9.7 Optional: Two-way tables

(Answers available in Sect. A.9)

Applying tattoos carries health risks as the skin is broken during application. An American study examined if a relationship existed between having hepatitis C and having tattoos (Haley and Fischer 2001).

To study this, 626 people were interviewed as part of an observational study, and asked about two issues: Whether they had hepatitis C (47 people) or not (579 people); and if they had a tattoo (113 people), or no tattoos (513 people).

  1. Which one of these five sets of hypotheses is not valid for this situation? Why?

    • \(H_0\): No association between having hepatitis C and having a tattoo in the population;
      \(H_1\): An association between having hepatitis C and having a tattoo in the population.
    • \(H_0\): The odds of having hepatitis C is the same with or without a tattoo in the population;
      \(H_1\): The odds of having hepatitis C is not the same with or without a tattoo in the population.
    • \(H_0\): The mean number of people having hepatitis C is the same for those with and without a tattoo in the population;
      \(H_1\): The mean number of people having hepatitis C is not the same for those with and without a tattoo in the population.
    • \(H_0\): The odds ratio of having hepatitis C, comparing those with or without a tattoo, is one in the population;
      \(H_1\): The odds ratio of having hepatitis C, comparing those with or without a tattoo, is not one in the population.
    • \(H_0\): The proportion having hepatitis C is the same with or without a tattoo in the population;
      \(H_1\): The proportion having hepatitis C is not the same with or without a tattoo in the population.
  2. Compute the percentage of people overall in the sample with a tattoo.

  3. Assuming the null hypothesis about the population is true, compute the number of people in the sample with hepatitis C that you would expect have a tattoo. Use the information above to answer the question.

  4. In the sample, 25 people had Hep. C and a tattoo. Use this information to create the two-way table summarising the data (Table 9.3).

TABLE 9.3: Five-year mortality for artifical limb users
Had Hep. C Did not have Hep. C Total
Had tattoo
Did not have tattoo
Total 626
  1. In the sample, what are the odds that someone has Hep. C, among those with a tattoo?
  2. In the sample, what are the odds that someone has Hep. C, among those without a tattoo?
  3. From the sample information, compute the odds ratio of having Hep. C, comparing students with a tattoo to those without a tattoo. Carefully explain what this value means.

References

Haley, Robert W., and R. Paul Fischer. 2001. “Commercial Tattooing as a Potentially Important Source of Hepatitis C Infection: Clinical Epidemiology of 626 Consecutive Patients Unaware of Their Hepatitis C Serologic Status.” Medicine 80: 134–51.