Three Sample Estimates of Fraction of Missing Information From Full Information Maximum Likelihood

In missing data analysis, the reporting of missing rates is insufficient for the readers to determine the impact of missing data on the efficiency of parameter estimates. A more diagnostic measure, the fraction of missing information (FMI), shows how the standard errors of parameter estimates increase from the information loss due to ignorable missing data. FMI is well-known in the multiple imputation literature (Rubin, 1987), but it has only been more recently developed for full information maximum likelihood (Savalei and Rhemtulla, 2012). Sample FMI estimates using this approach have since then been made accessible as part of the lavaan package (Rosseel, 2012) in the R statistical programming language. However, the properties of FMI estimates at finite sample sizes have not been the subject of comprehensive investigation. In this paper, we present a simulation study on the properties of three sample FMI estimates from FIML in two common models in psychology, regression and two-factor analysis. We summarize the performance of these FMI estimates and make recommendations on their application.

p is the number of variables; following the notation of Yuan and Bentler (2000), these parameters will be referred to as β. The most common saturated model is one where all means, variances, and covariances are freely estimated, so that β = ((vechΣ) , µ ) , where "vech" is the operator that stacks nonrendundant elements of a matrix into a vector columnwise. The number of parameters in the structured model is q < p * ; these parameters will be referred to as θ.
Denote byβ = ((vechΣ) ,μ ) the saturated FIML estimates, obtained by maximizing Equation 1 with respect to β; these saturated FIML estimates are also known as the "EM" means and "EM" covariance matrix (stacked in a vector). Denote . . , J enumerates the missing data patterns that are possible under a particular MAR mechanism, n j is the number of observations with the jth pattern, and J β,j is the pattern-specific observed information estimate, which is a function of the pattern-specific sample means and covariance matrix as well as of the population parameters µ and Σ (see eq. 5 in Savalei and Rosseel (in press) for an explicit expression).
The general expression for the observed information matrix under the structured model is J θ = ∆ J β ∆ − d β ⊗ I q H, where ∆ is the p * × q matrix of the SEM model first derivatives, H is the matrix of the SEM model second derivatives, I q is the identity APPENDICES 6 matrix of order q, and d β is the vector of first derivatives of the log-likelihood with respect to the saturated model.
The first and second derivatives in the general expression for J θ can be approximated using various methods, leading to three different definitions for the observed information matrix estimates: tractable and thus the estimateĴ θ , which relies on them, is harder to obtain analytically. However, because it is also the second derivative of the log-likelihood, most SEM software including lavaan obtain this estimate numerically rather than following the equation above. For this reason, we will refer to the estimateĴ θ as the numeric Hessian. On the other hand, the two approximationsĴ θ,h1 andJ θ are obtained analytically using equations above. These approximations essentially discard the second term in the expression forĴ θ , which depends on the first derivatives of the log-likelihood under the saturated model; these derivatives approach zero when the model is correct and this term tends to be very small relative to the first term. Thus, the analytic approximations assume the model is correct.

APPENDICES 7
To compute FMIs according to Equation 5, we need estimates of complete and incomplete data information matrices, but we only have incomplete data. Fortunately, estimates of the information matrix with complete data can be obtained by using complete data formulas in equations above, but evaluating them at FIML estimates. In particular, lavaan can take the general expression for the complete data fit function (which is just a transformation of the complete data log-likelihood), evaluate it at the FIML estimatesβ and obtain numeric second derivatives to yield the estimateĴ X,θ , the numeric Hessian estimating the would-be variability of parameter estimates if data were complete. Similarly, lavaan can take the expression for the complete data observed information matrix J X,θ , which, through its components, is a function ofx, S, µ, and Σ, and evaluate it either at {μ,Σ,μ,Σ}, resulting inJ Y,θ , or evaluate it at {μ,Σ,μ,Σ}, resulting inĴ X,θ,h1 .
Using the same type of information matrix estimates in the numerator and denominator of Equation 5 will yield three local FMI estimate types: