STATS 2 Notes Final Exam

Categorical Data Analysis Note -8

Logistic Regression Model with Categorical Predictor- II

• Single Dummy Variable Predictor

Consider the case that independent variable, x, is dichotomous and coded as 0 and 1,
so the difference in logit is

g(1) − g(0) = β0 + β1 (1) − (β0 + β1 (0)) = β1

• Now, the odds of the outcome being present for individuals with x = 1 , and for
individuals with x = 0 are
π(1) π(0)
odds for(x = 1) = odds for(x = 0) =
1 − π(1) 1 − π(0)

• The odds ratio, is defined as the ratio of the odds for x = 1 to the odds for x = 0 and
is defined as
π(1)
1−π(1)
OR = π(0)
1−π(0)
   
exp(β0 +β1 ) 1
1+exp(β0 +β1 )
/ 1+exp(β0 +β1 )
=    
exp(β0 ) 1
1+exp(β0 )
/ 1+exp(β0 )

exp(β0 + β1 )
= = exp(β0 + β1 − β0 )
exp(β0 )
= exp(β1 )

• The relationship between the odds ratio and the logistic regression coefficient when
independent variable is dichotomous.

OR = exp(β1 )

• It is an indicate to show how much more likely (or unlikely) is that outcome to be
present among those with x = 1 than among those with x = 0.

• Odds ratio can be estimated by the fact that

ln(OR)
d = β̂1