A medical researcher is interested in assessing the probability of some symptoms of coronary heart disease being present at a given age. For this reason, he uses the following data collected on 100 individuals. The response variable is CHD which is 1 if coronary heart disease is present and 0 otherwise. The following linear probability model and logit model have been estimated: Linear probability: = - 0.53796 + 0.02181Age
Logit: = For the linear probability model, 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 being present at a given age. For this reason, he uses the following data collected on 100 individuals.  The response variable is CHD which is 1 if coronary heart disease is present and 0 otherwise. The following linear probability model and logit model have been estimated: Linear probability:  = - 0.53796 + 0.02181Age
Logit:  =  For the linear probability model, 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 probability model, the estimated probability p is given as:

p = β0 + β1Age

where β0 is the intercept and β1 is the slope coefficient.

The estimated probability of coronary heart disease (CHD) is given as 11ea8309_a9aa_5d28_b223_d3c4ea139bb7_TB6618_11 = - 0.53796 + 0.02181Age.

We need to find the range of values of Age for which 11ea8309_a9aa_5d29_b223_6bb9757306b4_TB6618_11 can be regarded as the estimated probability of CHD.

In logistic regression, we estimate the log-odds of CHD as:

log(odds) = β0 + β1Age

where β0 is the intercept and β1 is the slope coefficient.

The estimated probability of CHD from the logit model is:

11ea8309_a9aa_5d29_b223_6bb9757306b4_TB6618_11 = 1 / (1 + exp(-11ea8309_a9aa_843a_b223_0114ce9f18b1_TB6618_11))

To find the range of values of Age for which 11ea8309_a9aa_843b_b223_3d26a4902dc4_TB6618_11 can be regarded as the estimated probability of CHD, we solve the equation:

11ea8309_a9aa_5d28_b223_d3c4ea139bb7_TB6618_11 = 1 / (1 + exp(-11ea8309_a9aa_843a_b223_0114ce9f18b1_TB6618_11))

0.02181Age - 0.53796 = -11ea8309_a9aa_843a_b223_0114ce9f18b1_TB6618_11

Age = (-0.53796 + 11ea8309_a9aa_843a_b223_0114ce9f18b1_TB6618_11) / 0.02181

We need to find the value(s) of Age for which 11ea8309_a9aa_843b_b223_3d26a4902dc4_TB6618_11 is within the range of 0 to 1.

On substituting the minimum and maximum values of 11ea8309_a9aa_843a_b223_0114ce9f18b1_TB6618_11 (from the logit model) into the above equation, we get:

Age(min) = (-0.53796 - 3.891) / 0.02181 = 24.7

Age(max) = (-0.53796 + 4.664) / 0.02181 = 70.5

Therefore, the range of values of Age for which 11ea8309_a9aa_843b_b223_3d26a4902dc4_TB6618_11 can be regarded as the estimated probability of CHD is from 24.7 to 70.5.

Hence, the answer is B.

A Medical Researcher Is Interested in Assessing the Probability of Some