Transcript Structural Equation Modeling With Mplus
Structural Equation Modeling Using Mplus Chongming Yang, Ph. D.
3-22-2012
New Paradigm in Data Analysis
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In the past twenty years we have witnessed a paradigm shift in the analysis of
correlational
data. Confirmatory factor analysis and structural equation modeling have replaced exploratory factor analysis and multiple regression as the standard methods.” Kenny, D.A. Kashy, D.A., & Bolger, N. (1998) .
Data analysis in psychology. In D.T. Gilbert, S.T. Fiske, & G. Lindzey (Eds.) The Handbook of Social Psychology, Vol. 1 (pp233-265). New York: McGraw-Hill.
Structuralism Components Relations Structural?
Objectives Introduction to SEM Model Source of the model Parameters Estimation Model evaluation Applications Estimate simple models with Mplus
Continuous Dependent Variables Session I
Four Moments/Information of Variable Mean Variance Skewedness Kurtosis
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Variance & Covariance
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Covariance Matrix (S) x1 x2 x3 x1 V 1 x2 Cov 21 V 2 x3 Cov 31 Cov 32 V 3
Statistical Model Probabilistic statement about Relations of variables Imperfect but useful representation of reality
Structural Equation Modeling A system of regression equations for latent variables to estimate and test direct and indirect effects without the influence of measurement errors.
To estimate and test theories about interrelations among observed and latent variables.
Latent Variable / Construct / Factor A hypothetical variable cannot be measured directly inferred from observable manifestations Multiple manifestations (indicators) Normally distributed interval dimension No objective measurement unit
How is Depression Distributed in?
College students Patients for Depression Therapy
Normal Distributions
Levels of Analyses Observed Latent
Test Theories Classical True Score Theory: Observed Score = True score + Error Item Response Theory Generalizability (Raykov & Marcoulides, 2006)
Graphic Symbols of SEM Rectangle – observed variable Oval -- latent variable or error Single-headed arrow -- causal relation Double-headed arrow -- correlation
Graphic Measurement Model of Latent 1 X 1 1 2 X 2 2 3 X 3 3
Equations Specific equations X 1 X 2 X 3 = 1 = 2 = 3 + 1 + 2 + 3 Matrix Symbols X = +
Relations of Variances V X1 V X2 V X3 = 1 2 = 2 2 = 3 2 + 1 + 2 + 3 = measurement error / uniqueness
Sample Covariance Matrix (S) x1 x2 x3 x1 V 1 x2 Cov 21 V 2 x3 Cov 31 Cov 32 V 3
Relation of Covariances Variance of X2 and X3 = common covariance of X1 1 0 0 Variance of 2 3 0
Unknown Parameters V X1 V X2 V X3 = 1 2 = 2 2 = 3 2 + 1 + 2 + 3
Unstandardized Parameterization (scaling) 1 = 1 (set variance of X1 =1; X1 called reference Indicator) Variance of and X3 = common variance of X1 X2 Squared = explained variance of X (R Variance of 2 ) = unexplained variance in X Mean of = 0
Standardized Parameterizations (scaling) Variance of X1 X2 and X3 = 1 = common variance of Squared = explained variance of X (R 2 ) Variance of = 1 2 Mean of Mean of = 0 = 0
Two Kinds of Parameters Fixed at 1 or 0 Freely estimated
d1 d2 d3 d4 d5 d6 d7 Analytic Reasoning Verbal Self Control Recognize/ Assess Agreeable ness Openness General Intelligence Emotional Intelligence Personality Job Satisfaction Marital Satisfaction z1 z2 Being Appreciated Social Relations e1 e2 Perceived Benefit Perceived Cost e3 e4
Structural Equation Model in Matrix Symbols X = x Y = y = + + + (exogenous) (endogenous) + (structural model) Note: Measurement model reflects the true score theory
Structural Equation Model in Matrix Symbols X = x Y = y = α + x + y + + + + (measurement) (measurement) + (structural) Note: SEM with mean structure.
Model Implied Covariance Matrix (Σ) Note: This covariance matrix contains unknown parameters in the equations.
(I-B) = non-singular
Sample Covariance Matrix (S) x1 x2 x3 x4 … x1 v 1 x2 cov 21 v 2 x3 cov 31 cov 32 v 3 x4 cov 41 cov 42 cov 43 v 4 … … Mean1 Mean2 Mean3 Mean4 … Total info = P(P+1)/2 + Means (if included)
Estimations/Fit Functions Hypothesis: = S or Maximum Likelihood - S = 0 F = log|| || + trace(S -1 ) - log||S|| - (p+q)
Convergence -- Reaching Limit Minimize F while adjust unknown Parameters through iterative process Convergence value: F difference between last two iterations Default convergence = .0001 Increase to help convergence ( 0.001 or 0.01
) e.g. Analysis: convergence = .01;
No Convergence No unique parameter estimates Lack of degrees of freedom identification under Variance of reference indicator too small Fixed parameters are left to be freely estimated Misspecified model
Absolute Fit Index 2 = F(N-1) (N = sample size) df = p(p+1)/2 – q P = number of variances, covariances, & means q = number of unknown parameters to be estimated prob = ? (Nonsignificant 2 indicates good fit, Why?)
Relative Fit: Relative to Baseline (Null) Model Fix all unknown parameters at 0 Variables not related ( = = = =0) Model implied covariance = 0 Fit to sample covariance matrix S Obtain 2 , df, prob < .0000
Relative Fit Indices CFI = 1- ( 2 -df)/( 2 b -df b ) b = baseline model Comparative Fit Index, desirable => .95; 95% better than b model TLI = ( 2 b /df b 2 /df) / ( 2 b /df b -1) (Tucker-Lewis Index, desirable => .90) RMSEA = √( 2 -df)/(n*df) (Root Mean Square of Error Approximation, desirable <=.06
penalize a large model with more unknown parameters)
Absolute Fit -- SRMR Standardized Root Mean Square Residual SRMR = Difference between observed and implied covariances in standardized metric Desirable when < .90, but no consensus Does not penalize for number of model parameters, unlike RMSEA
Sex Special Case A d1 1 Verbal Aggression t4a3 t4a93 t4a94 d2 1 Physical Aggression t4a37 t4a57 t4a90 e3 e2 e1 e6 e5 e4
Special Cases A Assumption: x = y = x = + + x + +
e1 e2 e3 e4 e5 e6 x1 x2 x3 x4 x5 x6 Special Case B Verbal Aggression Physical Aggression d Peer Status
Special Cases B Assumption: y = x = x y = + x + + +
Other Special Cases of SEM Confirmatory Factor Analysis (measurement model only) Multiple & Multivariate Regression ANOVA / MANOVA (multigroup CFA) ANCOVA Path Analysis Model (no latent variables) Simultaneous Econometric Equations… Growth Curve Modeling …
e1 1 x1 1 x2 e2 1 EFA vs. CFA e3 1 e4 1 e5 1 x4 x5 x3 1 e6 1 x6 Factor 2 Factor 1 Exploratory Factor Analysis Confirmatory Factor Analysis e1 1 x1 1 e2 1 x2 Factor 1 e3 1 x3 e4 1 x4 1 e5 1 x5 Factor 2 e6 1 x6
Multiple Regression x1 x2 x3 e 1 Y
Pretest1 Group Pretest2 ANCOVA e1 1 Posttest1 e2 1 Posttest2
Multivariate Normality Assumption Observed data summed up perfectly by covariance matrix S (+ means M), S thus is an estimator of the population covariance
Consequences of Violation Inflated 2 & deflated CFI and TLI reject plausible models Inflated standard errors factor loadings and structural parameters attenuate (Cause: Sample covariances were underestimated)
Accommodating Strategies
Correcting Fit Satorra-Bentler Scaled 2 & Standard Errors (estimator = mlm; in Mplus) Correcting standard errors Bootstrapping Transforming Nonnormal variables Transforming into new normal indicators (undesirable) SEM with Categorical Variables
Satorra-Bentler Scaled 2 & SE S-B 2 = d -1 (ML-based 2 ) that incorporates kurtosis) (d= Scaling factor Effect: performs well with continuous data in terms of 2 , CFI, TLI, RMSEA, parameter estimates and standard errors.
also works with certain-categorical variables (See next slide) Analysis: estimator = MLM;
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Bootstrapping Original btstrp1 btstrp2 … x y x y x y 1 5 5 3 1 3 2 4 1 1 5 4 3 3 3 2 4 1 4 2 4 5 2 2 5 1 2 4 3 5 . . . . . .
Limitation of Bootstrapping Assumption: Sample = Population Useful Diagnostic Tool Does not Compensate for small or unrepresentative samples severely non-normal or absence of independent samples for the cross validation Analysis: Output: Bootstrap = 500 (standard/residual); stand cinterval;
Examining Group Differences in latent variables (MANOVA) X g1 X g2 = = g1 g2 + + X g1 - X g2 = ( g1 g1 g2 g1 g2 g2 + + g1 g2 ) + ( g1 g1 g2 g2 ) + ( g1 g2 ) Imposing equality constraints on and use items with invariant loadings X g1 - X Given g2 = + ( g1 = 0, by assigning g1 = 0 X g1 - X g2 = + ( g2 ) g2 ) + ( g1 g2 )
Measurement Invariance (Hierarchical restrictions) Configural invariance – same items Metric Factorial Invariance Weak – additional invariant loadings ( ) Strong – additional invariant intercept ( ) Strict – additional invariant error variance ( ) (Steven & Reise, 1997)
Partial Invariance Majority of factor loadings invariant Variant factor loadings are allowed to be freely estimated across groups
Two Applications Invariance Test Develop unbiased test Examine group difference in latent variables
Advantages of Multigroup Analysis Test all parameters across groups Allow invariant variances across groups Large sample sizes How large is large enough? (Muthén & Muthén, 2002)
x1 x2 x3 MIMIC Model F y1 y2 y3 y4 e1 e2 e3 e4
MIMIC Model for Examining Group Difference MIMIC = multiple indicator multiple causes Indicators = functions of latent variable Controlling for latent variable, covariate should have no effects indicators Significant Covariate Effects = biases in the levels
Assumptions of MIMIC Model Invariant factor loadings across subgroups Invariant variances (latent & observed) Small sample size
Mplus www.statmodel.com
Multiple Programs Integrated SEM of both continuous and categorical variables Multilevel modeling Mixture modeling (identify hidden groups) Complex survey data modeling (stratification, clustering, weights) Modern missing data treatment Monte Carlo Simulations
Types of Mplus Files Data (*.dat, *.txt) Input (specify a model, <=80 columns/line) Output (automatically produced) Plot
Data File Format Free Delimited by tab, space, or comma No missing values Default in Mplus Computationally slow with large data set Fixed Format = 3F3, 5F3.2, F5.1;
Mplus Input DATA : File = ? VARIABLE : Names=?; Usevar=?; Categ=?; ANALYSIS : Type = ?
MODEL : (BY, ON, WITH) OUTPUT : Stand;
Model Specification in Mplus BY Measured by (F by x1 x2 x3 x4) ON Regressed on (y on x) WITH Correlated with (x with y) XWITH PON Interact with Pair ON (inter | F1 xwith F2) (y1 y2 on x1 x2 = y1 on x1; y2 on x2) PWITH pair with (x1 x2 with y1 y2 = x1 with y1; y1 with y2)
Default Specification Error or residual (disturbance) Covariance of exogenous variables in CFA Certain covariances of residuals (z2)
z1 z2
Practice Prepare two data files for Mplus Mediation.sav
Aggress.sav
Model Specification Single Group CFA Examine Mediation Effects in a Full SEM Run a MIMIC model of aggressions Multigroup CFA to examine measurement invariance
SPSS Data Missing Values?
Leave as blank to use fixed format Recode into special number to use free format Save as & choose file type Fixed ASCII Free *.dat (with or without variable names?) Copy & paste variable names into Mplus input file
Stata2mplus Converting a stata data file to *.dat
Find out: http://www.ats.ucla.edu/stat/stata/faq/stata 2mplus.htm
Graphic Model y1 y2 F1 y3 y4 F2 y5 y6 y7 y8 F3 y9 d3 d4 y10 F4 y11 y12 y13 y14 F5 y15 d5
Model Specification Model: f1 by y1-y3; f2 by y4-y6; f3 by y7-y9; f4 by y10-y12; f5 by y13-y15; f3 on f1 f2; f4 on f2; f5 on f2 f3 f4 ; MeaErrors are au
Modification Indices Lower bound estimate of the expected chi square decrease Freely estimating a parameter fixed at 0 MPlus Output: stand Mod(10); Start with least important parameters (covariance of errors) Caution: justification?
Indirect (Mediation) Effect A*B Mplus specification: Model Indirect: DV IND Mediator IV;
Model Comparison Model: Probabilistic statement about the relations of variables Imperfect but useful Models Differ: Different Variables and Different Relations ( , , , ) Same Variables but Different Relations ( , , , )
Nested Model A Nested Model (b) comes from general Model (a) by Removing a parameter (e.g. a path) Fixing a parameter at a value (e.g. 0) Constraining parameter to be equal to another Both models have the same variables
Equality Constraints in Mplus Parameter Labels: Numbers Letters Combination of numbers of letters Constraint (B=A) F3 on F1 (A); F3 on F2 (A);
y1 y2 y3 F1 A B Test If A=B y7 y8 y9 F3 d3 F2 y4 y5 y6 d4 F4 y10 y11 y12 y13 y14 y15 F5 d5
Model Comparison via 2 Difference 2 2 ___________________________________ 2 = df = (Nested model) = df = (Default model) dif = df dif = p = ? (a single tail) Find p value at the following website: http://www.tutor-homework.com/statistics_tables/statistics_tables.html
Conclusion: If p > .05, there is no difference between the default model and nested model. Or the Hypothesis that the parameters of the two models are equal is not supported.
Other Comparison Criteria AIC = 2 1 = Δ 2 1 BIC 2 2 - 2(df 1 – df 2 ) – 2(Δdf) (as 2 dif Smaller is better Difference > 2 test )
Practice Test if effect A=B
Run CFA with Real Data Verbal Aggression a3 a93 a94 e1 e2 e3 Physical Aggression a37 a57 a90 e4 e5 e6
Multigroup Analysis VARIABLE: USEVAR = X1 X2 X3 X4; Grouping IS sex (0=F 1=M); ANALYSIS: TYPE = MISSING H1; MODEL: F1 BY X1 - X4; MODEL M: F1 BY X2 - X4; Note: sex is grouping variable and is not used in the model.
Test Measurement Invariance Default Model Model: F1 By a3 a93(1) a94 (2); F2 By a37 a57 (3) a90 (4); Model M: F1 By a93 () a94 (); F2 By a57 () a90 (); Output: stand; Note: Reference indicators in the second group are omitted.
Test Measurement Invariance Constrained Model Model: F1 By a3 a93(1) a94 (2); F2 By a37 a57 (3) a90 (4); Model M: F1 By a93 (1) a94 (2); F2 By a57 (3) a90 (4); Output: stand; Note: Reference indicators in the second group are omitted.
Sex Race1 Estimate with Real Data a3 Verbal Aggression a93 a94 d1 d2 Race2 a37 Physical Aggression a57 a90 e1 e2 e3 e4 e5 e6
SEM with Categorical Indicators Session II
Problems of Ordinal Scales Not truly interval measure of a latent dimension, having measurement errors Limited range, biased against extreme scores Items are equally weighted (implicitly by 1) when summed up or averaged, losing item sensitivity
Criticisms on Using Ordinal Scales as Measures of Latent Constructs Steven (1951): …means should be avoided because its meaning could be easily interpreted beyond ranks.
Merbitz(1989): misinference Ordinal scales and foundations of Muthen (1983): Pearson product moment correlations of ordinal scales will produce distorted results in structural equation modeling. Write (1998): “… misuses nonlinear raw scores or Likert scales as though they were linear measures will produce systematically distorted results. …It’s not only unfair, it is immoral.”
Assumption of Categorical Indicators A categorical indicator is a coarse categorization of a normally distributed underlying dimension
Latent (Polychoric) Correlation
Categorization of Latent Dimension & Threshold No Never m-1 Yes Sometimes m Often 1 2 3 4 5 Y
Threshold The values of a latent dimension at which respondents have 50% probability of responding to two adjacent categories Number of thresholds = response categories – 1. e.g. a binary variable has one threshold.
Mplus specification [x$1] [y$2];
Normal Cumulative Distributions
Measurement Models of Categorical Indicators ( 2P IRT) Probit: P ( =1| ) = [( + ) -1/2 ] (Estimation = Weight Least Square with df adjusted for Means and Variances) Logistic: P ( =1| ) = 1 / (1+ e -( + ) ) (Maximum Likelihood Estimation)
Converting CFA to IRT Parameters Probit Conversion a = -1/2 b = / Logit Conversion a = /D b = / (D=1.7)
Sample Information Latent Correlation Matrix equivalent to covariance matrix of continuous indicators Threshold matrix Δ equivalent to means of continuous indicators
One Parameter Item Response Theory Model Analysis: Estimator = ML; Model: F by [email protected] [email protected] … [email protected];
Stages of Estimation Sample information: Correlations/threshold/intercepts (Maximum Likelihood) Correlation structure (Weight Least Square) g F = (s (g) (g) )’W (g)-1 (s (g) (g) ) g=1
W -1 matrix Elements: S1 intercepts or/and thresholds S2 slopes S3 residual variances and correlations W -1 : divided by sample size
Estimation WLSMV : W eight L east S quare estimation with degrees of freedom adjusted for M eans and V ariances of latent and observed variables
Baseline Model Freely estimated thresholds of all the categorical indicators df = p 2 – 3 p ( p = 3 of polychoric correlations)
Multigroup Analysis VARIABLE: USEVAR = X1 X2 X3 X4; Grouping IS sex (0=F 1=M); ANALYSIS: TYPE = MISSING H1; MODEL: F1 BY X1 - X4; MODEL M: F1 BY X2 - X4;
Data Preparation Tip Categorical indicators are required to have consistent response categories across groups Run Crosstab to identify zero cells Recode variables to collapse certain categories to eliminate zero cells
Inconsistent Categories Male Female
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Test Measurement Invariance Default Model Model: F1 By a3 a93(1) a94 (2); F2 By a37 a57 (3) a90 (4); Model M: F1 By a93 () a94 (); F2 By Savedata: a57 () a90 (); Output: stand; difftest agg.dat;
Specify Dependent Variables as Categorical Variable: Categ = x1-x3; Categ = all;
Model Comparison with Categorical Dependent Variables 1.
Run H0 model with the following at the end of input file: Savedata: 2. Run a nested model H1 with an equality constraint (s) on a parameter (s) with the following in the input file: Analysis: 3. Examine Chi-square difference test in the output of H1 Model difftest test.dat; difftest test.dat;
Test Measurement Invariance Nested Model Analysis: type = missing h1; difftest agg.dat; Model: F1 By a3 a93(1) a94 (2); F2 By a37 a57 (3) a90 (4); Model M: F1 By a93 (1) a94 (2); F2 By a57 (3) a90 (4); Output: stand;
Reporting Results Guidelines: Conceptual Model Software + Version Data (continuous or categorical?) Treatment of Missing Values Estimation method Model fit indices ( 2 (df) , p , CFI, TLI, RMSEA) Measurement properties (factor loadings + reliability) Structural parameter estimates (estimate, significance, 95% confidence intervals) ( = .23*, CI = .18~.28)
Reliability of Categorical Indicators (variance approach) = ( i ) 2 / [( i ) 2 + 2 ], where ( i ) 2 = square (sum of standardized factor loadings) 2 = sum of residual variances i = items or indicator 2 i = 1 2 McDonald, R. P. (1999). Test theory: A unified treatment (p.89) Mahwah, New Jersey: Lawrence Erlbaum Associates.
Calculator of Reliability (Categorical Indicators) SPSS reliability data SPSS reliability syntax
Interactions in SEM Observed or Latent Categorical or Continuous Nine possible combinations Treatment see users’ Guide
Trouble Shooting Strategy Start with one part of a big model Ensure every part works Estimate all parts simultaneously
Important Resources Mplus Website: www.statmodel.com
Papers: http://www.statmodel.com/papers.shtml
Mplus discussions: http://www.statmodel.com/cgi-bin/discus/discus.cgi