Pattern Mixture Models for Quantifying Missing Data Uncertainty in Longitudinal Invariance Testing

Abstract

Many psychology applications assess measurement invariance of a construct (e.g., depression) over time. These applications are often characterized by few time points (e.g., 3), but high rates of dropout. Although such applications routinely assume that the dropout mechanism is ignorable, this assumption may not always be reasonable. In the presence of nonignorable dropout, fitting a conventional longitudinal factor model (LFM) to assess longitudinal measurement invariance can yield misleading inferences about the level of invariance, along with biased parameter estimates. In this article we develop pattern mixture longitudinal factor models (PM-LFMs) for quantifying uncertainty in longitudinal invariance testing due to an unknown, but potentially nonignorable, dropout mechanism. PM-LFMs are a kind of multiple group model wherein observed missingness patterns define groups, LFM parameters can differ across these pattern-groups subject to identification constraints, and marginal inference about longitudinal invariance is obtained by pooling across pattern-groups. When dropout is nonignorable, we demonstrate via simulation that conventional LFMs can indicate longitudinal noninvariance, even when invariance holds in the overall population; certain PM-LFMs are shown to ameliorate this problem. On the other hand, when dropout is ignorable, PM-LFMs are shown to provide results comparable to conventional LFMs. Additionally, we contrast PM-LFMs to a latent mixture approach for accommodating nonignorable dropout—wherein missingness patterns can differ across latent groups. In an empirical example assessing longitudinal invariance of a harsh parenting construct, we employ PM-LFMs to assess sensitivity of results to assumptions about nonignorable missingness. Software implementation and recommendations for practice are discussed.

Keywords:

• longitudinal factor model
• longitudinal invariance
• nonignorable missing data
• pattern mixture model

ACKNOWLEDGMENTS

The author would like to thank Kristopher J. Preacher for helpful comments on a previous version of this article.

Notes

¹ Note that Hafez et al.’s (2015) model regresses $m_{it}$ only on ${\eta }_{i(t-1)}$ and involves p = 1, thus requiring T − 1 dimensions of integration.

² Although we use the term pattern mixture model here to be consistent with prior literature, note that the groups in this model are observed, rather than latent, unlike the conventional mixture models discussed later in this article and elsewhere in this special issue.

³ Imposing this equality constraint parallels the approach of Allison (1987), which has been employed previously in other modeling contexts.

⁴ The online Appendix is available at http://www.vanderbilt.edu/peabody/sterba/.

⁵ Many recent applications of invariance testing using LFM continue to analyze only complete-case data despite the fact that the FIML estimation methods used did not require this (e.g., Brydges et al., 2014; King, 2011; Mason et al., 2013; Mogos et al., 2015; Richerson et al., 2014; Varni et al., 2008; Wang et al., 2012; Wang & Su, 2013). This is akin to looking at results for only one specific missingness pattern-group and, as such, these authors would be unaware of whether results differ across missingness pattern-group.

⁶ Although not demonstrated here, when fitting a conventional LFM in the presence of nonignorable missingness, it would also be possible for non-MI across missingness group to masquerade marginally as MI across time.

⁷ Note that families came from 17 sites, but site codes were not made available in the data file “due to confidentiality concerns.” See http://doi.org/10.3886/ICPSR03804.v5.

⁸ Here we continue to define ${\mathbf{m}}_i$ as consisting of T – 1 dropout indicators, for consistency with the earlier presentation. If we were considering intermittent missingness for the set of J items, we could redefine our vector ${\mathbf{m}}_i$ of binary missingness indicators to be of dimension T × 1 where element $m_{it}$ = 1 when ${\mathbf{y}}_{it}$ is missing and $m_{it}$ = 0 when ${\mathbf{y}}_{it}$ is observed. (Similar results for the latent missingness class approach were obtained when ${\mathbf{m}}_i$ did vs. did not include intermittent missingness.) Finally, note that we are not discussing item-specific intermittent missingness here (e.g., missingness on Item 3 but not on Items 1, 2, and 4 at time t). This kind of missingness, if present, would still be assumed ignorable.