Logistic regression

Question 1

OLS assumptions violated

Answer 1

1) Non-normal error term (b/c bimodal distribution)
2) Non-constant error variance
3) Problem with range of outcome (OLS assumes -infinity to +infinity)

Question 2

Odds of observing outcome

Answer 2

Odds(1) = π / (1-π)

Odds bound (0 to infinity)

By taking natural logarithm of odds, transform response probability to be bound on (-infinity to +infinity)

Question 3

Logit link function

Answer 3

Logit(π / (1-π)) = log(e) (π / (1-π))

Question 4

Equation with 1 predictor

Answer 4

logit(π / (1-π)) = β0 + β1X1 + ε

Question 5

Interpreting model

Answer 5

If X is continuous, and β >0, say: Log odds of event occurring increase as X increases.

If X is continuous, and β

Question 6

Odds ratio

Answer 6

exp(β), or e&β

Interpret: As X increases by 1 unit, the odds of seeing the outcome increase (or decrease) by exp(β)

If exp(β) is 1, odds of observing outcome &y-1) is the same for individuals regardless of level of x.

If exp(β) 1, odds of observing outcome is exp(β)-1 higher for individuals with x=1 compared to individuals with x=0.

Question 7

Significance of predictors

Answer 7

Test in same fashion as the OLS model. β / se(β) as t-statistic or z-statistic.

Or calculate Wald chi square statistic: β^2 / (se(β))^2.

Question 8

Other link functions

Answer 8

Probit: Inverse of the cumulative normal distribution function.

Log-log link:

How to choose link function? If p(π) relatively large, doesn’t matter. If very rare events (p 95%), choice makes a difference.