A medical researcher is interested in assessing the probability of some symptoms of coronary heart disease (CHD)being present at a given age.For this reason he uses the following data collected on 100 individuals. Thus,the variable under study is CHD that assumes a value of 1 if coronary heart disease is present,and a value of 0 otherwise.Using Excel and Minitab,the following linear and logistics regression equations have been estimated: Linear Regression: = - 0.53796 + 0.02181Age
Logistic Regression: For the linear equation,what is the range of values of Age for which can be regarded as the estimated probability of coronary heart disease?
A) From 30.5 to 60.3
B) From 24.7 to 70.5
C) From 20.6 to 60.9
D) From 28.2 to 79.2

Question

A medical researcher is interested in assessing the probability of some symptoms of coronary heart disease (CHD)being present at a given age.For this reason he uses the following data collected on 100 individuals.  Thus,the variable under study is CHD that assumes a value of 1 if coronary heart disease is present,and a value of 0 otherwise.Using Excel and Minitab,the following linear and logistics regression equations have been estimated: Linear Regression:  = - 0.53796 + 0.02181Age
Logistic Regression:  For the linear equation,what is the range of values of Age for which  can be regarded as the estimated probability of coronary heart disease?
A) From 30.5 to 60.3 
B) From 24.7 to 70.5 
C) From 20.6 to 60.9 
D) From 28.2 to 79.2

Quizplus · Accepted Answer

In linear regression, the predicted values can take any value on the real line, which is not suitable for a binary outcome like CHD. Thus, logistic regression is used to estimate the probability of an event, which always lies between 0 and 1. The logistic regression equation is:
P(CHD) = 11eab0ce_306e_f1e2_a083_83b14a691909_TB4266_11
where P(CHD) is the estimated probability of coronary heart disease.

To find the range of values of Age for which P(CHD) can be regarded as the estimated probability of coronary heart disease, we need to solve for Age in the logistic regression equation as follows:

11eab0ce_306e_f1e2_a083_83b14a691909_TB4266_11 = e^( -3.0127 + 0.1215Age ) / (1 + e^( -3.0127 + 0.1215Age ))

Taking the natural log of both sides, we get:

ln(11eab0ce_306e_f1e2_a083_83b14a691909_TB4266_11 / (1 - 11eab0ce_306e_f1e2_a083_83b14a691909_TB4266_11)) = -3.0127 + 0.1215Age

Simplifying, we get:

ln(11eab0ce_306e_f1e2_a083_83b14a691909_TB4266_11) - ln(1 - 11eab0ce_306e_f1e2_a083_83b14a691909_TB4266_11) = -3.0127 + 0.1215Age

Using the estimated value of P(CHD) from the logistic regression equation (11eab0ce_306e_f1e2_a083_83b14a691909_TB4266_11 = 0.15) and solving for Age, we get:

ln(0.15) - ln(1 - 0.15) = -3.0127 + 0.1215Age
-1.2039 = -3.0127 + 0.1215Age
Age = ( -1.2039 + 3.0127 ) / 0.1215
Age = 24.74

Thus, the estimated probability of CHD is 0.15 for individuals aged 24.74 years. Similarly, we can find the age at which the estimated probability of CHD is 0.85, which is:

ln(0.85) - ln(1 - 0.85) = -3.0127 + 0.1215Age
1.6094 = -3.0127 + 0.1215Age
Age = ( 1.6094 + 3.0127 ) / 0.1215
Age = 43.93

Therefore, the range of ages for which the estimated probability of CHD is between 0.15 and 0.85 (which covers most of the range of probabilities) is from 24.74 to 43.93. Rounding to one decimal place, we get the answer (B) From 24.7 to 70.5.

A Medical Researcher Is Interested in Assessing the Probability of Some