Researchers often have questions about inter-relationships between observed variables (indicators) and latent variables (factors). The Multiple Indicators and Multiple Causes (MIMIC) model is one of the models to quench the thirst for such questions! The latest JASP release provides MIMIC models as part of the SEM module. This tutorial introduces the idea of MIMIC models and, with a simple example, explains how to fit and interpret them.
MIMIC models are used when researchers are interested in (1) identifying factors that are measured by multiple indicators and (2) examining predictors that cause those factors. In “MIMIC”,“MI” stands for “multiple indicators”. These multiple indicators measure a latent variable of interest. “MIC”, on the other hand, means “multiple causes”. These causes refer to observed variables that are assumed to predict the latent variable. For a more concrete understanding, consider the example below (see Hodge & Treiman, 1968 for more information about the sample and the research context; see Schumacker & Lomax, 2016; Jöreskog & Sörbom, 1996 for a detailed explanation about the MIMIC model applied in the social participation data).
In our social participation example, researchers are interested in investigating the effects of income, occupation (occup), and education (educ) in predicting social participation. Here, social participation cannot be measured directly. It is therefore operationalized as a latent variable measured by the number of church attendances (church), the number of voluntary organization memberships (member), and the number of friends seen (friends). Three variables, church, member, and friends, are (multiple) indicators of social participation (the “MI” part of of “MIMIC”). The other three variables, income, occup, and educ, are causes of social participation (“MIC”). We provide a dataset that was simulated based on Hodge and Treiman (1968). You can download the simulated dataset here.
Once you load the dataset into JASP, the spreadsheet looks like below. Note that the dataset was simulated based on a sample correlation matrix. Because of this, there are some seemingly nonsensical values (e.g. -0.026 number of friends seen) that can be safely ignored.
To proceed with the MIMIC model to analyze the data, open the MIMIC analysis under SEM > MIMIC. Then, drag the variables church, member, and friends to the Indicators section and income, occup, and educ to the Predictors section.
JASP gives the parameter estimates in the output panel on the right. The table of Predictor coefficients shows us how much each predictor predicts social participation. The table of Indicator coefficients provides the factor loadings. Please note that the factor variance is fixed to 1 for model identification. When you interpret these coefficients, just consider them as regression coefficients.
Let’s first interpret the values from the Table of Predictor coefficients. First, the predictor coefficient of income is estimated as 0.135. With every unit increase of income, social participation increases by 0.135 units. Given that social participation is a normally distributed variable with a variance of 1, we can we this as a moderate effect at most. Likewise, with a one-unit increase in education, there is a 0.391 increase in social participation. Note that only the effects of income and education are statistically significant, assuming an alpha-level of 0.05.
Next, take a look at the Table of Indicator coefficients. With a one-unit increase in social participation, church participation increases by 0.489 units. Next, with a one unit increase in social participation, voluntary organization memberships increase by 0.304 units. Finally, with a one unit increase in social participation, the number of friends seen increases by 0.114 units.
However, please note that these estimates are not standardized. That is, we cannot compare the relative strength of prediction among the three predictors. What should we do to examine the relative importance of predictors? Estimates should be standardized. In the control panel, click the Options tab. Next, check Standardized estimates.
The two tables under Parameter estimates present the standardized estimates. We interpret the All column under Standardized.
Between income and education, one standard deviation increase in income leads to a 0.232 standard deviation increase in social participation. On the other hand, one standard deviation increase in education leads to an increase of 0.334 standard deviations in social participation. We can conclude that education results in the biggest change of social participation per standard deviation increase among the three predictors. We can also check the R-squared under the Options tab. This displays an additional table with the proportion of explained variance by the latent factor in each of the indicator variables. In this case, 21.7% of the variance in church participation, 54.1% of the variance in voluntary organization memberships, and 16.1% of the variance in number of friends seen was explained by the latent social participation factor respectively.
So far, we have interpreted the coefficients from the output. We can also check whether our MIMIC model fits the data well, in general. As a default, JASP provides the model fit with the chi-square test in the output panel.
In the chi-square test of model fit, we reject our fitted model in favor of the saturated model if the p-value is lower than 0.05. According to the table of Chi Square test, the p-value of the Factor model is not significant although the p-value is slightly over 0.05. Therefore, we can say the model fits the data. It is common to supplement the chi-square test with other fit indices to assess overall model fit. To let JASP give various fit measures to us, check Additional fit measures under the Options tab.
Among various fit measures, we consider the Comparative Fit Index (CFI) and Tucker-Lewis Index (TLI) in the table of Fit indices and Root mean square error of approximation (RMSEA) and Standardized root mean square residual (SRMR) in the table of Other fit measures. The respective values of CFI and TLI are 0.969 and 0.939. These values are greater than 0.90, which indicates an appropriate model fit. For RMSEA and SRMR, the values are 0.045 and 0.023, respectively. Because the value of RMSEA is lower than 0.05 and that of SRMR is lower than 0.08, our model has a good fit to the data.
Last but not least, it would be great if we can plot a path diagram to inspect the estimates among variables visually. JASP allows you to easily do this! In the control panel, open the Plots tab, and check the Parameter estimates and Legend. The path plot will appear in the output panel.
The values at each arrow express the predictor coefficients, the indicator coefficients, the variance and covariance among predictors, the variance of social participation (denoted as Y), and the measurement error variances of indicators. Please note that predictor coefficients and indicator coefficients are unstandardized in this plot. Next to the plot, the legend is presented to identify the acronyms of variables in the path.
In this tutorial, we have introduced the MIMIC model, ways to implement it, and how to interpret the results. Cannot wait to apply MIMIC models to your substantive research questions? Download JASP now to MIMIC!
Hodge, R. W., & Treiman, D. J. (1968). Social participation and social status. American Sociological Review, 722-740.
Jöreskog, K. G., & Sörbom, D. (1996). LISREL 8: User’s reference guide (2nd ed.). Scientific Software International.
Schumacker, R. E., & Lomax, R. G. (2016). A beginner’s guide to structural equation modeling (4th ed.). Routledge.