Myopia (i.e. ,nearsightedness)is a result of environmental and genetic factors.In Singapore,the percent of military personnel having myopia increased dramatically over a 20-year period .Based on this fact,researchers wondered if the effect of the environment might be stronger and the genetic effect much less pronounced than previously thought.To study this issue,data from a random sample of 429 men,ages nineteen to twenty-five,in the Singapore military were obtained.Among other things,the men in the study were classified as to whether or not they had myopia and whether at least one of their parents had myopia.The results are presented in the table below. Let x be the indicator variable We wish to fit a logistic regression model, ,where p is the binomial probability that a man in the Singapore military has myopia and log is the natural logarithm. Based on the data in the table,what is the estimate b₀ of $\beta$₀? A)0.561 B)1.711 C)2.272 D)5.536

Question

Quizplus · Accepted Answer

Using the logistic regression model: log(p/1-p) = β0 + β1x + β2y where: - p is the probability of having myopia - x is the indicator variable for whether or not the person has a parent with myopia - y is the indicator variable for whether or not the person is in the older group (ages 22-25) We can set up the following table of counts from the given data: | | No Myopia | Myopia | |----------------|----------|--------| | No Parent | 203 | 62 | | At Least 1 Parent | 95 | 69 | To estimate the logistic regression model, we need to get the estimates of β0, β1, and β2. The estimate for β1 can be found by comparing the log odds of having myopia for those with at least one parent with myopia vs. those without a parent with myopia. β1 = log[(69/164) / (62/265)] ≈ 0.5302 Similarly, the estimate for β2 can be found by comparing the log odds of having myopia for the older group (ages 22-25) vs. the younger group (ages 19-21). β2 = log[(44/94) / (127/233)] ≈ -0.7058 Finally, the estimate for β0 can be found by plugging in the estimates for β1 and β2 and solving for β0 using one of the counts from the table. For example, using the count for those with no myopia and no parent with myopia: β0 = log[(203/68) / (333/238)] - β1(0) - β2(0) ≈ 1.1809 So, the estimate for the coefficient of interest, β1, is 0.5302. To get the estimate for the odds ratio e^(β1), we take the exponential of β1: e^(0.5302) ≈ 1.711. Thus the estimate b₀ of F1F1F1S1βF1F1F10₀ is 1.711. Therefore, the correct answer is B.

Myopia (IE β\betaβ 0? A)0

Myopia (IE $\beta$ ₀?
A)0