WWS 510

Logistic regression interpretation stata. Let us consider Example 16.1 in Wooldridge (2010), concerning school and employment decisions for young men. The data contain information on employment and schooling for young men over several years. We will work with the data for 1987. The outcome is status, coded 1=in school, 2=at home (meaning not in school and not working), and 3=working. The predictors are education, a quadratic on work experience, and an indicator for black. We read the data from the Stata website, keep the year 1987, drop missing values, label the outcome, and fit the model. The results agree exactly with Table 16.1 in Wooldridge (2010, page 645). Relative Probabilities. Let us focus on the coefficient of black in the work equation, which is 0.311. Exponentiating we obtain. Thus, the relative probability of working rather than being in school is 37% higher for blacks than for non-blacks with the same education and work experience. (Relative probabilities are also called relative odds.) A common mistake is to interpret this coefficient as meaning that the probability of working is higher for blacks. It is only the relative probability of work over school that is higher. To obtain a fuller picture we need to consider the second equation as well. The coefficient of black in the home equation is 0.813. Exponentiating, we obtain. Thus, the relative probability of being at home rather than in school for blacks is more than double the corresponding relative probability for non blacks with the same education and work experience. In short, black is associated with an increase in the relative probability of work over school, but also a much large increase in the relative probability of home over school. What happens with the actual probability of working depends on how these two effects balance out. Marginal Effects (Continuous) To determine the effect of black in the probability scale we need to compute marginal effects, which can be done using continuous or discrete calculations. The continuous calculation is based on the derivative of the probability of working with respect to a predictor. Let \( \pi_ = \Pr\ \) denote the probability that the i-th observation follows on the j-th category, which is given by \[ \pi_ = \frac > > \] where \( \beta_j = 0\) when j is the baseline or reference outcome, in this case school. Taking derivatives w.r.t. the k-th predictor we obtain, after some simplification \[ \frac > > = \pi_ ( \beta_ - \sum_r \pi_ \beta_ ) \] noting again that the coefficient is zero for the baseline outcome. To compute these we predict the probabilities and then apply the formula. We find that the average marginal effect of black on work is actually negative: -0.0406. This means that the probability of working is on average about four percentage points lower for blacks than for non-blacks with the same education and experience. Stata can do this calculation using the dydx() option of the margins command. Here's the marginal effect for work: This agrees exactly with our hand calculation. Note that Stata uses the derivative for continuous variables and a discrete difference for factor variables, which we consider next. Marginal Effects (Discrete) For the discrete calculation we compute predicted probabilities by setting ethnicity to black and then to non-black and averaging: We find that the average probability of working is 0.7275 if black and 0.7710 if not black, a difference of -0.0435, so the probability of working is on average just over four percentage points lower for blacks. Stata can calculate the predictive margins if you specify black as a factor variable when you fit the model, and then issue the command margins black . This only works for factor variables. The marginal effect can then be obtained as a discrete difference. These results agree exactly with our hand calculations. The take away conclusion here is that multinomial logit coefficients can only be interpreted in terms of relative probabilities, to reach conclusions about actual probabilities we need to calculate continuous or discrete marginal effects. © 2017 Germán Rodríguez, Princeton University.