Back to the Future: Modeling Time Dependence in Binary Data

TLDR

Standard practice models time dependence in binary data with time dummies or splines, yet spline complexity often leads researchers to use inappropriate knots and overlook substantive temporal dependence. The authors propose a simple alternative: including t, t², and t³ terms in the regression. The cubic polynomial approximation is trivial to implement, interpretable, avoids quasi‑complete separation, and readily accommodates nonproportional hazards. The cubic polynomial approach eliminates separation problems, outperforms time dummies, matches splines and autosmoothing in simulations, and when applied to Crowley and Skocpol’s data provides new support for the historical‑institutionalist perspective.

Abstract

Since Beck, Katz, and Tucker (1998), the standard method for modeling time dependence in binary data has been to incorporate time dummies or splined time in logistic regressions. Although we agree with the need for modeling time dependence, we demonstrate that time dummies can induce estimation problems due to separation. Splines do not suffer from these problems. However, the complexity of splines has led substantive researchers (1) to use knot values that may be inappropriate for their data and (2) to ignore any substantive discussion concerning temporal dependence. We propose a relatively simple alternative: including t, t 2 , and t 3 in the regression. This cubic polynomial approximation is trivial to implement—and, therefore, interpret—and it avoids problems such as quasi-complete separation. Monte Carlo analysis demonstrates that, for the types of hazards one often sees in substantive research, the polynomial approximation always outperforms time dummies and generally performs as well as splines or even more flexible autosmoothing procedures. Due to its simplicity, this method also accommodates nonproportional hazards in a straightforward way. We reanalyze Crowley and Skocpol (2001) using nonproportional hazards and find new empirical support for the historical-institutionalist perspective.