(MIMIC) MODEL
CHAPTER CONCEPTS MIMIC model Model specification Model identification Model estimation Model testing Model modification Model interpretation
MODEL SPECIFICATION
The term MIMIC refers to multiple indicators and multiple causes, and defines a particular type of SEM model. The MIMIC model involves using latent variables that are predicted by observed variables. We discuss an example by Jöreskog and Sörbom (1996a; 1996b, example 5.4, pp. 185–187). The MIMIC model shows a latent variable, social participation (social) that is defined by church attendance (church), memberships (member), and friends (friends). The social participation latent variable is predicted by the observed variables, income, occupation, and education. The MIMIC model for the social participation latent variable is dia-grammed in Figure 11.1.
The MIMIC model indicates a latent variable, social, which has arrows point-ing out to the three observed indicator variables (church, member, friends) with separate measurement error terms for each. This is the measurement part of the MIMIC model that defines the latent variable. In the MIMIC model, the latent variable, social, also has arrows pointing toward it from the three observed
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predictor variables, which have implied correlations among them (curved arrows).
This is the structural part of the MIMIC model that uses observed variables to predict a latent variable. The MIMIC model diagram also shows the prediction error for the latent variable, social. The important thing to remember in a MIMIC model is the direction of the arrows.
MODEL IDENTIFICATION
Model identification pertains to whether the estimates in the MIMIC model can be calculated, which is quickly gauged by the degrees of freedom. Do you recall how the degrees of freedom are determined? There are a total of 15 free param-eters to be estimated in the MIMIC model. The number of distinct values in the variance–covariance matrix, S, based on 6 observed variables is: p(p + 1)/2 = 6(6 + 1)/2 = 21. The degrees of freedom are computed by subtracting the number of free parameters from the number of distinct parameters in the matrix S, which is 21 − 15 = 6. The model is considered identified given the positive degrees of freedom.
MODEL ESTIMATION
The MIMIC model diagram provides the basis for specifying the SEM program, and determining whether the model is identified (positive degrees of freedom).
The SEM program will need to include the observed variables, sample size, cor-relation matrix (standardized variables), or covariance matrix, and the equations that reflect the MIMIC model to permit estimation of parameters in the model.
Model estimation involves selecting the estimation method for the parameters in the MIMIC model. The SEM program default is usually maximum likelihood
income
occup
educ
church
member
friends social
error err_c
err_m
err_f
Figure 11.1: SOCIAL MIMIC MODEL (AMOS)
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estimation, which may or may not be your choice. The SEM MIMIC model goodness-of-fit criteria follow from the chi-square computed to determine whether a reasonably good fit of the data to the MIMIC model exists. We are only report-ing the chi-square, RMSEA, and GFI values for determinreport-ing model fit.
MODEL TESTING
The χ2 = 12.04, df = 6, and p = .061, which suggests a reasonably good fit of the data to the MIMIC model. The goodness-of-fit index (GFI) suggested that 99%
of the variance–covariance in matrix S is reproduced by the MIMIC model. The standardized solution indicated factor loadings of .47 (church), .74 (member), and .40 (friends). However, the z value in the computer output dropped church as an important indicator variable in defining the latent variable, social. The observed variables, member (z = 6.718) and friends (z = 6.035), were therefore selected to define the latent variable social. The measurement equations from the computer output are listed below.
Measurement Equations
church = 0.466*social, Errorvar.= 0.783, R² = 0.217
Standerr (0.0575)
Z-values 13.621
P-values 0.000
member = 0.735*social, Errorvar.= 0.459, R² = 0.541 Standerr (0.109) (0.0753)
Z-values 6.718 6.102
P-values 0.000 0.000
friends = 0.402*social, Errorvar.= 0.839, R² = 0.161
Standerr (0.0665) (0.0577)
Z-values 6.035 14.526
P-values 0.000 0.000
Note: Because a matrix was used rather than raw data, standard error and z value are not output for the reference indicator variable, church. The HELP menu offers this explanation: LISREL for Windows uses a reference indicator (indicator with a unit factor loading) to set the scale of each of the endogen-ous latent variables of the model. If you do not specify reference indicators for the endogenous latent variables of your model, it will select a reference indica-tor for each endogenous latent variable of your model. Although it scales the factor loadings to obtain the appropriate estimates for the factor loadings of
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the reference indicators, it does not use the delta method to compute the cor-responding standard error estimates.
The observed independent variables (income, occup, and educ) in the MIMIC model were correlated amongst themselves as identified in the correlation matrix of the SEM program output:
income occup educ 1.000
.304 1.000 .305 .344 1.000
The structural equation indicated that the latent variable social had 26% of its variance predicted (R2 = .26), with 74% unexplained error variance due to ran-dom or systematic error, and variables not in the MIMIC model. The z values for the structural equation coefficients indicated that occup (occupation) didn’t statistically significantly predict social (z = parameter estimate divided by stand-ard error = .097/.056 = 1.73 is less than z = 1.96 at the .05 level of significance, two-tailed test, whereas income (z = 3.82) and educ (z = 4.93) were statistically sig-nificant at the .05 level of significance). The structural equation with coefficients, standard errors in parentheses, and associated z values are listed below.
Structural Equation
social = 0.232*income + 0.0973*occup + 0.334*educ, Errorvar.= 0.742, R² = 0.258
Standerr (0.0608) (0.0563) (0.0675) (0.170)
Z-values 3.821 1.728 4.938 4.353
P-values 0.000 0.084 0.000 0.000
MODEL MODIFICATION
The original MIMIC model was modified by dropping church as an indicator vari-able, and occup as a predictor variable. The MIMIC model diagram with these modifications appears in Figure 11.2. The model modification fit criteria are more acceptable, indicating an almost perfect fit of the data to the MIMIC model, because the Minimum Fit Function χ2 value was close to zero.
The modified social MIMIC model indicated a good data to model fit (χ2 = .19, df = 1, and p = .66). This modified model reflects dropping both a non-significant factor loading in the measurement model (church), and a non-significant
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predictor variable (occup) in the structural model. The computer output is now shown as:
Measurement Equations
member = 0.630*social, Errorvar.= 0.603, R² = 0.397
Standerr (0.0822)
Z-values 7.332
P-values 0.000
friends = 0.420*social, Errorvar.= 0.823, R² = 0.177 Standerr (0.0757) (0.0602)
Z-values 5.553 13.677
P-values 0.000 0.000
Note: Because a matrix was used rather than raw data, standard errors are not output for one of the reference indicator variables, member = 0.63 (social). The HELP menu offers further explanation as noted above.
The structural equation now indicated two statistically significant predictor vari-ables with R2 = .36. This also implies that 64% of the latent variable variance is left unexplained, mostly due to random or systematic error or other variables not included in the MIMIC model. Ideally we desire a higher R2 value to indicate more explanation of the variability in the latent variable, social. The computer output is shown as:
Structural Equations
social = 0.313*income + 0.424*educ, Errorvar.= 0.641, R² = 0.359 Standerr (0.0625) (0.0638) (0.189)
Z-values 5.014 6.653 3.393
P-values 0.000 0.000 0.001
income
educ
member
friends social
error
err_m
err_f
Figure 11.2: SOCIAL MIMIC MODIFIED MODEL (AMOS)
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SUMMARY
MIMIC models permit the specification of one or more latent variables in a measure-ment model with one or more observed variables as predictors of the latent variables in a structural model. This type of SEM model demonstrates how observed variables can be incorporated into a theoretical model and tested. In some cases, the researcher first establishes a good fit of the data to the measurement model, so a latent variable is well defined. Afterwards, it is straightforward to include observed predictor variables of the latent variable(s). We followed the five basic steps in the SEM: model specifica-tion, model identificaspecifica-tion, model estimaspecifica-tion, model testing, and model modification to obtain our best data to model fit. We interpreted both the model fit (chi-square test), as well as the statistical significance of the parameter estimates (z values).
It is important in MIMIC models to understand the direction of the paths in the theoretical model. This corresponds to how the equations are written in an SEM program. For illustration, we provide the following explanation. The meas-urement model statement has the indicator variables on the left-hand side of the equation (member, friends) when defining the latent variable social. The structural model has the observed predictor variables on the right-hand side of the equation (income, educ), which predict the dependent latent variable, social. The structural model statement is similar to how we write our prediction equations. If not done correctly, the path diagram should reveal the error in the drawing of the model.
EXERCISE
Create and run an SEM program given the MIMIC model in Figure 11.3. The MIMIC model includes the latent variable job satisfaction (satisfac), which is defined by two observed variables: peer rating and self rating. A person’s income level (income), which shift they worked (shift), and age (age) are observed predic-tor variables of job satisfaction.
Interpret the results including any model modification, significance of coefficients, and R2 value. The data set information is:
Observed Variables peer self income shift age Sample Size 530
Correlation Matrix 1.00
.42 1.00 .24 .35 1.00 .13 .37 .25 1.00 .33 .51 .66 .20 1.00
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SUGGESTED READINGS
anderson, K. g., smith, g. t., & Mccarthy, d. M. (2005). elementary school drink-ing: the role of temperament and learning. Psychology of Addictive Behaviors, 19(1), 21–27.
sanchez-perez, M., & iniesta-Bonillo, M. a. (2004, Winter). consumers felt com-mitment towards retailers: index development and validation. Journal of Business and Psychology, 19(2), 141–159.
shenzad, s. (2006). the determinants of child health in pakistan: an economic analysis. Social Indicators Research, 78, 531–556.
REFERENCES
Jöreskog, K., & sörbom, d. (1996a). LISREL 8: User’s reference guide. chicago, il: scientific software international.
Jöreskog, K., & sörbom, d. (1996b). LISREL 8: Structural equation modeling with the SIMPLIS command language. chicago, il: scientific software international.
Income
Shift
Peer Rating
Self Rating SatisfactionJob
error
err_p
err_s
Age
Figure 11.3: JOB SATISFACTION MIMIC MODEL (AMOS)