Shapiro-Francia Test for Assessing Normality From Normal Scores

A goodness-of-fit test for normality based on the correlation between the ordered sample values and the corresponding expected normal order statistics, particularly suited for larger sample sizes than the Shapiro-Wilk test.

Usage

shapiroFranciaTest(x)

Arguments

x

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

Value

A list with 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.

Details

Performs the Shapiro-Francia test for the composite hypothesis of normality, see e.g. Thode (2002, Sec. 2.3.2).

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).

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.

Based on code by Juergen Gross, adapted to conform to package standards.

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.

See also

shapiro.test() for performing the Shapiro-Wilk test for normality

Other test.normality: andersonDarlingTest(), cramerVonMisesTest(), jarqueBeraTest(), lillieTest(), pearsonTest()

Examples

shapiroFranciaTest(rnorm(100, mean = 5, sd = 3))
#> 
#> 	Shapiro-Francia normality test
#> 
#> data:  rnorm(100, mean = 5, sd = 3)
#> W = 0.98753, p-value = 0.4013
#> 
shapiroFranciaTest(runif(100, min = 2, max = 4))
#> 
#> 	Shapiro-Francia normality test
#> 
#> data:  runif(100, min = 2, max = 4)
#> W = 0.96044, p-value = 0.005673
#>