Foundations of Logistic Regression:
(1) What values can the response variable Y take in logistic regression, and hence what statistical distribution does Y follow?The response variable can take the value of either a 1 or a 0, and follows a binomial distribution.
(2) How are the parameters estimated in logistic regression? Is this different from how the parameters are estimated in Ordinary Least Squares (OLS) regression?Logistic regressionparameters are estimated utilizing the maximum likelihood method, which is the same underlying method for OLS regression. However, with logistic regression, an iterative method conducted via software because it is more complicated to estimate nonlinear parameters β0 and β1. This differs from OLS, because the OLS method is by differentiating the sum of squared deviations. This is an easier method because those deviations are linear in relation to β.
Coefficient estimates in logistic regression can also be found by utilizing the following methods
- noniterative weighted least squares
- discriminant function analysis
(3) How do we define a "residual" in logistic regression, and how is it computed?
In Logistic Regression, the Deviance fills the same role as the residual sumo f squares in linear regression.
This is computed by calculating what is known as the likelihood-ratio test, Illustrated below:
D = -2ln [(likelihood of the fitted model)/(likelihood of the saturated model)]
Model 1: Let's consider the logistic regression model, which we will refer to as Model 1, given by
log(pi / [1-pi]) = 0.25 + 0.32*X1 + 0.70*X2 + 0.50*X3 (M1),
where X3 is an indicator variable with X3=0 if the observation is from Group A and X3=1 if the observation is from Group B. The likelihood value for this fitted model on 100 observations is 0.0850.
(4) For X1=2 and X2=1 compute the log-odds for each group, i.e. X3=0 and X3=1. Group A (X3=0);
Group B (X3=1);
(5) For X1=2 and X2=1 compute the odds for each group, i.e. X3=0 and X3=1.
(6) For X1=2 and X2=1 compute the probability of an event for each group, i.e. X3=0 and X3=1.
(7) Using the equation for M1, compute the relative odds associated with X3, i.e. the relative odds of Group B compared to Group A.
(8) Use the odds for each group to compute the relative odds of Group B to Group A. How does this number compare to the result in Question #7. Does this make sense?
Model 2: Now let's consider an alternate logistic regression model, which we will refer to as Model 2, given by
log(pi / [1-pi]) = 0.25 + 0.32*X1 + 0.70*X2 + 0.50*X3 + 0.1*X4 (M2),
where X3 is an indicator variable with X3=0 if the observation is from Group A and X3=1 if the observation is from Group B. The likelihood value from fitting this model to the same 100 observations as M1 is 0.0910.
(9) Use the G statistic to perform a likelihood ratio test of nested models for M1 and M2. State the hypothesis that is being tested, compute the test statistic, and test the statistical significance using a critical value for alpha=0.05 from Table A.3 on page 375 in Regression Analysis By Example. From these results should we prefer M1 or M2?