To center or not to center? Investigating inertia with a multilevel autoregressive model

Abstract

Whether level 1 predictors should be centered per cluster has received considerable attention in the multilevel literature. While most agree that there is no one preferred approach, it has also been argued that cluster mean centering is desirable when the within-cluster slope and the between-cluster slope are expected to deviate, and the main interest is in the within-cluster slope. However, we show in a series of simulations that if one has a multilevel autoregressive model in which the level 1 predictor is the lagged outcome variable (i.e., the outcome variable at the previous occasion), cluster mean centering will in general lead to a downward bias in the parameter estimate of the within-cluster slope (i.e., the autoregressive relationship). This is particularly relevant if the main question is whether there is on average an autoregressive effect. Nonetheless, we show that if the main interest is in estimating the effect of a level 2 predictor on the autoregressive parameter (i.e., a cross-level interaction), cluster mean centering should be preferred over other forms of centering. Hence, researchers should be clear on what is considered the main goal of their study, and base their choice of centering method on this when using a multilevel autoregressive model.

Keywords: autoregressive models; centering; dynamics; inertia; multilevel models.

Figure 1

Illustration of within-person and between-person relationships between two variables. Each ellipse represents the data from a single person. Dashed lines represent the between-person slope (i.e., β_B), which may have a different sign as the within-person slope (Left panel), may be similar to the (average or fixed) within-person slope (Middle panel), or may be larger than the (average or fixed) within-person slope (Right panel).

Figure A1

Numerator of expectation, [that is, y = T(1 − ϕ)(1 − ϕ²_i) − (1 − 2ϕϕ_i + ϕ²_i)(1 − ϕ^T_i)] plotted against ϕ_i, for T = 40 and different values of ϕ (i.e., average ϕ_i). Note that only for ϕ = 0.95 the numerator becomes negative on a substantial portion of the interval (−1, 1). See text for implications.