Parameter uncertainty in structural equation models: Confidence sets and fungible estimates.

Abstract

Current concerns regarding the dependability of psychological findings call for methodological developments to provide additional evidence in support of scientific conclusions. This paper highlights the value and importance of two distinct kinds of parameter uncertainty which are quantified by confidence sets (CSs) and fungible parameter estimates (FPEs; T. Lee, MacCallum, & Browne, in press); both provide essential information regarding the defensibility of scientific findings. Using the structural equation model, we introduce a general perturbation framework based on the likelihood function that unifies CSs and FPEs and sheds new light on the conceptual distinctions between them. A targeted illustration is then presented to demonstrate the factors which differentially influence CSs and FPEs, further highlighting their theoretical differences. With three empirical examples on initiating a conversation with a stranger (Bagozzi & Warshaw, 1988), posttraumatic growth of caregivers in the context of pediatric palliative care (Cadell et al., 2014), and the direct and indirect effects of spirituality on thriving among youth (Dowling et al., 2004), we illustrate how CSs and FPEs provide unique information which lead to better informed scientific conclusions. Finally, we discuss the importance of considering information afforded by CSs and FPEs in strengthening the basis of interpreting statistical results in substantive research, conclude with future research directions, and provide example OpenMx code for the computation of CSs and FPEs.Keywords: confidence sets, fungible estimates, sensitivity analysis, profile likelihood, structural equation modeling CONFIDENCE SETS AND FUNGIBLE ESTIMATES 3 Parameter uncertainty in structural equation models:Confidence sets and fungible estimates Statistical practice in psychological science is undergoing reform in response to concerns over the dependability of findings (Harlow, Mulaik, & Steiger, 2016; Open Science Collaboration, 2015;Pashler & Wagenmakers, 2012;Simmons, Nelson, & Simonsohn, 2011;Sijtsma, 2015). In response to these concerns, the reporting of effect sizes or focal parameter estimates and their confidence intervals (CIs) have been recommended as best practice (Cumming, 2014; Wilkinson & the Task Force on Statistical Inference, 1999). Confidence intervals communicate precision in estimation, providing information to researchers on whether inferences about their commensurate parameters can be drawn. Note that confidence regions (CRs) are an extension of CIs from a single parameter to a set of multiple parameters, and we use the term confidence sets (CSs) to collectively refer to CIs and CRs. Additionally, recent quantitative developments show that examining parameter sensitivity (T. Lee & MacCallum, 2015) and fungible parameter estimates (FPEs; T. Lee et al., in press;MacCallum, Browne, & Lee, 2009;Waller, 2008) can add to substantive researchers' diagnostic toolkit in terms of generating more information about the validity of their statistical results. Rel...

Fungible parameter estimates in structural equation modeling.

“…Such an approach would introduce some differences from the approach presented in the current article. Pek (2012) has investigated relationships between fungible contours and confidence regions from such a perspective, and has also explored factors that may influence the size of each. Further discussion here is beyond the present scope but will be taken up in future articles based on the work by Pek.…”

“…For example, one could conjecture about such effects arising from factors such as model complexity, degree of correlations among variables, level of model fit, level of correlations among parameters, and a number of other factors. Pek (2012) has investigated some of these issues.…”

Fungible weights in logistic regression.

“…Over three decades ago, Green (1977) noted that “if a parameter has little effect on the fit of a model, then it provides a weak basis for scientific conclusions” (p. 264). More recently, this view has been echoed by several methodologists (Koopman, 1988; Lee & MacCallum, 2015; MacCallum et al, 2009, 2012; Pek, 2012; Waller, 2008; Waller & Jones, 2009) who describe novel techniques for quantifying parameter sensitivity in common multivariate methods. Within this line of research, Waller (2008; Waller & Jones, 2009) showed how fungible weights could be used to assess parameter sensitivity in linear regression.…”

“…The goal of this article is to fill this lacuna by describing two novel methods for evaluating parameter sensitivity in logistic regression. These methods are logical extensions of recent work on fungible regression weights (Waller, 2008; Waller & Jones, 2009) and fungible weights in latent variable models (see Lee & MacCallum, 2015; MacCallum, Browne, & Lee, 2009; MacCallum, Lee, & Browne, 2012; Pek, 2012). To set the stage for this work, we briefly review the mechanics of fungible weights in linear regression.…”

Summed versus estimated factor scores: Considering uncertainties when using observed scores.

“…We set ε to 0, 0.03, and 0.09, reflecting exact, close, and poor fit (cf. Pek & Wu, 2018). For imperfect models (i.e., ε>0), two types of model errors were specified.…”