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**|**HHS Author Manuscripts**|**PMC2905175

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- Abstract
- Introduction
- Setup and Notation
- Scaling Correction for the Difference Test
- Problem with the Current Scaled Difference Test
- A New Scaled Test Statistic d
- An Illustration
- Discussion
- References

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Psychometrika. Author manuscript; available in PMC 2010 July 16.

Published in final edited form as:

Psychometrika. 2010 June; 75(2): 243–248.

doi: 10.1007/s11336-009-9135-yPMCID: PMC2905175

NIHMSID: NIHMS127970

Albert Satorra, Universitat Pompeu Fabra, Barcelona;

See other articles in PMC that cite the published article.

A scaled difference test statistic ${\stackrel{~}{T}}_{d}$ that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (2001). The statistic ${\stackrel{~}{T}}_{d}$ is asymptotically equivalent to the scaled difference test statistic * _{d}* introduced in Satorra (2000), which requires more involved computations beyond standard output of SEM software. The test statistic ${\stackrel{~}{T}}_{d}$ has been widely used in practice, but in some applications it is negative due to negativity of its associated scaling correction. Using the implicit function theorem, this note develops an improved scaling correction leading to a new scaled difference statistic

Moment structure analysis is widely used in behavioural, social and economic studies to analyze structural relations between variables, some of which may be latent see, e.g., Bollen and Curran (2006), Grace (2006), Yuan and Bentler (2007), and references therein. In such analyses it frequently happens that two nested models *M*_{0} and *M*_{1} are compared using estimation methods that are non-optimal (asymptotically) given the distribution of the data; e.g. ML estimation is used when the data are not multivariate normal. In those circumstances, the usual chi-square difference test *T _{d}* =

Satorra and Bentler’s (2001) correction provided a simple procedure to obtain an approximate scaled chi-square statistic using hand calculations on regular output of SEM analysis; it has the drawback, however, that a positive value for the scaling correction is not assured. The present paper develops a simple procedure by which a researcher can compute the exact SB difference test statistic based only on output from standard SEM programs.

Throughout we adhere to the notation and results of Satorra and Bentler (2001). Let *σ* and *s* be *p*-dimensional vectors of population and sample moments respectively, where *s* tends in probability to *σ* as sample size *n* → +∞. Let $\sqrt{n}(s-\sigma )$ be asymptotically normally distributed with a finite asymptotic variance matrix Γ (*p* × *p*). Consider the model *M*_{0} : *σ* = *σ*(*θ*) for the moment vector *σ*, where *σ*(.) is a twice-continuously differentiable vector-valued function of *θ*, a *q*-dimensional parameter vector. Consider a discrepancy function *F* = *F* (*s, σ*) in the sense of Browne (1984), and the estimator based on *F* or on an (asymptotically equivalent) weighted least squares (WLS) analysis with weight matrix $V=\frac{1}{2}{\partial}^{2}F(s,\sigma )\u2215\partial \sigma \partial {\sigma}^{\prime}$ evaluated at *σ* = *s*.

Let *M*_{0} : *σ* = *σ*(*δ*), *a*(*δ*) = 0, and *M*_{1} : *σ* = *σ*(*δ*) be two nested models for *σ*. Here *δ* is a (*q*+*m*)-dimensional vector of parameters, and *σ*(.) and *a*(.) (an *m*-valued function) are twice-continuously differentiable vector-valued functions of *δ* ϴ_{1}, a compact subset of *R*^{q+m}. Our interest is in the test of a null hypothesis *H*_{0} : *a*(*δ*) = 0 against the alternative *H*_{1} : *a*(*δ*) ≠ 0.

For the developments that follow, we require the Jacobian matrices Π(*p*× (*q* + *m*)) := *σ*(*δ*)/*δ*’ and *A*(*m* × (*q* + *m*)) := *a*(*δ*)/*δ*’, which we assume to be regular at the true value of *δ*, say *δ*_{0}. We also assume that *A* is of full row rank. By using the implicit function theorem, associated to *M*_{0} (more specifically, to the restrictions *a*(*δ*) = 0), there exists (locally in a neighborhood of *δ*_{0}) a one-to-one function *δ* = *δ*(*θ*) defined in an open and compact subset S of *R ^{q}*, and a

$${c}_{1}\u2254\frac{1}{{r}_{1}}tr\left\{{U}_{1}\Gamma \right\}=\frac{1}{{r}_{1}}tr\left\{V\Gamma \right\}-\frac{1}{{r}_{1}}tr\left\{{P}^{-1}{\Pi}^{\prime}V\Gamma V\Pi \right\},$$

(1)

where

$${U}_{1}\u2254V-V\Pi {P}^{-1}{\Pi}^{\prime}V.$$

(2)

We refer to Satorra and Bentler (2001) for further details.

When both models *M*_{0} and *M*_{1} are fitted, for example by ML, then we can test the restriction *a*(*δ*) = 0, assuming *M*_{1} holds, using the chi-square difference test statistic *T _{d}* :=

$${\stackrel{\u2012}{T}}_{d}\u2254{T}_{d}\u2215{\widehat{c}}_{d},\text{where}\phantom{\rule{thickmathspace}{0ex}}{c}_{d}\u2254\frac{1}{m}tr\left\{{U}_{d}\Gamma \right\}$$

(3)

with

$${U}_{d}=V\Pi {P}^{-1}{A}^{\prime}{\left(A{P}^{-1}{A}^{\prime}\right)}^{-1}A{P}^{-1}{\Pi}^{\prime}V.$$

(4)

Here, *ĉ _{d}* denotes

A practical problem with the statistic * _{d}* is that it requires computations that are outside the standard output of current structural equation modeling programs. Furthermore, difference tests are usually hand computed from different modeling runs. Satorra and Bentler (2001) proposed a procedure to combine the estimates of the scaling corrections

- Obtain the unscaled and scaled goodness-of-fit tests when fitting
*M*_{0}and*M*_{1}respectively; that is,*T*_{0}and_{0}when fitting*M*_{0}, and*T*_{1}and_{1}when fitting*M*_{1}; - Compute the scaling corrections
*ĉ*_{0}=*T*_{0}/_{0},*ĉ*_{1}=*T*_{1}/_{1}, and the unscaled chi-square difference*T*=_{d}*T*_{0}-*T*_{1}and its degrees of freedom*m*=*r*_{0}-*r*_{1}; - Compute the scaled difference test statistic as$${\stackrel{~}{T}}_{d}\u2254{T}_{d}\u2215{\stackrel{~}{c}}_{d}\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}{\stackrel{~}{c}}_{d}=({r}_{0}{\widehat{c}}_{0}-{r}_{1}{\widehat{c}}_{1})\u2215m.$$

Here *r*_{0} and *r*_{1} are the respective degrees of freedom of the models *M*_{0} and *M*_{1}.

The basis for computing the scaling correction for the difference test statistic is the following alternative expression for *U _{d}* of (4) (see Satorra and Bentler, 2001, p. 510)

$${U}_{d}={U}_{0}-{U}_{1},$$

(5)

where *U*_{1} is given in (2) and *U*_{0} := *V* - *V*Π*H*(*H*’Π’*V*Π*H*)^{-1}*H*’Π’*V*. Since (5) implies

$$m{c}_{d}=tr\left\{{U}_{d}\Gamma \right\}=tr\left\{({U}_{0}-{U}_{1})\Gamma \right\}={r}_{0}{c}_{0}-{r}_{1}{c}_{1},$$

(6)

it follows that *c _{d}* = (

For an arbitrary matrix *V* > 0, (5) and (6) are exact equalities when the matrices *U _{d}*,

In order to be sure to avoid a negative value for ${\stackrel{~}{c}}_{d}$ and hence ${\stackrel{~}{T}}_{d}$, currently one would need to resort to computing * _{d}* using the (3). Unfortunately this is impractical or impossible for most applied researchers who only have access to standard SEM software.

Fortunately, as we show next, the exact value of * _{d}* can also be obtained from the standard output of SEM software, using a new hand computation.

Denote by *M*_{10} the fit of model *M*_{1} to a model setup with starting values taken as the final estimates obtained from model *M*_{0}, and with number of iterations set to 0. Consider *ĉ*_{1}^{(10)} := *T*_{1}^{(10)}/_{1}^{(10)}, where *T*_{1}^{(10)} and _{1}^{(10)} are the standard unscaled and scaled test statistic of this additional run. Note that the estimate *ĉ*_{1}^{(10)} uses model *M*_{1} but the matrices and *A* are now evaluated at _{0} := *δ*(), where is the estimate under *M*_{0}. Since now all the matrices involved in (5) are evaluated at *δ*^_{0}, the equality (6) holds exactly, and not only asymptotically, as when *U*_{0} is evaluated at *δ*^_{0} and *U*_{1} at *δ*^_{1}. The scaling correction that is now computed is

$${\widehat{c}}_{d}^{\left(10\right)}\u2254({r}_{0}{\widehat{c}}_{0}-{r}_{1}{\widehat{c}}_{1}^{\left(10\right)})\u2215m,$$

(7)

which, in view of (6), is the scaling correction of (3) when *U _{d}* is evaluated at the estimate

$${\stackrel{\u2012}{T}}_{d}^{\left(10\right)}\u2254({T}_{0}-{T}_{1})\u2215{\widehat{c}}_{d}^{\left(10\right)},$$

(8)

Clearly, ${\stackrel{\u2012}{T}}_{d}^{\left(10\right)}={\stackrel{\u2012}{T}}_{d}$ that is, ${\stackrel{\u2012}{T}}_{d}^{\left(10\right)}$ coincides numerically with the scaled statistic (3) proposed in Satorra (2000).

We use this data just for illustrative purposes, and because it provides an example where the standard scaling correction fails to be positive. We use a latent variable model discussed for this data by Bentler, Satorra and Yuan (2009). The Bonett-Woodward-Randall (2002) test shows that these data have significant excess kurtosis indicative of non-normality at a one-tail .05 level, so test statistics derived from ML estimation may not be appropriate and we do the scaling corrections.

The model considered is a structured means model, with the mean cigarette sales indirectly affecting the mean rates of the various cancers. The specified model is

$${V}_{j}={\lambda}_{j}F+{E}_{j},\phantom{\rule{1em}{0ex}}j=2,\dots ,5,\phantom{\rule{1em}{0ex}}F=\beta {V}_{1}+{D}_{1},\phantom{\rule{1em}{0ex}}{V}_{1}=\mu +{E}_{1},$$

where the *V _{j}*’s denote observed variables;

$${T}_{1}=107.398,\phantom{\rule{1em}{0ex}}{\stackrel{\u2012}{T}}_{1}=65.3524,\phantom{\rule{1em}{0ex}}{r}_{1}=9,\phantom{\rule{1em}{0ex}}{\widehat{c}}_{1}=1.6434,$$

along with the degrees of freedom *r*_{1} and the scaling correction *ĉ*_{1}. The model does not fit, though for the sake of the illustration we are aiming for, this is not of concern to us.

The same model is now fitted with the added restriction that the error variances of the kidney and leukemia cancers, *E*_{4} and *E*_{5}, are equal. This model gives the following statistics

$${T}_{0}=139.495,\phantom{\rule{1em}{0ex}}{\stackrel{\u2012}{T}}_{0}=97.4034,\phantom{\rule{1em}{0ex}}{r}_{0}=10,\phantom{\rule{1em}{0ex}}{\widehat{c}}_{0}=1.4322.$$

Our main interest lies in testing the difference between *M*_{0} and *M*_{1}, which we do with the chi-square difference test. The ML difference statistic is

$${T}_{d}=139.495-107.398=32.097,$$

which, with 1 degree of freedom (*m*), rejects the null hypothesis that the error variances for E4 and E5 are equal. Since the data is not normal, we compute the SB (2001) scaled difference statistic. This requires computing the scaling factor ${\stackrel{~}{c}}_{d}=({r}_{0}{\widehat{c}}_{0}-{r}_{1}{\widehat{c}}_{1})\u2215m$ given by

$$[10\left(1.4322\right)-9\left(1.6434\right)]\u22151=14.322-14.7906=-.4686.$$

The scaling factor ${\stackrel{~}{c}}_{d}$ is negative, so the SB difference test cannot be carried out; or, if carried out, it results in an improper negative chi-square value.

As described above, to compute the scaled statistic * _{d}* we implement (7) and (8). The output that is missing in the prior runs is the value of the SB statistic obtained at the final parameter estimates for model

$${T}^{\left(10\right)}=139.495,\phantom{\rule{1em}{0ex}}{\stackrel{\u2012}{T}}^{\left(10\right)}=94.9551,\phantom{\rule{1em}{0ex}}{r}_{1}=9,\phantom{\rule{1em}{0ex}}{\widehat{c}}^{\left(10\right)}=1.4691,$$

where as expected, *T*^{10} = *T*_{0} as reported above (i.e., the ML statistics are identical), and the value *ĉ*^{10} is hand-computed. As a result, we can compute

$${\widehat{c}}_{d}^{\left(10\right)}=({r}_{0}{\widehat{c}}_{0}-{r}_{1}{\widehat{c}}_{1}^{\left(10\right)})\u2215m=[\left(10\right)\left(1.4322\right)-\left(9\right)\left(1.4691\right)]=1.10,$$

which, in contrast to the SB (2001) computations, is positive. Finally, we can compute the proposed new SB corrected chi-square statistic as

$${\stackrel{\u2012}{T}}_{d}={\stackrel{\u2012}{T}}_{d}^{\left(10\right)}=({T}_{0}-{T}_{1})\u2215{\widehat{c}}_{d}^{\left(10\right)}=(139.495-107.398)\u22151.10=29.179,$$

which can be referred to a ${\chi}_{1}^{2}$ variate for evaluation.

The implicit function theorem was used to provide a theoretical basis for the development of a practical version of the computationally more difficult scaled difference statistic proposed by Satorra (2000).^{2} The proposed method is only marginally more difficult to compute than that of Satorra and Bentler (2001) and solves the problem of an uninterpretable negative *χ*^{2} difference test that applied researchers have complained about for some time.

Like the method it is replacing, the proposed procedure applies to a general modeling setting. The vector of parameters *σ* to be modeled may contain various types of moments: means, product-moments, frequencies (proportions), and so forth. Thus, this scaled difference test applies to methods such as factor analysis, simultaneous equations for continuous variables, log-linear multinomial parametric models, etc.. It can easily be seen that the procedure applies also in the case where the matrix Γ is singular, when the data is composed of various samples, as in multi-sample analysis, and to other estimation methods. It applies also to the case where the estimate of Γ reflects the fact that we have intraclass correlation among observations, as in complex samples. Hence this new statistic should be useful in a variety of applied modeling contexts. Simulation work will be needed to understand its virtues and limitations, relative to other alternatives, in such contexts.

^{*}Research supported by grants SEJ2006-13537 and PR2007-0221 from the Spanish Ministry of Science and Technology and by USPHS grants DA00017 and DA01070. This paper is in press in Psychometrika and presently available on line at www.springerlink.com.

^{1}For this particular example, the Appendix of Satorra and Bentler (2008) illustrates this procedure with EQS (Bentler, 2008). In the same reference, there is a second illustration with a larger degrees of freedom.

^{2}Satorra (2000) provides also Monte Carlo evidence—on a specific model context and various sample sizes - of the superiority of the scaling correction over other alternatives such as the adjusted (mean and variance corrected) statistic.

Albert Satorra, Universitat Pompeu Fabra, Barcelona.

Peter M. Bentler, University of California, Los Angeles.

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