Class Activity, November 14

The Framingham heart study

In this class activity, we revisit the data from the Framingham heart study. Recall the data contains the following variables:

  • cigsPerDay: The number of cigarettes smoked per day during the study period.
  • education: 1 = High School, 2 = Some College, 3 = College Degree, 4 = Advanced Degree.
  • age: The age of the individual in years.
  • diabetes: 1 if the individual has diabetes, 0 otherwise.
  • currentSmoker: 1 if the individual currently smokes, 0 otherwise

Previously, we fit a Poisson regression model for the number of cigarettes smoked by current smokers. But what if we donโ€™t observe whether someone is a current smoker or not? Then we could fit a ZIP model, where the latent variable \(Z_i\) would represent whether someone is a non-smoker (\(Z_i = 1\)) or a smoker (\(Z_i = 0\)).

In this class activity, we will work with the following model. Letting \(Y_i\) denote the number of cigarettes smoked per day,

\[P(Y_i = y) = \begin{cases} e^{-\lambda_i}(1 - \alpha_i) + \alpha_i & y = 0 \\ \dfrac{e^{-\lambda_i} \lambda_i^y}{y!}(1 - \alpha_i) & y > 0 \end{cases}\]

\[\log \left( \dfrac{\alpha_i}{1 - \alpha_i} \right) = \gamma_0 + \gamma_1 EducationSome_i + \gamma_2 EducationCollege_i + \\ \gamma_3 EducationAdv_i + \gamma_4 Diabetes_i + \gamma_5 Age_i\]

\[\log(\lambda_i) = \beta_0 + \beta_1 EducationSome_i + \beta_2 EducationCollege_i + \\ \beta_3 EducationAdv_i + \beta_4 Diabetes_i + \beta_5 Age_i\]

Questions

  1. Suppose researchers want to answer the following question: for smokers, does the number of cigarettes smoked per day depend on age? What would the null and alternative hypotheses be, in terms of one or more model parameters?

  2. Now the researchers want to answer a different question: is there a relationship between age and whether someone is a smoker? What would the null and alternative hypotheses be?

  3. Finally, the researchers want to know whether there is a relationship between education level and the number of cigarettes smoked. What would the null and alternative hypotheses be?