polychoric correlation

Ridge Structural Equation Modeling with Correlation Matrices forOrdinal and Continuous Data*

Ke-Hai Yuan,University of Notre Dame

Ruilin Wu, andBeihang University

Peter M. BentlerUniversity of California, Los Angeles

AbstractThis paper develops a ridge procedure for structural equation modeling (SEM) with ordinal andcontinuous data by modeling polychoric/polyserial/product-moment correlation matrix R. Ratherthan directly fitting R, the procedure fits a structural model to Ra = R + aI by minimizing thenormal-distribution-based discrepancy function, where a > 0. Statistical properties of theparameter estimates are obtained. Four statistics for overall model evaluation are proposed.Empirical results indicate that the ridge procedure for SEM with ordinal data has betterconvergence rate, smaller bias, smaller mean square error and better overall model evaluation thanthe widely used maximum likelihood procedure.

KeywordsPolychoric correlation; bias; efficiency; convergence; mean square error; overall model evaluation

1. IntroductionIn social science research data are typically obtained by questionnaires in which respondentsare asked to choose one of a few categories on each of many items. Measurements areobtained by coding the categories using 0 and 1 for dichotomized items or 1 to m for itemswith m categories. Because the difference between 1 and 2 cannot be regarded as equivalentto the difference between m 1 and m, such obtained measurements only possess ordinalproperties. Pearson product-moment correlations cannot reflect the proper association foritems with ordinal data. Polychoric correlations are more appropriate if the ordinal variablescan be regarded as categorization from an underlying continuous normal distribution. Undersuch an assumption, each observed ordinal random variable x is related to an underlyingcontinuous random variable z according to

*This research was supported by Grants DA00017 and DA01070 from the National Institute on Drug Abuse and a grant from theNational Natural Science Foundation of China (30870784). The third author acknowledges a financial interest in EQS and itsdistributor, Multivariate Software.Correspondence concerning this article should be addressed to Ke-Hai Yuan ([email protected]).

NIH Public AccessAuthor ManuscriptBr J Math Stat Psychol. Author manuscript; available in PMC 2011 December 08.

Published in final edited form as:Br J Math Stat Psychol. 2011 February ; 64(0 1): . doi:10.1348/000711010X497442.

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-PA Author Manuscript

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where 0 = < 1 < < m1 < m = are thresholds. All the continuous variablestogether form a vector z = (z1, z2, , zp) that follows a multivariate normal distributionNp(, ), where = 0 and = (ij) is a correlation matrix due to identificationconsiderations. When such an assumption holds, polychoric correlations are consistent,asymptotically normally distributed and their standard errors (SE) can also be consistentlyestimated (Olsson, 1979; Poon & Lee, 1987). On the other hand, the Pearson product-moment correlation is generally biased, especially when the number of categories is smalland the observed frequencies of the marginal distributions are skewed. Simulation studiesimply that polychoric correlations also possess certain robust properties when the underlyingcontinuous distribution departs from normality (see e.g., Quiroga, 1992).

Because item level data in social sciences are typically ordinal, structural equation modeling(SEM) for such data has long been developed. Bock and Lieberman (1970) developed amaximum likelihood (ML) approach to factor analysis with dichotomous data and a singlefactor. Lee, Poon and Bentler (1990) extended this approach to general SEM withpolytomous variables. Because the ML approach involves the evaluation of multipleintegrals, it is computationally intensive. Instead of ML, Christoffersson (1975) and Muthn(1978) proposed procedures of fitting a multiple-factor model using pairwise frequencies fordichotomous data by generalized least squares with an asymptotically correct weight matrix(AGLS). Muthn (1984) further formulated a general procedure for SEM with ordinal andcontinuous data using AGLS, which forms the basis of LISCOMP (an early version ofMplus, Muthn & Muthn, 2007). An AGLS approach for SEM with ordinal and continuousdata was developed in Lee, Poon and Bentler (1992), where thresholds, polychoric,polyserial and product-moment correlations were estimated by ML. Lee, Poon and Bentler(1995) further formulated another AGLS approach in which thresholds, polychoric,polyserial and product-moment correlations are estimated by ML using different subsets ofvariables; they named the procedure partition ML (see also Poon & Lee, 1987). Theapproach of Lee et al. (1995) has been implemented in EQS (Bentler, 1995) with variousextensions for better statistical inference. Jreskog (1994) also gave the technical details toSEM with ordinal variables, where thresholds are estimated using marginal frequencies andfollowed by the estimation of polychoric correlations using pairwise frequencies and holdingthe estimated thresholds constant. Jreskogs development formed the basis for ordinal datain LISREL1 (see e.g., Jreskog 1990; Jreskog & Srbom, 1996). Technical details of thedevelopment in Muthn (1984) were provided by Muthn and Satorra (1995). Recently,Bollen and Maydeu-Olivares (2007) proposed a procedure using polychoric instrumentalvariables. In summary, various technical developments have been made for SEM withordinal data, emphasizing AGLS.

Methods currently available in software are two-stage procedures where polychoric,polyserial and product-moment correlations are obtained first. This correlation matrix is thenmodeled with SEM using ML, AGLS, the normal-distribution-based GLS (NGLS), least

1In LISREL, the AGLS procedure is called weighted least squares (WLS) while GLS is reserved for the GLS procedure when theweight matrix is obtained using the normal distribution assumption as in covariance structure analysis (Browne, 1974). We will furtherdiscuss the normal-distribution-based GLS in the concluding section.

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squares (LS), and diagonally-weighted least squares (DWLS), as implemented in EQS,LISREL and Mplus. We need to note that the ML procedure in software fits a structuralmodel to a polychoric/polyserial/product-moment correlation matrix by minimizing thenormal distribution based discrepancy function, treating the correlation matrix as a samplecovariance matrix from a normally distributed sample, which is totally different from theML procedure considered by Lee et al. (1990). We will refer to this ML method in SEMsoftware as the ML method from now on. The main purpose of this paper is to develop aridge procedure for SEM with ordinal and continuous data that has a better convergencerate, smaller bias, smaller mean square error (MSE) and better overall model evaluation thanML. We next briefly review studies on the empirical behavior of several procedures toidentify the limitation and strength of each method and to motivate our study anddevelopment.

Babakus, Ferguson and Jreskog (1987) studied the procedure of ML with modeling fourdifferent kinds of correlations (product-moment, polychoric, Spearmans rho, and Kendallstau-b) and found that ML with polychoric correlations provides the most accurate estimatesof parameters with respect to bias and MSE, but it is also associated with mostnonconvergences. Rigdon and Ferguson (1991) studied modeling polychoric correlationmatrices with several discrepancy functions and found that distribution shape of the ordinaldata, sample size and fitting function all affect convergence rate. In particular, AGLS has themost serious problem of convergence and improper solutions, especially when the samplesize is small. ML generates the most accurate estimates at sample size n = 500; ML almostgenerates the most accurate parameter estimates at n = 300, as reported in Table 2 of thepaper. Potthast (1993) only studied AGLS estimators and found that the resulting SEs aresubstantially underestimated while the associated chi-square statistic is substantiallyinflated. Potthast also found that AGLS estimators contain positive biases. Dolan (1994)studied ML and AGLS with polychoric correlations and found that ML with polychoriccorrelations produces the least biased parameter estimates while AGLS estimators containsubstantial biases. DiStefano (2002) studied the performance of AGLS and also found thatthe resulting SEs are substantially underestimated while the associated chi-square statistic issubstantially inflated. There also exist many nonconvergence problems and impropersolutions. The literature also shows that, for ML with polychoric correlations, the resultingSEs and test statistic behave badly, because without correction the formula for SEs and teststatistics are not correct. Currently, when modeling polychoric/polyserial correlationmatrices, software has the option of calculating SEs based on a sandwich-type covariancematrix and using rescaled or adjusted statistics for overall model evaluation (e.g., EQS,LISREL, Mplus). These corrected versions are often called robust procedures in theliterature. A recent study by Lei (2009) on ML in EQS and DWLS in Mplus found thatDWLS has a better convergence rate than ML. She also found that relative biases of ML andDWLS parameter estimates were similar conditioned on the study factors. She concluded (p.505)

ML performed slightly better in standard error estimation (at smaller sample sizesbefore it started to over-correct) while robust WLS provided slightly better overallType I error control and higher power in detecting omitted paths. In cases whensample size is adequately large for the model size, especially when the model isalso correctly specified, it would matter little whether ML of EQS6 or robust WLSof Mplus3.1 is chosen. However, when sample sizes are very small for the modeland the ordinal variables are moderately skewed, ML with Satorra-Bentler scaledstatistics may be recommended if proper solutions are obtainable.

In summary, ML and AGLS are the most widely studied procedures, and the latter cannot betrusted with not large enough sample sizes although conditional on the correlation matrix itis asymptotically the best procedure. These studies indicate that robust ML and robust

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DWLS are promising procedures for SEM with ordinal data. Comparing robust ML androbust DWLS, the former uses a weight matrix that is determined by the normal distributionassumption while the latter uses a diagonal weight matrix that treats all the correlations asindependent. The ridge procedure to be studied can be regarded as a combination of ML andLS.

One problem with ML is its convergence rate. This is partially because the polychoric/polyserial correlations are obtained from different marginals, and the resulting correlationmatrix may not be positive definite, especially when the sample size is not large enough andthere are many items. As a matter of fact, the normal-distribution-based discrepancyfunction cannot take a correlation matrix that is not positive definite because the involvedlogarithm function cannot take non-positive values. When the correlation matrix is nearsingular and is still positive definite, the model implied matrix will need to mimic the nearsingular correlation (data) matrix so that the estimation problem becomes ill-conditioned(see e.g., Kelley, 1995), which results not only in slower or nonconvergence but alsounstable parameter estimates and unstable test statistics (Yuan & Chan, 2008). Such aphenomenon can also happen to the sample covariance matrix when the sample size is smallor when the elements of the covariance matrix are obtained by ad-hoc procedures (see e.g.,Wothke, 1993). Although smoothing the eigenvalues and imposing a constraint of positivedefinitiveness is possible (e.g., Knol & ten Berge, 1989), the statistical consequences havenot been worked out and remain unknown.

When a covariance matrix S is near singular, the matrix Sa = S+aI with a positive scalar awill be positive definite and well-conditioned. Yuan and Chan (2008) proposed to model Sarather than S, using the normal distribution based discrepancy function. They showed thatthe procedure results in consistent parameter estimates. Empirical results indicate that theprocedure not only converges better, at small sample sizes the resulting parameter estimatesare more accurate than the ML estimator (MLE) even when data are normally distributed.Compared to modeling sample covariance matrices, modeling correlations typicallyencounters more problems of convergence with smaller sample sizes, especially for ordinaldata that are skew distributed. Actually, both EQS and LISREL contain warnings aboutproper application of modeling polychoric correlations when sample size is small. Thus, themethodology in Yuan and Chan (2008) may be even more relevant to the analysis ofpolychoric/polyserial correlation matrices than to sample covariance matrices. The aim ofthis paper is to extend the procedure in Yuan and Chan (2008) to SEM with ordinal andcontinuous data by modeling polychoric/polyserial/product-moment correlation matrices.

When a p p sample covariance matrix S = (sij) is singular, the program LISREL providesan option of modeling S + a diag(s11, , spp), which is called the ridge option (Jreskog &Srbom, 1996, p. 24). With a correlation matrix R, the ridge option in LISREL fits thestructural model to R + aI, which is the same as extending the procedure in Yuan and Chan(2008) to correlation matrices. Thus, ridge SEM with ordinal data has already beenimplemented in LISREL. However, it is not clear how to properly apply this procedure inpractice, due to lack of studies of its properties. Actually, McQuitty (1997) conductedempirical studies on the ridge option in LISREL 8 and concluded that (p. 251) thereappears to be ample evidence that structural equation models should not be estimated withLISRELs ridge option unless the estimation of unstandardized factor loadings is the onlygoal. One of the contributions of this paper is to obtain statistical properties of ridge SEMwith ordinal data and to make it a statistically sound procedure. We will show that ridgeSEM with ordinal data enjoys consistent parameter estimates and consistent SEs. We willalso propose four statistics for overall model evaluation. Because ridge SEM is most usefulwhen the polychoric/polyserial/product-moment correlation matrix is near singular, whichtends to occur with smaller sample sizes, we will conduct Monte Carlo study to see how

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ridge SEM performs with respect to bias and efficiency of parameter estimates. We will alsoempirically identify the most reliable statistics for overall model evaluation and evaluate theperformance of formula-based SEs.

Section 2 provides the details of the development for model inference, including consistentparameter estimates and SEs as well as rescaled and adjusted statistic for overall modelevaluation. Monte Carlo results are presented in section 3. Section 4 contains a real dataexample. Conclusion and discussion are offered at the end of the paper.

2. Model InferenceLet R be a pp correlation matrix, including polychoric, polyserial and Pearson product-moment correlations for ordinal and continuous variables. Let r be the vector of all thecorrelations formed by the below-diagonal elements of R and be the populationcounterpart of r. Then it follows from Jreskog (1994), Lee et al. (1995) or Muthn andSatorra (1995) that,

(1)

where denotes convergence in distribution and is the asymptotic covariance matrix of that can be consistently estimated. For SEM with ordinal data, we will have a

correlation structure (). As mentioned in the introduction, popular SEM software has theoption of modeling R by minimizing

(2)

for parameter estimates . Such a procedure is just to replace the sample covariance matrix Sby the correlation matrix R in the most commonly used ML procedure for SEM. Equations(1) and (2) can be compared to covariance structure analysis when S is based on a samplefrom an unknown distribution. Actually, the same amount of information is provided in bothcases, where s need to be estimated using fourth-order moments. Similar to modelingcovariance matrices, (2) needs R to be positive definite. Otherwise, the term log |R1()| isnot defined.

For a positive a, let Ra = R + aI. Instead of minimizing (2), ridge SEM minimizes

(3)

for parameter estimates a, where a(a) = (a) + aI. We will show that a is consistent andasymptotically normally distributed. Notice that corresponding to Ra is a populationcovariance matrix a = + aI, which has identical off-diagonal elements with . Althoughminimizing (2) for with a = 0 for categorical data is available in software, we cannot findany documentation of its statistical properties. Our development will be for an arbitrarypositive a, including the ML procedure when a = 0. Parallel to Yuan and Chan (2008), thefollowing technical development will be within the context of LISREL models.

2.1 ConsistencyLet z = (x*, y*) be the underlying standardized population. Using LISREL notation, themeasurement model2 is given by

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where x = E(x*), y = E(y*), x and y are factor loading matrices; and are vectors oflatent constructs with E() = 0 and E() = 0; and and are vectors of measurement errorswith E() = 0, E() = 0, = E(), = E(). The structural model that describesinterrelations of and is

where is a vector of prediction errors with E() = 0 and = E(). Let = E(), theresulting covariance structure of z is (see Jreskog & Srbom, 1996, pp. 13)

Recall that = E(z) = 0 and that = Cov(z) is a correlation matrix when modelingpolychoric/polyserial/product-moment correlations. We have x = 0, y = 0,

and

where diag(A) means the diagonal matrix formed by the diagonal elements of A, and I andI are identity matrices with the same dimension as and , respectively. Thus, thediagonal elements of and are not part of the free parameters but part of the model of() through the functions of free parameters in x, y, B, , , , offdiag() andoffdiag(), where offdiag(A) implies the off-diagonal elements of A.

When () is a correct model for , there exist matrices , B(0), (0), (0), (0), and such that = (0), where 0 is the vector containing the

population values of all the free parameters in . Let a0 be the corresponding vector of a at, B(a) = B(0), (a) = (0), (a) = (0), (a) = (0),

and . In addition, let

2The model presented here is the standard LISREL model in which and are not correlated. Results in this paper still hold when and correlate.

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(4a)

and

(4b)

Thus, a0 = 0 and and are functions of a0 = 0 and a. Let the functions in (4a)and (4b) be part of the model of a(a). Then we have a = a(a0) = a(0). Notice thata() is uniquely determined by () and a. This implies that whenever () is a correctmodel for modeling , a(a) will be a correct model for modeling a. The above result alsoimplies that for a correctly specified (), except for sampling errors, the parameterestimates for x, y, B, , , and the off-diagonal elements of and when modelingRa will be the same as those when modeling R. The resulting estimates for the diagonals of

and of modeling Ra are different from those of modeling R by the constant a,which is up to our choice. Traditionally, the diagonal elements of and are estimates ofthe variances of measurement errors/uniquenesses. When modeling Ra, these can beobtained by

(5a)

and

(5b)

The above discussion implies that a of modeling Ra may be different from of modeling Rdue to sampling error, but their population counterparts are identical. We have the followingformal result.

Theorem 1Under the conditions (I) () is correctly specified and identified and (II) and is a compact subset of the Euclidean space , a is consistent for 0 regardless ofthe value of a.

The proof of the theorem is essentially the same as that for Theorem 1 in Yuan and Chan(2008) when replacing the sample covariance matrix there by the correlation matrix R. Yuanand Chan (2008) also discussed the benefit of modeling Sa = S + aI from a computationalperspective using the concept of condition number; the same benefit holds for modeling Ra.One advantage of estimation with a better condition number is that a small change in thesample will only cause a small change in a while a small change in the sample can cause agreat change in if R is near singular. Readers who are interested in the details are referredto Yuan and Chan (2008).

2.2 Asymptotic normalityWe will obtain the asymptotic distribution of a, which allows us to obtain its consistentSEs. For such a purpose we need to introduce some notation first.

For a symmetric matrix A, let vech(A) be a vector by stacking the columns of A and leavingout the elements above the diagonal. We define sa = vech(Ra), a() = vech[a()], s =

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vech(R), and () = vech[()]. Notice that sa and s are vectors of length p* = p(p + 1)/2while the r in (1) is a vector of length p* = p(p 1)/2. The difference between r and sa isthat sa also contains the p elements of a + 1 on the diagonal of Ra, and sa = s when a = 0. LetDp be the duplication matrix defined by Magnus and Neudecker (1999, p. 49), and

We will use a dot on top of a function to denote the first derivative or the gradient. Forexample, if contains q unknown parameters, a() = a()/ is a p*q matrix. Understandard regularity conditions, including that 0 is an interior point of , a satisfies

(6)

where

In equation (6), because a() and () only differ by a constant a, a() = () and sa a() = s (). So the effect of a on a in (6) is only through

(7)

where

Notice that the constant coefficient 1/[2(a + 1)2] in (7) does not have any effect on a. It isUa that makes a difference. When a = 0, a = is the ML parameter estimate. When a = ,

is the weight matrix corresponding to modeling R by least squares. Thus, ridge SEM can beregarded as a combination of ML and LS. We would expect that it has the merits of bothprocedures. That is, ridge SEM will have a better convergence rate and more accurateparameter estimates than ML and more efficient estimates than LS.

It follows from (6) and a Taylor expansion of ga(a) at 0 that

(8)

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where a() is q q matrix and each of its rows is evaluated at a vector that is between 0and a, and op(1) represents a quantity that converges to zero in probability as n increases.We also omitted the argument of the functions in (8) when evaluated at the population value0. Notice that the rows of a corresponding to the diagonal elements of a are zeros and thevector (sa a) constitutes of (r ) plus p zeros. It follows from (1) that

(9)

where * is a p* p* matrix consisting of and p rows and p columns of zeros. Obviously,* is singular. We may understand (9) by the general definition of a random variable, whichis just a constant when its variance is zero. It follows from (8) and (9) that

(10a)

where

(10b)

Let * be a consistent estimator of *, which can be obtained from a consistent plus prows and p columns of zeros. A consistent estimate = (ij) of can be obtained whenreplacing the unknown parameters in (10) by a and * by *. Notice that the in (10) isthe asymptotic covariance matrix. We will compare the formula-based SEs against empirical SEs at smaller sample sizes using Monte Carlo.

A consistent can be obtained by the approach of estimating equations (see e.g., Yuan &Jennrich, 1998). Actually, the estimates of given in Lee et al. (1992, 1995), Jreskog(1994) and Muthn and Satorra (1995) can all be regarded as using estimating equations.

2.3 Statistics for overall model evaluationThis subsection presents four statistics for overall model evaluation. When minimizing (3)for parameter estimates, we automatically get a measure of discrepancy between data andmodel, i.e., FM La(a). However, the popular statistic TM La = nFM La(a) does notasymptotically follow a chi-square distribution even when a = 0. Let

and

(11)

be the so-called reweighted LS statistic, which is in the default output of EQS whenmodeling the covariance matrix. Under the assumption of a correct model structure, thereexists

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(12)

Notice that a = a(0). It follows from (8) that

(13)

where

Combining (12) and (13) leads to

(14)

Notice that * in (9) has a rank of p*, there exists a p* p* matrix A such that AA = *.Let u ~ Np*(0, I), then it follows from (9) that

(15)

Combining (14) and (15) yields

(16)

Notice that (PaA)Wa(PaA) is nonnegative definite and its rank is p*q. Let 0 < 1 2 p*q be the nonzero eigenvalues of (PaA)Wa(PaA) or equivalently of . Itfollows from (16) that

(17)

where are independent and each follows . Unless all the js are 1.0, the distribution ofTMLa will not be . However, the behavior of TMLa might be approximately describedby a chi-square distribution with the same mean. Let

Then, as n ,

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approaches a distribution whose mean equals p* q. Thus, we may approximate thedistribution of TMLa by

(18)

parallel to the Satorra and Bentler (1988) rescaled statistic when modeling the samplecovariance matrix. Again, the approximation in (18) is motivated by asymptotics, we willuse Monte Carlo to study its performance with smaller sizes.

Notice that the systematic part of TRMLa is the quadratic form

which agrees with in the first moment. Allowing the degrees of freedom to beestimated rather than p* q, a statistic that agrees with the chi-square distribution in boththe first and second moments was studied by Satterthwaite (1941) and Box (1954), andapplied to covariance structure models by Satorra and Bentler (1988). It can also be appliedto approximate the distribution of TMLa. Let

Then TMLa/m1 asymptotically agrees with in the first two moments. Consistentestimates of m1 and m2 are given by

(19)

Thus, using the approximation

(20)

might lead to a better description of TMLa than (18). We will also study the performance of(20) using Monte Carlo in the next section.

In addition to TMLa, TRLSa can also be used to construct statistics for overall modelevaluation, as printed in EQS output when modeling the sample covariance matrix. LikeTMLa, when modeling Ra, TRLSa does not asymptotically follow a chi-square distributioneven when a = 0. It follows from (12) that the distribution of TRLSa can be approximatedusing

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(21)

or

(22)

The rationales for the approximations in (21) and (22) are the same as those for (18) and(20), respectively. Again, we will study the performance of TRRLSa and TARLSa using MonteCarlo in the next section.

We would like to note that the op(1) in equation (12) goes to zero as n . But statisticaltheory does not tell how close TMLa and TRLSa are at a finite n. The Monte Carlo study inthe next section will allow us to compare their performances and to identify the best statisticfor overall model evaluation at smaller sample sizes.

We also would like to note that the ridge procedure developed here is different from theridge procedure for modeling covariance matrices developed in Yuan and Chan (2008).When treating Ra as a covariance matrix in the analysis, we will get identical TMLa andTRLSa as defined here if all the diagonal elements of a(a) happen to be a + 1, which is truefor many commonly used SEM models. However, the rescaled or adjusted statistics will bedifferent. Similarly, we may also get identical estimates for factor loadings, but their SEswill be different when based on either the commonly used information matrix or thesandwich-type covariance matrix constructed using *.

3. Monte Carlo ResultsThe population z contains 15 normally distributed random variables with mean zero andcovariance matrix

where

with = (.60, .70, 0.75, .80, .90) and 0 being a vector of five zeros;

and is a diagonal matrix such that all the diagonal elements of are 1.0. Thus, z can beregarded as generated by a 3-factor model, and each factor has 5 unidimensional indicators.

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Three sets of conditions are used to obtain the observed variables. In condition 1, all the 15variables in x are dichotomous and are obtained using thresholds

which corresponds to 30%, 40%, 50%, 60%, 70%, 35%, 40%, 45%, 50%, 55%, 50%, 55%,60%, 65%, and 70% of zeros at the population level for the 15 variables, respectively. Incondition 2, for each factor the first two variables have five categories and the last threevariables are dichotomous. The thresholds for the 6 five-category variables are respectively1 = (0.52, 0.25, 0.25, 0.52), corresponding to proportions (30%, 10%, 20%, 10%, 30%)for the five categories of x1; 2 = (0.84, 0.25, 0.25, 0.84), corresponding to proportions(20%, 20%, 20%, 20%, 20%) for the five categories of x2; 6 = (0.84, 0.25, 0.25, 0.84),corresponding to proportions (20%, 20%, 20%, 20%, 20%) for the five categories of x6; 7 =(0.84, 0.52, 0.25, 0.25), corresponding to proportions (20%, 10%, 10%, 20%, 40%) forthe five categories of x7; 11 = (0.52, 0.25, 0.25, 0.52), corresponding to proportions(30%, 10%, 20%, 10%, 30%) for the five categories of x11; 12 = (0.25, 0.25, 0.52, 0.84),corresponding to proportions (40%, 20%, 10%, 10%, 20%) for the five categories of x12.The thresholds for the 9 dichotomous variables are = (0, 0.25, 0.52, 0.25, 0, 0.13, 0.25,0.39, 0.52), which correspond to 50%, 60%, 70%, 40%, 50%, 55%, 60%, 65%, 70% of zerosat the population level of x3, x4, x5, x8, x9, x10, x13, x14 and x15, respectively. In condition 3,the first two variables for each factor are continuously observed and the last three variablesare dichotomous using thresholds

which correspond to 50%, 60%, 70%, 40%, 50%, 55%, 60%, 65%, and 70% of zeros at thepopulation level of variables x3, x4, x5, x8, x9, x10, x13, x14 and x15, respectively.

Because it is with smaller sample sizes that ML encounter problems, we choose samplesizes3 n = 100, 200, 300 and 400. One thousand replications are used at each sample size.For each sample, we model Ra with a = 0, .1 and .2, which are denoted by ML, ML.1 andML.2, respectively. Our evaluation includes the number of convergence or converging rate,the speed of convergence, biases and SEs as well as mean square errors (MSE) of theparameter estimates; and the performance of the four statistics given in the previous section.We also compare SEs based on the covariance matrix in (10) against empirical SEs.

All the thresholds are estimated using the default probit function in SAS. Pearson product-moment correlations are obtained for continuously observed variables. Fisher scoringalgorithms are used to obtain polychoric/polyserial correlations (Olsson, 1979; Olsson,Drasgow & Dorans, 1982) and to solve equation (6) for structural model parameters a (Lee& Jennrich, 1979; Olsson, 1979). The convergence criterion in estimating the polychoric/polyserial correlations is set as |r(k+1) r(k)| < 104, where r(k) is the value of r after the kthiteration; the convergence criterion for obtaining a is set as

, where is the jth parameter after the kth iteration. Truepopulation values are set as the initial value in both estimation processes. We record the

3Actually, at sample size 400, we found that all replications converged with ML.

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estimation as unable to reach a convergence if the convergence criterion cannot be reachedafter 100 iterations. For each R, the in (1) is estimated using estimating equations.

Note that not all of the N = 1000 replications converge in all the conditions, all the empiricalresults are based on Nc converged replications for each estimation method. Let ij be theestimate of j in the ith converged replication. For each estimation method at a given samplesize we obtained

with , and

For the performance of the formula-based SE, we also obtained

where SEij is the square root of the jth diagonal element of /n in the ith convergedreplication. In contrast, the empirical SE, SEEj, is just the square root of Varj. Notice thatthere are 18 free model parameters: 15 factor loadings and 3 factor correlations. With 3 dataconditions, 3 estimation methods and 4 sample sizes, there will be too many tables toinclude in the paper if we report bias, MSE and SEs for each individual parameter. Instead,these are put on the web at www.anonymous.edu/ridge-item-SEM/. In the paper, for eachsample size and each estimation method, we report the averaged bias, variance and MSEgiven by

and the averaged difference

which will give us the information of the formula-based SEs when predicting empirical SEs.

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Results for data condition 1 with 15 dichotomous indicators are in Tables 1 to 5. Allreplications reached convergence when estimating R. But more than half replications cannotconverge when solving (6) with a = 0 and .1 at n = 100, as reported in the upper panel ofTable 1. When a = .2, the number of convergence doubles that when a = 0 at n = 100. Whenn = 200, there are still about one third of the replications cannot reach convergence at a = 0while all reach convergence at a = .2. Ridge SEM not only results in more convergences butalso converges faster, as reported in the lower panel of Table 1, where each entry is theaverage of the number of iterations for Nc converged replications.

Tables 2 contains the averaged bias, variance and MSE for the four sample sizes and threeestimation methods. Clearly, all the averaged biases are at the 3rd decimal place. Except forn = 100, all the other variances and MSEs are also at the 3rd decimal place. Because theNcs for the three different as for n = 100 and 200 are so different, it is had to compare theresults between different estimation methods. When n = 400, all the three methodsconverged for all the replications, the averaged bias, variance, and MSE all become smalleras a changes from 0 to .2. At n = 300, both ML.1 and ML.2 converged for all thereplications, the averaged bias, variance, and MSE corresponding to ML.2 are also smaller.These indicate that the ridge procedure with a proper a leads to less biased, more efficientand more accurate parameter estimates than ML. The corresponding biases, variances andMSEs for individual parameters are in Tables A1 to A4 at www.nd.edu~kyuan/ridge-item-SEM/. From these tables we may notice that most individual biases are also at the 3rddecimal place although they tend to be negative. We may also notice that estimates forsmaller loadings tend to have greater variances and MSEs.

Table 3 contains the empirical mean, standard deviation (SD), the number of rejection andthe rejection ratio of the statistics TRMLa and TAMLa, based on the converged replications.Each rejection is compared to the 95th percentiles of the reference distribution in (18) or(20). For reference, the mean and SD of TMLa are also reported. Both TRMLa and TAMLaover-reject the correct model although they tend to improve as n or a increases. Because thethree Ncs at n = 100 are very different, the rejection rates, means or SDs of the threeestimation methods are not comparable for this sample size. Table 3 also suggests that TMLacannot be used for model inference because its empirical means and SDs are far away fromthose of the nominal .

Table 4 contains the results of TRRLSa and TARLSa, parallel to those in Table 3. For n = 200,300 and 400, the statistic TRRLSa performed very well in mean, SD and rejection ratealthough there is a slight over-rejection at smaller n. While TRLSa monotonically decreaseswith a, TRRLSa is very stable when a changes. The statistic TARLSa also performed well witha little bit of under-rejection.Table 5 compares the averages of the formula-based SEs against the empirical ones. We use4 decimals to inform the fine differences among the three estimation methods. As expected,within each estimation method, both the averaged SEE and SEF become smaller as nincreases, as reflected by Table 5(a). They are also smaller at a given n when a changes from0 to .2, although at n = 100 and 200 they are based on different number of replications.Table 5(a) also implies that formula-based SEs tend to slightly under-predict empirical SEs.The results in Table 5(b) indicate that SEF predicts SEE better when either a or n increases.At n = 400, the under-prediction is between 1% and 2% with ML.2. It is interesting to notethat the averaged difference between SEF and SEE is smaller at n = 300 and a = .2 than thatat n = 400 and a = 0 or .1. The corresponding SEs for individual parameters are in Tables A5to A8 on the web, where almost all individual SEEs are slightly under-predicted by thecorresponding SEF s, with only a few exceptions mostly for factor correlations.

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Results for data condition 2 with 6 five-category and 9 dichotomous indicators are in Tables6 to 10. All replications converged when estimating R. But only 294 out of 1000 replicationsconverged when solving (6) with a = 0 at n = 100, as reported in the upper panel of Table 6.At n = 100, the number of convergences almost tripled when a = .1, and only 3 replicationscould not reach convergence with ML.2. Nonconvergences still exist for ML at n = 200 and300 while all replications converged for ML.1 and ML.2. The lower panel of Table 6 impliesthat ridge SEM also converges faster. It is interesting to see that ML at n = 100 in Table 6enjoys fewer convergences than that in Table 1. Further examination indicates that almostall the nonconvergences are due to nonpositive definite or near singular R. This is becausepositive definiteness is a function of all the elements in R. One element in R can change thesmallest eigenvalue from positive to negative.

Tables 7 contains the averaged bias, variance and MSE for data condition 2. At n = 400when all the three estimation methods converged on all replications, bias, variance and MSEall become smaller when a changes from 0 to .2. These quantities also become smaller as aincreases at n = 100, 200 and 300. These indicate that the ridge procedure with a proper aleads to less biased, more efficient and more accurate parameter estimates than ML.Compared to the results in Table 2, we notice that, except for the averaged bias of ML at n =100, all the other numbers in Table 7 are smaller. This indicates that having indicators withmore categories leads to less biased and more efficient parameter estimates for all the threeestimation methods. The exception for the averaged bias with ML at n = 100 is due to thevery different convergence rates. The corresponding biases, variances and MSEs forindividual parameters are in Tables A9 to A12 on the web. From these tables we may noticethat most individual biases are negative. We may also notice that estimates for smallerloadings do not tend to have greater variances or MSEs anymore, due to their correspondingindicators having more categories.

Parallel to Table 3, Table 8 contains the results for the statistics TRMLa and TAMLa, based onthe converged replications. Similar to those in Table 3, both TRMLa and TAMLa over-rejectthe correct model although they tend to improve as n or a increases. Among the statistics,TAMLa performed the best at a = .2. Table 8 also suggests that TMLa cannot be used formodel inference. Table 9 contains the results of TRRLSa and TARLSa, parallel to those inTable 4. Both TRRLSa and TARLSa continue to be stable when n or a changes. But TRRLSatends to reject the correct model more than in Table 4 while TARLSa slightly under-rejectsthe correct model. Tables 8 and 9 also suggest that TMLa and TRLSa tend to be smaller withindicators having more categories, but they are still too far away from the expected .

Table 10 contains the averages of empirical and the formula-based SEs. Similar to those inTable 5, the SE becomes smaller as either n or a increases. Table 10(a) also implies thatformula-based SEs tend to slightly under-predict empirical SEs. The results in Table 10(b)indicate that SEF predicts SEE better when either a or n increases. SEF also predicts SEEbetter under ML.2 at n = 300 than under ML at n = 400. Comparing the results in Table 10 tothose in Table 5, we notice that the formula-based SEs predict the empirical ones betterwhen indicators have more categories. The corresponding SEs for individual parameters arein Tables A13 to A16 on the web, where almost all individual SEEs are slightly under-predicted by the corresponding SEF s, with only a few exceptions.

Results for data condition 3 with 6 continuous and 9 dichotomous indicators are in Tables 11to 15. Similar to data conditions 1 and 2, all nonconvergences occurred when solving (6)with a = 0 at n = 100, 200, 300 and with a = .1 at n = 100, as reported in the upper panel ofTable 11. Obviously, ridge SEM not only results in more convergences but also convergesfaster, as reported in the lower panel of Table 1. Similar to being observed previously, thecondition with 6 continuous and 9 dichotomous indicators does not necessarily correspond

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to more positive definite R than the condition with 15 dichotomous indicators when n issmall. It is expected that both ML.1 and ML.2 perform well.

Tables 12 contains the averaged bias, variance and MSE. Similar to the two previousconditions, bias, variance and MSE all become smaller from a = 0 to a = .2. Compared to theresults in Tables 2 and 7, we notice that, except for the averaged bias of ML at n = 100, allthe other numbers in Table 12 are smaller. This is expected because more continuousindicators should correspond to less biased and more efficient parameter estimates. Theexception for the averaged bias with ML at n = 100 is due to the very different convergencerates. The corresponding biases, variances and MSEs for individual parameters are in TablesA17 to A20 on the web.

Table 13 gives the results of TRMLa and TAMLa, parallel to those in Tables 3 and 8. Similarto being observed earlier, both TRMLa and TAMLa over-reject the correct model althoughthey tend to improve as n or a increases. Table 14 contains the results of TRRLSa andTARLSa, parallel to those in Tables 4 and 9. Both TRRLSa and TARLSa continue to be stable tothe changes of n and a. But TRRLSa tends to reject the correct model more often than inprevious tables. TARLSa performed quite well, only slightly under-rejecting the correctmodel. Tables 13 and 14 suggest that TMLa and TRLSa tend to be smaller with morecontinuous indicators, but they are still too far away from the expected .

Table 15 compares SEF against SEE for data condition 3, parallel to Tables 5 and 10. Boththe averaged SEE and SEF become smaller as n increases or a changes from 0 to .2.Formula-based SEs still tend to slightly under-predict empirical SEs. The results in Table15(b) indicate that SEF predicts SEE better when either a or n increases. Again, the averageddifference between SEF and SEE under ML.2 at n = 300 is smaller than those under ML andML.1 at n = 400. Comparing the numbers in Table 15 with those in Tables 5 and 10 wefound that more continuous indicators not only lead to more efficient parameter estimatesbut also more accurate prediction of empirical SEs by formula-based SEs. Thecorresponding SEs for individual parameters are in Tables A21 to A24 on the web, wheremost individual SEEs are still slightly under-predicted by the corresponding SEF s.

In summary, for the three data conditions and four sample sizes, ML.2 performed better thanML.1, which was a lot better than ML with respect to convergence rate, convergence speed,bias, efficiency, as well as the accuracy of formula-based SEs.

4. An Empirical ExampleThe empirical results in the previous section indicate that, with a proper a, MLa performedmuch better than ML. This section further illustrates the effect of a on individual parameterestimates and test statistics using a real data example, where ML fails.

Eysenck and Eysenck (1975) developed a Personality Questionnaire. The Chinese version ofit was available through Gong (1983). This questionnaire was administered to 117 first yeargraduate students in a Chinese university. There are four subscales in this questionnaire(Extraversion/Introversion, Neuroticism/Stability, Psychoticism/Socialisation, Lie), and eachsubscale consists of 20 to 24 items with two categories. We have access to the dichotomizeddata of the Extraversion/Introversion subscale, which has 21 items. According to the manualof the questionnaire, answers to the 21 items reflect a respondents latent trait ofExtraversion/Introversion. Thus, we may want to fit the dichotomized data by a one-factormodel. The tetrachoric correlation matrix4 R was first obtained together with . However,R is not positive definite. Its smallest eigenvalue is .473. Thus, the ML procedure cannotbe used to analyze R.

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Table 16(a) contains the parameter estimates and their SEs when modeling Ra = R + aI witha = .5, .6 and .7. To be more informative, estimates of error variances (11 to 21,21) are alsoreported. The results imply that both parameter estimates and their SEs change little when achanges from .5 to .7. Table 16(b) contains the statistics TRMLa, TAMLa, TRRLSa and TARLSaas well as the associated p-values. The estimated degrees of freedom m2 for the adjustedstatistics are also reported. All statistics indicate that the model does not fit the data well,which is common when fitting practical data with a substantive model. The statistics TRMLaand TAMLa decrease as a increases, while TRRLSa and TARLSa as well as m2 are barelyaffected by a. Comparing the statistics implies that statistics derived from TMLa may not beas reliable as those derived from TRLSa. The p-values associated with TRRLSa are smallerthan those associated with TARLSa, which agrees with the results in the previous sectionwhere TARLSa tends to slightly under-reject the correct model.

5. Conclusion and DiscussionProcedures for SEM with ordinal data have been implemented in major software. However,there exist problems of convergence in parameter estimation and lack of reliable statisticsfor overall model evaluation, especially when the sample size is small and the observedfrequencies are skewed in distribution. In this paper we studied a ridge procedure pairedwith the ML estimation method. We have shown that parameter estimates are consistent,asymptotically normally distributed and their SEs can be consistently estimated. We alsoproposed four statistics for overall model evaluation. Empirical results imply that the ridgeprocedure performs better than ML in convergence rate, convergence speed, accuracy andefficiency of parameter estimates, and accuracy of formula-based SEs. Empirical results alsoimply that the rescaled statistic TRRLSa performed best at smaller sample sizes and TARLSaalso performed well for n = 300 and 400, especially when R contains product-momentcorrelations.

For SEM with covariance matrices, Yuan and Chan (2008) suggested choosing a = p/n.Because p/n 0 as n , the resulting estimator is asymptotically equivalent to the MLestimator. Unlike a covariance matrix that is always nonnegative definite, the polychoric/polyserial/product-moment correlation matrix may have negative eigenvalues that aregreater than p/n in absolute value, hence choosing a = p/n may not lead to a positive definiteRa, as is the case with the example in the previous section. In practice, one should choose ana that makes the smallest eigenvalue of Ra greater than 0 for a proper convergence whenestimating a. Once converged, a greater a makes little difference on parameter estimatesand test statistics TRRLSa and TARLSa, as illustrated by the example in the previous sectionand the Monte Carlo results in section 3. If the estimation cannot converge for an a thatmakes the smallest eigenvalue of Ra greater than, say, 1.0, then one needs to choose either adifferent set of starting values or to reformulate the model. A good set of starting values canbe obtained from submodels where analytical solutions exist. For example, when zi, zj andzk are unidimensional indicators for a factor , with loading i, j and k, respectively; thenij = ij and i = (ijik/jk)1/2. Thus, (rijrik/rjk)1/2 gives a good starting value for i. Whenall the correlations are positive, we may choose .5 for all the free parameters. Actually, allthe starting values of factor loadings for the example in the previous section are set at .5without any convergence problem with a = .5, .6 and .7 even when three of the 21 estimatesare negative. The product-moment correlations of the ordinal and continuous data can beused as the starting values when estimating the polychoric and polyserial correlations.

4Four of the 21 20/2 = 210 contingency tables contain a cell with zero observations, which was replaced by .1 to facilitate theestimation of R.

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We have only studied ridge ML in this paper, mainly because ML is the most popular andmost widely used procedure in SEM. In addition to ML, the normal-distribution-basedNGLS procedure has also been implemented in essentially all SEM software. The ridgeprocedure developed in section 2 can be easily extended to NGLS. Actually, the asymptoticdistribution in (10) also holds for the NGLS estimator after changing (a) in the definitionof Wa to Sa; the rescaled and adjusted statistics parallel to those in (21) and (22) for theNGLS procedure can be similarly obtained. Further Monte Carlo study on such an extensionis valuable.

Another issue with modeling polychoric/polyserial/product-moment correlation matrix is theoccurrence of negative estimates of error variances. Such a problem can be caused by modelmisspecification and/or small sample together with true error variances being small (see e.g,Kano, 1998; van Driel, 1978). Negative estimates of error variances can also occur with theridge estimate although it is more efficient than the ML estimator. This is because a() willbe also misspecified if () is misspecified, and true error variances corresponding to theestimator in (5) continue to be small when modeling Ra. For correctly specified models,negative estimates of error variances is purely due to sampling error, which should becounted when evaluating empirical efficiency and bias, as is done in this paper.

We have only considered the situation when z ~ N (, ) and when () is correctlyspecified. Monte Carlo results in Lee et al. (1995), Flora and Curran (2004), and Maydeu-Olivares (2006) imply that SEM by analyzing the polychoric correlation matrix with ordinaldata has certain robust properties when z ~ N (, ) is violated, which is a directconsequence of the robust properties possessed by R, as reported in Quiroga (1992). Theserobust properties should equally hold for ridge SEM because a is a continuous function ofR. When both () is misspecified and z ~ N (, ) does not hold, the twomisspecifications might be confounded. Test procedures for checking z ~ N (, ) underordinal data exist (see e.g., Maydeu-Olivares, 2006). Further study is needed for theirpractical use in SEM with a polychoric/polyserial/product-moment correlation matrix.

As a final note, the developed procedure can be easily implemented in a software thatalready has the option of modeling polychoric/polyserial/product-moment correlation matrixR by robust ML. With R being replaced by Ra, one only needs to set the diagonal of thefitted model to a + 1 instead of 1.0 in the iteration process. The resulting statistics TML,TRML, TAML and TRLS will automatically become TMLa, TRMLa, TAMLa and TRLSa,respectively. To our knowledge, no software currently generates TRRLSa and TARLSa.However, with TRLSa, m1 and m2, these two statistics can be easily calculated. Although Rais literally a covariance matrix, except unstandardized a when diag(Ra) = (a) happen tohold, treating Ra as a covariance matrix will not generate correct analysis (see e.g.,McQuitty, 1997).

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Christoffersson A. Factor analysis of dichotomized variables. Psychometrika. 1975; 40:532.DiStefano C. The impact of categorization with confirmatory factor analysis. Structural Equation

Modeling. 2002; 9:327346.Dolan CV. Factor analysis of variables with 2, 3, 5 and 7 response categories: A comparison of

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Eysenck, HJ.; Eysenck, SBG. Manual of the Eysenck personality questionnaire. San Diego: Educationand Industrial Testing Service; 1975.

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Table 1

Number of convergences and average number of iterations with 15 dichotomized indicators, 1000 replications.

n ML ML.1 ML.2

number of convergences

100 438 491 961

200 688 995 1000

300 988 1000 1000

400 1000 1000 1000

average number of iterations

100 18.767 12.112 10.286

200 9.416 8.085 7.405

300 7.886 6.949 6.480

400 7.046 6.348 5.996


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Tabl

e 2

Ave

rage

d bi

as1

03, var

ianc

e10

3 an

d M

SE1

03 w

ith 1

5 di

chot

omiz

ed in

dica

tors

, bas

ed o

n co

nver

ged

repl

icat

ions

.

n

ML

ML .

1M

L .2

Bias

Var

MSE

Bias

Var

MSE

Bias

Var

MSE

100

5.21

812

.249

12.3

058.

346

11.5

2311

.608

4.72

611

.148

11.1

75

200

5.31

95.

687

5.72

32.

930

5.35

35.

364

2.21

15.

250

5.25

6

300

3.05

23.

647

3.65

92.

020

3.48

13.

486

1.55

33.

420

3.42

3

400

2.57

42.

734

2.74

31.

898

2.62

72.

631

1.58

32.

587

2.59

0


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Yuan et al. Page 24

Tabl

e 3

Perfo

rman

ce o

f the

stat

istic

s TRM

La an

d T A

MLa

w

ith 1

5 di

chot

omiz

ed in

dica

tors

, bas

ed o

n co

nver

ged

repl

icat

ions

(df =

p*

q

= 8

7, R

N=r

eject

numb

er,R

R=r

eject

ratio)

.

nM

etho

d

T MLa

T RM

LaT A

MLa

Mea

nSD

Mea

nSD

RN

RR

Mea

nSD

RN

RR

100

ML

497.

3116

2.63

87.5

932

.53

9221

.0%

32.1

513

.63

5211

.9%

ML .

146

8.81

140.

4213

7.47

39.7

336

373

.9%

56.2

315

.99

284

57.8

%

ML .

229

0.74

80.3

312

1.35

32.7

157

259

.5%

51.2

013

.47

375

39.0

%

200

ML

708.

3022

8.50

133.

4041

.16

474

68.9

%71

.03

21.7

338

856

.4%

ML .

137

1.14

89.0

510

9.59

25.2

744

144

.3%

62.5

713

.85

308

31.0

%

ML .

223

6.76

43.2

599

.24

17.7

824

824

.8%

58.6

410

.07

130

13.0

%

300

ML

619.

5618

9.86

116.

8033

.29

505

51.1

%71

.41

19.7

241

041

.5%

ML .

133

5.20

62.2

599

.71

18.3

426

226

.2%

65.7

911

.75

177

17.7

%

ML .

222

5.60

37.4

794

.77

15.7

715

615

.6%

64.5

210

.34

959.

5%

400

ML

549.

6112

6.47

105.

0023

.18

365

36.5

%69

.97

15.0

727

427

.4%

ML .

132

0.01

55.3

795

.55

16.4

518

918

.9%

68.3

011

.44

123

12.3

%

ML .

221

9.34

35.1

692

.24

14.8

511

411

.4%

67.9

610

.62

707.

0%


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Yuan et al. Page 25

Tabl

e 4

Perfo

rman

ce o

f the

stat

istic

s TRR

LSa

and

T ARL

Sa w

ith 1

5 di

chot

omiz

ed in

dica

tors

, bas

ed o

n co

nver

ged

repl

icat

ions

(df =

p*

q

= 8

7, R

N=r

eject

numb

er,R

R=r

eject

ratio)

.

nM

etho

d

T RLS

aT R

RLSa

T ARL

Sa

Mea

nSD

Mea

nSD

RN

RR

Mea

nSD

RN

RR

100

ML

523.

6088

.61

90.4

513

.47

378.

4%32

.71

6.17

61.

4%

ML .

128

0.69

45.2

782

.46

12.8

114

2.9%

33.7

65.

191

0.2%

ML .

221

1.03

32.7

288

.19

13.5

164

6.7%

37.2

45.

475

0.5%

200

ML

443.

0569

.13

83.6

612

.52

162.

3%44

.63

7.05

60.

9%

ML .

129

6.88

48.2

387

.74

13.8

065

6.5%

50.1

37.

5923

2.3%

ML .

220

9.50

32.9

287

.83

13.6

165

6.5%

51.9

17.

6724

2.4%

300

ML

462.

7277

.62

87.5

113

.96

666.

7%53

.63

8.75

292.

9%

ML .

129

5.39

46.1

987

.88

13.6

767

6.7%

57.9

98.

7437

3.7%

ML .

220

9.27

31.9

187

.91

13.4

664

6.4%

59.8

68.

7937

3.7%

400

ML

457.

1475

.88

87.4

113

.94

595.

9%58

.29

9.18

313.

1%

ML .

129

2.46

45.1

087

.33

13.4

448

4.8%

62.4

39.

3231

3.1%

ML .

220

7.57

31.3

187

.29

13.2

546

4.6%

64.3

29.

4529

2.9%


NIH


NIH


NIH


Yuan et al. Page 26

Tabl

e 5

Perfo

rman

ce o

f sta

ndar

d er

rors

with

15

dich

otom

ized

indi

cato

rs, b

ased

on

conv

erge

d re

plic

atio

ns.

(a) A

vera

ged e

mpiri

cal s

tanda

rd er

rors

and f

ormu

la-ba

sed st

anda

rd er

rors

n

ML

ML .

1M

L .2

SEE

SEF

SEE

SEF

SEE

SEF

100

0.10

870.

0992

0.10

600.

0980

0.10

400.

0964

200

0.07

420.

0709

0.07

210.

0692

0.07

140.

0687

300

0.05

950.

0576

0.05

810.

0567

0.05

760.

0563

400

0.05

140.

0498

0.05

040.

0492

0.05

010.

0489

(b) A

vera

ged d

iffer

ence

betw

een e

mpiri

cal s

tanda

rd er

rors

and f

ormu

la-ba

sed st

anda

rd er

rors.

nM

LM

L .1

ML .

2

100

0.00

950.

0083

0.00

76

200

0.00

350.

0032

0.00

29

300

0.00

210.

0016

0.00

14

400

0.00

180.

0015

0.00

13


NIH


NIH


NIH


Yuan et al. Page 27

Table 6

Number of convergence and average number of iterations with 6 five-category and 9 dichotomous indicators,1000 replications.

n ML ML.1 ML.2

number of convergence

100 294 882 997

200 938 1000 1000

300 998 1000 1000

400 1000 1000 1000


100 16.320 10.249 9.002

200 8.453 7.317 6.776

300 7.119 6.433 6.030

400 6.514 5.958 5.622


NIH


NIH


NIH


Yuan et al. Page 28

Tabl

e 7

Ave

rage

d bi

as1

03, var

ianc

e10

3 an

d M

SE1

03, w

ith 6

five

-cat

egor

y an

d 9

dich

otom

ous i

ndic

ator

s, ba

sed

on co

nver

ged

repl

icat

ions

.

n

ML

ML .

1M

L .2

Bias

Var

MSE

Bias

Var

MSE

Bias

Var

MSE

100

6.28

710

.170

10.2

305.

424

9.22

29.

259

4.25

48.

934

8.95

7

200

3.71

44.

595

4.61

42.

306

4.36

14.

369

1.80

64.

255

4.26

1

300

2.40

22.

985

2.99

31.

629

2.85

42.

857

1.26

12.

798

2.80

0

400

2.01

02.

242

2.24

71.

478

2.15

32.

156

1.20

12.

115

2.11

7


NIH


NIH


NIH


Yuan et al. Page 29

Tabl

e 8

Perfo

rman

ce o

f the

stat

istic

s TRM

La an

d T A

MLa

w

ith 6

five

-cat

egor

y an

d 9

dich

otom

ous i

ndic

ator

s, ba

sed

on co

nver

ged

repl

icat

ions

(df =

p*

q

= 8

7,R

N=r

eject

numb

er, R

R=rej

ect ra

tio).

nM

etho

d

T MLa

T RM

LaT A

MLa

Mea

nSD

Mea

nSD

RN

RR

Mea

nSD

RN

RR

100

ML

504.

1916

4.86

123.

4343

.13

165

56.1

%42

.64

16.7

212

442

.2%

ML .

131

3.13

94.0

812

6.18

37.4

256

564

.1%

48.2

714

.05

378

42.9

%

ML .

218

3.64

38.0

510

6.38

21.6

338

338

.4%

42.5

78.

4318

018

.1%

200

ML

487.

3616

9.66

123.

4741

.20

522

55.7

%59

.69

19.7

840

443

.1%

ML .

124

9.16

49.1

010

1.26

19.6

230

430

.4%

53.8

810

.19

177

17.7

%

ML .

216

4.26

27.5

495

.18

15.9

219

019

.0%

52.9

28.

5787

8.7%

300

ML

413.

7711

0.85

106.

3127

.42

365

36.6

%59

.24

15.1

525

425

.5%

ML .

123

4.68

43.5

895

.79

17.7

319

019

.0%

58.5

810

.61

114

11.4

%

ML .

215

9.44

27.0

292

.41

15.7

112

912

.9%

58.9

89.

7175

7.5%

400

ML

383.

5581

.45

99.4

320

.54

270

27.0

%60

.00

12.3

117

417

.4%

ML .

122

7.84

39.0

693

.19

16.0

014

414

.4%

61.5

210

.35

888.

8%

ML .

215

6.81

25.1

890

.89

14.6

810

710

.7%

62.6

09.

8466

6.6%


NIH


NIH


NIH


Yuan et al. Page 30

Tabl

e 9

Perfo

rman

ce o

f the

stat

istic

s TRR

LSa

and

T ARL

Sa w

ith 6

five

-cat

egor

y an

d 9

dich

otom

ous i

ndic

ator

s, ba

sed

on co

nver

ged

repl

icat

ions

(df =

p*

q

= 8

7,R

N=r

eject

numb

er, R

R=rej

ect ra

tio).

nM

etho

d

T RLS

aT R

RLSa

T ARL

Sa

Mea

nSD

Mea

nSD

RN

RR

Mea

nSD

RN

RR

100

ML

361.

6398

.63

86.2

417

.23

279.

2%29

.19

5.85

41.

4%

ML .

121

4.81

33.1

986

.67

13.3

344

5.0%

33.2

15.

258

0.9%

ML .

215

2.91

24.0

888

.62

13.7

872

7.2%

35.4

85.

4013

1.3%

200

ML

339.

9857

.03

86.4

114

.10

555.

9%41

.88

7.40

131.

4%

ML .

121

6.18

34.8

787

.88

14.0

569

6.9%

46.7

87.

3515

1.5%

ML .

215

1.76

23.6

087

.94

13.6

763

6.3%

48.9

07.

3415

1.5%

300

ML

341.

3361

.49

87.8

015

.29

898.

9%48

.98

8.71

363.

6%

ML .

121

5.28

35.7

787

.87

14.5

675

7.5%

53.7

58.

7138

3.8%

ML .

215

1.60

24.4

687

.87

14.2

067

6.7%

56.0

88.

7535

3.5%

400

ML

337.

9858

.18

87.6

614

.79

808.

0%52

.92

8.99

373.

7%

ML .

121

4.29

34.2

187

.65

14.0

570

7.0%

57.8

79.

0934

3.4%

ML .

215

1.23

23.5

387

.65

13.7

466

6.6%

60.3

79.

1931

3.1%


NIH


NIH


NIH


Yuan et al. Page 31

Tabl

e 10

Perfo

rman

ce o

f sta

ndar

d er

rors

with

6 fi

ve-c

ateg

ory

and

9 di

chot

omou

s ind

icat

ors,

base

d on

conv

erge

d re

plic

atio

ns.

(a) A

vera

ged e

mpiri

cal s

tanda

rd er

rors

and f

ormu

la-ba

sed st

anda

rd er

rors

n

ML

ML .

1M

L .2

SEE

SEF

SEE

SEF

SEE

SEF

100

0.09

890.

0911

0.09

440.

0886

0.09

290.

0876

200

0.06

660.

0642

0.06

500.

0628

0.06

420.

0624

300

0.05

370.

0523

0.05

260.

0514

0.05

210.

0511

400

0.04

640.

0453

0.04

550.

0446

0.04

510.

0443

(b) A

vera

ged d

iffer

ence

betw

een e

mpiri

cal s

tanda

rd er

rors

and f

ormu

la-ba

sed st

anda

rd er

rors.

nM

LM

L .1

ML .

2

100

0.00

800.

0058

0.00

54

200

0.00

260.

0022

0.00

20

300

0.00

160.

0013

0.00

12

400

0.00

140.

0012

0.00

12


NIH


NIH


NIH


Yuan et al. Page 32

Table 11

Number of convergences and average number of iterations with 6 continuous and 9 dichotomized indicators,1000 replications.

n ML ML.1 ML.2

number of convergence

100 279 919 1000

200 946 1000 1000

300 999 1000 1000

400 1000 1000 1000


100 13.756 9.950 8.703

200 8.205 7.121 6.626

300 6.974 6.278 5.916

400 6.391 5.847 5.532


NIH


NIH


NIH


Yuan et al. Page 33

Tabl

e 12

Ave

rage

d bi

as1

03, var

ianc

e10

3 an

d M

SE1

03, w

ith 6

cont

inuo

us an

d 9

dich

otom

ized

indi

cato

rs, b

ased

on

conv

erge

d re

plic

atio

ns.

n

ML

ML .

1M

L .2

Bias

Var

MSE

Bias

Var

MSE

Bias

Var

MSE

100

7.93

99.

160

9.24

55.

000

8.49

08.

522

3.91

88.

196

8.21

6

200

3.23

24.

264

4.28

02.

114

4.00

44.

011

1.67

93.

896

3.90

0

300

2.11

02.

735

2.74

11.

408

2.60

32.

606

1.07

62.

546

2.54

8

400

1.77

12.

077

2.08

11.

323

1.98

61.

988

1.09

11.

945

1.94

7


NIH


NIH


NIH


Yuan et al. Page 34

Tabl

e 13

Perfo

rman

ce o

f the

stat

istic

s TRM

La an

d T A

MLa

w

ith 6

cont

inuo

us an

d 9

dich

otom

ized

indi

cato

rs, b

ased

on

conv

erge

d re

plic

atio

ns (d

f = p

*

q

= 8

7,R

N=r

eject

numb

er, R

R=rej

ect ra

tio).

nM

etho

d

T MLa

T RM

LaT A

MLa

Mea

nSD

Mea

nSD

RN

RR

Mea

nSD

RN

RR

100

ML

451.

8314

8.05

125.

8542

.36

164

58.8

%42

.90

16.1

711

039

.4%

ML .

127

6.10

80.0

912

3.91

35.3

355

860

.7%

45.7

112

.77

353

38.4

%

ML .

216

3.02

35.3

410

5.92

22.6

837

237

.2%

41.0

68.

5517

417

.4%

200

ML

427.

8314

9.16

120.

1439

.62

492

52.0

%54

.92

17.8

136

738

.8%

ML .

122

1.96

45.8

310

0.88

20.7

030

430

.4%

51.1

410

.24

184

18.4

%

ML .

214

6.32

26.0

195

.20

17.0

019

619

.6%

50.6

18.

7694

9.4%

300

ML

372.

1312

4.47

106.

1734

.89

366

36.6

%55

.62

18.1

524

224

.2%

ML .

120

9.60

41.5

895

.60

18.8

519

919

.9%

55.3

110

.71

121

12.1

%

ML .

214

2.08

25.6

392

.41

16.6

713

913

.9%

55.9

99.

8385

8.5%

400

ML

344.

3475

.63

99.1

421

.14

257

25.7

%55

.98

11.8

617

017

.0%

ML .

120

3.74

36.0

193

.15

16.4

815

815

.8%

57.9

110

.06

868.

6%

ML .

213

9.81

23.0

990

.95

15.1

110

710

.7%

59.2

19.

6061

6.1%


NIH


NIH


NIH


Yuan et al. Page 35

Tabl

e 14

Perfo

rman

ce o

f the

stat

istic

s TRR

LSa

and

T ARL

Sa w

ith 6

cont

inuo

us an

d 9

dich

otom

ized

indi

cato

rs, b

ased

on

conv

erge

d re

plic

atio

ns (d

f = p

*

q

= 8

7,R

N=r

eject

numb

er, R

R=rej

ect ra

tio).

nM

etho

d

T RLS

aT R

RLSa

T ARL

Sa

Mea

nSD

Mea

nSD

RN

RR

Mea

nSD

RN

RR

100

ML

308.

9686

.38

84.3

918

.34

3111

.1%

28.2

55.

967

2.5%

ML .

119

5.18

32.1

587

.73

14.4

368

7.4%

32.4

35.

4810

1.1%

ML .

213

7.51

22.7

889

.39

14.8

696

9.6%

34.6

85.

6514

1.4%

200

ML

308.

3356

.25

86.8

815

.26

737.

7%39

.84

7.45

151.

6%

ML .

119

4.32

33.3

288

.33

15.1

592

9.2%

44.8

07.

5433

3.3%

ML .

213

5.86

22.4

388

.40

14.7

085

8.5%

47.0

07.

5625

2.5%

300

ML

308.

4460

.13

88.0

916

.58

103

10.3

%46

.20

8.85

464.

6%

ML .

119

3.14

34.2

988

.11

15.5

786

8.6%

50.9

88.

8540

4.0%

ML .

213

5.47

23.2

188

.10

15.1

083

8.3%

53.3

88.

8938

3.8%

400

ML

305.

3254

.38

87.9

515

.30

878.

7%49

.68

8.72

373.

7%

ML .

119

2.26

31.5

887

.91

14.4

974

7.4%

54.6

58.

8537

3.7%

ML .

213

5.11

21.5

887

.90

14.1

472

7.2%

57.2

28.

9735

3.5%


NIH


NIH


NIH


Yuan et al. Page 36

Tabl

e 15

Perfo

rman

ce o

f sta

ndar

d er

rors

with

6 co

ntin

uous

and

9 di

chot

omiz

ed in

dica

tors

, bas

ed o

n co

nver

ged

repl

icat

ions

.

(a) A

vera

ged e

mpiri

cal s

tanda

rd er

rors

and f

ormu

la-ba

sed st

anda

rd er

rors

n

polychoric correlation

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