A researcher was interested in determining risk factors for high blood pressure (hypertension) among women. Data from a sample group of 680 women were collected. The following table gives the observed relationship between hypertension and smoking:

Let π be the probability of having hypertension, and suppose that the researcher used logistic regression to model the relationship between smoking and hypertension.
One model the researcher considered is
loge Ï€ /1 €“ Ï€ = β0 + βiXi
where X1 = smoking status (1 = smoker, 0 = nonsmoker). For this model, the estimated odds ratio for the effect of smoking on hypertension status can be computed by using the simple formula for an odds ratio in a 2 × 2 table€”namely, ad/bc, where a, b, c, and d are the cell frequencies in the preceding table,
a. Based on this information, compute the point estimate β1 of βi.
The researcher ultimately decided to use the following logistic regression model:
loge Ï€/1 €“ Ï€ = β0 + β1X1 + β2X2 + β3X1X2
where X1 = smoking status (1 = smoker, 0 = nonsmoker) and X2 = age. The following information was obtained:

The estimated covariance matrix for β0, β1, β2, and β3 is as follows:

b. What is the estimated logistic regression model for the relationship between age and hypertension for nonsmokers?
c. What is a 20-year-old smoker's predicted probability of having hypertension?
d. Estimate the odds ratio comparing a 20-year-old smoker to a 21-year-old smoker. Interpret this estimated odds ratio.
e. Find a 95% confidence interval for the population odds ratio being estimated in part (d).
f. The log likelihood (€“ 2 In )for the model consisting of the intercept, age, and smoking status was 308.00. Use this information, plus other information provided earlier, to perform an LR test of the null hypothesis that β3 = 0 in model 1.

Transcribed Image Text:

Hypertension Yes No Smokers Nonsmokers d28 c# 13 b # 271 d 368 41 639 Parameter Bo B2 Estimate SE -2.8 0.706 0.0004 0.0006 1.2 0.311 0.0001 0.0003 B, -2 In L 303.84 144 0.0001 0.000 00001 0.0967 0.1×10-8 2.0 x 10-8 1.0 × 10-s 3.0 × 10-8 9.0 × 10-8」