Mplus Users Guide v6.. - Muthén & Muthén

Special Modeling Issuesgiven x using the two thresholds t 1 and t 2 and the single probit regressioncoefficient b,P (u = 0 | x) = F (t 1 - b*x),P (u = 1 | x) = F (t 2 - b*x) - F (t 1 - b*x),P (u = 2 | x) = F (- t 2 + b*x).CALCULATING PROBABILITIES FROM LOGISTICREGRESSION COEFFICIENTSFollowing is a description of how to translate logistic regressioncoefficients to probability values. Also described is how to interpret thecoefficient estimates in terms of log odds, odds, and odds ratios. For atreatment of logistic regression for binary, ordered categorical (ordinal),and unordered categorical (nominal) variables, see Agresti (1996, 2002)and Hosmer and Lemeshow (2000).An odds is a ratio of two probabilities. A log odds is therefore the log ofa ratio of two probabilities. The exponentiation of a log odds is an odds.A logistic regression coefficient is a log odds which is also referred to asa logit.For a binary dependent variable u, the logistic regression modelexpresses the probability of u given x as,(1) P (u = 1 | x) = exp (a + b*x) / (1 + exp (a + b*x) )= 1 / (1 + exp (-a – b*x)),where P (u = 0 | x) = 1 – P (u = 1 | x). The probability expression in (1)results in the linear logistic regression expression also referred to as alog odds or logit,log [P (u = 1 | x) / P (u = 0 | x)] = log [exp (a + b*x)] = a + b*x,where b is the logistic regression coefficient which is interpreted as theincrease in the log odds of u = 1 versus u = 0 for a unit increase in x.For example, consider the x values of x 0 and x 0 + 1. The correspondinglog odds are,log odds (x 0 ) = a + b*x 0 ,441