Difference between Probit and Logit
- Economists prefer Probit because by assuming error term is standard normal distribution, findings can be generalized easily
- In industry/corporate and health, Logit is preferred
- when the error term has a logistic distribution, logit is better. When the error term has a standard normal distribution, probit is better.
Method of Maximum Likelihood Estimation
finds the coefficients of the independent variables that maximize prediction of the dependent variable; produces MLEs (coefficients)
Logit parameter interpretation methods
- Odds ratios
- Log odds
- marginal effects
Odds Ratio
P/(1-P); odds of success divided by odds of failure; not the same as PROBABILITY of success.
Odds Ratio Method
find e^B; e^B is your multiplicative factor of change. To find percent change: (e^B - 1)*100.
Odds Ratio Method Interpretation: age predicts divorce rate, coefficient = 3.12
For every additional unit of age, odds of divorce (success) changes by a multiplicative factor of 22.65, ceteris paribus.
Log Odds
log(odds); makes interpretation odds less than and greater than one.
Log Odds Method
no operation, coefficients translate directly to interpretation.
Log Odds Method Interpretation: age predicts divorce rate, coefficient = 3.12
For every additional unit of age, the log odds of divorce (success) change by a multiplicative factor of 3.12.
Marginal Effect Method
- Find mean values of all independent variables; categorical variables have a mean value of [percent frequency of class / 100 ]
- Find probability of success at the mean values: M = e^(B0 + B1X1mean + B2X2mean …); P = M/(1+M)
- plug coefficient into equation to find “Marginal Effect at the Mean” for a specific variable: MEM = (B * P) * (1 - P)
- (Optional) multiply the MEM * 100 to find its percent effect on the probability of success.
Marginal Effect Method Interpretation; age predicts divorce rate, MEM coefficient = 0.256
For every additional unit of age, the probability of divorce (success) increases by a multiplicative factor of 0.256, ceteris paribus.
Probit parameter interpretation methods
- z-score
- marginal effect
z-score method
no operation; coefficients used directly.
z-score method interpretation: age predicts divorce rate, coefficient = 3.12
For every additional unit of age, the z-score of divorce (success) increases by a factor of 3.12, ceteris paribus.
Marginal Effect Method
- Z = B0 + B1X1 + B2X2… Find Z at the mean by plugging average from each independent variable.
- Plug Z into the nasty phi equation. o(Z) = (1/root(2pie))e^-(Z^2/2)
- o(Z) * B; just o(Z) times the coefficient.
Marginal Effect Method Interpretation: age predicts divorce rate, MEM divorce rate = 0.112
For every additional of age, the probability of divorce (success) changes by a multiplicative factor of 0.112, ceteris paribus.
Linear Regression for binary categorical dependent variables
can be used when avg. success probability is between 3% and 98%. The middle of the curve is close enough to straight. Otherwise, the main problem is that its predictions for certain values might not make sense.
