A Multilevel AR(1) Model: Allowing for Inter-Individual Differences in Trait-Scores, Inertia, and Innovation Variance

Abstract

In this article we consider a multilevel first-order autoregressive [AR(1)] model with random intercepts, random autoregression, and random innovation variance (i.e., the level 1 residual variance). Including random innovation variance is an important extension of the multilevel AR(1) model for two reasons. First, between-person differences in innovation variance are important from a substantive point of view, in that they capture differences in sensitivity and/or exposure to unmeasured internal and external factors that influence the process. Second, using simulation methods we show that modeling the innovation variance as fixed across individuals, when it should be modeled as a random effect, leads to biased parameter estimates. Additionally, we use simulation methods to compare maximum likelihood estimation to Bayesian estimation of the multilevel AR(1) model and investigate the trade-off between the number of individuals and the number of time points. We provide an empirical illustration by applying the extended multilevel AR(1) model to daily positive affect ratings from 89 married women over the course of 42 consecutive days.

Notes

Instead of using the innovation variance in the multivariate normal distribution, we could have decided to use the logarithm of this variance to ensure that no negative variances can occur. However, we believe this to be less intuitive than considering the variance itself, and moreover we do not expect computational problems because innovation variances are expected to be clearly larger than zero in the data.

In MLWin (Rasbash, Charlton, Browne, Healy, & Cameron, 2009), this second limitation can be circumvented by using syntax, but this implies one can no longer make use of the user friendly interface of the program.

Note that this approach of using the ML estimates in the IW prior is equivalent to the training sample approach suggested by O’Hagan (1995), where part of the data is used to obtain a prior for the analysis of the rest of the data.

This is relevant in the creation of the data sets as negative variances will make it impossible to generate data. However, it has no further practical consequences.

We do not have an analytical expression of the within-person variance. Instead, we simulated a data set consisting of 10,000,000 persons, given a particular set of parameter values, and determined the average within-person variance. This way we determined that when φ = .20, τ²_φ = .01, σ² = 3.00, and , the total within-person variation is equal to 3.198, 3.170, and 3.141 for values of.6, 0, and -.6 respectively.

During data generation it was evaluated whether all parameter values fell into a permissible range, that is, no individual innovation variances smaller than 0 and no values of φ_i greater than ∣1∣ to ensure stationarity. If parameter values fell outside these ranges, those data were discarded and a new data set was generated; however this was rarely needed (i.e., for less than 4–5% of generated data sets).

A problem we encountered with the use of the ML estimates in the IW prior is that when the ML estimates are very inaccurate, the WinBUGS analysis crashes. This occurred in one out of every 300 to 500 data sets. In practice, if this problem occurs, the user should change the scale values; in the current simulation study, we solved this by preventing the τ²_φ estimate in the scale matrix (R) of the IW prior to become too small (by substituting the value.005 for the ML estimate of τ²_φ if this estimate is smaller than this boundary value) and by producing a new data set in case WinBUGS crashed.

The reason for this large standard error is that it had to be computed from the standard errors of c and φ, since μ is not directly estimated in this approach: To this end, we used the following equation from Mood, Graybill, and Boes (1985) for the variance of a quotient . This extra estimation step forms an additional source of uncertainty, leading to large standard errors and thus coverage rates that are always 1.

Tables containing these correlations and the individual coverage rates can be obtained from the first author.