A Simple Risk-Adjusted Exponentially Weighted Moving Average

Abstract

In such contexts as medical monitoring there is a need for a simple procedure to estimate the level of a smoothly changing dynamic process while adjusting for risk factors associated with heterogeneous nonnormal observations. Standard methods exist for when the data are normal, and simple methods have been developed for nonnormal data without covariates. The dynamic generalized linear model (DGLM) may be considered a standard for nonnormal data with covariates, but perhaps one that is overcomplicated for many contexts. We propose adapting the standard exponentially weighted moving average (EWMA) to take into account the effect of risk factors. By approximating the correct exponential family likelihood, we derive a risk-adjusted EWMA (RA–EWMA) that is essentially a standard EWMA applied to “pseudo-observations,” which are the original observations adjusted for differential risk. The RA–EWMA can be expressed as a type of filter, in which current estimate = previous estimate + discounted predictive error. We review Bayesian state-space models related to the EWMA and examine the properties of each. The RA–EWMA and DGLM are compared algebraically, through simulation and an example. In the example, the expected mortality after surgery by a particular surgeon is allowed to vary over time and is estimated through the RA–EWMA. Each binary observation is adjusted for the effect of patient-specific risk factors to standardize information fed into the estimator for the expected mortality for a “baseline” patient. The tool used for adjustment can be used to provide future patient-specific preoperative risk assessments. We conclude that the RA–EWMA performs similarly to the DGLM, is conceptually and computationally simpler by virtue of having fewer and simpler stages, and has an intuitive appeal to a wide variety of stakeholders.