View source for D'Agostino's K-squared test

{{Short description|Goodness-of-fit measure in statistics}}
{{DISPLAYTITLE:D'Agostino's ''K''-squared test}}
In [[statistics]], '''D'Agostino's ''K''<sup>2</sup> test''', named for [[Ralph D'Agostino (statistician)|Ralph D'Agostino]], is a [[goodness-of-fit]] measure of departure from [[normal distribution|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<sub>i</sub>'' } denotes a sample of ''n'' observations, ''g''<sub>1</sub> and ''g''<sub>2</sub> are the sample [[skewness]] and [[kurtosis]], ''m<sub>j</sub>''’s are the ''j''-th sample [[central moment]]s, and <math style="position:relative;top:-.3em">\bar{x}</math> is the sample [[mean]]. Frequently in the literature related to [[normality tests|normality testing]], the skewness and kurtosis are denoted as {{radic|''β''<sub>1</sub>}} and ''β''<sub>2</sub> respectively. Such notation can be inconvenient since, for example, {{radic|''β''<sub>1</sub>}} can be a negative quantity.

The sample skewness and kurtosis are defined as
: <math>\begin{align}
    & 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{align}</math>

These quantities [[consistent estimator|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 ''μ''<sub>1</sub>, variances ''μ''<sub>2</sub>, skewnesses ''γ''<sub>1</sub>, and kurtosis ''γ''<sub>2</sub>. This has been done by {{harvtxt|Pearson|1931}}, who derived the following expressions:{{better source needed|reason=need more accessible source so that quoted expression can be checked|date=November 2010}}

: <math>\begin{align}
    & \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{align}</math>
and
: <math>\begin{align}
    & \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{align}</math>
For example, a sample with size {{nowrap|''n'' {{=}} 1000}} drawn from a normally distributed population can be expected to have a skewness of {{nowrap|0, SD 0.08}} and a kurtosis of {{nowrap|0, SD 0.15}}, where SD indicates the standard deviation.{{citation needed|date=January 2012}}

== Transformed sample skewness and kurtosis ==
The sample skewness ''g''<sub>1</sub> and kurtosis ''g''<sub>2</sub> are both asymptotically normal. However, the rate of their convergence to the distribution limit is frustratingly slow, especially for ''g''<sub>2</sub>. For example even with {{nowrap|1=''n'' = 5000}} observations the sample kurtosis ''g''<sub>2</sub> 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 ''g''<sub>1</sub> and ''g''<sub>2</sub> in a way that makes their distribution as close to standard normal as possible.

In particular, {{harvtxt|D'Agostino|Pearson|1973}} suggested the following transformation for sample skewness:
: <math>
    Z_1(g_1) = \delta \operatorname{asinh}\left( \frac{g_1}{\alpha\sqrt{\mu_2}} \right),
  </math>
where constants ''α'' and ''δ'' are computed as
: <math>\begin{align}
    & W^2 = \sqrt{2\gamma_2 + 4} - 1, \\
    & \delta = 1 / \sqrt{\ln W}, \\
    & \alpha^2 = 2 / (W^2-1),
  \end{align}</math>
and where ''μ''<sub>2</sub> = ''μ''<sub>2</sub>(''g''<sub>1</sub>) is the variance of ''g''<sub>1</sub>, and ''γ''<sub>2</sub> = ''γ''<sub>2</sub>(''g''<sub>1</sub>) is the kurtosis — the expressions given in the previous section.

Similarly, {{harvtxt|Anscombe|Glynn|1983}} suggested a transformation for ''g''<sub>2</sub>, which works reasonably well for sample sizes of 20 or greater:
: <math>
    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\},
  </math>
where
: <math>
    A = 6 + \frac{8}{\gamma_1} \left( \frac{2}{\gamma_1} + \sqrt{1+4/\gamma_1^2}\right),
  </math>
and ''μ''<sub>1</sub> = ''μ''<sub>1</sub>(''g''<sub>2</sub>), ''μ''<sub>2</sub> = ''μ''<sub>2</sub>(''g''<sub>2</sub>), ''γ''<sub>1</sub> = ''γ''<sub>1</sub>(''g''<sub>2</sub>) are the quantities computed by Pearson.

== Omnibus ''K''<sup>2</sup> statistic ==
Statistics ''Z''<sub>1</sub> and ''Z''<sub>2</sub> can be combined to produce an omnibus test, able to detect deviations from normality due to either skewness or kurtosis {{harv|D'Agostino|Belanger|D'Agostino|1990}}:
: <math>
    K^2 = Z_1(g_1)^2 + Z_2(g_2)^2\,
  </math>

If the [[null hypothesis]] of normality is true, then ''K''<sup>2</sup> is approximately [[chi-squared distribution|''χ''<sup>2</sup>-distributed]] with 2 degrees of freedom.

Note that the statistics ''g''<sub>1</sub>, ''g''<sub>2</sub> are not independent, only uncorrelated. Therefore, their transforms ''Z''<sub>1</sub>, ''Z''<sub>2</sub> will be dependent also {{harv|Shenton|Bowman|1977}}, rendering the validity of ''χ''<sup>2</sup> approximation questionable. Simulations show that under the null hypothesis the ''K''<sup>2</sup> test statistic is characterized by
<!-- each experiment was based on 1,000,000 simulations -->
{|class="wikitable" style="text-align:right"
|-
!
! expected value
! standard deviation
! 95% quantile
|-
|style="text-align:left"| ''n'' = 20
| 1.971
| 2.339
| 6.373
|-
|style="text-align:left"| ''n'' = 50
| 2.017
| 2.308
| 6.339
|-
|style="text-align:left"| ''n'' = 100
| 2.026
| 2.267
| 6.271
|-
|style="text-align:left"| ''n'' = 250
| 2.012
| 2.174
| 6.129
|-
|style="text-align:left"| ''n'' = 500
| 2.009
| 2.113
| 6.063
|-
|style="text-align:left"| ''n'' = 1000
| 2.000
| 2.062
| 6.038
|-
| ''χ''<sup>2</sup>(2) distribution
| 2.000
| 2.000
| 5.991
|}

== See also ==
* [[Shapiro–Wilk test]]
* [[Jarque–Bera test]]

== References ==
{{Refbegin}}
* {{cite journal
  | title = Distribution of the kurtosis statistic ''b''<sub>2</sub> for normal statistics
  | first1 = F.J.
  | last1 = Anscombe
  | first2 = William J.
  | last2 = Glynn
  | year = 1983
  | journal = [[Biometrika]]
  | volume = 70
  | issue = 1
  | pages = 227–234
  | jstor = 2335960
  | doi=10.1093/biomet/70.1.227
  }}
* {{cite journal
  | title = Transformation to normality of the null distribution of ''g''<sub>1</sub>
  | first = Ralph B.
  | last = D'Agostino
  | journal = [[Biometrika]]
  | volume = 57
  | issue = 3
  | year = 1970
  | pages = 679–681
  | jstor = 2334794
  | doi=10.1093/biomet/57.3.679
  }}
* {{cite journal
  | title = Tests for Departure from Normality. Empirical Results for the Distributions of b<sub>2</sub> and √b<sub>1</sub>
  | first1 = Ralph B.
  | last1 = D'Agostino
  | first2 = E. S.
  | last2 = Pearson
  | journal = [[Biometrika]]
  | volume = 60
  | issue = 3
  | year = 1973
  | pages = 613–622
  | jstor = 2335012
  }}
* {{cite journal
  | title = A suggestion for using powerful and informative tests of normality
  | first1 = Ralph B.
  | last1 = D'Agostino
  | first2 = Albert
  | last2 = Belanger
  | first3 = Ralph B. Jr.
  | last3 = D'Agostino
  | journal = [[The American Statistician]]
  | volume = 44
  | issue = 4
  | year = 1990
  | pages = 316–321
  | jstor = 2684359
  | doi = 10.2307/2684359
  | url = http://www.cee.mtu.edu/~vgriffis/CE%205620%20materials/CE5620%20Reading/DAgostino%20et%20al%20-%20normaility%20tests.pdf
  | url-status = dead
  | archiveurl = https://web.archive.org/web/20120325140006/http://www.cee.mtu.edu/~vgriffis/CE%205620%20materials/CE5620%20Reading/DAgostino%20et%20al%20-%20normaility%20tests.pdf
  | archivedate = 2012-03-25
  }}
* {{cite journal
  | last = Pearson | first = Egon S. | authorlink = Egon Pearson
  | title = Note on tests for normality
  | year = 1931
  | journal = [[Biometrika]] | volume = 22 | issue = 3/4
  | pages = 423–424

  | jstor = 2332104 | doi=10.1093/biomet/22.3-4.423
  }}
* {{cite journal
  | title = A bivariate model for the distribution of √b<sub>1</sub> and b<sub>2</sub>
  | last1 = Shenton
  | first1 = L.R.
  | last2 = Bowman
  | first2 = Kimiko O. | author2-link = Kimiko O. Bowman
  | year = 1977
  | journal = Journal of the American Statistical Association
  | volume = 72
  | issue = 357
  | pages = 206–211
  | jstor = 2286939
  | doi=10.1080/01621459.1977.10479940
  }}
{{Refend}}

{{Statistics |inference}}

{{DEFAULTSORT:D'agostino'S K-Squared Test}}
[[Category:Parametric statistics]]
[[Category:Normality tests]]