The Usefulness and Uselessness of the Decomposition of Tobit Coefficients

McDonald and Moffitt's (1980) decomposition method for Tobit coefficients disaggregates the total changes in a censored outcome variable into the changes in the probability of attaining noncensored values and the changes in the noncensored values themselves. McDonald and Moffitt, however, did not highlight two limitations of this decomposition, and sociological applications of the Tobit decomposition that do not acknowledge these limitations have been misleading. First, the decomposition ratio of the two effects depends on the choice of an observation point in the sample. Some sociological studies invalidly generalized a decomposition at the mean point of the sample to the whole sample. Second, once the observation point is fixed, the decomposition ratio is the same across all coefficients or across all predictors. That is, the two effects remain proportionally constant, but studies often erroneously assume they differ across predictors. Further implications are discussed in relation to other selection bias models.