test in Problem 7.56 and provide a two-tailed p-value. 7.58 Provide a 95% CI for the change in LVM over 4 years based on the data in Table 7.6. 7.59 Since this was a pilot study, the main question of interest is how many subjects would be needed to detect an increase of 10 g in mean LVM over 4 years using a two- sided test with a = 0.05 and power = 80%? Hint: Assume that the estimated variance of change in LVM in the pilot study is the true variance of change in LVM. - Nutrition, Cardiovascular Disease Previous studies have shown that supplementing the diet with oat bran may lower serum-cholesterol levels. However, it is not known whether the cholesterol is reduced by a direct effect of oat bran or by replacing fatty foods in the diet. To address this question, a study was performed to compare the effect of dietary supplementation with high-fiber oat bran (87 g/day) to dietary supplementation with a low-fiber refined wheat product on the serum cholesterol of 20 healthy partici- pants ages 23-49 years [3]. Each subject had a cholesterol level measured at baseline and then was randomly assigned to receive either a high-fiber or a low-fiber diet for 6 weeks. A 2-week period followed during which no supplements were taken. Participants then took the alternate supplement for a 6-week period. The results are shown in Table 7.7. Change (g) Difference (high fiber- baseline) - 24 -14 (-21,-7) 56 -2 33 18 65 9 -16 10 18.9 26.4 Difference (low fiber- baseline) 1 -13 (-20, -6)