Introduction to Mplus

Piotr Wilkpiotr.wilk@schulich.uwo.ca

May 12, 2010

SPONSORED BY:Research Data Centre

Population and Life Course Studies PLCSInterdisciplinary Development Initiative

OVERVIEW

• Mplus modeling framework • Mplus language• Examples of research using Mplus

WHAT IS MPLUS?

• Statistical modeling program for structural equation modeling... and more

• Extremely flexible modeling framework:– Multiple types of data formats (variables)– Multiple types of statistical models

(relationships)• Code-based path-centric specification

MODELING FRAMEWORK

MODELING FRAMEWORK

• Describes structure of the (rectangles and circles)

• Describes relationships between variables (arrows)

• Acknowledges complex data structures:– Multilevel and multiple population data

DATA STRUCTURE

• Observe variables (rectangles)• Latent variables (circles)• Combinations

OBSERVED OUTCOME VARIABLES

• Continuous (y)• Categorical (u):

– Censored– Binary– Ordered categorical (ordinal)– Unordered categorical (nominal)– Counts

• Combinations: – Single model (y with u)

LATANT VARIABLES

• Continuous latent variables (f):– Continuous indicators– Categorical indicators

• Categorical latent variables (c)– Measurement model– Group membership

RELATIONSHIPS: OBSERVED VARIABLES

• Linear regression for continuous outcomes• Probit or logistic regression for binary

outcomes• Poisson or zero-inflated regression for

count outcomes• Simultaneous modeling of several related

relationships:– Path analysis

RELATIONSHIPS: LATANT VARIABLES

• Continuous latent variables:– Structural equation modeling

• Categorical latent variables:– Mixture modeling– Latent class analysis

• Latent variable interactions

RELATIONSHIPS: ALL VARIABLES

• Ability to combine all types of variables and all types of relationships into a single analytical framework

MODELING POSSIBILITIES:EXAMPLES

• Complex survey data• Multiple group analysis• Multilevel modeling• Mixture modeling • Latent class analysis• Longitudinal data analysis• Modeling with missing data• Monte Carlo simulations• And more…

COMPLEX SURVEY DATA

• Adjustment of standard errors:– Takes into account stratification and/or non-

independence of observations – Unequal probabilities of selection (sampling

weights)• Multilevel framework:

– Specify a separate model for each level of the multilevel data

• Both approaches can be combined

MULTILEVEL MODELING

• Multilevel models separate the overall variance into two sources:– Within (individual-level variation)– Between (group-level variation)

• Allows random intercepts and random slopes

• Random effects can be specified for any relationship

MIXTURE MODELING

• Modeling with categorical latent variables• Represent subpopulations where

population membership not known but inferred from the data

LATENT CLASS ANALYSIS

• A special case of mixture modeling• Explains relationships among observed

dependent variables• Provides classification of individuals into

more homogenous sub-groups

LONGITUDINAL DATA ANALYSIS

• Broad class of statistical methods for longitudinal data

• Latent growth curve analysis – Resembles classic confirmatory factor analysis

• Multilevel modeling

MODELING WITH MISSING DATA

• Several options for estimating models with missing data

• Estimation based on two assumptions:– Missing completely at random– Missing at random

• Non-ignorable missing data modeling:– Categorical outcomes as indicators of

missingness• Generates and analyzes multiple data sets

using multiple imputation• Computes bootstrapped standard errors

MONTE CARLO SIMULATIONS

• Extensive Monte Carlo facilities for data generation and data analysis

• Generates several types of data based on specified parameters

• Can be used for power analysis• Other Monte Carlo features:

– Saving generated data and parameter estimates

– Analytical results from each replication can be saved in an external file

OTHER USEFUL FEATURES

• Indirect effects (specific paths)• Bootstrap standard errors and confidence

intervals• Robust estimation of standard errors and

chi-square tests for model fit• And more…

COMMAND STRUCTURE

• Mplus is a command-based program• There are nine sets of Mplus commands:

– TITLE:– DATA:– VARIABLE:– DEFINE: – ANALYSIS:– MODEL:– OUTPUT:– SAVEDATA:– PLOT: – MONTECARLO:

GENERAL RULES

• All commands must begin on a new line and must be followed by a colon (:)

• Some commands have numerous subcommands

• Semicolons (;) separate subcommands• Individual lines of code cannot exceed 80

characters • Not case sensitive (only variable names

are case sensitive) • Exclamation mark in front (!) serves as a

comment character

TITLE COMMAND

• Specifies a title that will be printed on each page of the output file

• No limit on length

DATA COMMAND

• Specifies where the data file is located and the format of the data

• Records may be in free format or fixed format

• Accepts covariance or correlation matrices • Data files from other statistical packages

have to be converted: – SAS and SPSS: fixed format ASCII file – STATA: stata2mplus function

DEFINE COMMAND

• Allows for transformation and creation of new variables

• Supports a large number of transformation functions

• Allows for conditional statements– Selection of observations

ANALYSIS COMMAND

• Specifies analysis type(s) and estimation procedure

• Many estimation options are available • Some analyses require additional

commands

MODEL COMMAND:OVERVIEW

• Specifies the parameters of the model • Models are built in terms of relationships

between variables: – Variable RELATIONSHIP Variable

MODEL COMMAND:RELATIONSHIPS

• BY keyword ("measured by"): – Define the latent variables

• ON keyword (“regressed on”): – Structural path between variables

• WITH keyword (“correlated with”): – correlation between two variables

MODEL COMMAND:PARAMETERS

• Variances: – Variable name without brackets

• [Means] or thresholds [catvar$1]: – Variable name inside square brackets

• {Scale factors}: – Variable name in curly brackets

OUTPUT COMMAND

• Specifies optional outputs to be generated• Mplus creates an output file using the

extension .out (text file) • Specific elements of output can be

included or suppressed

SAVEDATA COMMAND

• Determines what to save in new text files• Analysis dependent outputs

– Datasets– Parameter estimates – Latent class memberships– Cook’s distances or “influence” statistics

PLOT COMMAND

• Provides graphical displays of observed data and results:– Histograms / scatterplots– Individual observed and estimated values– Sample and estimated means and

proportions/probabilities• Available for:

– Total sample– By group / class– Adjusted for covariates

• Editing and exporting of plots

DEFAULTS

• The command language is set up with defaults to minimize the amount of text

• Version specific defaults• Example: Missing data

– Mplus assumes that there are no missing values or that FIML estimation (missing values are missing at random)

– Listwise deletion must be specified under the DATA command

SUMMARY: PROS

• Many great features not available in other packages

• Ability to combine various data types • Path-centric specification:

– Relatively intuitive and easy to learn – Extensions to larger models are easy to

implement • Commitment to development • Excellent support

SUMMARY: CONS

• Cost: Mplus is a commercial package• Annual fee: Support and updates • Matrix specification is not supported • No data management beyond Monte Carlo

capabilities, transformations, and selection of observations

ADDITIONAL RESOURCES

• Technical and theoretical support:– Homepage: www.statmodel.com– Discussion forum: www.statmodel.com/cgi-

bin/discus/discus.cgi• Online manuals and tutorials • Other websites:

http://www.ats.ucla.edu/stat/mplus/

MPLUS COMMERCIAL VERSION

• Current version: 6.0 (new!) • Base Program: 595 USD • Mixture "add-on": 745 USD• Multilevel "add-on": 745 USD• Combination "add-on": 895 USD

MPLUS DEMO VERSION

• Free version of the software• www.statmodel.com/demo.shtml• Limit on the number of variables

– 2 independent variables– 6 dependent variables

CONCLUSION

• Advantages and disadvantage of using only one program

• Each program has strengths and weaknesses

• Use the correct one for the problem at hand

EXAMPLES

• Path analysis (3.11)• Structural equation modeling (511)• Latent growth curve analysis

– Quadratic growth (6.9)– Paralleled processes (6.13)

• Mixture modeling (7.1)• Advanced models

– Latent class growth curves analysis – Complier average causal effect

PATH ANALYSIS

PATH ANALYSIS

TITLE: Path analysis with continuous dependent variablesDATA:

FILE IS ex3.11.dat;VARIABLE:

NAMES ARE y1-y3 x1-x3;MODEL: y1 y2 ON x1 x2 x3;

y3 ON y1 y2 x2;

STRUCTURAL EQUATION MODEL

STRUCTURAL EQUATION MODEL

TITLE: SEM with continuous indicators DATA: FILE IS ex5.11.dat;VARIABLE: NAMES ARE y1-y12;MODEL: f1 BY y1-y3;

f2 BY y4-y6;f3 BY y7-y9;f4 BY y10-y12;f4 ON f3;f3 ON f1 f2;

LATENT GROWTH MODEL

LATENT GROWTH MODEL

TITLE: Quadratic growth model DATA: FILE IS ex6.9.dat;VARIABLE: NAMES ARE y11-y14;MODEL: i s q | y11@0 y12@1 y13@2 y14@3;PLOT: Type is Plot3;Series = y11 (0)

y12 (1)y13 (2)y14 (3);

LATENT GROWTH MODEL

LATENT GROWTH MODEL

TITLE: Growth model for two parallel processesDATA: FILE IS ex6.13.dat;VARIABLE: NAMES ARE y11- y24;MODEL:

i1 s1 | y11@0 y12@1 y13@2 y14@3;i2 s2 | y21@0 y22@1 y23@2 y24@3;s1 ON i2;s2 ON i1;

MIXTURE MODEL

MIXTURE MODEL

TITLE: Mixture regression analysisDATA: FILE IS ex7.1.dat;VARIABLE: NAMES ARE y x1 x2;

CLASSES = c (2);ANALYSIS: TYPE = MIXTURE;MODEL:

%OVERALL%y ON x1 x2;c ON x1;%c#2%y ON x2;y;

GROWTH MIXTURE MODEL

COMPLIER AVERAGE CAUSAL EFFECT

Outcome

Covariates

Compliance

Treatment