Calculate univariate or multivariate (Mardia's test) skew and kurtosis for a vector, matrix, or data.frame
skew.RdFind the skew and kurtosis for each variable in a data.frame or matrix. Unlike skew and kurtosis in e1071, this calculates a different skew for each variable or column of a data.frame/matrix. mardia applies Mardia's tests for multivariate skew and kurtosis
skew(x, na.rm = TRUE,type=3)
kurtosi(x, na.rm = TRUE,type=3)
mardia(x,na.rm = TRUE,plot=TRUE)Arguments
.
Details
given a matrix or data.frame x, find the skew or kurtosis for each column (for skew and kurtosis) or the multivariate skew and kurtosis in the case of mardia.
As of version 1.2.3,when finding the skew and the kurtosis, there are three different options available. These match the choices available in skewness and kurtosis found in the e1071 package (see Joanes and Gill (1998) for the advantages of each one).
If we define \(m_r = [\sum(X- mx)^r]/n\) then
Type 1 finds skewness and kurtosis by \(g_1 = m_3/(m_2)^{3/2} \) and \(g_2 = m_4/(m_2)^2 -3\).
Type 2 is \(G1 = g1 * \sqrt{n *(n-1)}/(n-2)\) and \(G2 = (n-1)*[(n+1)g2 +6]/((n-2)(n-3))\).
Type 3 is \(b1 = [(n-1)/n]^{3/2} m_3/m_2^{3/2}\) and \(b2 = [(n-1)/n]^{3/2} m_4/m_2^2)\).
For consistency with e1071 and with the Joanes and Gill, the types are now defined as above.
However, from revision 1.0.93 to 1.2.3, kurtosi by default gives an unbiased estimate of the kurtosis (DeCarlo, 1997). Prior versions used a different equation which produced a biased estimate. (See the kurtosis function in the e1071 package for the distinction between these two formulae. The default, type 1 gave what is called type 2 in e1071. The other is their type 3.) For comparison with previous releases, specifying type = 2 will give the old estimate. These type numbers are now changed.
Value
- skew
if input is a matrix or data.frame, skew is a vector of skews
- kurtosi
if input is a matrix or data.frame, kurtosi is a vector of kurtosi
- bp1
Mardia's bp1 estimate of multivariate skew
- bp2
Mardia's bp2 estimate of multivariate kurtosis
- skew
Mardia's skew statistic
- small.skew
Mardia's small sample skew statistic
- p.skew
Probability of skew
- p.small
Probability of small.skew
- kurtosis
Mardia's multivariate kurtosis statistic
- p.kurtosis
Probability of kurtosis statistic
- D
Mahalanobis distance of cases from centroid
References
Joanes, D.N. and Gill, C.A (1998). Comparing measures of sample skewness and kurtosis. The Statistician, 47, 183-189.
L.DeCarlo. 1997) On the meaning and use of kurtosis, Psychological Methods, 2(3):292-307,
K.V. Mardia (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57(3):pp. 519-30, 1970.
Note
The mean function supplies means for the columns of a data.frame, but the overall mean for a matrix. Mean will throw a warning for non-numeric data, but colMeans stops with non-numeric data. Thus, the function uses either mean (for data frames) or colMeans (for matrices). This is true for skew and kurtosi as well.
Note
Probability values less than 10^-300 are set to 0.
See also
describe, describe.by, mult.norm in QuantPsyc, Kurt in QuantPsyc
Examples
round(skew(attitude),2) #type 3 (default)
#> [1] -0.36 -0.22 0.38 -0.05 0.20 -0.87 0.85
round(kurtosi(attitude),2) #type 3 (default)
#> rating complaints privileges learning raises critical advance
#> -0.77 -0.68 -0.41 -1.22 -0.60 0.17 0.47
#for the differences between the three types of skew and kurtosis:
round(skew(attitude,type=1),2) #type 1
#> [1] -0.38 -0.23 0.40 -0.06 0.21 -0.91 0.89
round(skew(attitude,type=2),2) #type 2
#> [1] -0.40 -0.24 0.42 -0.06 0.22 -0.96 0.94
mardia(attitude)
#> Call: mardia(x = attitude)
#>
#> Mardia tests of multivariate skew and kurtosis
#> Use describe(x) the to get univariate tests
#> n.obs = 30 num.vars = 7
#> b1p = 20.09 skew = 100.45 with probability <= 0.11
#> small sample skew = 113.23 with probability <= 0.018
#> b2p = 61.91 kurtosis = -0.27 with probability <= 0.79
x <- matrix(rnorm(1000),ncol=10)
describe(x)
#> vars n mean sd median trimmed mad min max range skew kurtosis
#> X1 1 100 -0.01 1.06 0.01 0.00 1.23 -3.67 2.61 6.29 -0.26 0.37
#> X2 2 100 -0.05 0.91 -0.07 -0.06 0.93 -2.58 1.91 4.49 0.03 -0.18
#> X3 3 100 -0.04 1.09 -0.05 0.01 1.09 -4.68 2.12 6.80 -0.82 2.01
#> X4 4 100 0.15 1.01 0.11 0.15 0.96 -2.82 3.07 5.90 -0.05 0.40
#> X5 5 100 0.01 1.07 -0.05 0.01 1.15 -2.30 2.26 4.56 0.04 -0.70
#> X6 6 100 0.09 0.97 0.10 0.11 0.89 -2.65 2.37 5.02 -0.24 0.00
#> X7 7 100 0.04 1.04 -0.04 0.04 0.88 -2.27 2.85 5.13 0.08 -0.11
#> X8 8 100 0.02 1.06 -0.05 -0.03 1.06 -2.58 2.56 5.14 0.25 -0.29
#> X9 9 100 0.16 0.99 0.15 0.16 1.10 -2.12 3.01 5.13 0.09 -0.35
#> X10 10 100 0.01 0.94 -0.02 0.00 0.98 -2.05 2.96 5.02 0.25 0.01
#> se
#> X1 0.11
#> X2 0.09
#> X3 0.11
#> X4 0.10
#> X5 0.11
#> X6 0.10
#> X7 0.10
#> X8 0.11
#> X9 0.10
#> X10 0.09
mardia(x)
#> Call: mardia(x = x)
#>
#> Mardia tests of multivariate skew and kurtosis
#> Use describe(x) the to get univariate tests
#> n.obs = 100 num.vars = 10
#> b1p = 14.29 skew = 238.23 with probability <= 0.19
#> small sample skew = 246.71 with probability <= 0.1
#> b2p = 116.7 kurtosis = -1.06 with probability <= 0.29