Nonparametric and Semiparametric Mixed Model Methods for Phase I Profile Monitoring

Profile monitoring is an approach in quality control best used where the process data follow a profile (or curve). The majority of previous studies in profile monitoring focused on the parametric modeling of either linear or nonlinear profiles, with both fixed and random-effects, under the assumption of correct model specification. Our work considers those cases where the parametric model for the family of profiles is unknown or, at least uncertain. Consequently, we consider monitoring profiles via two methods, a nonparametric (NP) method and a semiparametric procedure that combines both parametric and NP profile fits. We refer to our semiparametric procedure as mixed model robust profile monitoring (MMRPM). Also, we incorporate a mixed model approach to both the parametric and NP model fits to account for the autocorrelation within profiles and to deal with the collection of profiles as a random sample from a common population. For each case, we propose two Hotelling’s T² statistics for use in Phase I analysis to determine unusual profiles, one based on the estimated random effects and one based on the fitted values and obtain the corresponding control limits. Our simulation results show that our methods are robust to the common problem of model misspecification of the user’s proposed parametric model. We also found that both the NP and the semiparametric methods result in charts with good abilities to detect changes in Phase I data, and in charts with easily calculated control limits. The proposed methods provide greater flexibility and efficiency when compared to parametric methods commonly used in profile monitoring for Phase I that rely on correct model specification, an unrealistic situation in many practical problems in industrial applications. An example using our techniques is also presented.