D'Agostino's K-squared test

In statistics, D'Agostino's K² test, named for Ralph D'Agostino, is a goodness-of-fit measure of departure from normality, that is the test aims to gauge the compatibility of given data with the null hypothesis that the data is a realization of independent, identically distributed Gaussian random variables. The test is based on transformations of the sample kurtosis and skewness, and has power only against the alternatives that the distribution is skewed and/or kurtic.

Skewness and kurtosis

In the following, { x_i } denotes a sample of n observations, g₁ and g₂ are the sample skewness and kurtosis, m_j’s are the j-th sample central moments, and ${\bar {x}}$ is the sample mean. Frequently in the literature related to normality testing, the skewness and kurtosis are denoted as √β₁ and β₂ respectively. Such notation can be inconvenient since, for example, √β₁ can be a negative quantity.

The sample skewness and kurtosis are defined as

${\begin{aligned}&g_{1}={\frac {m_{3}}{m_{2}^{3/2}}}={\frac {{\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{3}}{\left({\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}\right)^{3/2}}}\ ,\\&g_{2}={\frac {m_{4}}{m_{2}^{2}}}-3={\frac {{\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{4}}{\left({\frac {1}{n}}\sum _{i=1}^{n}\left(x_{i}-{\bar {x}}\right)^{2}\right)^{2}}}-3\ .\end{aligned}}$

These quantities consistently estimate the theoretical skewness and kurtosis of the distribution, respectively. Moreover, if the sample indeed comes from a normal population, then the exact finite sample distributions of the skewness and kurtosis can themselves be analysed in terms of their means μ₁, variances μ₂, skewnesses γ₁, and kurtosis γ₂. This has been done by Pearson (1931), who derived the following expressions:

${\begin{aligned}&\mu _{1}(g_{1})=0,\\&\mu _{2}(g_{1})={\frac {6(n-2)}{(n+1)(n+3)}},\\&\gamma _{1}(g_{1})\equiv {\frac {\mu _{3}(g_{1})}{\mu _{2}(g_{1})^{3/2}}}=0,\\&\gamma _{2}(g_{1})\equiv {\frac {\mu _{4}(g_{1})}{\mu _{2}(g_{1})^{2}}}-3={\frac {36(n-7)(n^{2}+2n-5)}{(n-2)(n+5)(n+7)(n+9)}}.\end{aligned}}$

and

${\begin{aligned}&\mu _{1}(g_{2})=-{\frac {6}{n+1}},\\&\mu _{2}(g_{2})={\frac {24n(n-2)(n-3)}{(n+1)^{2}(n+3)(n+5)}},\\&\gamma _{1}(g_{2})\equiv {\frac {\mu _{3}(g_{2})}{\mu _{2}(g_{2})^{3/2}}}={\frac {6(n^{2}-5n+2)}{(n+7)(n+9)}}{\sqrt {\frac {6(n+3)(n+5)}{n(n-2)(n-3)}}},\\&\gamma _{2}(g_{2})\equiv {\frac {\mu _{4}(g_{2})}{\mu _{2}(g_{2})^{2}}}-3={\frac {36(15n^{6}-36n^{5}-628n^{4}+982n^{3}+5777n^{2}-6402n+900)}{n(n-3)(n-2)(n+7)(n+9)(n+11)(n+13)}}.\end{aligned}}$

For example, a sample with size n = 1000 drawn from a normally distributed population can be expected to have a skewness of 0, SD 0.08 and a kurtosis of 0, SD 0.15, where SD indicates the standard deviation.

Transformed sample skewness and kurtosis

The sample skewness g₁ and kurtosis g₂ are both asymptotically normal. However, the rate of their convergence to the distribution limit is frustratingly slow, especially for g₂. For example even with n = 5000 observations the sample kurtosis g₂ has both the skewness and the kurtosis of approximately 0.3, which is not negligible. In order to remedy this situation, it has been suggested to transform the quantities g₁ and g₂ in a way that makes their distribution as close to standard normal as possible.

In particular, D'Agostino & Pearson (1973) suggested the following transformation for sample skewness:

$Z_{1}(g_{1})=\delta \operatorname {asinh} \left({\frac {g_{1}}{\alpha {\sqrt {\mu _{2}}}}}\right),$

where constants α and δ are computed as

${\begin{aligned}&W^{2}={\sqrt {2\gamma _{2}+4}}-1,\\&\delta =1/{\sqrt {\ln W}},\\&\alpha ^{2}=2/(W^{2}-1),\end{aligned}}$

and where μ₂ = μ₂(g₁) is the variance of g₁, and γ₂ = γ₂(g₁) is the kurtosis — the expressions given in the previous section.

Similarly, Anscombe & Glynn (1983) suggested a transformation for g₂, which works reasonably well for sample sizes of 20 or greater:

$Z_{2}(g_{2})={\sqrt {\frac {9A}{2}}}\left\{1-{\frac {2}{9A}}-\left({\frac {1-2/A}{1+{\frac {g_{2}-\mu _{1}}{\sqrt {\mu _{2}}}}{\sqrt {2/(A-4)}}}}\right)^{\!1/3}\right\},$

where

$A=6+{\frac {8}{\gamma _{1}}}\left({\frac {2}{\gamma _{1}}}+{\sqrt {1+4/\gamma _{1}^{2}}}\right),$

and μ₁ = μ₁(g₂), μ₂ = μ₂(g₂), γ₁ = γ₁(g₂) are the quantities computed by Pearson.

Omnibus K² statistic

Statistics Z₁ and Z₂ can be combined to produce an omnibus test, able to detect deviations from normality due to either skewness or kurtosis (D'Agostino, Belanger & D'Agostino 1990):

$K^{2}=Z_{1}(g_{1})^{2}+Z_{2}(g_{2})^{2}\,$

If the null hypothesis of normality is true, then K² is approximately χ²-distributed with 2 degrees of freedom.

Note that the statistics g₁, g₂ are not independent, only uncorrelated. Therefore, their transforms Z₁, Z₂ will be dependent also (Shenton & Bowman 1977), rendering the validity of χ² approximation questionable. Simulations show that under the null hypothesis the K² test statistic is characterized by

	expected value	standard deviation	95% quantile
n = 20	1.971	2.339	6.373
n = 50	2.017	2.308	6.339
n = 100	2.026	2.267	6.271
n = 250	2.012	2.174	6.129
n = 500	2.009	2.113	6.063
n = 1000	2.000	2.062	6.038
χ2(2) distribution	2.000	2.000	5.991

See also

• Shapiro–Wilk test
• Jarque–Bera test

References

• Anscombe, F.J.; Glynn, William J. (1983). "Distribution of the kurtosis statistic b₂ for normal statistics". Biometrika. 70 (1): 227–234. doi:10.1093/biomet/70.1.227. JSTOR 2335960.
• D'Agostino, Ralph B. (1970). "Transformation to normality of the null distribution of g₁". Biometrika. 57 (3): 679–681. doi:10.1093/biomet/57.3.679. JSTOR 2334794.
• D'Agostino, Ralph B.; Pearson, E. S. (1973). "Tests for Departure from Normality. Empirical Results for the Distributions of b₂ and √b₁". Biometrika. 60 (3): 613–622. JSTOR 2335012.
• D'Agostino, Ralph B.; Belanger, Albert; D'Agostino, Ralph B. Jr. (1990). "A suggestion for using powerful and informative tests of normality" (PDF). The American Statistician. 44 (4): 316–321. doi:10.2307/2684359. JSTOR 2684359. Archived from the original (PDF) on 2012-03-25.
• Pearson, Egon S. (1931). "Note on tests for normality". Biometrika. 22 (3/4): 423–424. doi:10.1093/biomet/22.3-4.423. JSTOR 2332104.
• Shenton, L.R.; Bowman, Kimiko O. (1977). "A bivariate model for the distribution of √b₁ and b₂". Journal of the American Statistical Association. 72 (357): 206–211. doi:10.1080/01621459.1977.10479940. JSTOR 2286939.