Statistical methods are commonly used to monitor quality in manufacturing processes. We consider a set of problems where the probability that a unit is defect, i.e., the defect probability, is influenced by a large number of sub-processes, and where the overall aim is to monitor the defect probability and link abnormalities in quality to observed production variables. We developed a simulation framework for studying the performances of statistical methods used to solve the considered problem. Fourteen prediction procedures were obtained by combining six prediction methods (linear regression, logistic regression, LASSO, penalized logistic regression, support vector machines, gradient boosting decision trees) and two pre-processing procedures. These prediction procedures were evaluated on four types of simulated datasets with different relationships between the explanatory variables and the defect probabilities. Additionally, two established methods for variable selection were compared to a novel method called mixed moments selection (MMS). MMS was more robust than the other methods, performed well on all dataset types, and can easily be combined with any type of prediction method. Additionally, it was shown that it can be advantageous to complement the original explanatory variables with their squared values prior to analyzing the data. Overall, a procedure combining MMS, including additional quadratic terms and using PLR had the best performance. The proposed framework can be applied to evaluate any type of prediction procedure for the general problem we consider. This would increase the understanding of different procedures and facilitate the selection of procedures for a specific problem.