Asymptotic Robustness Study of the Polychoric Correlation Estimation
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Abstract
Asymptotic robustness against misspecification of the underlying distribution for the polychoric correlation estimation is studied. The asymptotic normality of the pseudo-maximum likelihood estimator is derived using the two-step estimation procedure. The t distribution assumption and the skew-normal distribution assumption are used as alternatives to the normal distribution assumption in a numerical study. The numerical results show that the underlying normal distribution can be substantially biased, even though skewness and kurtosis are not large. The skew-normal assumption generally produces a lower bias than the normal assumption. Thus, it is worth using a non-normal distributional assumption if the normal assumption is dubious.
Keywords
underlying distribution asymptotic covariance matrix non-normality pseudo-maximum likelihood1 Introduction
Structural equation models (SEMs) are widely used in social sciences to model latent structures. Typically, normal distributions are assumed for both latent variables and error terms. However, observed measures in surveys are often ordinal. For example, a five-point Likert scale is commonly used in psychometric studies. Conceptually, categorical data should not be incorporated into a SEM by assuming they are continuous. There have been numerous advances in the literature on SEMs with respect to analysing ordinal data as they are. The observed ordinal data are usually assumed to be counterparts of some underlying continuous distributions. A typical choice of the underlying distributions is the standard normal distribution. Olsson (1979) studied the one-step maximum likelihood estimator (MLE) and the two-step MLE of the polychoric correlation coefficient. All parameters (i.e. thresholds and polychoric correlation) are estimated simultaneously for the one-step MLE, whereas the thresholds are estimated from the marginals and the polychoric correlation is computed based on the threshold estimates for the two-step MLE. Olsson showed that under the normality assumption, the one- and the two-step MLEs produce similar polychoric correlation estimates and similar variance estimates. Jöreskog (1994) derived the estimator of the asymptotic covariance matrix of the polychoric correlation estimators for the two-step maximum likelihood procedure (for a more compact expression, see Christoffersson & Gunsjö, 1996, and related references).
The underlying normality assumption is questionable. For example, the underlying normality assumption in the Life Orientation Test dataset (Scheier & Carver, 1985) was rejected by Maydeu-Olivares (2006). In yet another example, income is commonly used in the socio-economic status studies (e.g. Chateau, Metge, Prior, & Soodeen, 2012; Hodge & Treiman, 1968; Scharoun-Lee, Adair, Kaufman, & Gordon-Larsen, 2009). A Pareto distribution is classically used to model income (Arnold, 2008). Using a normal distribution to model income is dubious because the income is bounded by a lower limit. The question regarding income, however, is commonly categorized in a questionnaire: for example, see the National Longitudinal Study of Adolescent Health dataset (Carolina Population Center, 2009) used by Scharoun-Lee et al. (2009). Thus, “income" is an ordinal indicator with a non-normal underlying distribution. The consequences of violating the underlying normality assumption have been investigated (e.g. Flora & Curran, 2004; Lee & Lam, 1988; Quiroga, 1992). Flora and Curran (2004) generated non-normal data from the Fleishman–Vale–Maurelli method (Fleishman, 1978; Vale & Maurelli, 1983) in which a standard univariate normal random variable is polynomially transformed to introduce skewness and kurtosis. The authors found that the polychoric correlation estimates are only slightly biased when the underlying distribution has a skewness of 0.75 or 1.25 and a kurtosis of 1.75 or 3.75. They found, however, that the polychoric correlation is not robust against extreme underlying non-normality (e.g. skewness = 5 and kurtosis = 50). Lee and Lam (1988) generated non-normal data from an elliptical t distribution and an elliptical contaminated normal distribution and noted that the polychoric correlation estimates based on the normality assumption are fairly robust against non-normal underlying distributions. The study of Quiroga (1992) was conducted using non-normal data from an underlying bivariate skew-normal distribution and from the Fleishman–Vale–Maurelli method. The author also suggests that the polychoric correlation estimator is robust to non-normality. These studies share two features in common. First, they assume that the underlying distribution is normal to investigate the effect of underlying non-normality. So, a non-normal distribution assumption has not been systematically studied. Second, they are simulation studies. To our knowledge, there are no robustness studies on polychoric correlations from a theoretical standpoint.
Because the polychoric correlation is not distribution-free, tests of the underlying normality assumption are desired. For example, LISREL (Jöreskog & Sörbom, 1996) uses a likelihood ratio test to assess underlying normality, which is equivalent to a Pearson \(\chi ^2\). Maydeu-Olivares, Forero, Gallardo-Pujol, and Renom (2009) and Maydeu-Olivares and Joe (2005, 2006) introduced a variant of the Pearson’s \(\chi ^2\) that is more suitable for the two-step MLE of the polychoric correlation. LISREL (Jöreskog & Sörbom, 1996) also provides the root-mean-square error of approximation (RMSEA) to assess the underlying normality assumption.
If the normality assumption fails, a new assumption of distribution is needed. Quiroga (1992) studied a new underlying distributional assumption whose marginal distributions are weighted averages of a univariate skew-normal distribution and a standard univariate normal distribution. Through an empirical example, the author showed that the polychoric correlation estimates based on the new assumption of distribution produce a smaller \(\chi ^2\) test statistic. The normality assumption has also been criticized in the item response theory and alternative distributions have been studied to account for the underlying non-normality (e.g. see Bolfarine & Bazán, 2010; Lucke, 2014; Woods & Thissen, 2006).
The purpose of this paper is twofold. First, we study robustness against misspecification of the underlying distribution from a theoretical perspective. The effect of distributional misspecification under the two-step maximum likelihood procedure is investigated. Because the two-step MLE is computationally easier (Olsson, 1979) and is implemented in LISREL, we focus only on the two-step MLE for its simplicity and popularity. Second, the underlying distribution is not restricted to a standard normal distribution. The t distribution and the skew-normal distribution are used as alternatives in the present study. In particular, the skew-normal distribution has been applied in the item response theory as an alternative to the normality assumption (e.g. see Azevedo, Bolfarine, & Andrade, 2011; Bázan, Branco, & Bolfarine, 2006; Molenaar, 2015; Molenaar, Dolan, & de Boeck, 2012; Santos, Azevedo, & Bolfarine, 2013). Because the underlying distribution cannot be fully determined from ordinal data, we attempt to pinpoint potential alternatives for the bivariate normal distribution assumption.
The remainder of this paper is organized as follows. General theories are presented, followed by numerical examples to illustrate our ideas. A brief conclusion ends the paper.
2 General Theory
2.1 Two-Step Estimation
2.1.1 Threshold Estimation
2.1.2 Polychoric Correlation Coefficient Estimation
Theorem 1
That is, \(\hat{\rho }\) is a consistent estimator of \(\rho ^{*}\) that minimizes the probabilistic divergence between H and F (Kullback, 1959) in the sense of the Kullback–Leibler information. This minimized divergence implies similarities of H and F in terms of cell probabilities.
The assumption in Theorem 1 requires uniqueness of the maximum. In so doing, we rule out all cases with local maxima. If we have several stationary points, we can then only conclude that one of the stationary points minimizes the Kullback–Leibler information.
2.1.3 Asymptotic Variance of Polychoric Correlations
Theorem 2
2.1.4 Estimating the Asymptotic Covariance Matrix
2.2 A Variant of Two-Step Estimation
3 Numerical Examples
A numerical study is conducted in this section to examine the asymptotic bias under different distributional assumptions. Asymptotic limits of PMLE for polychoric correlation coefficients are numerically computed.
3.1 Distributional Assumption
Four experiments are conducted in which different true underlying distributions are investigated.
3.1.1 Experiment 1: Elliptical Distribution
- 1.
Normal distributions: \(q\left( z \right) = \exp \left( -z/2 \right) \);
- 2.
t(v) distributions with degrees of freedom v: \(q\left( z \right) = \left( 1+z/v \right) ^{-(v+2)/2}\);
- 3.
Bivariate uniform distributions: \(q\left( z \right) = 2\text {I}_{ \{z\le 1\} }\) with \(\text {I}\) being an indicator function;
- 4.
Bivariate Logistic distributions: \(q\left( z \right) = 4\exp \left( -z \right) /\left[ 1+\exp \left( -z \right) \right] ^2\);
- 5.
Bivariate exponential power distributions: \(q\left( z \right) = 2\exp \left( -z^{\beta }/2 \right) /\left( 2^{1/\beta } \Gamma (1+1/\beta )\right) \).
For some members of the elliptical distribution family, the marginal distribution is still elliptical but not of the same type (Gómez, Gómez-villegas, & Marín, 2003). The bivariate uniform distribution, the logistic distribution, and the exponential power distribution possess such properties. The support of the bivariate uniform distribution is not the whole Cartesian plane, whereas the other distributions have the whole Cartesian plane as their support. The exponential power distribution includes the normal distribution \((\beta =1)\) and the Laplace distribution \((\beta =1/2)\) as special cases.
3.1.2 Experiment 2: Skew-Normal Distribution
An elliptical distribution is symmetric. Qui-roga (1992) reported that kurtosis does not have strong effects on the polychoric correlation but that skewness increases the bias. The above elliptical distributions examine various values of kurtosis. The following distributions introduce nonzero values of skewness.
A natural generalization of a standard normal distribution is the univariate skew-normal distribution proposed by Azzalini (1985) and extended by Azzalini and Valle (1996) to a multivariate skew-normal distribution. The bivariate density function, covariance matrix and marginal density function are shown in Eqs. (8), (9), and (10), respectively. The ranges of the skewness and excess kurtosis are \((-0.9953,0.9953)\) and [0, 0.8692), respectively (Azzalini & Capitanio, 2014, p. 32). This range is close to the low skewness and low kurtosis case in Flora and Curran (2004). The reader can refer to Azzalini (2005) for an overview of the skew-normal distribution and to Azzalini and Capitanio (2014) for the expressions of skewness and excess kurtosis. Note that the bivariate skew-normal distribution proposed by Azzalini and Valle (1996) is different from the skew-normal distribution in Quiroga (1992). The specification in Azzalini and Valle (1996) is used in the present study for its connection with the skew-t(v) distribution in the next experiment.
3.1.3 Experiment 3: Skew-t(v) Distribution
3.1.4 Experiment 4: Other Distributions
3.2 Numerical Design
Three combinations of categories are used. First, both U and V have five categories with cell probabilities (0.1, 0.2, 0.4, 0.2, 0.1) and (0.1, 0.1, 0.3, 0.3, 0.2), respectively. Second, both U and V have three categories with cell probabilities (0.2, 0.5, 0.3) and (0.1, 0.3, 0.6), respectively. Third, U has three categories with cell probabilities (0.2, 0.5, 0.3) and V has five categories with cell probabilities (0.1, 0.1, 0.3, 0.3, 0.2).
In Experiment 1, \(\beta \) in the exponential power distribution is \(\beta =0.3,0.4,0.5,0.6\). In Experiments 2 and 3, three values of \(\alpha _1\) are considered (\(\alpha _1=0.1,0.5,1\)) and 20 evenly spaced values of \(\alpha _2\) are considered ranging from 0.5 to 10 for both skew-normal and skew-t(v) distributions. Thus, different combinations of univariate skewness and kurtosis are investigated. In Experiments 1, 2 and 3, the degrees of freedom for the t(v) and skew-t(v) distributions are 4, 6, 8, and 10. In Experiment 4, parameters for the Pareto distribution are \(\theta _1=\theta _2=3\).
For all experiments, two values of \(\rho \) are used: \(\rho _{0}=0.4,0.6\). For the purpose of illustration, the assumed underlying distributions are bivariate normal, skew-normal, and t(v) distributions. The normal assumption consists of only one unknown parameter of interest, \(\rho \). The skew-normal assumption consists of three parameters: \(\alpha _1, \alpha _2\), and \(\omega \) that determine the correlation coefficient. The degrees of freedom in the t(v) are prefixed to be 4, 6, 8, 10 and the correlation coefficient is the only parameter of interest. The expressions of the partial derivatives of the skew-normal distribution and t distribution can be found in the supplementary materials.
3.3 Numerical Results
3.3.1 Experiment 1
3.3.2 Experiment 2
As expected, the polychoric correlation is consistently estimated when the true and assumed underlying distributions are both skew-normal (Figure 2). The normal assumption produces negatively biased correlation estimates. It can be moderately or strongly biased unless both \(\alpha _1\) and \(\alpha _2\) are small. Recall that \(\alpha _1\) and \(\alpha _2\) control the skewness and kurtosis of the underlying distribution. Small values of \(\alpha _1\) and \(\alpha _2\) only introduce a small departure from the bivariate normal distribution. All the t distribution assumptions produce similar RBs relative to the normal assumption. Under both the normal and t(v) distribution assumptions, three categories in both ordinal variables generally lead to a higher magnitude of the RB value than five categories in both variables. For example, the RB with five-category variables does not exceed \(-15\) when \(\alpha _1=1\) and \(\rho _0=0.4\), whereas the RB with three-category variables frequently exceeds \(-25\) under the same condition (Figure 2). As the true value of the correlation increases while the other conditions remain the same, the RB generally becomes smaller (see Figures 7, 8, and 9 in the supplementary materials).
On the other hand, although the skew-normal assumption consistently estimates the polychoric correlation in Experiment 2, the numerical difficulties (such as non-convergence and local maximizer) are encountered in the present study. The fit function \(L\left( \varvec{\theta } \right) \) can be fairly flat (see Figure 13 in the supplementary materials as an illustration). A bad choice of the starting value for the numerical optimization process can lead to the aforementioned issues. Thus, 20 starting values are employed. As a result, the skew-normal assumption is computationally much more intensive than the normal assumption.
3.3.3 Experiment 3
3.3.4 Experiment 4
Relative bias (RB) and root-mean-squared error of approximation (RMSEA) of polychoric correlations in Experiment 4.
\(m_{U}\) | \(m_{V}\) | \(\rho _{0}\) | Assumed distribution | |||||
---|---|---|---|---|---|---|---|---|
Normal | t(4) | t(6) | t(8) | t(10) | Skew-normal | |||
RB | ||||||||
3 | 3 | 0.4 | \(-\)47.18 | \(-\)45.98 | \(-\)46.29 | \(-\)46.47 | \(-\)46.59 | \(-\)32.89 |
0.6 | \(-\)51.23 | \(-\)50.17 | \(-\)50.42 | \(-\)50.57 | \(-\)50.68 | \(-\)38.78 | ||
3 | 5 | 0.4 | \(-\)38.23 | \(-\)37.83 | \(-\)37.60 | \(-\)37.60 | \(-\)37.65 | \(-\)32.12 |
0.6 | \(-\)43.50 | \(-\)43.12 | \(-\)42.92 | \(-\)42.93 | \(-\)42.97 | \(-\)37.99 | ||
5 | 5 | 0.4 | \(-\)36.26 | \(-\)38.19 | \(-\)36.94 | \(-\)36.48 | \(-\)36.28 | \(-\)31.60 |
0.6 | \(-\)41.96 | \(-\)43.35 | \(-\)42.36 | \(-\)42.01 | \(-\)41.86 | \(-\)37.41 | ||
RMSEA | ||||||||
3 | 3 | 0.4 | 0.04 | 0.07 | 0.05 | 0.05 | 0.04 | 0.00 |
0.6 | 0.05 | 0.08 | 0.06 | 0.06 | 0.06 | 0.01 | ||
3 | 5 | 0.4 | 0.04 | 0.05 | 0.04 | 0.04 | 0.04 | 0.01 |
0.6 | 0.05 | 0.06 | 0.05 | 0.05 | 0.05 | 0.01 | ||
5 | 5 | 0.4 | 0.03 | 0.04 | 0.04 | 0.03 | 0.03 | 0.01 |
0.6 | 0.04 | 0.05 | 0.04 | 0.04 | 0.04 | 0.01 |
3.4 Asymptotic Variance
4 Conclusion and Discussion
In this paper, we study robustness of polychoric correlation estimation against misspecification of underlying distributions. The asymptotic polychoric correlation and its asymptotic (co)variance are derived under the conditions of the support of assumed distributions. Unlike the continuous case, the correlation structure is not asymptotically unbiased any more. Although the bias is sometimes small, a large bias can occur, especially when the true underlying distribution is skewed but a bivariate normal or t distribution is assumed. It is seen from the numerical example that the skew-normal assumption performs as well as the conventional normal assumption when the true underlying distribution is a t distribution and improves the normal assumption when skewness exists.
Both Flora and Curran (2004) and Quiroga (1992) found that the normal assumption is robust against non-normal data generated from the Fleishman–Vale–Maurelli method. For example, the largest skewness and kurtosis considered in Flora and Curran (2004) are 1.25 and 3.75, respectively. The RB is lower than 10 in most conditions and is lower than 5 when the number of categories is five and \(\rho _0=0.49\) (Flora & Curran, 2004, Table 2). Our results show that the polychoric correlation can be largely underestimated using the normal assumption when the true underlying distribution is a skew-normal distribution skewness and kurtosis of which are bounded by some small values. The bias becomes even higher when the true underlying distribution is skew-t(4) or a Pareto distribution in which cases the kurtosis is not well defined. Although the skew-normal assumption is also largely biased sometimes, it greatly improves the conventional normal assumption. Still, the skew-normal assumption has a much higher variance than the normal assumption when the number of categories is small. Thus, the volatility is high under the skew-normal assumption. Obviously, more studies are needed to investigate small sample volatility in order to provide suggestions for practice.
Lee and Lam (1988) suggested using the correct underlying distributional assumption to estimate more accurately the polychoric correlation if the ordinal data are asymmetric. Because the ordinal data indicate the loss of information when comparing with continuous data, we cannot have visual inspections of the underlying distribution. If the tests of the underlying distribution were rejected, the underlying distributional assumption is questionable, and an alternative distributional assumption should be used. In practice, several assumptions of underlying distribution can be tested and then the most plausible one chosen.
The normal distribution is a special case of the skew-normal distribution. We have shown that both distributions consistently estimate the polychoric correlation when the true distribution is normal. Thus, the skew-normal assumption, which is able to model skewness and kurtosis, is a natural extension to the conventional normal assumption and frequently outperforms the normal assumption. However, three parameters are simultaneously estimated in the skew-normal distribution. Because the thresholds are determined through \(\alpha _1, \alpha _2\), and \(\omega \), the gradient and Hessian matrix involve derivatives of the thresholds with respect to \(\alpha _1, \alpha _2\), and \(\omega \). Accordingly, it is computationally more difficult than the normal assumption. Besides, non-convergence and local optimizers are encountered in the present study and multiple starting values are used to obtain the correlation estimate.
Although only the t and skew-normal assumptions are illustrated as non-normal alternatives in the present study, other distributions that are differentiable with respect to unknown parameters can be used to estimate the correlation coefficient by the aid of Theorem 1 or Eq. (6). Its asymptotic variance and covariance can be estimated using Theorem 2 or Eq. (7). For example, the logistic distribution can be assumed in the two-step estimation and the skew-t distribution can be assumed in the variant of the two-step estimation. It will be of interest to derive analytical expressions for the skew-elliptical distribution family that consists of the skew-normal and skew-t distributions. Our numerical results demonstrate that the skew-normal assumption generally improves the conventional normal assumption in the imaginary case where n is infinite. It is worthy to conduct a simulation study to investigate the small sample bias in estimating the correlation coefficient and its effects on the bias of parameters in a SEM with ordinal data.
Notes
Acknowledgments
The research reported in this article has been supported by the Swedish Research Council (VR) under the program: Structural Equation Modeling with Ordinal Variables, 421-2011-1727.
Open Access
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Supplementary material
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