A Bayesian Vector Autoregressive Model with Nonignorable Missingness in Dependent Variables and Covariates: Development, Evaluation, and Application to Family Processes

Abstract

Intensive longitudinal designs involving repeated assessments of constructs often face the problems of nonignorable attrition and selected omission of responses on particular occasions. However, time series models, such as vector autoregressive (VAR) models, are often fit to these data without consideration of nonignorable missingness. We introduce a Bayesian model that simultaneously represents the over-time dependencies in multivariate, multiple-subject time series data via a VAR model, and possible ignorable and nonignorable missingness in the data. We provide software code for implementing this model with application to an empirical data set. Moreover, simulation results comparing the joint approach with two-step multiple imputation procedures are included to shed light on the relative strengths and weaknesses of these approaches in practical data analytic scenarios.

Keywords:

• Intensive longitudinal data
• Bayesian vector autoregressive model
• Multiple imputation
• Nonignorable missing data

Notes

¹ To start the procedure, all missing observations are filled in using random draws with replacement from all observed values. Then, for the first iteration, ${\varphi }_1^1$ is drawn from the distribution $P({\varphi }_1^1|{\mathbf{y}}_1^{obs},{\mathbf{Y}}_{-1}^0,{\mathbf{X}}^0,\mathbf{R})$. Missing values in ${\mathbf{y}}_1$, ${\mathbf{y}}_1^{miss}$, are then filled in by drawing values from $P({\mathbf{y}}_1|{\mathbf{y}}_1^{obs},{\mathbf{Y}}_{-1}^0,{\mathbf{X}}^0,\mathbf{R},{\varphi }_1^1)$, the posterior predictive distribution of ${\mathbf{y}}_1$ conditioned on ${\mathbf{y}}_1^{obs},{\mathbf{Y}}_{-1}^0,{\mathbf{X}}^0,\mathbf{R}$, and ${\varphi }_1^1$. Here, we use superscript $k$ to denote data sets and parameter estimates from the $k^{th}$ iteration, with $k=0$ denoting the original data sets or initial starting values of the parameters. Similar procedure is subsequently performed to generate predicted values for ${\mathbf{y}}_2^{miss}$, only that imputed values for ${\mathbf{y}}_1$ from the previous step are used in the prediction process. The first iteration ends when missing observations are filled in for all variables. The procedure is repeated for $K$ iterations to result in one set of data with imputed values. The whole procedure is repeated multiple times to generate multiple imputed data sets and correspondingly, multiple sets of parameter and standard error estimates for subsequent pooling (van Buuren, 2012).

² The mixture models factor the full-data model as:

$P(\mathbf{Y},\mathbf{R}|\mathbf{X},\omega )=P(\mathbf{Y}|\mathbf{R},\mathbf{X},\omega )P(\mathbf{R}|\mathbf{X},\omega ),$

With this model, the relations between $\mathbf{Y}$ and $\mathbf{X}$ are conditioned on different missing data patterns. In contrast, the shared parameter approach assumes a multilevel structure and models random effects $\mathbf{b}$ jointly with $\mathbf{Y}$ and $\mathbf{R}$ with following general model:

$P(\mathbf{Y},\mathbf{R}|\mathbf{X},\omega )=\int P(\mathbf{Y},\mathbf{R},\mathbf{b}|\mathbf{X},\omega )d\mathbf{b}.$

³ The version of the R package dynr we used in this study contained a small error in handling the uncertainty of the missing data, which resulted in slight overestimation of the process noise variances, and higher corresponding biases for these particular parameters under the Partial MI approach. Without this error, biases for all process noise-related parameters would likely be even lower under the Partial MI, but conclusions involving other parameters should remain the same.

⁴ The standard deviation of all point estimates on a parameter across MC runs.

⁵ Since neither the LD method nor the two-step partial MI method explicitly specify and estimate a missingness model, no missing data parameter estimates were available from these methods.

Additional information

Funding

This work was supported by the National Center for Advancing Translational Sciences [UL TR000127]; The Intensive Longitudinal Health Behavior Cooperative Agreement Program funded by the National Institutes of Health under Award Number U24AA027684; National Institutes of Health [R01GM105004]; National Science Foundation [IGE-1806874]; and Penn State Quantitative Social Sciences Initiative.