SFTEST

Updated 2023-10-26 14:17:57.167000

Syntax

SELECT [westclintech].[wct].[SFTEST] (
  <@x, float,> 
 ,<@statistic, nvarchar(4000),>)

Description

Use the aggregate function SFTEST to perform the Shapiro-Francia test for the composite hypothesis of normality. 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).

Arguments

@x

the sample values. @x is an expression of type float or of a type that can be implicitly converted to float.

@statistic

identifies the return value as either the Shapiro-Francia statistic (w) or the p-value (p).

Return Type

float

Remarks

If @x is null then @x is not included in the aggregate.

@statistic must be invariant for the GROUP.

SFTEST is an aggregate function and follows the same conventions as all other aggregate functions in SQL Server.

@x must have at least 3 rows.

Examples

In this example, we have two samples, which we have identified as sample 1 and sample 2 and we perform the Shapiro-Francia test on both in a single SELECT, grouping the results by sample number.

SELECT n.s as sample_num,

       wct.SFTEST(x, 'w') as w_observed,

       wct.SFTEST(x, 'p') as p_value

FROM

(

    VALUES

        (1, 0.11),

        (2, 3.49),

        (1, 7.87),

        (2, 1.36),

        (1, 4.61),

        (2, 1.14),

        (1, 10.14),

        (2, 2.92),

        (1, 7.95),

        (2, 2.55),

        (1, 3.14),

        (2, 1.46),

        (1, 0.46),

        (2, 1.06),

        (1, 4.43),

        (2, 5.27),

        (1, 0.21),

        (2, -1.11),

        (1, 4.75),

        (2, 3.48),

        (1, 0.71),

        (2, 1.10),

        (1, 1.52),

        (2, 0.88),

        (1, 3.24),

        (2, -0.51),

        (1, 0.93),

        (2, 1.46),

        (1, 0.42),

        (2, 0.52),

        (1, 4.97),

        (2, 6.20),

        (1, 9.53),

        (2, 1.69),

        (1, 4.55),

        (2, 0.08),

        (1, 0.47),

        (2, 3.67),

        (1, 6.66),

        (2, 2.81)

) n (s, x)

GROUP BY n.s;

This produces the following result

{"columns":[{"field":"sample_num","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"},{"field":"w_observed","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"},{"field":"p_value","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"sample_num":"1","w_observed":"0.917024201827551","p_value":"0.0810926381951407"},{"sample_num":"2","w_observed":"0.956959220606241","p_value":"0.411915065401292"}]}