Nonlinear Estimation Procedures - Maximum Likelihood and Probit/Logit Models

The maximum likelihood function has been "worked out" for probit and logit regression models. Specifically, the loss function for these models is computed as the sum of the natural log of the logit or probit likelihood L1 so that:

log(L1) = Sin= 1 [yi*log(pi ) + (1-yi )*log(1-pi )]

where

log(L1)

is the natural log of the (logit or probit) likelihood (log-likelihood) for the current model

yi

is the observed value for case i

pi

is the expected (predicted or fitted) probability (between 0 and 1)

The log-likelihood of the null model (L0), that is, the model containing the intercept only (and no regression coefficients) is computed as:

log(L0) = n0*(log(n0/n)) + n1*(log(n1/n))

where

log(L0)

is the natural log of the (logit or probit) likelihood of the null model (intercept only)

n0

is the number of observations with a value of 0 (zero)

n1

is the number of observations with a value of 1

n

is the total number of observations

These formulas are provided here for your reference; maximum likelihood estimation will automatically be used for probit and logit models so you do not need to type in this complex formula into the loss function editor.

Note: Generalized Linear/Nonlinear Model (GLZ). You can also use the Generalized Linear/Nonlinear Model (GLZ) module to analyze binary response variables. GLZ is an implementation of the generalized linear model and allows you to compute a standard, stepwise, or best subset multiple regression analysis with continuous as well as categorical predictors, and for binomial or multinomial dependent variables (probit regression, binomial and multinomial logit regression; see also Link Functions). In general, the estimation algorithms implemented in the Generalized Linear Models (GLZ) module are more efficient, and STATISTICA only includes these models here for compatibility purposes.