Question

confidence bands for the Exercise yocardial infarc 13.8 In a paper in American Statistician, Hauck [1983] derived logistic response curve. He illustrated the method with data from the Ontario Heart Collaborative Study. The logistic model dealt with the risk of m tion (MI) during a study period of four years. A logistic model based on the two most important variables, smoking (%) and serum triglyceride level (X2), was calculated to be logit(P) =-2.2791 + 0.7682x, + 0.001952(X2-100) where P is the probability of an MI during the four-year observation period. The variable X1 had values X10 (nonsmoker) and X1 1 (smoker). As in ordinary regression, the confidence band for the entire line is narrowest at the means of Xi and (X2-100) and spreads out the farther you go from the means. (See the paper for more details.) (a) The range of values of triglyceride levels is assumed to be from 0 to 550. Graph the probability of MI for smokers and nonsmokers separately. (b) The standard errors of regression coefficients for smoking and serum triglycende are 0.3137 and 0.001608, respectively. Test their significance.

Answer 1

a) Let's run the following code in R to graph the probability of MI for smokers and non-smokers separately:

Code:

X2 = seq(from=0,to=550,by=1)
p1 = -2.2791 + 0.7682*1 + 0.001952*(X2-100)
p2 = -2.2791 + 0.7682*0 + 0.001952*(X2-100)
py.s = exp(p1)/(1+exp(p1))
py.ns = exp(p2)/(1+exp(p2))
plot(x=X2,y=py.s)
plot(x=X2,y=py.ns)

Output:

For Smokers:

py.s 0.15 0.20 0.25 0.30 0.35 8

Non smokers:

髦3 CN 100 20N) 300 400 500 0 X2

b) We have to assume the sample size as we don't know the value. Let's take N = 100

	Coefficient	Standard Error	t	p-value
Smoking	0.7682	0.3137	2.448836468	0.016
Serum	0.001952	0.001608	1.213930348	0.229

Hence, the smoking variable is significant here and not the serum triglyceride variable.