Introduction to Structural Equation Modeling Dominique Zephyr Applied Statistics Lab

Transcription

1 Applied Statistics Lab Introduction to Structural Equation Modeling Dominique Zephyr Applied Statistics Lab

2 SEM Model Education 2.6 Income Charac. of Individuals 1 5.2e HRQL.41 LSATISFY GENHLTH 2.4 PHYSHLTH.61 MENTHLTH

3 What is SEM? SEM is a very powerful multivariate technique for theory testing. Systems of linear equations that describe a network of relations among variables. Structural equation models go beyond ordinary regression models to incorporate multiple independent and dependent variables as well as hypothetical latent constructs.

4 Advantages of SEM Explore multivariate relationships in an integrated manner Uses CFA to correct for measurement error Ability to model mediating variables Test coefficients across multiple between-subjects groups Ability to handle difficult data Longitudinal with auto-correlated error Multi-level data Non-normal data Incomplete data Cause-effect Experimental data still needed

5 Variable Types Latent versus observed Observed variables are directly measured in a study. Latent variables are not measured directly in a study. They are assumed to bring about the observed responses. Exogenous versus endogenous Exogenous variables have causes that are assumed to be external to the model. Exogenous variables can only have double headed arrows (i.e., correlation) going into them. Endogenous variables are predicted by other variables in the model. Endogenous variables will have a directed arrow entering into them (i.e., prediction) both from the substantive predictors and a residual term that represents the variance not explained by the predictors..

6 SEM Graphic Vocabulary Latent variables, factors, constructs Observed variables, measures, indicators, manifest variables Direct effects Reciprocal effects Correlation or covariance

7 Types of SEM Models Path Analysis: A model with only observed variables Confirmatory Factor Analysis: A model with no directed arrow going into a latent variable Structural Equation Model: A model with at least one directed arrow going into a latent variable

8 Path Analysis Education D1 Income Life Satisfaction BMI

9 Path Analysis: Direct and Indirect Effect Education b b+c=indirect a= direct effect D1 Income c Life satisfaction D3 BMI

10 Measurement Model: Confirmatory Factor Analysis Observed or manifest variables D1 HRQL GENHLTH PHYSHLTH MENHLTH e1 e2 e3 Latent construct or factor

11 Full SEM model 3 7 Education Income Charac. of Individuals LSATISFY 1 2 HRQL GENHLTH PHYSHLTH MENTHLTH 4 5 6

12 How SEM Works You supply two main things Formal specification of model Observed relationship between variables (i.e., a covariance or correlation matrix You also need to supply the number of participants or cases Model implies a set of covariances Software tries to reproduce observed covariance matrix It does this by estimating parameters in the model Software produces two main things: Parameter estimates Information about how well it did in reproducing the covariance matrix

13 Five Steps to Modeling 1. Model Specification (i.e., of observed variables as indicators of factors and of paths). 2. Model Identification (Evaluating Identification). 3. Model Estimation (i.e., running the model setup to solve the equations). 4. Model Evaluation (Evaluating goodness of fit and other results, e.g., Lagrange test for modification) 5. Model Re-specification (i.e., modifications ending in a revised model)

14 Specification Theorize your model What observed variables? How many observed variables? What latent variables? How many latent variables? Relationship between latent variables? Relationship between latent variables and observed variables? Correlated errors of measurement?

15 Wilson & Cleary Model of Health-Related Quality of Life (1995)

16 Full SEM model 3 7 Education Income Charac. of Individuals LSATISFY 1 2 HRQL GENHLTH PHYSHLTH MENTHLTH 4 5 6

17 Identification Refers to the relationship between what will be estimated (the parameters) and the information used to derive these estimates If a model is identified it is possible to calculate (estimate) a unique value for every parameter Just identified: # equations = # unknowns Over-identified: # equations > # unknowns If not, the model is unidentified or underidentified Model will be unidentified if # equations < # unknowns Can also be empirically underidentified depending on data e.g., with high multicollinearity, it s as if you have fewer observed variables

18 Estimation Education 2.6 Income Charac. of Individuals 1 5.2e HRQL.41 LSATISFY GENHLTH 2.4 PHYSHLTH.61 MENTHLTH

19 Model Fit Statistics Goodness-of-fit tests based on predicted vs. observed covariances: 1. χ2 tests d.f.=(# non-redundant components in S) (# unknown parameter in the model) Null hypothesis: lack of significant difference between Σ(θ) and S Sensitive to sample size Sensitive to the assumption of multivariate normality Χ2 tests for difference between NESTED models 2. Root Mean Square Error of Approximation (RMSEA). A population index, insensitive to sample size. No specification of baseline model is needed Test a null hypothesis of poor fit <0.10 good, <0.05 very good 3. Standardized Root Mean Residual (SRMR) Squared root of the mean of the squared standardized residuals SRMR = 0 indicates perfect fit, <.05 good fit, <.08 adequate fit

20 Model Fit Statistics Goodness-of-fit tests comparing the given model with an alternative model 1. Comparative Fit Index (CFI; Bentler 1989) compares the existing model fit with a null model which assumes uncorrelated variables in the model (i.e. the "independence model") Interpretation: % of the covariation in the data can be explained by the given model CFI ranges from 0 to 1, with 1 indicating a very good fit; acceptable fit if CFI> The Tucker-Lewis Index (TLI) or Non-Normed Fit Index (NNFI) Relatively independent of sample size NNFI >=.95 indicates a good model fit, <0.9 poor fit

21 Goodness of Fit Misfit chi2 (6)= p > chi2=0.000 RMSEA=0.047 SRMR=0.024 Baseline comparison CFI=0.975 Comparative fit index TLI=0.977 Tucker-Lewis index

22 Respecification Theory! Dimensionality? Correct pattern of loadings? Correlated errors of measurement? Other paths? Modification Indexes Residuals