Tuyns et al. (1977) carried out a case-control study of esophageal cancer in the region known as Ille-et-Vilaine in Brittany, France. The referring data set is oesoph_new.dta, and use logistic regression models to answer each of the following questions. For each question, carefully state the appropriate logistic regression model and relevant hypothesis, both in the contest of the problem and in terms of model parameters. Use both the Wald and likelihood ratio methods to carry out any hypothesis tests, and provide relevant estimated Odds Ratios (with 95% confidence intervals) where appropriate.

a. Investigate the relationship between alcohol consumption and incidence of esophageal cancer. Treat alcohol consumption as a dichotomous variable (> 80 g/day vs. < 80 g/day), ignoring age.

b. Investigate the relationship between alcohol consumption and incidence of oesophageal cancer, controlling for the potential confounding effects of age. Treat alcohol consumption as a dichotomous variable (> 80 g/day vs. < 80 g/day), and age as a dichotomous variable (25 to 54 years old or 55 to 75+ years old). Give your assessment of the extent of confounding by age using the models fit in (a) and (b).

c. Investigate the evidence of interaction between age and alcohol consumption in relation to incidence of esophageal cancer. Treat alcohol consumption and age as dichotomous variables as in (b).

d. Investigate the relationship between alcohol consumption and incidence of esophageal cancer. First, treat alcohol consumption as a categorical variable with four categories (0 to 39 g/day, 40 to 79 g/day, 80 to 119 g/day, and > 120 g/day), by using indicator variables for the various categories (select 0 to 39 g/day as the reference group); second, treat alcohol consumption as an ordered variable by appropriately coding the four categories of
2 consumption. Compare the two analyses and discuss whether an increasing trend in risk, as alcohol consumption increases, adequately fits the pattern of risks for the four categories.

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To answer each of the questions, we will use logistic regression models. Logistic regression is a statistical method used to model the relationship between a binary dependent variable (esophageal cancer incidence in this case) and one or more independent variables (alcohol consumption and age in this case).

a. Investigating the relationship between alcohol consumption and the incidence of esophageal cancer without considering age:
- Logistic regression model:
Log(odds) = β0 + β1*Alcohol
where Alcohols is a binary variable indicating whether alcohol consumption is greater than 80 g/day (> 80 g/day = 1, < 80 g/day = 0).

The relevant hypothesis is:
H0: β1 = 0 (There is no relationship between alcohol consumption and esophageal cancer incidence)
Ha: β1 ≠ 0 (There is a relationship between alcohol consumption and esophageal cancer incidence)

To test this hypothesis, you can use the Wald test or the likelihood ratio test. Both methods will provide p-values to determine the significance of the relationship. Additionally, the estimated Odds Ratio (with 95% confidence interval) for alcohol consumption can be obtained from the logistic regression model.

b. Investigating the relationship between alcohol consumption and esophageal cancer incidence while controlling for age:
- Logistic regression model:
Log(odds) = β0 + β1*Alcohol + β2*Age
where Alcohol is still a binary variable indicating alcohol consumption (> 80 g/day = 1, < 80 g/day = 0), and Age is another binary variable indicating age group (25 to 54 years old = 1, 55 to 75+ years old = 0).

The relevant hypothesis is:
H0: β1 = 0 (There is no relationship between alcohol consumption and esophageal cancer incidence, after accounting for age)
Ha: β1 ≠ 0 (There is a relationship between alcohol consumption and esophageal cancer incidence, after accounting for age)

Like in (a), you can use the Wald test or the likelihood ratio test to test the hypothesis and obtain p-values. The estimated Odds Ratio (with 95% confidence interval) for alcohol consumption can be obtained from the logistic regression model, along with the estimated Odds Ratio for age.

Assessment of the extent of confounding by age can be done by comparing the estimates from models (a) and (b). If the estimate for alcohol consumption changes substantially between the two models, it suggests that age is a confounding variable, meaning age influences both alcohol consumption and esophageal cancer incidence.

c. Investigating the evidence of interaction between age and alcohol consumption in relation to esophageal cancer incidence:
- Logistic regression model:
Log(odds) = β0 + β1*Alcohol + β2*Age + β3*(Alcohol*Age)
where Alcohol and Age are the same binary variables as in (b), and Alcohol*Age is the interaction term between alcohol consumption and age.

The relevant hypothesis is:
H0: β3 = 0 (There is no interaction between age and alcohol consumption in relation to esophageal cancer incidence)
Ha: β3 ≠ 0 (There is an interaction between age and alcohol consumption in relation to esophageal cancer incidence)

To test this hypothesis, you can again use the Wald test or the likelihood ratio test. The p-value will determine the significance of the interaction. The estimated Odds Ratio (with 95% confidence interval) for alcohol consumption, age, and the interaction term can be obtained from the logistic regression model.

d. Investigating the relationship between alcohol consumption and esophageal cancer incidence with alcohol consumption as a categorical variable:
- Logistic regression model:
Log(odds) = β0 + β1*Alcohol1 + β2*Alcohol2 + β3*Alcohol3
where Alcohol1, Alcohol2, and Alcohol3 are indicator variables for the four categories of alcohol consumption (0 to 39 g/day, 40 to 79 g/day, 80 to 119 g/day, and > 120 g/day) with 0 to 39 g/day as the reference group.

The relevant hypothesis is:
H0: β1 = β2 = β3 = 0 (There is no relationship between alcohol consumption categories and esophageal cancer incidence)
Ha: At least one βi ≠ 0 (There is a relationship between alcohol consumption categories and esophageal cancer incidence)

To test this hypothesis, you can use the Wald test or the likelihood ratio test. The p-value will determine the significance of the relationship. The estimated Odds Ratios (with 95% confidence intervals) for each alcohol consumption category can be obtained from the logistic regression model.

Additionally, you can treat alcohol consumption as an ordered variable by coding the four categories accordingly. This will create a trend variable that represents an increasing trend in alcohol consumption. You can compare the two analyses to see if an increasing trend in risk, as alcohol consumption increases, adequately fits the pattern of risks for the four categories.