DPHS 568 Biostatistics in Dentistry, Summer 2007, Homework #2

Due July 30, 2006 (beginning of class). If more than one sheet, please staple.

1. Fifty draws are made at random from an urn containing 4 balls. The balls are numbered "1", "3", "3", "4". After each draw, the number on the ball drawn is noted, and then the ball is replaced into the urn. One of the histograms below is the histogram for the contents of the urn. Another is the histogram for the actual numbers drawn in the fifty draws. The remaining histogram is the probability histogram for the average of the draws. Which is which? Explain your reasoning.

2. Suppose that a multicenter clinical trial to assess possible anti-cariogenic effects of Vitamin D has enrolled participants in four different cities. Suppose that in reality the effect of Vitamin D on caries is no different than placebo. If the study computes hypothesis tests at the α= 0.05 significance level separately at each city, what is the probability that this method will report a "statistically significant" effect of Vitamin D compared to placebo in at least one city?

3. Suppose 80 hypertensive people are given an antihypertensive drug and the drug is effective in 17 of the people. The drug is considered effective if it lowers the diastolic blood pressure by at least 10 mm Hg as judged by a repeated measurement one month after starting on the drug. a. What is the best estimate of the probability of the drug being effective? b. Suppose we know that 12% of all hypertensive patients who are given placebo medication will have their blood pressure lowered by 10 mm Hg after 1 month. Can we carry out some procedure to be sure that we are not simply observing the placebo effect?

4. Suppose we decide that a better measure of the effectiveness of the drug in problem #3 is the mean decrease in blood pressure rather than the yes/no measure used above. Suppose that the mean difference in blood pressure (after – before) is X  6.3 mm Hg, and the sample standard deviation is s = 11 mm Hg. a. What is the standard error of the mean? b. Compute a 95% confidence interval for the mean difference in blood pressure. c. Use a hypothesis test to assess whether the drug lowers blood pressure on average. d. What complementary information do the hypothesis test and the confidence interval give you? e. Now suppose that investigators also gave placebo to 80 other people and measured their changes in blood pressure. The mean difference in blood pressure in the placebo group was X  0.2 mm Hg with standard deviation s = 17 mm Hg. Test the hypothesis that the effect of the active drug differs from placebo.

5. A study was undertaken 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 25 healthy subjects. 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 “washout” period followed during which no supplements were taken. Subjects then took the alternate supplement for a 6-week period. The results are shown in Table 1.

Table 1: Serum-cholesterol levels before and during high-fiber/low fiber supplementation* Difference Difference Difference High Low Baseline (high fiber – (high fiber – (low fiber – fiber fiber low fiber) baseline) baseline) Total cholesterol 186 ± 31 171 ± 25 173 ± 28 -1 (-8, +5) -14 (-21, -8) -13 (-19, -7) (mg/dL) * Plus-minus values are mean ± sd. Values in parentheses are 95% confidence limits.

a. Test the hypothesis that the low-fiber diet has an effect on cholesterol levels as compared with baseline (report your results as p < .05 or p > .05). b. Test the hypothesis that the high-fiber diet has an effect on cholesterol levels as compared with baseline (report your results as p < .05 or p > .05). c. Test the hypothesis that the high-fiber diet has a differential effect on cholesterol levels compared to a low-fiber diet (report your results as p < .05 or p > .05). d. What is the approximate standard error of the mean for the high-fiber compared to the low-fiber diet (that is, with respect to the mean difference between high-fiber and low-fiber results)? e. Estimate how many subjects would be needed so that a two-sided test with significance level = 0.05 would have a 80% chance of finding a significant difference in the cholesterol lowering between the high- and low-fiber diets if the high-fiber diet changes cholesterol by 5 mg/dl more than the low-fiber diet.

6. Revisit the Mercury vapor data from homework #1. Perform a statistical procedure to determine whether or not the difference seen in urinary Hg between the cases and controls can be attributable to random chance. You may assume that the urinary-Hg data are Normally distributed. 7. In a study to compare the accuracy of two electronic apex locators in determining the working length of teeth after removal of the root canal obturation materials, 20 extracted straight, single-rooted teeth were used. The root canals were instrumented and the actual working length was determined. For each tooth the WL was then evaluated using the two competing electronic methods. For each tooth, the differences (D) between the working length and the electronically determined lengths were calculated and are presented in table 1 below. Use an appropriate statistical test to determine whether there is good evidence that one method is better (smaller D) than the other.

Table 2: Difference between actual and electronically- determined lengths (mm) tooth A B 1 0.0 0.5 2 -0.5 -1.0 3 0.0 0.5 4 0.0 0.0 5 0.0 -1.0 6 -0.5 -0.5 7 0.5 -0.5 8 0.0 0.0 9 0.0 -0.5 10 0.5 -1.0 11 0.0 0.0 12 0.0 -0.5 13 0.0 -1.0 14 0.0 0.0 15 0.0 -1.0 16 0.5 -0.5 17 0.5 0.0 18 0.0 -0.5 19 0.5 0.0 20 0.0 0.0