Test of Kurtosis
kurtosis.test.Rd
Computes \(G_{2}\), the expected population kurtosis of the values in x using
kurtosis()
, performs a t-test of its significance and calculates a confidence interval.
Arguments
- x
a
numeric
vector.- alternative
a character string specifying the alternative hypothesis, must be one of
"two.sided"
(default),"greater"
or"less"
. You can specify just the initial letter.- se_method
a
character
string specifying the method of calculating the standard error; must be one of"Cramer"
(default), or"simple"
. You can specify just the initial letter.- conf.level
the confidence level required; default 0.95.
- n
an
integer
, the number of observations.
Value
A list with class "htest"
containing the following components: -
- statistic
the value of the t-statistic.
- parameter
the degrees of freedom for the t-statistic.
- p.value
the p-value for the test.
- conf.int
confidence interval of the kurtosis (95% or other specified level).
- estimate
the estimate of kurtosis.
- alternative
a character string describing the alternative hypothesis.
- method
the character string "Kurtosis with t-test" and the standard error method used.
- data.name
a character string giving the name of the data.
Details
The t-statistic is given by the estimated population kurtosis
, \(G_{2}\),
divided by its standard error, \(SE_{G_{2}}\), where: -
$$SE_{G_{2}} = \displaystyle 2(SE_{G_{1}}) \sqrt{\frac{(n^{2} - 1)}{(n-3)(n+5)}}$$
(see e.g., Joanes and Gill, 1998; Wright and Herrington 2011), or alternatively its approximation,
\(\sqrt (24 / n_x)\), and the associated probability is derived from the t-distribution with
\(n_{x}-2\) degrees of freedom. The t-test is conducted according to Crawley (2012),
except that the default here is a two-tailed test. The corresponding confidence interval is calculated similarly
from the quantiles of the t-distribution using both the alternative
and conf.level
arguments.
kurt.test()
is an alias for kurtosis.test()
.
Note
The confidence interval is poorly described in the available literature, seems somewhat controversial and should be used with caution.
References
Crawley, Michael J. (2012) The R Book. John Wiley & Sons, Incorporated. ISBN:9780470973929. p.350-352. doi:10.1002/9781118448908
Joanes, D.N., and Gill, C.A. (1998). Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society. Series D (The Statistician) 47(1): 183–189. doi:10.1111/1467-9884.00122
Wright, D.B., and Herrington, J.A. (2011). Problematic standard errors and confidence intervals for skewness and kurtosis. Behavior Research Methods 43(1): 8-17. doi:10.3758/s13428-010-0044-x
See also
Other skewness:
kurtosis()
,
skewness()
,
skewness.test()
Examples
## Heights of 100 randomly selected male university students, adapted from Spiegel and Stephens
## (Theory and Problems of Statistics. 4th edn. McGraw-Hill. 1999. ISBN 9780071755498).
table(heights)
#> heights
#> 61 64 67 70 73
#> 5 18 42 27 8
kurtosis.test(heights)
#>
#> Kurtosis with t-test (Cramer stderr)
#>
#> data: heights
#> t = -0.43724, df = 98, p-value = 0.6629
#> alternative hypothesis: true kurtosis is not equal to 0
#> 95 percent confidence interval:
#> -1.1583796 0.7400855
#> sample estimates:
#> kurtosis
#> -0.2091471
#>
length(heights) |> stderr_kurtosis()
#> [1] 0.4783311
kurtosis.test(heights, se_method = "simple")
#>
#> Kurtosis with t-test (simple stderr)
#>
#> data: heights
#> t = -0.42692, df = 98, p-value = 0.6704
#> alternative hypothesis: true kurtosis is not equal to 0
#> 95 percent confidence interval:
#> -1.1813336 0.7630395
#> sample estimates:
#> kurtosis
#> -0.2091471
#>
length(heights) |> stderr_kurtosis(se_method = "simple")
#> [1] 0.4898979
## Litter sizes in albino rats (n = 815), data from King (1924; Litter production and
## the sex ratio in various strains of rats. The Anatomical Record 27(5), 337-366).
table(litter_sizes)
#> litter_sizes
#> 1 2 3 4 5 6 7 8 9 10 11 12
#> 7 33 58 116 125 126 121 107 56 37 25 4
kurtosis.test(litter_sizes)
#>
#> Kurtosis with t-test (Cramer stderr)
#>
#> data: litter_sizes
#> t = -2.7832, df = 813, p-value = 0.005507
#> alternative hypothesis: true kurtosis is not equal to 0
#> 95 percent confidence interval:
#> -0.8119700 -0.1403447
#> sample estimates:
#> kurtosis
#> -0.4761573
#>
length(litter_sizes) |> stderr_kurtosis()
#> [1] 0.1710811