We find that in zero-truncated count data (y=1,2,...), individuals often gain information at first observation (y=1), leading to a common but unaddressed phenomenon of "one-inflation". The current standard, the zero-truncated negative binomial (ZTNB) model, is misspecified under one-inflation, causing bias and inconsistency. To address this, we introduce the one-inflated zero-truncated negative binomial (OIZTNB) regression model. The importance of our model is highlighted through simulation studies, and through the discovery of one-inflation in four datasets that have traditionally championed ZTNB. We recommended OIZTNB over ZTNB for most data, and provide estimation, marginal effects, and testing in the accompanying R package oneinfl.PDF: One-inflated zero-truncated count regression models.pdf
