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1. Refer to the table provided to answer the questions below. The study compared independent groups.

1. a) Use the information provided to calculate the 95% confidence interval for the difference in energy intake between males and females. Assume a pooled variance (for simplicity, it is fine for you to just average the two variances since the sample sizes in the groups are almost identical). Use a T-value rather than a Z-value when building the confidence interval. (5 points)

1. b) Circle all the statistical tests that would be appropriate for comparing the energy intake between the two groups, assuming that energy intake is reasonably normally distributed. There is more than one correct answer. (5 points).

Two-sample t-test       Paired t-test                        ANOVA

Cox regression            Linear regression        Chi-square test

McNemar’s test          Rate ratio

2. The following table shows beta coefficients from a linear regression model. The outcome variable is serum vitamin D (ng/mL).

1. a) What is the predicted vitamin D level for a person who has a BMI of 25, drinks no alcohol or coffee, exercises 2 hours per week, and is a nonsmoker? (5 points)
2. b) Which of the following is associated with the biggest decrease in vitamin D? (5 points)

1. A 5-kg/m² increase in BMI.
2. Being a smoker versus a non-smoker.

iii. Drinking 2 fewer cups of coffee per day.

1. Drinking 10 more units of alcohol per week.

3. Researchers studied 150 collegiate runners over four years. The runners had their bone density measured at the beginning of the study, and yearly thereafter, for a total of 600 bone density measurements. The researchers wanted to know whether runners who had a history of playing ball sports (e.g., soccer, basketball) prior to college had higher bone density than those who had never played ball sports. They presented the following analysis, including a p-value from a two-sample t-test.

*P-value is from a two-sample t-test comparing the two groups.

1. a) What is wrong with their analysis? Explain briefly. (5 points)

1. b) If the researchers correct their analysis, will the resulting p-value be smaller or larger. Explain briefly (5 points)

4. Read the following abstract:

Background: Women runners are at high-risk of stress fractures. Several previous studies have found a relationship between dietary factors and stress fractures. Methods: We performed a case-control study to try to identify nutritional risk factors for stress fractures in women runners. Cases (n=50) were women runners who had sustained a stress fracture within the last three months. Controls (n=50) were women runners who had never sustained a stress fracture. Women filled out both a food frequency questionnaire and a 24-hour dietary recall. From each instrument separately, we estimated the daily intake of 50 vitamins, minerals, and macronutrients. We compared all 100 nutrients between the two groups using two-sample t-tests. Results: Women in the two groups were similar in age, weight, BMI, weekly mileage, and best mile time. We found that vitamin K intake, as measured on the food frequency questionnaire, was significantly lower in the case group than the control group (p=.023). We also found that selenium intake, as measured on the 24-hour dietary recall, was significantly higher in the case group compared with the control group (p=.03). Conclusions:  To prevent stress fractures, women runners should increase their intake of vitamin K and decrease their intake of selenium.

Which of the following statistical pitfalls (from Unit 6) most undermines the authors’ conclusions? (5 points)

1. a) Failure to prove an effect is not proof of no effect
2. b) There is a multiple testing issue
3. c) Statistical significance does not imply clinical significance
4. d) You need to compare effect sizes, not p-values between two groups.

5. Read the following abstract:

Background: Women runners are at high-risk of low bone density and stress fractures. Observational studies have found a relationship between vitamin D and bone density in runners, but no randomized trials have tested this hypothesis. Methods: We performed a randomized placebo-controlled trial to determine whether vitamin D supplementation can increase bone density in women. We randomly assigned 100 women runners to take a vitamin D supplement (1000 IU/day) and 100 women runners to take a placebo pill daily. We compared bone density before and after supplementation using a paired t-test. Results: Women in the two groups were balanced with regards to age, weight, height, BMI, weekly mileage, and best mile time. After three months of supplementation, there was a significant increase in bone density in the vitamin-D supplemented group (p<.05), but not in the placebo group (p>.05). Conclusions:  To increase bone density and prevent stress fractures, women runners should take a daily vitamin D supplement.

Which of the following statistical pitfalls (from Unit 6) most undermines the authors’ conclusions? (5 points)

1. a) Correlation is not causation
2. b) There is a multiple testing issue
3. c) Statistical significance does not imply clinical significance
4. d) You need to compare effect sizes, not p-values between two groups.

6. What is the approximate linear regression model that corresponds to the following picture? Assume that the model contains only the variables pictured in the graph. Cognitive change score is the outcome variable. High/low complexity is a binary variable (low complexity = reference group); and social isolation score is a continuous variable (standardized score of 0 = reference value). (10 points)

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