Missing Data

This page summarizes missing-data information from an Mplus output: variable-level missingness, missing-data patterns, covariance coverage, longitudinal attrition, and parameter-level fraction of missing information (FMI). For imputed analyses, FMI and relative efficiency are read from the output. For FIML analyses, FMI is obtained from the second run generated by the page.

What to request from Mplus. OUTPUT: PATTERNS SAMPSTAT; — PATTERNS prints the missing data patterns and their frequencies, and Mplus prints the covariance coverage of a run with missing data on its own. For the fraction of missing information of an analysis of imputed data, DATA IMPUTATION: or TYPE = IMPUTATION, whose results table carries the column Mplus heads Rate of Missing. For the fraction of missing information of an analysis fitted by full information maximum likelihood, add RESIDUAL to the OUTPUT command as well: the page then writes the second run that the procedure of Savalei and Rhemtulla (2012) needs. A warning printed by Mplus is reported and does not remove a result the file supports; what removes a result is a quantity that is missing or inadmissible.

What to do

  1. Add PATTERNS to the OUTPUT command of the analysis, and RESIDUAL as well if the analysis is fitted by full information maximum likelihood; then run it.
  2. Drop the .out file on the box below.
  3. If the variables are the waves of a longitudinal design, tick the box under the file: the page then reads the order in which they are printed as the order in time and reports the attrition.
  4. Read the tables. What is missing from the file is named rather than assumed.
  5. For a FIML analysis, save the data file and the input the page writes at the end, run that input, and drop its output on the second box: the fraction of missing information of every parameter follows.
Drop the Mplus .out file here, or press Enter to choose one.

The file is read in your browser and is not sent anywhere.

Mplus stops with a message below .10 by default, which is the value here.
Read only when the file is an analysis of imputed data.

What the file reports

The tables appear here once an output file has been read.

The fraction of missing information of a FIML analysis

Imputed analyses report the missing-information rate directly. FIML analyses do not. Following Savalei and Rhemtulla (2012), parameter-level FMI is obtained from the ratio of observed-data and complete-data standard errors. The page generates the complete-data run from the model-implied means and covariances and computes FMI when that second output is loaded.

The second run appears here once an output fitted by full information maximum likelihood has been read above.

Mplus reads a covariance matrix given to it as scaled by n−1. The correction matters with a small sample and is the one the authors apply.

Methods and formulas

What the page reads

Three sections of the output, and nothing else. SUMMARY OF MISSING DATA PATTERNS gives the patterns themselves, printed as a column of crosses and dots for each pattern, and the frequency of each; the covariance coverage table gives, for every pair of variables, the proportion of cases observed on both, and on its diagonal the proportion observed on each variable alone; and the results table of an analysis of imputed data gives the column Mplus heads Rate of Missing. When a section is absent the page says which one and what would have printed it.

Coverage

Pairwise covariance coverage is the proportion of cases observed on both variables. Low coverage can make covariance estimates imprecise; Mplus stops estimation when coverage falls below the value set by COVERAGE (default .10). The page lists pairs below the selected threshold and reports the minimum coverage.

Patterns, monotonicity and attrition

A pattern is a distinct arrangement of observed and missing values. The number of patterns, the frequency of the complete pattern and the frequency of the largest incomplete ones describe the shape of the missingness in a way that a single percentage cannot. In a longitudinal design the patterns also say what kind of attrition there is. Missingness is monotone when a case that is missing at one wave is missing at every later wave, which is dropout; it is intermittent when a case comes back after a wave it missed. The distinction matters because the two are not handled alike: monotone missingness admits methods that non-monotone missingness does not, and intermittent missingness is often informative about the reason for the absence. The page classifies each pattern by reading the printed order of the variables as the order in time, which is why that reading has to be confirmed with the box above.

The fraction of missing information

With multiple imputation the uncertainty of a parameter has two parts: the variance of its estimate within each imputed data set, averaged over imputations, and the variance of the estimate across imputations. The second is the part the missing data are responsible for. Mplus prints their ratio in the column Rate of Missing:

FMI=BB+W

Here B is the variance of the point estimate across imputations and W the variance of the point estimate within an imputed data set, averaged over the imputations (Muthén, n.d.). Rubin's (1987) fraction of missing information carries in addition a correction for the number of imputations and for the degrees of freedom, γ=r+2/(ν+3)r+1 with r=(1+1/m)B/W; the two agree closely and the page reports the value the file carries rather than recomputing it, because B and W are not given separately.

The efficiency of an estimate based on m imputations, relative to an infinite number of them, is

RE=(1+FMIm)−1

Here m is the number of imputations and FMI the fraction of missing information of that parameter (Rubin, 1987). The page reports it for the number of imputations set above, and the number of imputations that would be needed to reach an efficiency of 99 percent for the parameter with the largest fraction, which is the one that governs the choice.

Under full information maximum likelihood no imputations are drawn, so there is no rate of missing to read, and the quantity is obtained another way. Savalei and Rhemtulla (2012, Equation 4) write it for the jth parameter as

λj=1−SEj,C2SEj,O2

SEj,O is the standard error from the observed-data analysis and SEj,C the corresponding complete-data standard error. Following Savalei and Rhemtulla, the latter is obtained from a second ML run fitted to the means and covariances implied by the first run, using the same sample size and INFORMATION = OBSERVED. The page writes that second input from the echoed model and implied moments, applies the required n/(n−1) covariance rescaling, and matches parameters across runs by heading and parameter name.

Mplus prints the standard error, estimate and estimate/SE ratio to three decimals. When the ratio permits a more precise reconstruction of the standard error, the page uses it; otherwise it uses the printed SE. A reconstruction inconsistent with the printed precision is discarded.

Beside each fraction the page reports the three readings the authors give (2012, p. 488): the effective sample size nj*=n(1−λj), which is the number of complete cases that would have estimated the parameter as precisely; the sample size n/(1−λj) that would have been needed to reach with these data the precision complete data would have given; and the width inflation factor 1/1−λj, the factor by which the confidence interval is wider than it would have been.

The two-run FMI calculation is limited to single-level continuous-variable models estimated with ML. MLR, weighted-least-squares estimators, multilevel, complex-survey and mixture models are not supported because their standard errors cannot be reconstructed from the moment input used here. The implementation follows the Savalei–Rhemtulla sample estimator. Finite-sample FMI estimates can be negative; these are sampling artifacts rather than negative population fractions. Chen and Savalei (2021) compare alternative sample estimators and recommend their third estimator for general use.

Fraction of missing information is not replaced by the proportion of missing values. FMI depends on how much information about a parameter is lost after accounting for the variables that remain observed, so it need not track the raw missing-data proportion.

Self-test

Append ?selftest=1 to check missing-data patterns, covariance coverage, attrition classification, relative efficiency, FMI calculations, implied-moment parsing, generated second-run syntax, and unsupported inputs.

References