During the last forty-five years, the LISREL model, methods and software have become synonymous with structural equation modeling (SEM). SEM allows researchers in the social sciences, management sciences, behavioral sciences, biological sciences, educational sciences and other fields to empirically assess their theories. These theories are usually formulated as theoretical models for observed and latent (unobservable) variables. If data are collected for the observed variables of the theoretical model, the LISREL program can be used to fit the model to the data.
Today, however, LISREL is no longer limited to SEM. LISREL 11 includes the 64-bit statistical applications LISREL, PRELIS, MULTILEV, SURVEYGLIM and MAPGLIM.
LISREL is a 64-bit application for standard and multilevel structural equation modeling. These methods are available for the complete and incomplete complex survey data on categorical and continuous variables as well as complete and incomplete simple random sample data on categorical and continuous variables.
MULTILEV is a 64-bit application that fits multilevel linear and nonlinear models to multilevel data from simple random and complex survey designs. It allows for models with continuous and categorical response variables.
MGLIM is a 64-bit application that uses adaptive quadrature to fit generalized linear models with categorical, count and non-normally distributed outcome variables to multilevel data.
|PRELIS is a 64-bit application for data manipulation, data transformation, data generation, computing moment matrices, computing estimated asymptotic covariance matrices of sample moments, imputation by matching, multiple imputation, multiple linear regression, logistic regression, univariate and multivariate censored regression, and ML and MINRES exploratory factor analysis.
SURVEYGLIM is a 64-bit application that fits Generalized LInear Models (GLIMs) to data from simple random and complex survey designs. Models for the Multinomial, Bernoulli, Binomial, Negative Binomial, Poisson, Normal, Gamma, and Inverse Gaussian sampling distributions are available.
LISREL 11 introduces several new features that were not available in previous versions.
The *.PTH or path diagram file is now self contained, allowing users to share these files with fellow researchers. It offers a cleaner display and users will no longer be prompted to save this file if no changes have been made to the path diagram. Path diagrams for adaptive quadrature analyses now include the display of the – 2 ln L and number of parameters estimated (nfree) on the path diagram, allowing comparison of nested models through the calculation of a chi-square statistic to evaluate improvement in model fit over the models.
To avoid accidentally running the wrong program, only the Run LISREL button will be enabled for files with file extension *.lis (LISREL syntax), *.spl (SIMPLIS syntax), *.lpj (LISREL syntax generated through the GUI), and *.spj (SIMPLIS syntax generated through the GUI). The Run PRELIS button will become active when a *.prl file (PRELIS, Multilevel, Multilevel GLIM, Survey GLIM syntax files) is active. If a user used a different file extension for a syntax file, for example *.inp, both the Run LISREL and Run PRELIS options will be disabled and the user would have to rename the syntax file to have the appropriate file extension.
LISREL 11 allows users to use variable names up to 16-characters long. In the sections to follow, the rules for variable naming and examples of use are given.
Variable names are case sensitive.
When a blank space is used as part of the name, the entire name should be enclosed in single quotes. For example, the name ‘Visual Percept’ will work, but Visual Percept (without quotes) will not as LISREL will assume the blank space in the name to be the space between two successive variable names. Likewise, ‘Visual Perception’ will not work as the name is 17 characters long.
All variables, observed or latent, can have names up to 16-characters long.
The use of special characters, such as $, * , + etc. are allowed provided the name is enclosed in quotes. A name such as Visual-Percept will not work due to the inclusion of “-”. To use this name, it should be given as ‘Visual-Percept’.
When neither blank spaces or special characters are used as part of a variable name, no quotes are needed. For example, VisualPerception can successfully be used as a variable name.
Labels can carry over lines, with a maximum of 256 characters per line.
USING IMPORTED DATA
If data are imported from an external file and variables have names longer than 16 characters, LISREL will truncate the names to 16 characters. Should the first 16 characters of multiple variables in the imported data be the same, LISREL will stop with an error message indicating duplication.
USING RAW DATA
If raw data or correlation matrices are used, observed variable names should be given as Observed Variables in SIMPLIS and using the LA command in LISREL. Latent variable names can also be read from an external file in the same way as in previous versions. The best way to read names from an external file is to leave a space between variable names.
LISREL EXAMPLE: ANALYSIS OF READER RELIABILITY IN ESSAY SCORING
In an experiment (Votaw, 1948) to establish methods of obtaining reader reliability in essay scoring, 126 examinees were given a three-part English Composition examination. Each part required the examinee to write an essay, and for each examinee, scores were obtained on the following:
the original part 1 essay, represented by the variable ‘Original part1’
Scores were assigned by a group of readers using procedures designed to counterbalance certain experimental conditions. The investigator would like to know whether, on the basis of this sample of size 126, the four scores can be used interchangeably or whether scores on the copies (2) and (3) are less reliable than the originals (1) and (4).
The covariance matrix of the four measurements is given in the command file below. The hypothesis to be tested in this example is that a one-factor congeneric measurement model describes these data well.
The LISREL command file for this analysis is (EX31A_16.LIS):
Analysis of Reader Reliability in Essay Scoring Votaw’s Data
The DA command specifies four observed variables and a sample size of 126; the MA default is assumed, so the covariance
In the results of this analysis, the goodness-of-fit statistic
Degrees of Freedom for (C1)-(C2) 2
indicates that the hypothesis is acceptable.
The path diagram obtained for this model is shown below.
The results under the hypothesis are given in the table below. The three columns of this table can be read off directly from
Table: Essay scoring data: results for congeneric model
Inspecting the different ’s, it is evident that these are different even taking their respective standard errors of estimate into
LISREL EXAMPLE: READING VARIABLE NAMES FROM EXTERNAL FILE
In some cases, it is useful to place the variable names in an external file rather than in the syntax file itself. An example of such an analysis is discussed in this section.
The example is based on data from Duncan, Haller & Portes (1968). Of interest is the way in which a person’s peers (e.g., best friends) influence his or her decisions (e.g., choice of occupation). We anticipate that the relation between respondent’s ambition (RespAmbition) and best friend’s ambition (BFriendAmbition) must be reciprocal. As a test of this model, a sample of Michigan high-school students were paired with their best friends and measured on a number of background variables. In addition, scaled measures of occupational and educational aspiration were obtained to serve as indicators of a latent variable AMBITION.
The observed measures in the study are:
x2 = respondent’s intelligence (RespIntelligence)
The correlation matrix to be analyzed here is stored in the file EX55.COR. Syntax is given in the file EX55A_16.LIS:
Peer Influences on Ambition: Model with BE(2,1) = BE(1,2) and PS(2,1) = 0
The LAB file, containing the variable names, is as follows:
Note that the names are given in free format, separated by a blank between each pair of names. The labels continue in the second line of the file. The path diagram for this model is shown below.
MULTILEVEL SEM ANALYSIS WITH STRUCTURED MEANS
In this example, we consider a multilevel SEM analysis with structured means. The between-schools model is a one factor CFA model with a fixed factor variance, a latent mean, equal intercepts and equal measurement error covariances while the within-schools model is a one factor CFA model with equal measurement error covariances.
The dataset maths.lsf is based on a longitudinal study and consists of data from 1721 students nested within 55 schools. This dataset is based on the datasets eg1.sav, eg2.sav and eg3.sav described in Chapter 4 of Raudenbush, S, Bryk, A, Cheong, Y.F., Congdon, R & Du Toit (2011).
The following variables are available:
SchoolID Cluster (level 2) ID
The model to be fitted is described in the syntax file math_trend3a_16.lis.
! STRUCTURED MEANS
The path diagram for this model is shown below, followed by selected output.
From the estimated lambdas it appears that the difference in estimates is not linear as these estimates increase monotonically. This would indicate that a linear growth curve model over the period during which measurements were made would probably be more appropriate if we wanted to fit a regression model to these data. The fit statistics given above indicate that the model provides an adequate description of the data.
A section of the output is given below.
Condition Number = 4.930
Group2 : Within Schools
LISREL Estimates (Maximum Likelihood)
CONVERTING LSF FILES
This example reads data from an external LSF file. In this case, the names of the variables need to be changed within the LSF file. There are two ways to do so.
USING A TXT FILE
This option shows how a txt file can be used to update names in the LSF file. It also illustrates the correct way of adding longer variable names to raw data.
To change the variable names, start by opening the older format maths.lsf file used in versions up to LISREL 10.3. Select the Export option from the File menu and export the contents of maths.lsf to the file maths.raw. This file is shown below:
Rename the variables as shown below. As most of the names have blanks as part of the names, use single quotes around the variable names as shown.
To create an LSF with 16-character variable names, reimport this data into LISREL and save it as maths_16.lsf. Remember to address the presence of any missing data when importing. For example, here -9 is defined as a global missing value.
USING AN LSF FILE
Open the LSF file
Next, use the Save As option on the File menu to save the LSF file in the new 16-character format:
Doing so will lead to the display of a small dialog box on which the user can select the old or new format.
After opting to save it in the new format, use the Define Variables option from the Data menu to access the Define Variables dialog box.
Finally, change the names of the variables on the Define Variables dialog box by using the Rename option. Click OK when done.
FIML AND MISSING DATA EXAMPLE: THE ASSESSMENT OF INVARIANCE
In practice, many multivariate data sets are observations from several groups. Examples of these groups are genders, languages, political parties, countries, faculties, colleges, schools, etc. For these data sets, it is often of interest to determine whether or not the parameters of the structural equation model for the observed variables are invariant across the groups. The statistical methods for multiple group structural equation modeling may be used to determine whether or not these parameters are invariant across the groups.
LISREL may be used to fit multiple group structural equation models to multiple group data. Traditional statistical methods such as Maximum Likelihood (ML), Robust Maximum Likelihood (RML), Weighted Least Squares (WLS), Diagonally Weighted Least Squares (DWLS), Generalized Least Squares (GLS) and Un-weighted Least Squares (ULS) are available for complete multiple group data while the Full Information Maximum Likelihood (FIML) method is available for incomplete multiple group data. The ML, RML, WLS, DWLS, GLS and ULS methods for multiple group structural equation modeling are described in Jöreskog & Sörbom (1999) while the FIML method is described in Du Toit & Du Toit (2001).
In this example, the FIML estimation method for incomplete data of LISREL is used to fit a measurement model to a multivariate data sets consisting of the simulated scores of a sample of 1250 boys and 1250 girls on six psychological tests. The raw data are given in the LISREL System File LIS11_MG_BOYS_GIRLS_16.
Variables of interest are:
Visual perception scores (VisualPerception)
Tests of spatial visualization (‘Spatial Visual’)
Test of spatial orientation (‘Spatial Orient’)
Paragraph completion score (ParagraphComp)
Sentence completion score (SentenceComplete)
Word meaning score (‘Word Meaning’)
The invariance of a model is often of interest if the sample data consist of data from different groups such as males and females, different political parties, freshmen, sophomores, juniors and seniors, etc. In this section, we illustrate how LISREL can be used to assess various levels of invariance across groups.
Configural invariance is achieved if the model of interest fits across the groups. Although the model is the same across groups, the unknown parameters of the model are assumed to be different across the groups. The multiple group (global) Chi-square test statistic for this multiple group model is used to assess configural invariance. The measurement model in Figure 1 will now be used to illustrate how the multiple group feature of LISREL may be used to assess the configural invariance of the measurement model in Figure 1 across gender.
FIGURE 1: MEASUREMENT MODEL
The syntax for this analysis is shown below (LIS11_EX5.SPL):
Group 1: Boys
Lines 11-18 specify the measurement model for boys. Lines 15-18 specify that the variance and covariance parameters of the model are different across the two groups. Line 19 requests the results in terms of the parameter matrices of the LISREL model for the measurement model in Figure 1. In addition, 3 decimal places (ND=3) and the completely standardized solutions (SC) are specified.
The path diagram and estimates obtained for this model are given below. The large p-value for the Chi-square test statistic value and corresponding small RMSEA value imply that the data supports the configural invariance of the measurement model in Figure 1 across boys and girls.
ADAPTIVE QUADRATURE EXAMPLE
We now use adaptive quadrature and a probit link function in this analysis based on the six ordinal variables described above. Aish & Jöreskog (1990) analyzed data on political attitudes. Their data consist of 16 ordinal variables measured on the same people at two occasions. Six of the 16 variables were considered to be indicators of political Efficacy. The attitude questions corresponding to these six variables are:
People like me have no say in what the government does (’NOSAYINMATTERS’)
Voting is the only way that people like me can have any say about how the government runs things (VOTING)
Sometimes politics and government seem so complicated that a person like me cannot really understand what is going on (COMPLEX)
I don’t think that public officials care much about what people like me think (NOCARE4PEOPLE)
Generally speaking, those we elect to Parliament lose touch with the people pretty quickly (TOUCH)
Parties are only interested in people’s votes but not in their opinions (INTEREST_LEVEL)
Permitted responses to these questions were agree strongly, agree, disagree, disagree strongly, don’t know and no answer.
The model fitted to the data is given in the file efficacy2a_16.spl.
Efficacy: Model 1 Estimated by FIML
Eight quadrature points are specified. Again, in order to create a new LSF file with 16-character names, we export the data from the old LSF file, amend the names as needed, and create a new LSF file. Note that the new LSF file is not downward compatible and can only be read by LISREL 11. In contrast, LSF files made by previous versions can still be opened and used in LISREL 11.
Portions of the output are given below:
VOTING = 0.377*Efficacy, Errorvar.= 1.000, R² = 0.124
COMPLEX = 0.601*Efficacy, Errorvar.= 1.000, R² = 0.265
NOCARE4PEOPLE = 1.656*Efficacy, Errorvar.= 1.000, R² = 0.733
TOUCH = 1.185*Efficacy, Errorvar.= 1.000, R² = 0.584
INTEREST_LEVEL = 1.361*Efficacy, Errorvar.= 1.000, R² = 0.649
Number of quadrature points = 8
When a cumulative log-log link function is used instead of a probit link function, the deviance statistic for that model is found to be 20069.22 with the same number of estimated parameters.
This indicates that the probit model fits the data better than the cumulative log-log model.
The following path diagram is obtained for this analysis:
TWO-STAGE MULTIPLE IMPUTATION SEM FOR ORDINAL VARIABLES
Suppose that the rows of X (n x p) are n observations of p ordinal variables x1 x2 , ,…, xp with m categories. Suppose further that these p ordinal variables are the result of the discretization of the underlying p continuous standard normal variables z1, z2,…, zp as such that z – , N (0,P and
where P denotes the population correlation matrix of z and
where ϕ (⋅) denotes the probability density function of the standard normal
where Φ−¹ (⋅) denotes the inverse of the cumulative distribution function of the standard normal
The polychoric correlation matrix, R , is a consistent estimator of the population correlation matrix P . The model for the bivariate marginal of variables Xi and Xj is
where ϕ2 (u, v, ρij ) denotes the probability density function of the bivariate standard normal distribution with correlation ρij . The maximization of the bivariate likelihood function is equivalent to minimization of the discrepancy function is equivalent to minimization of the discrepancy function
where (Olsson (1979))
where ϕ₂ (⋅) denotes the density function of the bivariate normal distribution. The information (Jöreskog, 1994) is given by
The Fisher scoring algorithm is used to minimize F (⋅) with respect to ρij . Let θ = ρij . If θ denotes the t ᵗʰ successive approximation to θ , then the (t +1)st approximation is obtained from
Iteration is terminated when the absolute gradient value is below
Typical elements of αij , βi , and β are given by
Structural equation models for ordinal variables can be fitted to the polychoric correlation matrix and the estimated asymptotic covariance matrix of the polychoric correlations by using the robust DWLS, WLS, or ULS methods (Chung and Cai (2019)).
Suppose now that the n observations of the p ordinal variables include missing data values with k missing data value
Suppose that the rows of Z(n × p) are n observations of the p underlying continuous variables
The initial estimate for the M-step is the sample covariance matrix, S , of the complete data or Ip if the number of complete observations is too small. In the E-step, the conditional covariance matrices of the missing variables given the observed variables for the missing data value patterns are computed and used to compute an updated estimate Σ of Σ . Iteration of the consecutive M and E steps is terminated when the absolute difference between Σ and Σ is below the tolerance limit ε = 10−⁵.
Let the rows of Zi (n × p) contain the observed and imputed data values for the standard normal variables z1, z2 ,…, zp . The observed data for the ordinal variables are obtained from the corresponding observed data values of X . The missing data values of X are then replaced by the values obtained from using
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Additional Multilevel material not contained in the Multilevel Modeling Guide:
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