SFTEST_q

Updated 2023-10-26 14:26:14.197000

Syntax

SELECT [westclintech].[wct].[SFTEST_q](
  <@X_RangeQuery, nvarchar(max),>
 ,<@Statistic, nvarchar(4000),>)

Description

Use the scalar function SFTEST_q 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

@Statistic

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

@X_RangeQuery

a string which contains the SQL to select the sample values. The resultant table must be of an expression of type float or of a type that can be implicitly converted to float.

Return Type

float

Remarks

If any x-value returned by @X_RangeQuery IS NULL, it is not included in the calculation.

@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 put into a temporary table. We perform the Shapiro Francia test on both in a single SELECT returning a row for each sample number.

/*Insert test data into a temp table*/

SELECT *

INTO #sf

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);

/*Perform the Shapiro Francia Test on the temp table*/

SELECT a.s as sample_num,

       wct.SFTEST_q('SELECT x from #sf where s = ' + cast(a.s as varchar), 'w') 

                 as w_observed,

       wct.SFTEST_q('SELECT x from #sf where s = ' + cast(a.s as varchar), 'p') 

                 as p_value

FROM

(SELECT DISTINCT s FROM #sf) a(s)

GROUP BY a.s;

/*Clean up the temp table*/

DROP TABLE #sf;

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":"0.081092638195141"},{"sample_num":"2","w_observed":"0.956959220606241","p_value":"0.411915065401288"}]}