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Sociological Methods & Research
DOI: 10.1177/0049124105280200 2005; 34; 240
Sociological Methods Research
Ke-Hai Yuan, Peter M. Bentler and Wei Zhang
Analysis: The Univariate Case and Its Multivariate Implication
The Effect of Skewness and Kurtosis on Mean and Covariance Structure
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and Covariance Structure Analysis
The Univariate Case and Its Multivariate Implication
KE-HAI YUAN
University of Notre Dame
PETER M. BENTLER
University of California, Los Angeles
WEI ZHANG
University of Notre Dame
The maximum likelihood (ML) method, based on the normal distribution assumption, is widely used in mean and covariance structure analysis. With typical nonnormal data, the ML method will lead to biased statistics and inappropriate scientific conclusions. This article develops a simple but informative case to show how ML results are influ-enced by skewness and kurtosis. Specifically, the authors discuss how skewness and kurtosis in a univariate distribution affect the standard errors of the ML estimators, the covariances between the estimators, and the likelihood ratio test of hypotheses on mean and variance parameters. They also describe corrections that have been devel-oped to allow appropriate inference. Enough details are provided so that this material can be used in graduate instruction. For each result, the corresponding results in the higher dimensional case are pointed out, and references are provided.
Keywords:likelihood ratio statistic; nonnormal data; sandwich-type covariance matrix; Wald statistics
1. INTRODUCTION
Mean and covariance structure analysis is becoming increasingly popular in social and behavioral sciences (Bollen 2002; Boomsma
AUTHORS’ NOTE: The research was supported by Grants DA00017 and DA01070 from the National Institute on Drug Abuse and NSF Grant DMS04-37167. We thank three referees for their constructive comments, which led to an improved version of the article.
SOCIOLOGICAL METHODS & RESEARCH, Vol. 34, No. 2, November 2005 240-258 DOI: 10.1177/0049124105280200
Ó2005 Sage Publications
2000; MacCallum and Austin 2000). The most widely used method for estimation and testing is normal theory-based maximum likeli-hood (ML). In this method, parameter estimates are obtained by maximizing the likelihood function derived from the multivariate normal distribution. Standard errors of the maximum likelihood estimators (MLE) are based on the covariance matrix that is obtained by inverting the associated information matrix. Overall model evaluation is accomplished by referring the likelihood ratio (LR) statistic to a chi-square distribution. Fit indices are also related to, or derived from, the LR statistic. Although data in practice are seldom normally distributed (Micceri 1989), researchers commonly use the ML method without checking the distribution assumption. One possible reason is that ML is the default method in almost all the structural equation modeling (SEM) software. Another reason may be that the effects of nonnormally distributed data on standard errors of the MLEs and on the LR statistic are not well understood by applied researchers. Actually, even more technically oriented publications do not emphasize limitations of the normal theory ML approach with nonnormal data (see, e.g., reviews by Breckler 1990; MacCallum and Austin 2000).
this case can be used as a teaching tool in SEM courses for graduate students in the social and behavioral sciences.
In the one-dimensional case, the interesting parameters are the population mean and variance. The effect of nonnormal data on sta-tistical inference for these two parameters can be totally character-ized by skewness and kurtosis. The concepts of skewness and kurtosis in the one-dimensional case are well known to graduate students in social sciences (see, e.g., Tabachnick and Fidell 2001:73-5). The other concepts involved in this article are partial derivatives, the law of large numbers, and the central limit theorem. We will provide the necessary steps for each result so that a quanti-tative graduate student will be able to check or derive it. For each result, we will also point out the parallel higher dimensional result in the SEM literature.
In section 2, we study the effect of nonnormal data on the iances and covariance of the MLEs of the population mean and var-iance. In section 3, we study the effect of nonnormal data on the LR and related statistics. In section 4, we present an example illustrat-ing the effect of nonnormal data on the distributions of the MLEs and the LR statistics. A discussion with a further guide to the litera-ture is provided in section 5.
2. THE NORMAL THEORY BASED MAXIMUM LIKELIHOOD ESTIMATOR
Let y1, y2,. . ., yn be a random sample from a population y with
Givenyi, the likelihood function based onyi∼N(µ, σ2)is
Li(µ, σ2)=L(µ, σ2|yi)= 1
(2πσ2)1/2exp{−(yi−µ)2/(2σ2)},
which is just the normal density function withyiknown. The corre-sponding log-likelihood functionli=log(Li)is
li(µ, σ2)= −
which maximize the log-likelihood function
l(µ, σ2)= n
i=1
li(µ, σ2): (1)
Closely related to the log-likelihood function is the so-called infor-mation matrix
as functions of a more basic set of parameters. Then elements of −I will be just the expectation of the second derivative of the log-likelihood function with respect to a pair of the parameters.
Letθ=(µ, σ2)0andθ^=(y, s 2)0. When data are normally distrib-uted, standard asymptotic statistical theory (see, e.g., Ferguson 1996:121) tells us that, asn→∞,
ffiffiffi n p
(θ^−θ)!L N(0,), (2)
where !L means ‘‘converging in distribution.’’ This means that, with a largen, the distribution of the left side of (2) can be approxi-mately described by a normal random vector with mean zero and covariance matrix . Furthermore, this covariance matrix is the inverse of the information matrix
= ω11 ω12
Because is a diagonal matrix, µ^ andσ^2 are asymptotically inde-pendent. Of course, they are also independent with any finite sample sizes due to y∼N(µ, σ2) (see, e.g., Casella and Berger 2002:218; Hays 1994:250). Such a result holds also in higher dimensional normal data. That is, when the mean and covariance structures do not have overlapping parameters, parameter estimates in the mean struc-ture are asymptotically independent of parameter estimates in the variance-covariance structure (see Yuan and Bentler forthcoming).
We next study the distribution of θ^=(µ,^ σ^2)0 when data are nonnormally distributed. Notice that
s2=1
Because (y−µ) approaches zero in probability and ffiffiffi n p
(y−µ) is bounded in probability (see, e.g., Bishop, Fienberg, and Holland 1975:476), ffiffiffi
n p
~
θ=(y, σ~2)0. It follows from (4) and the well-known Slutsky’s (1925) theorem1 that ffiffiffi
n
Applying the central limit theorem to the right side of (5) leads to
Bentler (1993); Browne and Arminger (1995); and Yuan and Bentler (1997a, 1998a, 1998b, 2000b).
It follows from (6) that the asymptotic distribution ofσ^2depends onσ2 andβbut notγ. In contrast, the asymptotic distribution of
^
µ
does not depend on either γ or β. This is also true in the higher dimensional case. Results in Yuan and Bentler (1999a, 2000a, 2002a) imply that the asymptotic distributions of the covariance parameter estimates, the commonly used sample correlation coeffi-cients, and sample reliability coefficients depend on only the joint fourth-order moments or kurtoses of the variables. Equation (6) also tells us thatµ^andσ^2 are no longer asymptotically independent when
γ6¼0. This is also true in the higher dimensional case when not all the third-order moments are zero, where mean and covariance para-meter estimates are not asymptotically independent even when they do not have overlapping parameters (Yuan and Bentler forthcoming).
3. THE NORMAL THEORY-BASED LIKELIHOOD RATIO TEST
We first consider the distribution of the LR statistic whenµis a free parameter. The null hypothesis isH0:σ2=σ20. Notice that whenH0 is true, theσ2
0 will equal the σ2 in section 2, which will also be the scenario we consider in this section. The behavior of the LR statistic with misspecified models was studied in Shapiro (1983); Satorra and Saris (1985); Steiger, Shapiro, and Browne (1985); Satorra (1989); Yuan and Hayashi (2003); Yuan (2005); and Yuan and Bentler (forthcoming) Yuan, Hayashi and Bentler (2005).
Using the log-likelihood function in (1), we obtain the LR statistic as
It is obvious that (7) is just the univariate version of the normal theory-based discrepancy function in covariance structure analysis (see equa-tion 4.67 of Bollen 1989:107). Notice that
and(s2/σ2
where nrn approaches zero in probability when n→∞. Putting (8) into (7), we get
It follows from (6) that
ffiffiffi
So the distribution of the LR statistic is proportional to kurtosis.
When data are normally distributed, β=3 and TML! L
χ2
1. A correct hypothesis σ2
0 can be easily rejected when we refer theTML in (7) to
χ2
1 while β >3. Similarly, a wrong hypothesis might not be rejected whenβ <3, even whennis large. In the higher dimensional case, the LR statistic is also proportional to the common kurtosis when data are elliptically symmetric (Browne 1984; Shapiro and Browne 1987), and
TML may still not depend on skewness when the marginal kurtosis is heterogeneous (Kano, Berkane, and Bentler 1990) or even when data are skewed (Yuan and Bentler 1999b).
With a consistent estimator ofπ22, we can rescaleTMLto
TR= 2σ^4
^
π22
It is obvious thatσ^4/π^
22converges in probability to 1/(β−1)and
TR! L
χ21:
In the multivariate case, the statisticTR is just the Satorra and Bentler (1988, 1994) rescaled statistic.
Notice that
The Wald-type statistic for testingσ2=σ2 0is
As long asπ^22is consistent forπ22, the asymptotic distribution ofTW is χ2
1, which does not depend on the underlying distribution of y. Such a property is commonly called asymptotically distribution free (ADF) in the SEM literature. Two estimates of π22 are available. One is
The other one is
^ asymptotically equivalent. The well-known ADF statistic (Browne 1984) in covariance structure analysis corresponds to a multivariate version ofTW withπ^22=π^
(1)
in Yuan and Bentler (1997b) corresponds to TW with π^22=π^ (2) 22. Yuan and Bentler (1998c) provided a corrected residual-based ADF statistic, which is also a multivariate version ofTWwithπ^22=π^ It follows from (6) that
xn=Π−1/2 ffiffiffi n p
(θ^−θ0)!L N(0,I2),
whereI2is the 2-by-2 identity matrix. Let
W= 1/σ
It follows from (8) and (13) that
TML=n exist eigenvectorsv1andv2 such that
Notice that the eigenvalues ofΠ1/2W Π1/2equal the eigenvalues of
The determinant equation determining the eigenvalues is
|W1/2Π W1/2−I2|=0, Solving this equation, we have
such a case, µ^ and σ^2 are asymptotically independent. A rescaled statistic that removes the effect ofβis given by
TR=n
2. A parallel statistic to (14) can also be constructed in the higher dimensional case, although we are not aware of the existence of such a development.
When data are skewed or γ 6¼0, the two eigenvalues are not equal. We might consider simultaneously removing the effect of skewness and kurtosis by constructing the rescaled statistic
TR=
a chi-square distribution (Satorra and Bentler 1994; Yuan and Bentler 2000b). Even with only a covariance structure model, the rescaled statistic parallel to theTR in (9) may not follow a chi-square distribu-tion either due to the heterogeneity of the eigenvalues, although its distribution is still chi-square under various conditions (see Yuan and Bentler 1999b). Similarly, ADF-type statistics can be constructed in testing H0: (µ, σ2)0=(µ0, σ20)0, and we leave it to readers to work out the details. See Browne and Arminger (1995) and Yuan and Bentler (1997b, 1999c) for the higher dimensional case.
4. A NUMERICAL EXAMPLE
Neumann (1994) studied the relationship of alcohol and psychological symptoms. His data set consists ofp=10 variables andn=335 cases. We will use the family history of psychopathology variable to illustrate the effect of skewness and kurtosis. For this variable, the MLEs ofµ
andσ2 are, respectively,µ^=y=1:361 andσ^2=s2=2:302; the sam-ple skewness and kurtosis are, respectively,
^
γ= 1
n n
i=1
(yi−y) 3/s3=2:001 and β^=8:766:
Note that both γ^ and β^−3 are significantly different from zero (see, e.g., Snedecor and Cochran 1989, Table A19), indicating that the sample most likely comes from a nonnormal distribution. Our purpose here is to illustrate the effect ofγandβon the asymptotic distributions ofµ^andσ^2and statistics for testingµ=µ
0andσ2=σ20, not to elabo-rate on the substantive side of the data.
Assuming the sample is fromN(µ, σ2), the asymptotic distribu-tion of(µ,^ σ^2)0is given by (2) with
^
= 2:302 0
0 10:598
:
Admitting that the data may not be normally distributed, the asymp-totic distribution of(µ,^ σ^2)0is given by (6) with
^
Π= 2:302 6:989 6:989 41:154
If based on (2), the estimated standard error ofσ^2is(10:598/335)1/2= 0:178. If based on (6), the estimated standard error of σ^2 is
(41:154/335)1/2=0:350, almost double that based on (2). Of course,
^
µ and σ^2 are no longer asymptotically independent in this example when using (6). Although the confidence interval forσ2based on (2) is much shorter than that based on (6), the shorter interval is a misleading result due to the nonnormality of the data.
Turning to hypothesis testing, suppose the null hypothesis is
H0:(µ, σ2)=(1:2,1:8). When testingσ2=1:8 alone, the LR statis-tic in (7) isTML=11:021, which is highly significant when referred toχ2
1. The rescaled statistic in (9) isTR=2:838; the Wald statistic in (10), using theπ^(221) in (11), isT(W1)=2:051; and the Wald statistic in (10), using theπ^(222)in (12) isTW(2)=2:039. None is statistically signif-icant at theα=0:05 level when referred toχ2
1. Note that T (1) W and
T(W2) have a tiny difference in this example because of p=1 and a relatively large sample size. In the higher dimensional case, their dif-ference can be huge (Yuan and Bentler 1997b, 1998c), and T(W2) is recommended for more reliable inference with smaller samples.
When testing(µ, σ2)=(1:2,1:8)simultaneously, the LR statis-tic in (13) is TML=15:845, which is highly significant when referred to χ2
2. The rescaled statistic in (15) is given by
TR=4:153, which is no longer statistically significant at the
α=0:05 level when referred toχ2
2. Note that theTR in (15) unli-kely followsχ2
2due to the significance ofγ^. However, referringTR to a chi-square distribution does make the inference more reliable. More empirical results about these statistics in higher dimensional cases can be found in Hu, Bentler, and Kano (1992) and Yuan and Bentler (1998c).
The rescaled statistic in (14) isTR=7:662, which is statistically significant at theα=0:05 level when referred toχ2
2. However, this
TR is not justified whenγ^is statistically significant.
5. DISCUSSION
Motivated by the gap in teaching resources and the technical litera-ture of SEM, this article provides a simplified version of SEM with nonnormal data. Some of the material in this article may be just a trivial exercise to quantitative graduate students. For students or researchers who do not have much quantitative training, the material should better facilitate an understanding of the effect of nonnormal data on standard errors and test statistics in mean and covariance structure analysis. Of course, fit indices defined throughTMLwill be equally affected by skewness or kurtosis (see Yuan 2005 and Yuan and Marshall 2004). Readers with a solid quantitative background may further read the literature in the higher dimensional case cited in sections 2 and 3. We hope the article will help to solidify an under-standing of the effect of nonnormal data on SEM inference, espe-cially in graduate education.
Proper procedures have to be used to get reliable inferences with nonnormal data. Although we do not have space to discuss these, we would like to note that truly robust methods, not depending on ML, do exist. These methods minimize the effect of bad data not only on standard errors and test statistics but also on parameter estimates and power evaluations (Yuan and Bentler 1998a, 1998b, 2000c; Yuan, Bentler, and Chan 2004; Yuan, Chan, and Bentler 2000; Yuan and Hayashi 2003; Yuan, Marshall, and Weston 2002). It is well known that Mardia’s (1970, 1974) measure of multivariate kurtosis is a generalization of the univariate kurtosis β−3. When the sample multivariate kurtosis is significantly greater than that of the multivari-ate normal distribution, a robust procedure might be necessary. In small samples, the significance of Mardia’s coefficient can be evalu-ated using the simulation approach of Bonett, Woodward, and Randall (2002). In addition to nonnormal data, a small sample size also tends to cause the significance of the statistic TML with correctly specified models. Remedies in this direction are addressed in Bentler and Yuan (1999) and Yuan and Bentler (1999c).
skewness and kurtosis do not vanish when data are missing or when data are obtained under hierarchical sampling schemes. The same principles apply—that is, normal theory-based ML standard errors and test statistics will be biased under nonnormality. Solutions to this problem for the missing data case were developed by Arminger and Sobel (1990) and Yuan and Bentler (2000b). Solutions for multilevel data were provided by Poon and Lee (1994) and Yuan and Bentler (2002b, 2003).
There is a related literature called asymptotic robustness theory. This is concerned with the validity of normal theory-based methods with large-sample nonnormal data (Amemiya and Anderson 1990; Anderson and Amemiya 1988; Browne and Shapiro 1988; Mooijaart and Bentler 1991; Satorra and Bentler 1990; Shapiro 1987; Yuan and Bentler 1999a, 1999b). Unfortunately, the conditions for asymptotic robustness depend on both the data and the model, and there is no effective way to verify these conditions at present. It is not appropriate to blindly trust that a researcher’s given data and model satisfy these conditions.
In practice, TML with empirical data is often statistically signifi-cant when referred to a chi-square distribution. However, a smallp
value associated with TML may not be due to a bad model and/or too much power (i.e., a huge sample size). It may be due to viola-tion of assumpviola-tions or bad data (Yuan and Bentler 2001). The newer statistics2 described in this article do not require making the stringent multivariate normality assumption. These statistics not only are liable to make a good model more acceptable statistically but also should lead to more accurate scientific conclusions.
NOTES
1. The theorem states that, ifxn!
L
x,anconverges in probability toaandbnconverges
in probability tob,anxn+bn!
L
ax+b.
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Ke-Hai Yuan is an associate professor in quantitative psychology at the University of Notre Dame. His research interests are in the areas of psychometric theory and applied multivariate statistics. He received the Cattell award for early career outstanding multi-variate research from the Society of Multimulti-variate Experimental Psychology in 2002.
Peter M. Bentler is a Distinguished Professor of Psychology and Statistics at UCLA who has been an elected president of the Psychometric Society, the Society of Multivari-ate Experimental Psychology, and the Division of Evaluation, Measurement, and Statis-tics of the American Psychological Association. He directs the Center for Collaborative Research on Drug Abuse.
