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| In [[statistics]], '''D’Agostino’s ''K''<sup>2</sup> test''' is a [[goodness-of-fit]] measure of departure from [[normal distribution|normality]], that is the test aims to establish whether or not the given sample comes from a normally distributed population. 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.
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| == Skewness and kurtosis ==
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| In the following, let { ''x<sub>i</sub>'' } denote 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]]. (Note that quite frequently in the literature related to [[normality tests|normality testing]] the skewness and kurtosis are denoted as √''β''<sub>1</sub> and ''β''<sub>2</sub> respectively. Such notation is less convenient since for example √''β''<sub>1</sub> can be a negative quantity).
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| The sample skewness and kurtosis are defined as
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| : <math>\begin{align}
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| & 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}}\ , \\
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| & 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\ .
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| \end{align}</math>
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| 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 kurtoses ''γ''<sub>2</sub>. This has been done by {{harvtxt|Pearson|1931}}, who derived the following expressions:{{better source|reason=need more accessible source so that quoted expression can be checked|date=November 2010}}
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| : <math>\begin{align}
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| & \mu_1(g_1) = 0, \\
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| & \mu_2(g_1) = \frac{ 6(n-2) }{ (n+1)(n+3) }, \\
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| & \gamma_1(g_1) \equiv \frac{\mu_3(g_1)}{\mu_2(g_1)^{3/2}} = 0, \\
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| & \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) }.
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| \end{align}</math>
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| and
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| : <math>\begin{align}
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| & \mu_1(g_2) = - \frac{6}{n+1}, \\
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| & \mu_2(g_2) = \frac{ 24n(n-2)(n-3) }{ (n+1)^2(n+3)(n+5) }, \\
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| & \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)}}, \\
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| & \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) }.
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| \end{align}</math>
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| 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}}
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| == Transformed sample skewness and kurtosis ==
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| 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.
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| In particular, {{harvtxt|D’Agostino|1970}} suggested the following transformation for sample skewness:
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| : <math>
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| Z_1(g_1) = \delta\cdot \ln\!\left( \frac{g_1}{\alpha\sqrt{\mu_2}} + \sqrt{\frac{g_1^2}{\alpha^2\mu_2} + 1}\right),
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| </math>
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| where constants ''α'' and ''δ'' are computed as
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| : <math>\begin{align}
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| & W^2 = \sqrt{2\gamma_2 + 4} - 1, \\
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| & \delta = 1 / \sqrt{\ln W}, \\
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| & \alpha^2 = 2 / (W^2-1), \\
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| \end{align}</math>
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| 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.
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| Similarly, {{harvtxt|Anscombe|Glynn|1983}} suggested a transformation for ''g''<sub>2</sub>, which works reasonably well for sample sizes of 20 or greater:
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| : <math>
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| 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\},
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| </math>
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| where
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| : <math>
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| A = 6 + \frac{8}{\gamma_1} \left( \frac{2}{\gamma_1} + \sqrt{1+4/\gamma_1^2}\right),
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| </math>
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| 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.
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| == Omnibus ''K''<sup>2</sup> statistic ==
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| 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}}:
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| : <math>
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| K^2 = Z_1(g_1)^2 + Z_2(g_2)^2\,
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| </math>
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| 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.
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| 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
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| <!-- each experiment was based on 1,000,000 simulations -->
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| {|class="wikitable" style="text-align:right"
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| |-
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| !
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| ! expected value
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| ! standard deviation
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| ! 95% quantile
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| |-
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| |style="text-align:left"| ''n'' = 20
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| | 1.971
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| | 2.339
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| | 6.373
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| |-
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| |style="text-align:left"| ''n'' = 50
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| | 2.017
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| | 2.308
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| | 6.339
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| |-
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| |style="text-align:left"| ''n'' = 100
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| | 2.026
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| | 2.267
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| | 6.271
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| |-
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| |style="text-align:left"| ''n'' = 250
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| | 2.012
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| | 2.174
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| | 6.129
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| |-
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| |style="text-align:left"| ''n'' = 500
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| | 2.009
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| | 2.113
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| | 6.063
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| |-
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| |style="text-align:left"| ''n'' = 1000
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| | 2.000
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| | 2.062
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| | 6.038
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| |-
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| | ''χ''<sup>2</sup>(2) distribution
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| | 2.000
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| | 2.000
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| | 5.991
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| |}
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| ==References==
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| {{Refbegin}}
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| * {{cite journal
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| | title = Distribution of the kurtosis statistic ''b''<sub>2</sub> for normal statistics
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| | first1 = F.J.
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| | last1 = Anscombe
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| | first2 = William J.
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| | last2 = Glynn
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| | year = 1983
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| | journal = [[Biometrika]]
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| | volume = 70
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| | issue = 1
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| | pages = 227–234
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| | jstor = 2335960
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| | ref = CITEREFAnscombeGlynn1983
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| }}
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| * {{cite journal
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| | title = Transformation to normality of the null distribution of ''g''<sub>1</sub>
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| | first = Ralph B.
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| | last = D’Agostino
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| | journal = [[Biometrika]]
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| | volume = 57
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| | issue = 3
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| | year = 1970
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| | pages = 679–681
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| | jstor = 2334794
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| | ref = CITEREFD.E2.80.99Agostino1970
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| }}
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| * {{cite journal
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| | title = A suggestion for using powerful and informative tests of normality
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| | author1 = D’Agostino, Ralph B.
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| | author2 = Albert Belanger
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| | author3 = Ralph B. D’Agostino, Jr
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| | journal = [[The American Statistician]]
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| | volume = 44
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| | issue = 4
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| | year = 1990
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| | pages = 316–321
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| | jstor = 2684359
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| |url=http://www.cee.mtu.edu/~vgriffis/CE%205620%20materials/CE5620%20Reading/DAgostino%20et%20al%20-%20normaility%20tests.pdf
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| | ref = CITEREFD.E2.80.99AgostinoBelangerD.E2.80.99Agostino1990
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| }}
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| * {{cite journal
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| | last = Pearson | first = Egon S. | authorlink = Egon Pearson
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| | title = Note on tests for normality
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| | year = 1931
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| | journal = [[Biometrika]] | volume = 22 | issue = 3/4
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| | pages = 423–424
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| | ref = harv
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| | jstor = 2332104
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| }}
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| * {{cite journal
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| | title = A bivariate model for the distribution of √b<sub>1</sub> and b<sub>2</sub>
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| | last1 = Shenton
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| | first1 = L.R.
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| | last2 = Bowman
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| | first2 = K.O.
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| | year = 1977
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| | journal = Journal of the American Statistical Association
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| | volume = 72
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| | issue = 357
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| | pages = 206–211
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| | ref = CITEREFShentonBowman1977
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| | jstor = 2286939
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| }}
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| {{Refend}}
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| {{DEFAULTSORT:D'agostino'S K-Squared Test}}
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| [[Category:Parametric statistics]]
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| [[Category:Normality tests]]
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