Test of Skewness

Computes $G_{1}$, the expected population skewness of the values in x using skewness(), performs a t-test of its significance and calculates a confidence interval.

Usage

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

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

stderr_skewness(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 skewness (95% or other specified level).
estimate: the estimate of skewness.
alternative: a character string describing the alternative hypothesis.
method: the character string "Skewness 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 skewness, $G_{1}$, divided by its standard error, $SE_{G_{1}}$, where: -

$$SE_{G_{1}} = \displaystyle \sqrt{\frac{6n(n - 1)}{(n-2)(n+1)(n+3)}}$$

(see e.g., Joanes and Gill, 1998; Wright and Herrington 2011), or alternatively its approximation, $\sqrt (6 / n_x)$, 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.

skew.test() is an alias for skewness.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 
 skewness.test(heights) 
#> 
#> 	Skewness with t-test (Cramer stderr)
#> 
#> data:  heights
#> t = -0.45492, df = 98, p-value = 0.6502
#> alternative hypothesis: true skewness is not equal to 0
#> 95 percent confidence interval:
#>  -0.5888187  0.3692019
#> sample estimates:
#>   skewness 
#> -0.1098084 
#> 
 length(heights) |> stderr_skewness()
#> [1] 0.2413798
 skewness.test(heights, se_method = "simple")
#> 
#> 	Skewness with t-test (simple stderr)
#> 
#> data:  heights
#> t = -0.44829, df = 98, p-value = 0.6549
#> alternative hypothesis: true skewness is not equal to 0
#> 95 percent confidence interval:
#>  -0.5959017  0.3762849
#> sample estimates:
#>   skewness 
#> -0.1098084 
#> 
 length(heights) |> stderr_skewness(se_method = "simple")
#> [1] 0.244949

 ## 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 
 skewness.test(litter_sizes) 
#> 
#> 	Skewness with t-test (Cramer stderr)
#> 
#> data:  litter_sizes
#> t = 2.0204, df = 813, p-value = 0.04367
#> alternative hypothesis: true skewness is not equal to 0
#> 95 percent confidence interval:
#>  0.004927511 0.341148438
#> sample estimates:
#> skewness 
#> 0.173038 
#> 
 length(litter_sizes) |> stderr_skewness()
#> [1] 0.08564453

 ## Compare a range of distributions, each with the three possible alternative hypotheses
 list(
   uniform = runif(30),
   normal = rnorm(30),
   lognormal = rlnorm(30),
   poisson = rpois(30, lambda = 10),
   negbinom = rnbinom(30, mu = 4, size = 2)
 ) |>
 lapply(\(distrib)
     c("less", "two.sided","greater") |>
     setNames(nm = _) |>
     lapply(\(altern)
         with(skewness.test(distrib, altern),
             data.frame(
                 Lower = conf.int[1],
                 Upper = conf.int[2],
                 Skewness = estimate,
                 t = statistic,
                 df = parameter,
                 p = p.value,
                 sig = starsig(p.value),
                 row.names = NULL
             )
         )
     ) |>
     bind_rows(.id = "Alternative")) |>
 bind_rows(.id = "Distribution")
#>    Distribution Alternative      Lower     Upper   Skewness          t df
#> 1       uniform        less       -Inf 0.5953722 -0.1308276 -0.3064651 28
#> 2       uniform   two.sided -1.0052771 0.7436218 -0.1308276 -0.3064651 28
#> 3       uniform     greater -0.8570275       Inf -0.1308276 -0.3064651 28
#> 4        normal        less       -Inf 0.9638922  0.2376923  0.5567968 28
#> 5        normal   two.sided -0.6367571 1.1121417  0.2376923  0.5567968 28
#> 6        normal     greater -0.4885076       Inf  0.2376923  0.5567968 28
#> 7     lognormal        less       -Inf 5.0202253  4.2940255 10.0588006 28
#> 8     lognormal   two.sided  3.4195761 5.1684749  4.2940255 10.0588006 28
#> 9     lognormal     greater  3.5678256       Inf  4.2940255 10.0588006 28
#> 10      poisson        less       -Inf 0.9089245  0.1827247  0.4280345 28
#> 11      poisson   two.sided -0.6917247 1.0571741  0.1827247  0.4280345 28
#> 12      poisson     greater -0.5434752       Inf  0.1827247  0.4280345 28
#> 13     negbinom        less       -Inf 2.5659773  1.8397774  4.3096983 28
#> 14     negbinom   two.sided  0.9653280 2.7142269  1.8397774  4.3096983 28
#> 15     negbinom     greater  1.1135776       Inf  1.8397774  4.3096983 28
#>               p sig
#> 1  3.807590e-01  NS
#> 2  7.615179e-01  NS
#> 3  6.192410e-01  NS
#> 4  7.089551e-01  NS
#> 5  5.820899e-01  NS
#> 6  2.910449e-01  NS
#> 7  1.000000e+00  NS
#> 8  8.455014e-11 ***
#> 9  4.227507e-11 ***
#> 10 6.640498e-01  NS
#> 11 6.719005e-01  NS
#> 12 3.359502e-01  NS
#> 13 9.999089e-01  NS
#> 14 1.822470e-04 ***
#> 15 9.112352e-05 ***