Question

Suppose we fitted the following model: logit(P( Y = 1 )) = alpha +betaX, where X is age in years and Y is a binary variable with 1 = dead and 0 = alive. Which of the following is FALSE?

i.) exp(beta) is the odds ratio of death for a one year increase in age.

ii.) exp( alpha+ beta * 20) / (1 + exp( alpha+ beta * 20)) is the probability of death when age is 20.

iii.) beta is the difference in odds of death for a one year increase in age.

iv.) beta represents the association between age and death.

Please provide which of the above is a FALSE statement and the reason why you think the other three options are TRUE.

Answer 1

1. is false

2 is true as the logistic equation is log(p/(1-p))=a=b*x which means p/(1-p)=exp(a+b*x) which means p/(1-p)+1= 1+exp(a+b*x)

which means(1/(1-p))=1+exp(a+b*x) and thus 1-p=1/(1+exp(a+b*x)) and hence p for x=20 comes out to be 2.

3 is true. If you increase age by 1 then a+b*(x+1) =a+b*x+b increases by b and hence difference in odds of death is just equal to beta.

4 is true as Logistic regression is build to find out if there exists any association between age and death and this is driven by beta as beta=0 implies no association etc.