# Programming Col Sig (Correlations): Classical Student Test

This article demonstrates how to script the **Classical Student Test **formula** **for two different Correlation outputs: First, the significance test of a correlation between two variables (with N respondents in common). Second, the significance test of a correlation between two variables (in two independent populations).

The QES file and portfolio containing the examples discussed can be downloaded **here**.

**Sig test of a correlation between two variables (with N respondents in common).****Sig test of a correlation between two variables (in two independent populations).**

### Sig test of a correlation between two variables (with N respondents in common)

The formula is:

To test whether a correlation r between 2 variables with N respondents in common is significantly different from zero.

**t** is the sigma value

- If
**t**> 1.645, significance at 90% - If
**t**> 1.960, significance at 95% - If
**t**> 2.576, significance at 99%

The test follows a reduced normal centred law on which a Student's test can be applied.

An example correlation table definition is shown below:

The result:

The **Significance** **[value]** calculation is done using calculation arithmetic:

Dim r1 = CurrentTable.GetCell(CurrentCell.X,CurrentCell.Y-1).Value Dim n1 = CurrentCell.Row.Question.Data.FilterbyXY().Size() Dim Denom1 = 1 - Pow(r1,2) Dim tValue = r1 * (Pow( ( (n1-2) / Denom1 ) , 0.5)) If r1 <> 1 Then Return tValue Else Return nr Endif

The conditional formatting is set on this calculation as follows:

The **Significance** **[+/-]** calculation can then be done using calculation arithmetic:

Dim r1 = CurrentTable.GetCell(CurrentCell.X,CurrentCell.Y-2).Value Dim s1 = CurrentTable.GetCell(CurrentCell.X,CurrentCell.Y-1).Value

'+++ If ((s1 >= 1.645 and s1 < 1.96)) and r1<>1 Then Return "+" Endif If ((s1 >= 1.96 and s1 < 2.576)) and r1<>1 Then Return "++" Endif If (s1 >= 2.576) and r1<>1 Then Return "+++" Endif

'--- If ((s1 <= -1.645 and s1 > -1.96)) and r1<>1 Then Return "-" Endif If ((s1 <= -1.96 and s1 > -2.576)) and r1<>1 Then Return "--" Endif If (s1 <= -2.576) and r1<>1 Then Return "---" Endif

If you just want the colouring to show the significance you can hide both these calculations and set the conditional formatting to take effect on 'All calculations'.

For more information: https://courses.lumenlearning.com/introstats1/chapter/testing-the-significance-of-the-correlation-coefficient/.

### Sig test of a correlation between two variables (in two independent populations)

The formula is:

Where r_{1} is the correlation between 2 variables on a sample size of N_{1 }and r_{2} is the correlation between the same 2 variables on a second independent sample size of N_{2.}

Again **t** is the sigma value . . .

- If
**t**> 1.645, significance at 90% - If
**t**> 1.960, significance at 95% - If
**t**> 2.576, significance at 99%

The test follows a reduced normal centred law on which a Student's test can be applied.

The table set-up is similar with the inbuilt **Correlation** and **N** (count) calculations used. However, in order to provide the independent sample columns we put the variable, **i2. Age**, in the edges and select the option break option shown below:

The result:

The calculation which returns the col sig letters is **T-Test**.

The other calculation, **T-values**, returns the sigma values of the current column against all the others in an array. This can be useful for checking e.g.

The calculation arithmetic script for **T-Test** is:

Dim N1 = CurrentCell.Row.Question.Data.FilterByXY().Size() Dim R1 = CurrentTable.GetCell(CurrentCell.X,CurrentCell.Y-1).Value Dim strColsig = "" Dim i = 0 For i = currentTable.StartX to currentTable.MaxX Dim N2 = CurrentCell.Row.Question.Data.FilterByXY(i).Size() Dim R2 = CurrentTable.GetCell(i,CurrentCell.Y-1).Value Dim dDenomP = ( ( (1+R1) * (1-R2) ) / ( (1-R1) * (1+R2) ) ) Dim dNumeratorP = ( ( 1 / (N1-3) ) + ( 1 / (N2-3) ) ) Dim dSigma = dDenomP.Log10() / ( 2 * ( dNumeratorP.Pow(0.5) ) )

' The test is done at 90% and the letter is shown with a minus sign if significantly lower If dSigma.Abs() > 1.65 and R1 > R2 Then strColsig = strColsig + CurrentTable.GetColSigLetter(i,2) ElseIf dSigma.Abs() > 1.65 and R2 > R1 Then strColsig = strColsig + CurrentTable.GetColSigLetter(i,-2) Endif Next i Return strColsig

The conditional formatting this time is taken care of by a cleaning script (General tab > Settings > Sorting).

```
' Conditional formatting
Dim i = 0
Dim j = 0
Dim k = 0
Dim h = 0
Dim arrColSigLetters = {""}
For i = currentTable.StartX to currentTable.MaxX
arrColSigLetters = arrColSigLetters + CurrentTable.GetColSigLetter(i)
Next i
arrColSigLetters = arrColSigLetters.RemoveAt(1)
For k = currentTable.StartX to currentTable.MaxX
For j = currentTable.StartY to currentTable.MaxY
For h = 1 to arrColSigLetters.Count
If Instr((arrColSigLetters[h]).ToString(), CurrentTable.GetCell(k,j).Text) and Instr("-", CurrentTable.GetCell(k,j).Text) Then
CurrentTable.GetCell(k,j).SetBackColor(244,204,204)
ElseIf Instr((arrColSigLetters[h]).ToString(), CurrentTable.GetCell(k,j).Text) Then
CurrentTable.GetCell(k,j).SetBackColor(202,237,226)
EndIf
Next h
Next j
Next k
```

For more information: https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/Tests_for_Two_Correlations.pdf.