D'Agostino's K-squared test

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 ${\displaystyle {\bar {x}}}$ 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

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

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 kurtoses γ₂. This has been done by Pearson (1931), who derived the following expressions:

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

and

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

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.

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 (1970) harvtxt error: no target: CITEREFD’Agostino1970 (help) suggested the following transformation for sample skewness:

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

where constants α and δ are computed as

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

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:

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

where

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

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

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) harv error: no target: CITEREFD’AgostinoBelangerD’Agostino1990 (help):

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

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

• Shapiro–Wilk test
• Jarque–Bera test