Mplus is a statistical modeling program that provides researchers with a flexible tool to analyze their data. Mplus offers researchers a wide choice of models, estimators, and algorithms in a program that has an easy-to-use interface and graphical displays of data and analysis results. Mplus allows the analysis of both cross-sectional and longitudinal data, single-level and multilevel data, data that come from different populations with either observed or unobserved heterogeneity, and data that contain missing values. Analyses can be carried out for observed variables that are continuous, censored, binary, ordered categorical (ordinal), unordered categorical (nominal), counts, or combinations of these variable types. In addition, Mplus has extensive capabilities for Monte Carlo simulation studies, where data can be generated and analyzed according to most of the models included in the program.

The Mplus modeling framework draws on the unifying theme of latent variables. The generality of the Mplus modeling framework comes from the unique use of both continuous and categorical latent variables. Continuous latent variables are used to represent factors corresponding to unobserved constructs, random effects corresponding to individual differences in development, random effects corresponding to variation in coefficients across groups in hierarchical data, frailties corresponding to unobserved heterogeneity in survival time, liabilities corresponding to genetic susceptibility to disease, and latent response variable values corresponding to missing data. Categorical latent variables are used to represent latent classes corresponding to homogeneous groups of individuals, latent trajectory classes corresponding to types of development in unobserved populations, mixture components corresponding to finite mixtures of unobserved populations, and latent response variable categories corresponding to missing data.

Mplus is a statistical software program that is popular amongst social science researchers for its unique features that provides researchers with a variety of sophisticated analytical tools while being user friendly. Mplus provides the user with a comprehensive statistical toolbox to analyze various models from the most basic (e.g., simple regression analysis) to advanced research models (e.g., latent growth modeling) (Maydeu-Olivares 2000; Vandenberg 2006) with both cross-sectional and longitudinal data sets (Muthén and Muthén 2017). This distinctive program is also favored for its extensive capacities in performing Monte Carlo Simula- tions (Wang and Wang 2019). Mplus users also favor this software for its capacity to use continuous, categorical latent variables, or a combination of both.

Mplus is a powerful statistical program that performs robust estimations of chi-square tests for model fit using a likelihood approach in which means and variances are adjusted. Standard errors are computed using bootstrapping, and non-normality of outcomes is taken into consideration (Muthén and Muthén 2017). At the end of each data analy- sis process, the diagram feature of Mplus offers a graphi- cal display of the analyzed model (e.g., plots of individual observed and estimated values). These graphical displays can be saved for future use in various formats such as DIB, EMF, or JPEG.

One of the other unique features of Mplus is its wide capacity to deal with missing data. Newman (2014) describes three patterns of missing data. To determine the best technique to deal with missing data, one must first assess the pattern of the missing values. Data are missing in a MCAR (missing completely at random) pattern when missing values are not dependent on the observed values or missing values. MAR (missing at random) happens when missing values are dependent on the observed values but not on the missing values. MNAR (missing not at random) is when missing values are dependent on the missing val- ues themselves. Mplus provides researchers with a user- friendly solution when dealing with missing data by offer- ing maximum likelihood estimations under MCAR, MAR, and NMAR (Maydeu-Olivares 2000; Muthén and Muthén 2017). Mplus uses an observed information matrix to com- pute standard errors for the parameter estimates (Kenward and Molenberghs 1998).

Social science researchers use data modeling to demon- strate a variety of relationships between sets of variables. Indeed, Mplus is a well-designed program for analyzing both single and multilevel models (Maydeu-Olivares 2000) that contain a combination of relationships within variables (individual-level) and the relationships between variables (cluster-level). Three-level models can be tested when one of the levels relates to recurrent measures over time (Maydeu- Olivares 2000).

The coding language of Mplus is different from that of competitive programs (e.g., LISREL, AMOS, and EQS) and requires researchers to learn a new coding language (Kel- loway 2014; Maydeu-Olivares 2000; Vandenberg 2006). Whereas first time users might feel intimated by this limita- tion at first, they will soon find it an easy task (Vandenberg 2006) as Mplus users can analyze a majority of complicated models using only a few Mplus commands. For instance, Mplus uses only 10 general commands for data analysis: TITLE, DATA (required), VARIABLE (required), DEFINE, ANALYSIS, MODEL, OUTPUT, SAVEDATA, PLOT, and MONTECARLO. Though specifying complex models using these commands may seem difficult, the challenge is not so much with using Mplus as a platform but with the complex- ity of the research model itself (Maydeu-Olivares 2000).

Several resources are available to help researchers with writing a suitable code within Mplus to test a model. For example, one such useful resources is https://offbeat.group.shef.ac.uk/FIO/mplusmedmod.htm, a webpage designed by Chris Stride that aims at providing coding instructions to Mplus users, especially those who are interested in testing moderation, mediation, and moderated mediation models with both categorical and continuous variables (Stride et al. 2015). This webpage contains Mplus codes and examples that could be used for analyzing numerous combinations of mediators and moderators. Moreover, codes are provided for both observed variables and latent variables.

Overall, it may be said that Mplus offers a superb soft- ware package that is distinctive, comprehensive, and user- friendly. In addition to the external resources available (e.g., books, workshops, and webpages), Mplus license holders receive strong support from the publisher. Usually, requests and questions are responded within a few hours of submis- sion with adequate details (Vandenberg 2006). Furthermore, the Mplus Web site (www.statmodel.com) offers numerous examples, questions and answers, and training courses.

The purpose of modeling data is to describe the structure of data in a simple way so that it is understandable and interpretable. Essentially, the modeling of data amounts to specifying a set of relationships between variables. The figure below shows the types of relationships that can be modeled in Mplus. The rectangles represent observed variables. Observed variables can be outcome variables or background variables. Background variables are referred to as x; continuous and censored outcome variables are referred to as y; and binary, ordered categorical (ordinal), unordered categorical (nominal), and count outcome variables are referred to as u. The circles represent latent variables. Both continuous and categorical latent variables are allowed. Continuous latent variables are referred to as f. Categorical latent variables are referred to as c.

The arrows in the figure represent regression relationships between variables. Regressions relationships that are allowed but not specifically shown in the figure include regressions among observed outcome variables, among continuous latent variables, and among categorical latent variables. For continuous outcome variables, linear regression models are used. For censored outcome variables, censored (tobit) regression models are used, with or without inflation at the censoring point. For binary and ordered categorical outcomes, probit or logistic regressions models are used. For unordered categorical outcomes, multinomial logistic regression models are used. For count outcomes, Poisson and negative binomial regression models are used, with or without inflation at the zero point.

Models in Mplus can include continuous latent variables, categorical latent variables, or a combination of continuous and categorical latent variables. In the figure above, Ellipse A describes models with only continuous latent variables. Ellipse B describes models with only categorical latent variables. The full modeling framework describes models with a combination of continuous and categorical latent variables. The Within and Between parts of the figure above indicate that multilevel models that describe individual-level (within) and cluster- level (between) variation can be estimated using Mplus.

Ellipse A describes models with only continuous latent variables. Following are models in Ellipse A that can be estimated using Mplus:

• Regression analysis

• Path analysis

• Exploratory factor analysis

• Confirmatory factor analysis

• Item response theory modeling

• Structural equation modeling

• Growth modeling

• Discrete-time survival analysis

• Continuous-time survival analysis

• Time series analysis

Observed outcome variables can be continuous, censored, binary, ordered categorical (ordinal), unordered categorical (nominal), counts, or combinations of these variable types.

Special features available with the above models for all observed outcome variables types are:

• Single or multiple group analysis

• Missing data under MCAR, MAR, and NMAR and with multiple imputation

• Complex survey data features including stratification, clustering, unequal probabilities of selection (sampling weights), subpopulation analysis, replicate weights, and finite population correction

• Latent variable interactions and non-linear factor analysis using maximum likelihood

• Random slopes

• Individually-varying times of observations

• Linear and non-linear parameter constraints

• Indirect effects including specific paths

• Maximum likelihood estimation for all outcomes types

• Bootstrap standard errors and confidence intervals

• Wald chi-square test of parameter equalities

• Factor scores and plausible values for latent variables

Ellipse B describes models with only categorical latent variables. Following are models in Ellipse B that can be estimated using Mplus:

• Regression mixture modeling

• Path analysis mixture modeling

• Latent class analysis

• Latent class analysis with covariates and direct effects

• Confirmatory latent class analysis

• Latent class analysis with multiple categorical latent variables

• Loglinear modeling

• Non-parametric modeling of latent variable distributions

• Multiple group analysis

• Finite mixture modeling

• Complier Average Causal Effect (CACE) modeling

• Latent transition analysis and hidden Markov modeling including mixtures and covariates

• Latent class growth analysis

• Discrete-time survival mixture analysis

• Continuous-time survival mixture analysis

Observed outcome variables can be continuous, censored, binary, ordered categorical (ordinal), unordered categorical (nominal), counts, or combinations of these variable types. Most of the special features listed above are available for models with categorical latent variables. The following special features are also available.

• Analysis with between-level categorical latent variables

• Tests to identify possible covariates not included in the analysis that influence the categorical latent variables

• Tests of equality of means across latent classes on variables not included in the analysis

• Plausible values for latent classes

Kelloway, E.K. 2014. *Using Mplus for structural equation modeling: A researcher’s guide*. Thousand Oaks: Sage.

Kenward, M.G., and G. Molenberghs. 1998. Likelihood based frequen- tist inference when data are missing at random. *Statistical Science *13 (3): 236–247.

Maydeu-Olivares, A. 2000. Review of MPLUS. *Multivariate Behavio- ral Research *35 (4): 501–505

Muthén, L.K., and B.O. Muthén. 2017. *Mplus user’s guide*, 8th ed. Los Angeles, CA: Muthén & Muthén.

Newman, D.A. 2014. Missing data: Five practical guidelines. *Organi- zational Research Methods *17 (4): 372–411.

Stride, C. B., Gardner S., Catley. N., & Thomas, F. (2015). *Mplus code for mediation, moderation, and moderated mediation mod- els*. Retrieved March 24, 2020, from https://www.offbeat.group.shef.ac.uk/FIO/mplusmedmod.htm

Vandenberg, R.J. 2006. Software review: Mplus 3.0. *Organizational Research Methods *9 (3): 408–412.

Wang, J., and X. Wang. 2019. *Structural equation modeling: Applica- tions using Mplus*. New York: Wiley.

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