Sta-209 (Spring 2025) Homework #7

Directions:

• Submit your assignment via P-web.
• Submit only a compiled R Markdown document (pdf, word, or html output are all okay, but you may need to “zip” an html file)
  • If you want to compile to a pdf you can install the tinytext package by running install.packages('tinytex') followed by tinytex::install_tinytex()
• Only submit your .Rmd file if you are unable to compile it due to errors (in the future you will be penalized for this)

Question #1

In a previous assignment you worked with the American Community Survey (ACS) data, which are a component of the US Census administered to a random sample of US addresses on a rolling basis. When the mailed version is combined with in-person visits and telephone calls the survey has a 95% response rate. The data linked below are a random sample of employed individuals drawn from a recent ACS (2020 Census):

acs = read.csv("https://remiller1450.github.io/data/EmployedACS.csv")

The ACS data linked above includes the following variables:

• Sex - “1” for males and “0” for females
• Age - age in years
• Married - “1” for married individuals and “0” for unmarried individuals
• Income - annual income (thousands of dollars)
• HoursWk - average hours worked per week
• Race - self-described race
• USCitizen - citizenship status, “1” for US citizens and “0” for non-citizens
• HealthInsurance - “1” if the individual has health insurance, “0” otherwise
• Language - “1” if the individual’s first/native language is English, “0” otherwise

In this question you will perform several hypothesis tests. For each hypothesis test (Parts A-C) you should include the following steps:

1. State the null and alternative hypotheses using either words or statistical notation.
2. Use either StatKey or an appropriate R function to find the \(p\)-value.
3. Provide a one-sentence conclusion summarizing the results of your hypothesis test. Be sure to follow the guidelines from this week’s lecture slides.

• Part A: According to Wikipedia, 78% of US adults speak English as their native language. Use a hypothesis test to determine whether the ACS sample provides evidence that refutes this claim.
• Part B: Perform a hypothesis test to evaluate whether these data provide compelling statistical evidence that married individuals are more likely to have health insurance than unmarried individuals.
• Part C: According to “worddata.info” (which seems like a questionable source to me) the average personal income in the United States is $64,000. Use a hypothesis test to determine whether the ACS sample provides sufficient evidence to refute this claim.

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Question #2

Rosiglitazone is the active ingredient in the controversial type 2 diabetes medicine Avandia and has been linked to an increased risk of serious cardiovascular problems such as stroke, heart failure, and death. A common alternative treatment is Pioglitazone, the active ingredient in a diabetes medicine called Actos. In a nationwide retrospective observational study of 227,571 Medicare beneficiaries aged 65 years or older, it was found that 2,593 of the 67,593 patients using Rosiglitazone and 5,386 of the 159,978 using Pioglitazone had serious cardiovascular problems. These data are summarized in the contingency table below.

Treatment	No CV problems	CV Problems	Total
Pioglitazone	154592	5386	159978
Rosiglitazone	65000	2593	67593
Total	219592	7979	227571

• Part A: Below are several statements about this study that are either true or false. Identify which statements are false and briefly explain why each is false. Note that some statements might reach the correct conclusion using incorrect reasoning. In these circumstances the entire statement should be considered false.
  • I: Since more than 50% of patients with cardiovascular problems were on Pioglitazone we can conclude that the risk of cardiovascular problems is higher for Pioglitazone than it is for Rosiglitazone.
  • II: The data suggest that diabetic patients who are taking Rosiglitazone appear to be likely to have cardiovascular problems (3.8% risk) than patients on Pioglitazone (3.4% risk).
  • III: A statistical test is needed to evaluate whether the observed difference in the risk of cardiovascular problems is due to the medication or could be explained by chance.
  • IV: If a statistical test performed on these data yielded a \(p\)-value less than 0.01 we can conclude that Rosiglitazone causes an increase in the rate of cardiovascular problems relative to Pioglitazone.
• Part B: If the type of treatment and having cardiovascular problems were independent, how many patients in the Rosiglitazone group would we expect to have had cardiovascular problems?
• Part C: Using both words and statistical symbols, write the null and alternative hypotheses for a statistical test that investigates whether there is an association between type of treatment and the risk of cardiovascular problems.
• Part D: Use prop.test() to perform a statistical test of the hypotheses you provided in Part C. Report the \(p\)-value of the test.
• Part E: Provide a one-sentence conclusion summarizing the results of the test you performed in Part D. Be careful to follow the guidelines covered in this week’s lecture slides.

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Question #3

In modern biomedical studies it is relatively common to record measurements for thousands of genetic features at once. Suppose a cancer researcher collects data on 2000 genes, and 20 of these genes are truly related to the cancer that the researcher is studying. For each of the 2000 genes, the researcher performs a hypothesis test and compares the \(p\)-value to a decision threshold of \(\alpha = 0.05\).

• Part A: Out of all of the hypothesis tests performed by the researcher, how many Type 1 errors do you expect? Hint: Remember that Type 1 errors can only occur when the null hypothesis is correct.
• Part B: Suppose the researcher’s statistical tests have 80% power, meaning they will correctly reject a false null hypothesis 80% of the time. Considering this information, how many Type 2 errors do you expect out of all of the hypothesis tests performed by this researcher. Hint: Remember that Type 2 errors can only occur when the null hypothesis is incorrect.
• Part C: Suppose the researcher identifies 109 statistically significant genes using \(\alpha = 0.05\). What do you estimate the false discovery rate to be for this study? Hint: You should use the number of Type 1 errors from Part A in your calculation.
• Part D: Consider a procedure that controls the false discovery rate at 5% and one that controls the family-wise Type 1 error rate at 5%. Which procedure do you expect to identify more of the 20 genes that are truly related to this cancer as being “significant”? Briefly explain your answer.
• Part E: Suppose a 95% confidence interval is calculated for the association between each gene and cancer status. How many of these intervals would you expect to contain the true association for their corresponding gene?