Test of Kurtosis

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.

Usage

kurtosis.test(
  x,
  alternative = c("two.sided", "less", "greater"),
  se_method = c("Cramer", "simple"),
  conf.level = 0.95
)

kurt.test(
  x,
  alternative = c("two.sided", "less", "greater"),
  se_method = c("Cramer", "simple"),
  conf.level = 0.95
)

stderr_kurtosis(n, se_method = c("Cramer", "simple"))

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

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