Abstract Additive isotonic models generalize linear models by replacing lines with isotonic (nondecreasing) transformations. Fitted transformations of several explanatory variables are added together and then transformed by a known function to yield fitted values of the response variable. The isotonic transformations are chosen to minimize an explicit criterion, such as the negative log-likelihood, by an algorithm that optimizes one transformation at a time while adjusting for the current fitted values of the others, cycling until the criterion converges. This approach can be used in various situations, notably for generalizing ordinary linear regression and linear logistic regression. At each step of the algorithm, the needed optimal isotonic transformation is found using a simple generalization of the standard pool-adjacent-violators algorithm (Ayer, Brunk, Ewing, Reid, and Silverman 1955). The fitted transformations are always made up of flat steps, so the technique is useful for finding optimal strati...