ULTIMO RILASCIO: June 2021

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.

INTRODUCTION 

 

LISREL 11 introduces several new features that were not available in previous versions.
Variable names can now be up to 16 characters long compared to 8 characters in previous versions, offering considerable flexibility in the naming of variables. The length of the path names that LISREL can accommodate has also been extended to 192 characters. The path diagrams can also accommodate the longer variable names in the display.

 

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.
Two-stage multiple imputation SEM for ordinal variables is now available. In previous versions of LISREL, the MCMC multiple imputation method for continuous variables was used to impute missing data values for ordinal variables under the assumption of underlying normal distributions. The new MCMC multiple imputation method for ordinal variables avoids the underlying normality assumption. For more information on the advantages of the two-stage multiple imputation SEM for ordinal variables see Chuang & Cai (2019). This procedure yields more reliable fit statistics for ordinal variables compared to the previous versions.

 

NAMING CONVENTION

 

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’
a handwritten copy of the original part 1 essay (‘Written copy’)
a carbon copy of the handwritten copy in (2) (‘Carbon copy’), and
the original part 2 essay, represented by the variable ‘Original part2’.

 

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
Congeneric model estimated by ML
DA NI=4 NO=126
LA
‘ORIGINAL PART1’ ‘WRITTEN COPY’ ‘CARBON COPY’ ‘ORIGINAL PART2’
CM
25.0704
12.4363 28.2021
11.7257 9.2281 22.7390
20.7510 11.9732 12.0692 21.8707
MO NX=4 NK=1 LX=FR PH=ST
LK
‘Essay ability’
PD
OU SE ND=2

 

The DA command specifies four observed variables and a sample size of 126; the MA default is assumed, so the covariance
matrix will be analyzed. Labels for the input variables follow the LA command. The CM command indicates that a covariance
matrix is to be input. Because an external file is not specified, the matrix follows in the command file. A format statement
does not appear, so the input is in free format. The MO command specifies four x-variables and one latent variable; the
elements of λ are all free (LX = FR), and the latent variable is standardized (PH = ST). A label for the latent variable follows
the LK command. Note that all observed variable names are given in single quotes, as all names have a blank space between
parts of the name. The same holds true for the latent variable ‘Essay ability’.

 

In the results of this analysis, the goodness-of-fit statistic

 

Degrees of Freedom for (C1)-(C2)                            2
Maximum Likelihood Ratio Chi-Square (C1)         2.298 (P = 0.3169)
Browne’s (1984) ADF Chi-Square (C2_NT)           2.236 (P = 0.3270)

 

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
the output for the ML solution. The reliabilities in column 3 appear where the output says “squared multiple correlations for
x-variables”.

 

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
account. Comparing the reliabilities in the last column, one sees that they are high for scores (1) and (4) and low for scores
(2) and (3). Thus, it appears that scores obtained from originals are more reliable than scores based on copies.

 

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)
x3 = respondent’s socioeconomic status (RespSocEconStat)
x4 = best friend’s socioeconomic status (BFriendSES)
x5 = best friend’s intelligence (BFriendIntel)
y1 = respondent’s occupational aspiration (RespOccupAspire)
y2 = respondent’s educational aspiration (RespEducAspire)
y3 = best friend’s educational aspiration (BFriendEducAsp)
y4 = best friend’s occupational aspiration (BFriendOccupAsp)
n1 = respondent’s ambition (RespAmbition)
n2 = best friend’s ambition (BFriendAmbition)

 

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
DA NI=10 NO=329
LA FI=EX55_16.LAB
KM FI=EX55.COR
SELECT
4 5 10 9 2 1 3 8 6 7
MO NY=4 NE=2 NX=6 FIXED-X BE=FU PS = DI
LE
RespAmbition BFriendAmbition
FR LY(2,1) LY(3,2) BE(1,2)
FI GA(5)-GA(8)
VA 1 LY(1) LY(8)
EQ BE(1,2) BE(2,1)
OU SE TV EF SS

 

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
ChildID                     Student number
Retained 1                If retained in the same grade at least once
‘Maths score 1’        Score in IRT metric on mathematics test on first measurement occasion
‘Maths score 2’        Score in IRT metric on mathematics test on second measurement occasion
‘Maths score 3’        Score in IRT metric on mathematics test on third measurement occasion
‘Maths score 4’        Score in IRT metric on mathematics test on fourth measurement occasion
‘Maths score 5’        Score in IRT metric on mathematics test on fifth measurement occasion
Gender                      1 if female, 0 if male
‘Ethnicity 1’              1 if African American, 0= other
‘Ethnicity 2’              1 if Hispanic, 0= other
Size                             School level variable, number of students in school
‘Low Income’           School level variable, percent of students from low income families
Mobility                     School level variable, percent of students moving during academic year

 

The model to be fitted is described in the syntax file math_trend3a_16.lis.

 

! STRUCTURED MEANS
! In this example, we specify equal non-random intercepts and a latent mean for the one
! factor CFA model between schools (eq tx(5) tx(4) tx(3) tx(2) tx(1); ka=fr).
TI
DA NI=14 NO=0 NG=2 MA=CM MI=-9
RA FI=maths_16.lsf
$CLUSTER SchoolID
SE
4 5 6 7 8 /
mo nx=5 nk=1 tx=fr lx=fu,fr td=sy,fi ka=fr ph=sy,fi
lk
Achieve
va 0.001 ph(1,1)
eq tx(5) tx(4) tx(3) tx(2) tx(1)
fr td(1,1) td(2,2) td(3,3) td(4,4) td(5,5)
fr td(2,1) td(3,2) td(4,3) td(5,4)
eq td(2,1) td(3,2) td(4,3) td(5,4)
pd
ou
Group2 : Within Schools
DA NI=14 NO=0 NG=2 MA=CM MI=-9
RA FI=maths_16.LSF
SE
4 5 6 7 8 /
mo nx=5 nk=1 tx=fi lx=fu,fr td=sy,fi ka=fi ph=sy,fr
lk

Achieve
va 0.0 ka(1)
va 0.0 tx(5) tx(4) tx(3) tx(2) tx(1)
fi lx(1,1)
va 1.0 lx(1,1)
fr td(1,1) td(2,2) td(3,3) td(4,4) td(5,5)
fr td(2,1) td(3,2) td(4,3) td(5,4)
eq td(2,1) td(3,2) td(4,3) td(5,4)
OU ND=3 MI

 

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
LISREL Estimates (Maximum Likelihood)

LAMBDA-X

Achievement
————

Mathematics1

7.205

(1.792)

4.021

Mathematics2

10.386

(1.670)

6.219

Mathematics3

12.624

(1.754)

7.197

Mathematics4

14.306

(1.894)

7.552

Mathematics5

16.740

(2.190)

7.642

Group2 : Within Schools
Number of Iterations = 54

 

LISREL Estimates (Maximum Likelihood)

LAMBDA-X

Achievement

————

Mathematics1

1.000

Mathematics2

1.233
(0.055)
22.400

Mathematics3

1.583
(0.071)
22.162

Mathematics4

1.519
(0.066)
23.044

Mathematics5

1.458
(0.064)
22.618

PHI

Achievement
————
0.375
(0.035)
10.794

 

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
$GROUPS=Gender
Raw Data from File LIS11_MG_BOYS_GIRLS_16.LSF
Latent Variables: Visual Verbal
VisualPerception = 1*Visual
‘Spatial Visual’ ‘Spatial Orient’ = Visual
ParagraphComp = 1*Verbal
SentenceComplete ‘Word Meaning’ = Verbal
Group 2: Girls
Raw Data from File LIS11_MG_BOYS_GIRLS_16.LSF
VisualPerception = 1*Visual
‘Spatial Visual’ ‘Spatial Orient’ = Visual
ParagraphComp = 1*Verbal
SentenceComplete ‘Word Meaning’ = Verbal
Set the Variance of Verbal Free
Set the Variance of Visual Free
Set the Covariance of Visual Verbal Free
Set the Error Variance of VisualPerception – ‘Word Meaning’ Free
LISREL Output: ND=3 SC
Path Diagram
End of Problem

 

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
Raw Data from file EFFICACY_16.LSF
$ADAPQ(8) PROBIT
Latent Variable Efficacy
Relationships
NOSAYINMATTERS – INTEREST_LEVEL = Efficacy
Path Diagram
End of Problem
End of Problem

 

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:

 

Measurement Equations
NOSAYINMATTERS = 0.739*Efficacy, Errorvar.= 1.000, R² = 0.353
Standerr (0.0407)
Z-values 18.154
P-values 0.000

 

VOTING = 0.377*Efficacy, Errorvar.= 1.000, R² = 0.124
Standerr (0.0324)
Z-values 11.643
P-values 0.000

 

COMPLEX = 0.601*Efficacy, Errorvar.= 1.000, R² = 0.265
Standerr (0.0375)
Z-values 16.042
P-values 0.000

 

NOCARE4PEOPLE = 1.656*Efficacy, Errorvar.= 1.000, R² = 0.733
Standerr (0.103)
Z-values 16.007
P-values 0.000

 

TOUCH = 1.185*Efficacy, Errorvar.= 1.000, R² = 0.584
Standerr (0.0632)
Z-values 18.754
P-values 0.000

 

INTEREST_LEVEL = 1.361*Efficacy, Errorvar.= 1.000, R² = 0.649
Standerr (0.0744)
Z-values 18.290

P-values 0.000

 

Number of quadrature points = 8
Number of free parameters = 24
Number of iterations used = 7
-2lnL (deviance statistic) = 19934.56514
Akaike Information Criterion 19982.56514
Schwarz Criterion 20113.22711

 

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 

 

POLYCHORIC CORRELATIONS

 

Suppose that the rows of X (n 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
−∞ = τi 0 < τi1 < τi 2
< τim = ∞ are parameters known as thresholds. The model for the univariate marginal of variable xi is

 

 

where ϕ (⋅) denotes the probability density function of the standard normal
distribution. The maximum likelihood estimator of τ ik
(Jöreskog, 1994) is given by

 

where Φ−¹ (⋅) denotes the inverse of the cumulative distribution function of the standard normal
distribution and pi1, pi 2, …, Pim denote the marginal sample proportions for xi .

 

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 τi and τ j denote the maximum likelihood estimators of the m −1 thresholds of variables Xand Xj , respectively. The gradient of F (⋅) (Olsson (1979)) may be expressed as

 

 

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
the tolerance limit ε = 10−³ .

 

 

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)).

 

MULTIPLE IMPUTATION

 

Suppose now that the n observations of the p ordinal variables include missing data values with k missing data value
patterns. The EM algorithm and the MMMC method for multiple imputation of incomplete data are intended for continuous variables and cannot readily be applied to ordinal variables. However, they can be applied to the underlying continuous variables Z1, Z2,…, Zp associated with the ordinal variables X1, X2,…, Xp. Although no observations for these continuous variables are available, these variables are assumed to have a multivariate standard normal distribution with a population covariance matrix Σ . As a result, we can simulate data from this distribution by using the polychoric correlation matrix of
the complete data of the variables if the number of complete cases is large enough and use either the EM algorithm or the MCMC algorithm to impute the missing data values for the underlying continuous variables. After imputation, the estimated thresholds can be used to replace the missing data values for the corresponding ordinal variables.

 

Suppose that the rows of Z(n × p) are n observations of the p underlying continuous variables
Z1, Z2 ,…, Zp simulated from the N (0, Σ) distribution and that Zₒ denotes the observed data values that corresponds with the observed data values of X . The EM algorithm (Dempster, Laird, and Rubin
1977) can be used to compute the maximum likelihood estimate of Σ . The minus two observed-data log
likelihood may be expressed as

 

 

The initial estimate for the M-step is the sample covariance matrix, S , of the complete data or Iif 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
the corresponding imputed data values of Z and the estimated thresholds.

The Software includes product activation designed to prevent unauthorized use and copying. This technology may cause your computer to automatically connect to the Internet and may prevent uses of the Software that are not permitted. Do not activate the Software provided under this license agreement (“Agreement”), until you have carefully read this Agreement. By activating the Software, using the activation code provided, you agree to be bound by the terms and conditions of this Agreement. If you do not agree to the terms and conditions of this Agreement, do not activate the Software. The effective date for this Agreement shall be the day you activate the licensed Software. You may have another agreement with SSI that supplements or supersedes all or portions of this Agreement.
Scientific Software International, Inc. (‘SSI’) grants to licensee the non-exclusive, non-transferable right to download, install, use, or otherwise benefit from the functionality or intellectual property of the copyrighted Software only in accordance with the terms of this Agreement. This license will continue for the specified duration or in perpetuity unless you violate the terms below or destroy all your copies of the Software. You may not sell, rent, or lease this license. The term ‘Software’ includes all computer code, activation codes and other materials, including any updates and upgrade thereto, that are provided to you under this Agreement.

 

SCOPE AND DURATION
The scope and duration (time period) of your license depends on the type of license you obtained from SSI, as specified in the accompanying invoice. The various license types are set forth below.

 

I SINGLE USER LICENSE
A single user license permits activation on one computer for one specific end user and is a perpetual license.

 

II CONCURRENT USER LICENSE
 Multiple concurrent user licenses permit activations on multiple computers for multiple end users. If the number of computers and the number of different end users exceed the number of concurrent user licenses, license metering software is required to restrict the simultaneous access to the Software to the number of concurrent user licenses.
 A fixed number (n) of concurrent user licenses permits activations on n computers for any number of different end users.
 An individual concurrent user license for one specific end user permits the end user activations on multiple computers as long as the specific end user is the only end user of the Software on all the computers involved.
 Concurrent user licenses are perpetual licenses.

 

III RENTAL LICENSE
A rental license permits activation on one computer for one specific end user and expires either 183 days or 365 days after activation.

 

IV SINGLE 12 MONTH NON-ACADEMIC LICENSE
A single 12 month non-academic license permits activation on one computer for one specific end user and expires 365 days after activation.

 

GENERAL TERMS
Title to the Software and all copies thereof remain with SSI and its suppliers. The Software is licensed, not sold. The Software is copyrighted and is protected by United States copyright laws and international treaty provisions. You will not remove any copyright notice from the Software. You agree to prevent any unauthorized copying of the Software. Except as expressly provided herein, no license or right is granted to you directly or by implication, inducement, estoppel, or otherwise, specifically SSI does not grant any express or implied right to you under SSI patents, copyrights, trademarks, or trade secret information. You may not reverse engineer, disassemble, decompile, or otherwise attempt to discover the source code of the Software.
SSI agrees to respond to brief technical questions received via email from end users with active licenses. The Software is guaranteed to replicate output generated from sample files provided with the Software. SSI reserves the right to determine if and/or when an error in Software coding exists, as well as when, how, and whether you will be sent a refund or a free correction. SSI warrants that the Software does not infringe any patent, trademark, trade secret, copyright, or other proprietary right of any other person or entity. Except as set forth herein, the Software is supplied without warranty of any kind, expressed or implied, including but not limited to the implied warranties of merchantability and fitness for a particular purpose. Neither SSI nor the author of the Software warrant that the functions contained in the Software will meet customer’s requirements or that operation of the Software will be error free and/or without Interruption.
In no event will SSI be liable to you for any damages, including any lost profits, lost savings, or other incidental or consequential damages arising out of the use of or the inability to make use of the Software, even if SSI has been advised of the possibility of such damages, or for any other claim by any other party. This Agreement is intended to be the entire agreement between you and SSI with respect to matters contained herein, and supersedes all prior or contemporaneous agreements and negotiations with respect to those matters. No waiver of any breach or default shall constitute a waiver of any subsequent breach or default. If any provision of this Agreement is determined by a court to be unenforceable, you and SSI will deem the provision to be modified to the extent necessary to allow it to be enforced to the extent permitted by law, or if it cannot be modified, the provision will be severed and deleted from this Agreement, and the remainder of the Agreement will continue in effect. Any change, modification or waiver to this Agreement must be in writing and signed by an authorized representative of you and SSI.

 

MICROSOFT WINDOWS

System Requirements

 

Operating System: Windows 7/8/8.1/10

Memory (RAM): 1 GB of RAM required.

Hard Disk Space: 100 MB of free space required.

Processor: Intel Pentium 4 or later.

 

Note:

 

Currently Lisrel offers no native Mac version of any of our programs currently available. However, if you have a dual-boot set-up or a Windows emulator, any of our programs should run without issues. Note that Lisrel can only provides support for using the software and are unable to provide guidance on setting up a Mac to allow for a program to be usable. Parallels Desktop is the most frequently used of the emulators. VirtualBox is a free alternative from Oracle, but it can be very slow compared to Parallels.


© 2022 Scientific Software International, Inc.

Software originalmente sviluppato per la stima di modelli di equazioni strutturali (SEM). Oggi comprende molte altre applicazioni statistiche.