Anderson-Darling test for normality

Performs the Anderson-Darling test for the composite hypothesis of normality, see e.g. Thode (2002, Sec. 5.1.4).

ad.test(x)

Arguments

x

a numeric vector of data values, the number of which must be greater than 7. Missing values are allowed.

Details

The Anderson-Darling test is an EDF omnibus test for the composite hypothesis of normality. The test statistic is $$ A = -n -\frac{1}{n} \sum_{i=1}^{n} [2i-1] [\ln(p_{(i)}) + \ln(1 - p_{(n-i+1)})], $$ where \(p_{(i)} = \Phi([x_{(i)} - \overline{x}]/s)\). Here, \(\Phi\) is the cumulative distribution function of the standard normal distribution, and \(\overline{x}\) and \(s\) are mean and standard deviation of the data values. The p-value is computed from the modified statistic \(Z=A (1.0 + 0.75/n +2.25/n^{2})\)\ according to Table 4.9 in Stephens (1986).

Value

A list with class “htest” containing the following components:

statistic

the value of the Anderson-Darling statistic.

p.value

the p-value for the test.

method

the character string “Anderson-Darling normality test”.

data.name

a character string giving the name(s) of the data.

References

Stephens, M.A. (1986): Tests based on EDF statistics. In: D'Agostino, R.B. and Stephens, M.A., eds.: Goodness-of-Fit Techniques. Marcel Dekker, New York.

Thode Jr., H.C. (2002): Testing for Normality. Marcel Dekker, New York.

Author

Juergen Gross

Note

The Anderson-Darling test is the recommended EDF test by Stephens (1986). Compared to the Cramer-von Mises test (as second choice) it gives more weight to the tails of the distribution.

See also

shapiro.test for performing the Shapiro-Wilk test for normality. cvm.test, lillie.test, pearson.test, sf.test for performing further tests for normality. qqnorm for producing a normal quantile-quantile plot.

Examples

ad.test(rnorm(100, mean = 5, sd = 3))
#> 
#> 	Anderson-Darling normality test
#> 
#> data:  rnorm(100, mean = 5, sd = 3)
#> A = 0.84479, p-value = 0.02864
#> 
ad.test(runif(100, min = 2, max = 4))
#> 
#> 	Anderson-Darling normality test
#> 
#> data:  runif(100, min = 2, max = 4)
#> A = 1.8342, p-value = 0.0001015
#>