Beta Regression with Autoregressive Errors for Interrupted Time Series Analysis of Proportion and Rate Outcomes: A Simulation Study

Abstract

Interrupted time series analyses (ITSA) of proportion and rate outcomes are frequently estimated using ordinary least squares regression despite the bounded nature of these outcomes. When methods appropriate for bounded outcomes are used, the standard approach is a quasi-likelihood generalized linear model (GLM) with heteroskedasticity- and autocorrelation-consistent (HAC) standard errors. However, no existing estimator jointly models the beta-distributed conditional density and autoregressive (AR) error structure. We introduce betark, a Stata implementation of a joint conditional maximum likelihood estimator for beta regression with AR(k) errors based on a recursive substitution that yields a closed-form conditional beta likelihood with autoregressive dependence of arbitrary order. Unlike two-stage approaches, betark jointly estimates the mean, precision, and AR(k) coefficients in a single likelihood, so reported standard errors directly account for autocorrelation without separate correction. A Monte Carlo study compared betark with a quasi-binomial GLM using Newey-West HAC standard errors across AR(1)-AR(3) processes, three series lengths, and four effect sizes in a single-group ITSA design. Both methods were essentially unbiased, but betark produced better-calibrated inference than GLM+HAC in most scenarios, with the largest gains under highly persistent autocorrelation, where GLM+HAC Type I error exceeded 60% for short series. Misspecifying the AR order by one lag and varying the starting mean and pre-intervention trend had only modest effects on performance. However, betark's own Type I error remained elevated under highly persistent AR(3) processes even for the longest series examined.