The Nonsingularity of Γ in Covariance Structure Analysis of Nonnormal Data

Abstract

Covariance structure analysis of nonnormal data is important because in practice all data are nonnormal. When applying covariance structure analysis to nonnormal data, it is generally assumed that the asymptotic covariance matrix Γ for the nonredundant terms in the sample covariance matrix S is nonsingular. It is shown this need not be the case, which raises a question of how restrictive this assumption may be and how difficult it may be to verify it. It is shown that Γ is nonsingular whenever sampling is from a nonsingular distribution, including any distribution defined by a density function. In the discrete case necessary and sufficient conditions are given for the nonsingularity of Γ, and it is shown how to demonstrate Γ is nonsingular with high probability. Thus, the nonsingularity of Γ assumption is mild and one should feel comfortable about making it. These observations also apply to the asymptotic covariance matrix Γ that arises in structural equation modeling.

Appendix: Proof of Theorem 1 and Singular Γ Example

1.1 A.1 Proof of Theorem 1

Let e _i =x _i −μ. Then $e_{i}-\bar{e} =x_{i}-\bar{x}$ and

$$S_n=\frac{1}{n}\sum(e_i-\bar{e}) (e_i-\bar{e})'=\frac{1}{n}\sum e_ie_i'-\bar{e}\bar{e}' $$

and

$$\sqrt{n}S_n=\frac{1}{\sqrt{n}}\sum e_ie_i'- \sqrt{n}\bar{e}\bar{e}'. $$

Since $\bar{e}=\bar{x}-\mu\stackrel{p}{\rightarrow}0$ and $\sqrt {n}\bar{e}=\sqrt{n}(\bar{x}-\mu )\stackrel{\mathcal{D}}{\rightarrow}N(0,\varSigma)$, $\sqrt{n}\bar{e}\bar{e}'\stackrel{p}{\rightarrow}0$. Thus,

$$\sqrt{n}S_n\stackrel{a}{=}\frac{1}{\sqrt{n}}\sum e_ie_i' $$

and

$$ \sqrt{n}(s_n-\sigma)\stackrel{a}{=}\frac{1}{\sqrt{n}} \sum \bigl(\operatorname{vech} \bigl(e_ie_i' \bigr)-\sigma\bigr). $$

(A.1)

Note that the $\operatorname{vech}(e_{i}e_{i}')$ are independent and identically distributed, σ is the common expected value of the $\operatorname{vech} (e_{i}e_{i}') =\operatorname{vech}((x_{i}-\mu)(x_{i}-\mu)')$, and Γ is the common covariance matrix of the $\operatorname{vech}(e_{i}e_{i}')$. It follows from the central limit theorem that the right-hand side of (A.1) converges in distribution to N(0,Γ), and hence that

$$\sqrt{n}(s_n-\sigma)\stackrel{\mathcal{D}}{\rightarrow}N(0, \varGamma). $$

Moreover,

$$\varGamma= \operatorname{cov}\bigl(\operatorname {vech}\bigl(e_ie_i' \bigr)\bigr) = \operatorname{cov}\bigl(\operatorname{vech}\bigl((x_i-\mu ) (x_i-\mu)'\bigr)\bigr)=\operatorname{cov} \bigl(\operatorname{vech} \bigl((x-\mu) (x-\mu)'\bigr)\bigr). $$

1.2 A.2 What Happens When Γ Is Singular?

Using a very simple example, we will show that Browne’s (1984) extensively used goodness of fit test for CSA can fail completely when Γ is singular. Consider random sampling from the array in Figure 1. We have shown that the asymptotic covariance matrix Γ for the sample covariances s generated in this way is singular. Consider a covariance structure

$$\gamma(\theta)=\left ( \begin{array}{c} 2/3 \\ \theta\\ 1/4 \end{array} \right ). $$

When θ=0 this is equal to σ. Using a least squares estimate $\hat{\theta}$ of θ, Browne’s statistic for testing the goodness of fit of γ(θ) is

$$T=n\bigl(s-\gamma(\hat{\theta})\bigr)'U\bigl(U'\hat{ \varGamma}U\bigr)^{-1}U'\bigl(s-\gamma (\hat{\theta})\bigr), $$

where U is an orthogonal complement of

$$\varDelta=\left ( \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right ), $$

the Jacobian of γ(θ), and $\hat{\varGamma}$ is a consistent estimator for Γ. We have used the estimator $\hat{\varGamma}$ given by Satorra and Bentler (1990, Formula 2.4).

If Γ were nonsingular, T would have an asymptotic χ ² distribution with two degrees of freedom. But in this example Γ is singular. To investigate the performance of Browne’s test in this situation we generated N=1000 samples of size n=100, and for each a value of T was computed. Figure 2 is a Q–Q plot of the values of T on the corresponding quantiles of the $\chi^{2}_{2}$. Clearly, the distribution of T differs greatly from $\chi^{2}_{2}$. Moreover, in 78 of the 1,000 trials T could not even be computed because $\hat {\varGamma}$ was too nearly singular. On these trials, the computer output correctly suggests that Γ is singular. On 92 % of the trials, however, no such warning is produced.

Figure 1.

Q–Q plot when Γ is singular.

One wonders what would happen in this example if Γ were nonsingular. Assume the array in Figure 1 were replaced by a 3 by 3 array with x and y values equal to 1, 2, 3. If these carry equal probability mass, Γ is nonsingular and the Q–Q plot becomes that displayed in Figure 3.

Figure 2.

Q–Q plot when Γ is nonsingular.

Clearly, T now is very nearly $\chi^{2}_{2}$ distributed. Moreover, there were no problems in computing the values of T.