Shapiro-Francia Test for Normality

Performs the Shapiro-Francia test for the composite hypothesis of normality.

ShapiroFranciaTest(x)

Arguments

x: a numeric vector of data values, the number of which must be between 5 and 5000. Missing values are allowed.

Details

The test statistic of the Shapiro-Francia test is simply the squared correlation between the ordered sample values and the (approximated) expected ordered quantiles from the standard normal distribution. The p-value is computed from the formula given by Royston (1993).

Value

A list of class htest, containing the following components:

statistic: the value of the Shapiro-Francia statistic.
p.value: the p-value for the test.
method: the character string “Shapiro-Francia normality test”.
data.name: a character string giving the name(s) of the data.

References

Royston, P. (1993): A pocket-calculator algorithm for the Shapiro-Francia test for non-normality: an application to medicine. Statistics in Medicine, 12, 181–184.

Thode Jr., H.C. (2002): Testing for Normality. Marcel Dekker, New York. (2002, Sec. 2.3.2)

Author

Juergen Gross <gross@statistik.uni-dortmund.de>

Note

The Shapiro-Francia test is known to perform well, see also the comments by Royston (1993). The expected ordered quantiles from the standard normal distribution are approximated by qnorm(ppoints(x, a = 3/8)), being slightly different from the approximation qnorm(ppoints(x, a = 1/2)) used for the normal quantile-quantile plot by qqnorm for sample sizes greater than 10.

Examples

ShapiroFranciaTest(rnorm(100, mean = 5, sd = 3))
#> 
#> 	Shapiro-Francia normality test
#> 
#> data:  rnorm(100, mean = 5, sd = 3)
#> W = 0.99345, p-value = 0.8471
#> 
ShapiroFranciaTest(runif(100, min = 2, max = 4))
#> 
#> 	Shapiro-Francia normality test
#> 
#> data:  runif(100, min = 2, max = 4)
#> W = 0.95661, p-value = 0.003319
#>