ISYE 6414 - Final questions and answers all are graded A+

Logistic Regression - Answer-Commonly used for modeling binary response data. The response variable is a binary variable, and thus, not normally distributed. In logistic regression, we model the probability of a success, not the response variable. In this model, we do not have an error term g-function - Answer-We link the probability of success to the predicting variables using the g link function. The g function is the s-shape function that models the probability of success with respect to the predicting variables The link function g is the log of the ratio of p over one minus p, where p again is the probability of success Logit function (log odds function) of the probability of success is a linear model in the predicting variables The probability of success is equal to the ratio between the exponential of the linear combination of the predicting variables over 1 plus this same exponential Odds of a success - Answer-This is the exponential of the Logit function Logistic Regression Assumptions - Answer-Linearity: The relationship between the g of the probability of success and the predicted variable, is a linear function. Independence: The response binary variables are independently observed Logit: The logistic regression model assumes that the link function g is a logit functionLinearity Assumption - Answer-The Logit transformation of the probability of success is a linear combination of the predicting variables. The relationship may not be linear, however, and transformation may improve the fit The linearity assumption can be evaluated by plotting the logit of the success rate versus the predicting variables. If there's a curvature or some non-linear pattern, it may be an indication that the lack of fit may be due to the non-linearity with respect to some of the predicting variables Logistic Regression Coefficient - Answer-We interpret the regression coefficient beta as the log of the odds ratio for an increase of one unit in the predicting variable We do not interpret beta with respect to the response variable but with respect to the odds of success The estimators for the regression coefficients in logistic regression are unbiased and thus the mean of the approximate normal distribution is beta. The variance of the estimator does not have a closed form expression Model parameters - Answer-The model parameters are the regression coefficients. There is no additional parameter to model the variance since there's no error term. For P predictors, we have P + 1 regression coefficients for a model with intercept (beta 0). We estimate the model parameters using the maximum likelihood estimation approach Response variable - Answer-The response data are Bernoulli or binomial with one trial with probability of success MLE - Answer-The resulting log-likelihood function to be maximized, is very complicated and it is nonlinear in the regression coefficients beta 0, beta 1, and beta pMLE has good statistical properties under the assumption of a large sample size i.e. large N For large N, the sampling distribution of MLEs can be approximated by a normal distribution The least square estimation for the standard regression model is equivalent with MLE, under the assumption of normality. MLE is the most applied estimation approach Parameter estimation - Answer-Maximizing the log likelihood function with respect to beta0, beta1 etc in closed (exact) form expression is not possible because the log likelihood function is a non-linear function in the model parameters i.e. we cannot derive the estimated regression coefficients in an exact form