Measures of Multivariate Kurtosis

The CALIS Procedure

Measures of Multivariate Kurtosis

In many applications, the manifest variables are not even approximately multivariate normal. If this happens to be the case with your data set, the default generalized least-squares and maximum likelihood estimation methods are not appropriate, and you should compute the parameter estimates and their standard errors by an asymptotically distribution-free method, such as the WLS estimation method. If your manifest variables are multivariate normal, then they have a zero relative multivariate kurtosis, and all marginal distributions have zero kurtosis (Browne 1982). If your DATA= data set contains raw data, PROC CALIS computes univariate skewness and kurtosis and a set of multivariate kurtosis values. By default, the values of univariate skewness and kurtosis are corrected for bias (as in PROC UNIVARIATE), but using the BIASKUR option enables you to compute the uncorrected values also. The values are displayed when you specify the PROC CALIS statement option KURTOSIS.

Corrected Variance for Variable z_j

$\sigma_j^2 = {1 \over N-1} {\sum_i^N(z_{ij} - \overline{z_j})^2}$
Corrected Univariate Skewness for Variable z_j

$\gamma_{1(j)} = {N \over (N - 1)(N - 2)} { {\sum_i^N (z_{ij} - \overline {z_j})^3} \over \sigma_j^3}$
Uncorrected Univariate Skewness for Variable z_j

$\gamma_{1(j)} = { {N \sum_i^N (z_{ij} - \overline {z_j})^3} \over \sqrt{N [\sum_i^N (z_{ij} - \overline {z_j})^2]^3 } }$
Corrected Univariate Kurtosis for Variable z_j

$\gamma_{2(j)} = {N(N + 1) \over (N - 1)(N - 2)(N - 3)} { {\sum_i^N (z_{ij} - \overline {z_j})^4} \over \sigma_j^4} - { 3(N - 1)^2 \over (N - 2)(N - 3)}$
Uncorrected Univariate Kurtosis for Variable z_j

$\gamma_{2(j)} = { N {\sum_i^N (z_{ij} - \overline {z_j})^4} \over [\sum_i^N (z_{ij} - \overline {z_j})^2]^2 } - 3$
Mardia's Multivariate Kurtosis

$\gamma_2 = {{1 \over N} {\sum_i^N[(z_i - \overline z)^' S^{-1} (z_i - \overline z)]^2} - n(n + 2) }$
Relative Multivariate Kurtosis

$\eta_2 = { {\gamma_2 + n(n + 2)} \over {n(n + 2)} }$
Normalized Multivariate Kurtosis

$\kappa_0 = { \gamma_2 \over \sqrt{8n(n + 2) / N} }$
Mardia Based Kappa

$\kappa_1 = { \gamma_2 \over n(n + 2) }$
Mean Scaled Univariate Kurtosis

$\kappa_2 = { {1 \over 3n} {\sum_j^n \gamma_{2(j)}} }$
Adjusted Mean Scaled Univariate Kurtosis

$\kappa_3 = { {1 \over 3n} {\sum_j^n \gamma^*_{2(j)}} }$
with
$\gamma^*_{2(j)} = \{ \matrix{ \gamma_{2(j)} , & if \gamma_{2(j)} \gt {-6 \over n+2} \cr & \cr { -6 \over n+2} , & {otherwise} \cr } .$

If variable Z_j is normally distributed, the uncorrected univariate kurtosis $\gamma_{2(j)}$ is equal to 0. If Z has an n-variate normal distribution, Mardia's multivariate kurtosis $\gamma_2$ is equal to 0. A variable Z_j is called leptokurtic if it has a positive value of $\gamma_{2(j)}$ and is called platykurtic if it has a negative value of $\gamma_{2(j)}$ . The values of $\kappa_1$ , $\kappa_2$ , and $\kappa_3$ should not be smaller than a lower bound (Bentler 1985):

$\hat{\kappa} \geq {-2 \over n + 2}$

PROC CALIS displays a message if this happens.

If weighted least-squares estimates (METHOD=WLS or METHOD=ADF) are specified and the weight matrix is computed from an input raw data set, the CALIS procedure computes two further measures of multivariate kurtosis.

Multivariate Mean Kappa

$\kappa_4 = {1 \over m} {\sum_i^n \sum_j^i \sum_k^j \sum_l^k \hat{\kappa}_{ij,kl}} - 1$
where
$\hat{\kappa}_{ij,kl}=\frac{s_{ij,kl}} { s_{ij}s_{kl} + s_{ik}s_{jl} + s_{il}s_{jk} }$
and m=n(n+1)(n+2)(n+3)/24 is the number of elements in the vector s_ij,kl (Bentler 1985).
Multivariate Least-Squares Kappa

$\kappa_5 = {s_4^' s_2 \over s_2^' s_2} - 1$
where
$s_{ij,kl} = {1 \over N} \sum_{r=1}^N{(z_{ri} - \overline{z_i}) (z_{rj} - \overline{z_j})(z_{rk} - \overline{z_k}) (z_{rl} - \overline{z_l})}$

s₄ is the vector of the s_ij,kl, and s₂ is the vector of the elements in the denominator of $\hat{\kappa}$ (Bentler 1985).

The occurrence of significant nonzero values of Mardia's multivariate kurtosis $\gamma_2$ and significant amounts of some of the univariate kurtosis values $\gamma_{2(j)}$ indicate that your variables are not multivariate normal distributed. Violating the multivariate normality assumption in (default) generalized least-squares and maximum likelihood estimation usually leads to the wrong approximate standard errors and incorrect fit statistics based on the $\chi^2$ value. In general, the parameter estimates are more stable against violation of the normal distribution assumption. For more details, refer to Browne (1974, 1982, 1984).

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