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XLeratorDB/statistics Documentation

SQL Server LINEST function


LINEST_q
Updated: 19 January 2017

Use the table-valued function LINEST_q to calculate the Ordinary Least Squares (OLS) solution for a series of x- and y-values. The OLS solution calculates a line that best fits the data supplied to the function. The LINEST_q function returns the statistics that describe the calculated solution, including the coefficients (m), the standard of the coefficients (se), the t statistic for each coefficient (tstat) and the associated p-values (pval), the coefficient of determination (rsq), the adjusted r-square value (rsqa) and the modified r-square value (rsqm), the standard error of the y estimate (sey), the F-observed value (F), the residual degrees of freedom (df), the regression sum of squares (ss_reg), and the residual some of squares. For more information on the calculation of these values, see Ordinary Least Squares.
In the case where we have one column of x-values and a column of y-values, the OLS solution is immediately recognizable as the formula for a line:

For the purpose of this function, we would re-write the equation as

The value for slope, then, is stored in m0.
 
For purposes of multi-linear regression, where there are multiple columns of x-values (and still a single column of y-values), the formula for the solution is described by the following equation:
 
 
Where n is the number of x-columns.
The function expects the input to be in row-column, or spreadsheet, format, rather than in third-normal form. Input into the function is a string which contains the SQL which when executed returns a resultant table comprising the input into the linear regression calculations, the specification of a 1-based index identifying the column reference for the y-values, and a bit value which specifies whether or not the y-intercept runs through 0.
LINEST_q automatically detects collinearity and removes the right-most co-linear column resulting in a regression coefficient of 0 for that column.
Syntax
SELECT * FROM [wct].[LINEST_q](
  <@Matrix_RangeQuery, nvarchar(max),>
 ,<@Y_ColumnNumber, int,>
 ,<@Lconst, bit,>)
Arguments
@Matrix_RangeQuery
The SELECT statement, as a string, which, when executed, creates the resultant table of x- and y-values which will be used in the calculation.
@y_ColumnNumber
the index into the array identifying the column containing the y-values. The index value must be between 1 and n, where n is the number of columns in the resultant table created by the @Matrix_RangeQuery. @y_ColumnNumber must be of the type int or of a type that implicitly converts to int.
@LConst
A logical value specifying whether to force the y-intercept value (m0) equal to zero. @LConst must be of the type bit or of a type that implicitly converts to bit.
Return Type
RETURNS TABLE (
       [stat_name] [nvarchar](10) NULL,
       [idx] [int] NULL,
       [stat_val] [float] NULL,
       [col_name] [nvarchar](128) NULL
)
Table Description
stat_name
Identifies the statistic being returned:

m
the estimated coefficient
se
the standard error of the estimated coefficient
tstat
the t statistic
pval
the p-value (t distribution) for the t statistic
rsq
the coefficient of determination (r2)
rsqa
adjusted r square
rsqm
multiple r square
sey
the standard error for the y estimate
f
the f-observed value
df
the residual degrees of freedom
ss_reg
the regression sum of squares
ss_resid
the residual sum of squares

idx
Identifies the subscript for the estimated coefficient, the standard error of the estimated coefficient, the t statistic and the p-values. For example, the stat_name m with an idx of 0, specifies that the stat_val is for m0, or the y-intercept (which is b in y = mx + b). An idx of 1 for the same stat_name identifies m1.
The stat_name se with and idx of 0 identifies the standard error of the m0 coefficient (which is sometimes referred to as the standard error of b or seb).
Idx values are only supplied for the m, se, tstat, and pval stat_names. All others will have an idx of NULL.
stat_val
the calculated value of the statistic.
col_name
the column name from the resultant table produced by the dynamic SQL. col_name values are produced only for the m, se, tstat, and pval statistics; all other stat_names have NULL for col_name.
Remarks
·         If @Lconst is NULL then @LConst is set to 'True'.
·         If @LConst is true than the number of rows must be greater than the number of columns.
·         If @Lconst is false than the number of rows must be greater than or equal to the number of columns.
·         For simpler queries, you can try the LINEST function.
Examples
Example #1
We create a table and put some data in it.
We put the x- and y-data into a temp table, #xy
SELECT
   *
INTO
   #xy
FROM (VALUES
    (6.1,1.21,4.35,5.42,6.45,138.08)
   ,(7.95,0.97,2.79,4.73,8.14,80.3)
   ,(8.53,9.73,9.7,1.16,9.05,284.45)
   ,(7.4,1.61,9.9,8.8,4.38,226.66)
   ,(7.42,4.58,0.06,8.97,8.75,112.37)
   ,(6.19,3.56,9.69,8.7,5.67,168.73)
   ,(8.44,3.85,8.23,1.05,0.92,160.38)
   ,(0.84,3.86,7.85,2.14,3.03,129.26)
   ,(0.37,7.33,5.07,8.06,3.25,170.39)
   ,(7.48,0.68,8.34,2.98,2.81,188.53)
   ,(5.33,3.51,7.03,6.49,7.54,131.1)
   )n(x1,x2,x3,x4,x5,y)
This is what the data look like.
To invoke the table-valued function LINEST_q, we enter the following SQL.
SELECT
   *
FROM
   wct.LINEST_q(
       'SELECT * FROM #xy', --@Matrix_RangeQuery
       6,            --@Y_columnNumber
       'True'        --@Lconst
       )
The following results are produced.
The results are returned in 3rd normal form. You can use standard SQL commands to re-format the results. For example, if you wanted to produce the the coefficients in a format similar to output of the Excel Data Analyis Regression Tool, you could use the following SQL.
;WITH mycte as (
   SELECT *
   FROM wct.LINEST_q('SELECT * FROM #xy',6,1)p
)
SELECT
    d.col_name
   ,d.m
   ,d.se
   ,d.tstat
   ,d.pval
   ,m-wct.T_INV_2T(.05,m.stat_val)*se as [Lower Confidence Level]
   ,m+wct.T_INV_2T(.05,m.stat_val)*se as [Upper Confidence Level]
FROM mycte p
PIVOT(MAX(stat_val) FOR stat_name in(m,se,tstat,pval))d
CROSS JOIN mycte m
WHERE stat_name = 'df'
AND d.col_name IS NOT NULL
This produces the follwing result.
Similarly, if you wanted to reformat the results to produce the equivalent of the ANOVA table from the Excel Data Analysis Regression tool, you could use the following SQL.
;WITH mycte as (
   SELECT
        df as [Regression df]
       ,Obs - df - 1 as [Residual df]
       ,ss_reg as [Regression SS]
       ,ss_resid as [Residual SS]
       ,F
   FROM (
       SELECT
           stat_name
          ,stat_val
       FROM
          wct.LINEST_q('SELECT * FROM #xy',6,1)p
       WHERE
          stat_name in('ss_reg','ss_resid','F','df')
       UNION
       SELECT
          'Obs'
          ,COUNT(*)
       FROM
          #xy)d
   PIVOT(MAX(stat_val) FOR stat_name in(df,F,ss_reg,ss_resid,Obs))p
   ) 
SELECT
    'Regression'
   ,[Regression df] as DF
   ,[Regression SS] as SS
   ,[Regression SS]/[Regression df] as MS
   ,F
   ,wct.F_DIST_RT(F,[Regression df],[Residual df]) as [Significance F] 
FROM
   mycte
UNION ALL
SELECT
    'Residual'
   ,[Residual df]
   ,[Residual SS]
   ,[Residual SS]/[Residual df] as [Residual MS]
   ,NULL
   ,NULL
FROM
   mycte
UNION ALL
SELECT
    'Total'
   ,[Regression df] + [Residual df]
   ,[Regression SS] + [Residual SS]
   ,NULL
   ,NULL
   ,NULL
FROM
   mycte
This produces the following result
Since the function executes the @Matrix_RangeQuery values, we did not need to put the x- and y-values into a temporary table and could have simply inlcuded them in @Matrix_RangeQuery, like this.
SELECT
   *
FROM
   wct.LINEST_q(
       'SELECT
          *
       FROM (VALUES
           (6.1,1.21,4.35,5.42,6.45,138.08)
          ,(7.95,0.97,2.79,4.73,8.14,80.3)
          ,(8.53,9.73,9.7,1.16,9.05,284.45)
          ,(7.4,1.61,9.9,8.8,4.38,226.66)
          ,(7.42,4.58,0.06,8.97,8.75,112.37)
          ,(6.19,3.56,9.69,8.7,5.67,168.73)
          ,(8.44,3.85,8.23,1.05,0.92,160.38)
          ,(0.84,3.86,7.85,2.14,3.03,129.26)
          ,(0.37,7.33,5.07,8.06,3.25,170.39)
          ,(7.48,0.68,8.34,2.98,2.81,188.53)
          ,(5.33,3.51,7.03,6.49,7.54,131.1)
          )n(x1,x2,x3,x4,x5,y)'   --@Matrix_RangeQuery
       ,6            --@Y_columnNumber
       ,'True'       --@Lconst
       )
Example #2
Using the same data as Example #1, if we wanted to calculate the coefficients without a y-intercept, we would modify function call to make the last parameter FALSE. This time, however, we will put the results into a temporary table, #L0, and reformat the results using the #L0 table.
SELECT
   *
INTO
   #L0
FROM
   wct.LINEST_q(
       'SELECT * FROM #xy', --@Matrix_RangeQuery
       6,            --@Y_columnNumber
       'False'       --@Lconst
       ) 
The #L0 table should contain the following data.
We can use the same technique as in Example #1 to reformat the coefficient statistics.
SELECT
    d.col_name
   ,d.m
   ,d.se
   ,d.tstat
   ,d.pval
   ,m-wct.T_INV_2T(.05,m.stat_val)*se as [Lower Confidence Level]
   ,m+wct.T_INV_2T(.05,m.stat_val)*se as [Upper Confidence Level]
FROM
   #L0 p
PIVOT(MAX(stat_val) FOR stat_name in(m,se,tstat,pval))d
CROSS JOIN #L0 m
WHERE
   m.stat_name = 'df'
   AND d.col_name IS NOT NULL
This produces the following result.
Note that even though we specified no intercept, an intercept row is still created with a regression coefficient of 0 (keeping the results consistent with the Excel Data Analysis Regression). It is simple enough to exclude this from the output (like R does) simply by adding another conidition to the WHERE clause.
WHERE
   m.stat_name = 'df'
   AND d.col_name IS NOT NULL
   AND d.col_name <> 'Intercept'
The following SQL can be used to reproduce the ANOVA table.
;WITH mycte as (
   SELECT
        df as [Residual df]
       --,Obs - df - 1 as [Regression df]
       ,Obs - df as [Regression df]
       ,ss_reg as [Regression SS]
       ,ss_resid as [Residual SS]
       ,F
   FROM (
       SELECT
           stat_name
          ,stat_Val
       FROM
          #L0
       WHERE
          stat_name in('ss_reg','ss_resid','F','df')
       UNION
       SELECT
          'Obs'
          ,COUNT(*)
       FROM
          #xy)d
   PIVOT(MAX(stat_val) FOR stat_name in(df,F,ss_reg,ss_resid,Obs))p
   ) 
SELECT
    'Regression'
   ,[Regression df] as DF
   ,[Regression SS] as SS
   ,[Regression SS]/[Regression df] as MS
   ,F
   ,wct.F_DIST_RT(F,[Regression df],[Residual df]) as [Significance F] 
FROM
   mycte
UNION ALL
SELECT
    'Residual'
   ,[Residual df]
   ,[Residual SS]
   ,[Residual SS]/[Residual df] as [Residual MS]
   ,NULL
   ,NULL
FROM
   mycte
UNION ALL
SELECT
    'Total'
   ,[Regression df] + [Residual df]
   ,[Regression SS] + [Residual SS]
   ,NULL
   ,NULL
   ,NULL
FROM
   mycte
This produces the following result.
Note that the calculation of the regression degrees of freedom needs to be adjusted to reflect the absence on intercept. It also worthwhile noting that the calculation in this SQL produces a Significance F that agrees with R while Excel produces a different value (which seems to be caused be subtracting 1 from the residual degrees of freedom).
Example #3
Here’s another example. Instead of just having a matrix of x- and y-values in our table, we are going to add a column, testid, which is a way of grouping x- and y-values together for purposes of doing the LINEST_q calculation. This allows us to compute the ordinary least sqaures values for multiple sets of data in a single SELECT statement.
Second, we take the x-values and raise them to the powers of 2, 3, 4, 5 thus solving for the equation:
y = B0 + B1x + B2x2 + B3x3 + B4x4 + B5x5
SELECT
       *
INTO
       #xy
FROM (VALUES
        ('Wampler5',7590001,0)
       ,('Wampler5',-20479994,1)
       ,('Wampler5',20480063,2)
       ,('Wampler5',-20479636,3)
       ,('Wampler5',25231365,4)
       ,('Wampler5',-20476094,5)
       ,('Wampler5',20489331,6)
       ,('Wampler5',-20460392,7)
       ,('Wampler5',18417449,8)
       ,('Wampler5',-20413570,9)
       ,('Wampler5',20591111,10)
       ,('Wampler5',-20302844,11)
       ,('Wampler5',18651453,12)
       ,('Wampler5',-20077766,13)
       ,('Wampler5',21059195,14)
       ,('Wampler5',-19666384,15)
       ,('Wampler5',26348481,16)
       ,('Wampler5',-18971402,17)
       ,('Wampler5',22480719,18)
       ,('Wampler5',-17866340,19)
       ,('Wampler5',10958421,20)
       ,('Wampler4',75901,0)
       ,('Wampler4',-204794,1)
       ,('Wampler4',204863,2)
       ,('Wampler4',-204436,3)
       ,('Wampler4',253665,4)
       ,('Wampler4',-200894,5)
       ,('Wampler4',214131,6)
       ,('Wampler4',-185192,7)
       ,('Wampler4',221249,8)
       ,('Wampler4',-138370,9)
       ,('Wampler4',315911,10)
       ,('Wampler4',-27644,11)
       ,('Wampler4',455253,12)
       ,('Wampler4',197434,13)
       ,('Wampler4',783995,14)
       ,('Wampler4',608816,15)
       ,('Wampler4',1370781,16)
       ,('Wampler4',1303798,17)
       ,('Wampler4',2205519,18)
       ,('Wampler4',2408860,19)
       ,('Wampler4',3444321,20)
       ,('Wampler3',760,0)
       ,('Wampler3',-2042,1)
       ,('Wampler3',2111,2)
       ,('Wampler3',-1684,3)
       ,('Wampler3',3888,4)
       ,('Wampler3',1858,5)
       ,('Wampler3',11379,6)
       ,('Wampler3',17560,7)
       ,('Wampler3',39287,8)
       ,('Wampler3',64382,9)
       ,('Wampler3',113159,10)
       ,('Wampler3',175108,11)
       ,('Wampler3',273291,12)
       ,('Wampler3',400186,13)
       ,('Wampler3',581243,14)
       ,('Wampler3',811568,15)
       ,('Wampler3',1121004,16)
       ,('Wampler3',1506550,17)
       ,('Wampler3',2002767,18)
       ,('Wampler3',2611612,19)
       ,('Wampler3',3369180,20)
       ,('Wampler2',1,0)
       ,('Wampler2',1.11111,1)
       ,('Wampler2',1.24992,2)
       ,('Wampler2',1.42753,3)
       ,('Wampler2',1.65984,4)
       ,('Wampler2',1.96875,5)
       ,('Wampler2',2.38336,6)
       ,('Wampler2',2.94117,7)
       ,('Wampler2',3.68928,8)
       ,('Wampler2',4.68559,9)
       ,('Wampler2',6,10)
       ,('Wampler2',7.71561,11)
       ,('Wampler2',9.92992,12)
       ,('Wampler2',12.75603,13)
       ,('Wampler2',16.32384,14)
       ,('Wampler2',20.78125,15)
       ,('Wampler2',26.29536,16)
       ,('Wampler2',33.05367,17)
       ,('Wampler2',41.26528,18)
       ,('Wampler2',51.16209,19)
       ,('Wampler2',63,20)
       ,('Wampler1',1,0)
       ,('Wampler1',6,1)
       ,('Wampler1',63,2)
       ,('Wampler1',364,3)
       ,('Wampler1',1365,4)
       ,('Wampler1',3906,5)
       ,('Wampler1',9331,6)
       ,('Wampler1',19608,7)
       ,('Wampler1',37449,8)
       ,('Wampler1',66430,9)
       ,('Wampler1',111111,10)
       ,('Wampler1',177156,11)
       ,('Wampler1',271453,12)
       ,('Wampler1',402234,13)
       ,('Wampler1',579195,14)
       ,('Wampler1',813616,15)
       ,('Wampler1',1118481,16)
       ,('Wampler1',1508598,17)
       ,('Wampler1',2000719,18)
       ,('Wampler1',2613660,19)
       ,('Wampler1',3368421,20)
       )n(testid,y,x)
Let’s say wanted to run LINEST_q for the all the data where the testid is equal to Wampler3. We could simply enter the following statement.
SELECT
   *
FROM
wct.LINEST_q('SELECT y,x,POWER(x,2),POWER(x,3),POWER(x,4),POWER(x,5) FROM #xy WHERE testid = ''Wampler3''',1,'True')
This produces the following result.
You will notice that the column names for the calculated columns are Column1, Column2, Column3, and Column4. This is because we have not assigned them a name and these are the default column names. Even though Column1 does not refer to the first column in the resultant table from our dynamic SQL, it is the first column with no name. To get more descriptive column names, you can simply assign the names in the @ColumnNames variable as in this example.
SELECT
   *
FROM
wct.LINEST_q('SELECT y,x,POWER(x,2) as [x^2],POWER(x,3) as [x^3],POWER(x,4) as [x^4],POWER(x,5) as [x^5] FROM #xy WHERE testid = ''Wampler3''',1,'True')
This produces the following result.
In the following SQL we return selected values for each test.
SELECT
    p.testid
   ,p.rsq
   ,p.rsqa
   ,p.rsqm
   ,p.F
   ,p.df
   ,p.ss_reg
   ,p.ss_resid
FROM (
   SELECT n.testid,k.stat_name,k.stat_val
   FROM (SELECT DISTINCT testid FROM #xy)n
CROSS APPLY wct.LINEST_q('SELECT y,x,POWER(x,2) as [x^2],POWER(x,3) as [x^3],POWER(x,4) as [x^4],POWER(x,5) as [x^5] FROM #xy WHERE testid = ''' + n.testid + '''',1,'True')k
   )d
PIVOT(SUM(stat_val) FOR stat_name IN(F,df,ss_reg,ss_resid,rsq,rsqm,rsqa))p
This produces the following result.
In this SQL we select the coefficient statistics for each of the tests.
SELECT
    p.testid
   ,p.col_name
   ,p.m
   ,p.se
   ,p.tstat
   ,p.pval
FROM (
   SELECT
        n.testid
       ,k.stat_name
       ,k.idx
       ,k.col_name
       ,k.stat_val
   FROM (
       SELECT DISTINCT testid
       FROM #xy)n
CROSS APPLY wct.LINEST_q('SELECT y,x,POWER(x,2) as [x^2],POWER(x,3) as [x^3],POWER(x,4) as [x^4],POWER(x,5) as [x^5] FROM #xy WHERE testid = ''' + n.testid + '''',1,'True')k
   WHERE k.idx IS NOT NULL
   )d
PIVOT(SUM(stat_val) FOR stat_name IN(m,se,tstat,pval))p
ORDER BY
    testid
   ,idx
This produces the following result.
See Also

 

 



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