As we have been learning about chi-square. I have mentioned a finding called Cramér's V.
Cramér's V is the preferred measure of effect size for a chi-square
Cramér's V is a measure of association between two nominal variables.
The range of the possible values will be between 0 and +1 inclusive,
and it should come as no surprise that Cramér's V cannot be negative because
chi-square uses squared values and the minimum value for a chi-square is 0.
Cramér's V is based on the Pearson chi-square statistic
where a +1 equals a perfect relationship, and 0 equals no relationship
What makes Cramér's V so useful is that it can be used with various chi-square analyses from a simple one-way
chi-square up to any number of rows and columns.
The formula for Cramér's V is
the square root of chi-square, divided by n times t
Where n = sample size,
and t = the smaller of (r - 1) or (c - 1)
Where r = the number of rows and c = the number of columns
However, there is a much more simple formula that you can use for a one-way chi-square,
and it is based on another measure called phi
Phi is a Greek letter that rhymes with my, and pi, and chi
Phi is a measure of association that adjusts the chi-square statistic
by the sample size if you are reporting Phi it should be because you are using a cross tabulation.
The lowest possible value for Phi is 0 and the maximum depends upon the dimensions of the crosstab table.
So, for example, for a 3x4 table, you take the smaller of 3 and 4, which is, of course, 3 -1.
The maximum value therefore is 2.
You may also see phi square reported as a measure of association as well
So here is how these three converge.
Phi, phi-squared, and Cramér's V are actually quite similar.
In fact, when there is no relationship between the variables, in other words, when chi-square equals 0,
then Cramér's V, phi, and phi-squared will also all equal zero.
Now, the formula for Phi is much more simple.
It is the square root of chi-square divided by n.
But, of course, we want to report
Cramér's V because Cramér's V is the preferred measure of effect size for chi-square.
The good news is that the output for Cramér's V is similar to Phi.
And in fact, when you use a one-way chi-square or a 2x2 crosstabulation
then Cramér's V will equal Phi,
and you can request Cramér's V in SPSS when you do a chi-square using crosstabs menu.
I have another video about crosstabulation.
Therefore, when you do a one-way or 2x2 chi-square
Just use the Phi formula, but report it as Cramér's V.
This allows you to use the simple formula for reporting effect size,
of course, for a more complex chi-square or cross tabulation use the actual Cramér's V formula.