LOGISTIC REGRESSION Binary dependent variable (pass-fail) Odds ratio: p/(1-p) eg. 1/9 means 1 time in 10 pass, 9 times fail Log-odds ratio: y = ln[p/(1-p)] produces a new variable that has known characteristics (normal ogive): Probability of occurrence 0 1 predictor

LOGISTIC MODEL y = b 1 x + b 0 Looks like regular regression, uses maximum likelihood iterative solution to get estimates of b-weights b 1 interpretation: change in 1 SD in x produces b1 change in logit (log-odds)

LOGISTIC MODEL For a b-weight of 2, the odds change by a factor of e 2 or ( ) or If the original odds were 1:1 (equivalent to an intercept of 0, or not knowing who is in one group or the other), then the predictor, say TAKS score, changes them to over 7:1 predicting whether a child will pass or fail by increasing the TAKS score 1 SD (if TAKS is in standardized form with mean 0 and SD 1).

MULTIPLE LOGISTIC MODEL y = b 1 x 1 + b 2 x 2 + … + b 0 Each b-weight is the partial contribution to changing the log-odds

LOGISTIC MODEL FIT Likelihood function L is used, -2lnL is chi- square distributed Changes in the log likelihood function are equivalent to changes in chi square for the number of degrees of freedom added (or subtracted) from a regression’s predictors Then a fit of R 2 = lnL model – lnL null lnL model –lnL perfect Which is chi-square with p (#predictors) df

LOGISTIC MODEL FIT Nagelkerke R-square statistic Based on Likelihood function compares obtained R-square to maximum possible R-square: R 2 N = [ 1 – {L(0)/L(fit)} 2/n ]/ { 1 – L(0) 2/n }