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XLeratorDB/statistics Documentation

SQL Server Shapiro-Francia test


SFTEST_q

Updated: 27 June 2012


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).
Syntax
SELECT [wctStatistics].[wct].[SFTEST_q](
  <@X_RangeQuery, nvarchar(max),>
 ,<@Statistic, nvarchar(4000),>)
Arguments
@X_RangeQuery
a string which contains the SQL to select the sample values. The resultant table must be ofan expression of type float or of a type that can be implicitly converted to float.
@statistic
identifies the return value as either the Shapiro-Wilk statistic (w) or the p-value (p).
Return Types
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
sample_num             w_observed                p_value
----------- ---------------------- ----------------------
          1      0.917024201827551      0.081092638195141
          2      0.956959220606241      0.411915065401288
 


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