From the way we define the phi and Cramers V and from my theoretical knowledge, I expect Phi coefficient and Cramer's V to bare positive values. In a recent analysis using SPSS 22.0, Negative values were displayed. Is this correct technically? What can be the best possible interpretation?
Hello Philip,
The SPSS algorithms also imply that a negative value would not occur.
Is it possible that you are using weighted data? That can cause odd behavior in some subprograms of spss.
If you were to furnish the data values for the table, someone might be able to help you sort out the appropriate value for phi/V.
Good luck with your work.
I don't know about the output for SPSS, but in general, phi can be positive or negative. That is, a 2 x 2 table with the following values (2, 10, 10, 2) will have the opposite sign of a table with the following values (10, 2, 2, 10). Cramer's V is usually defined as positive, even for 2 x 2 tables.
Here is SPSS code for Salvatore S. Mangiafico's example.
* Generate some data to illustrate.
DATA LIST LIST / table r c n (4F2.0).
BEGIN DATA
1 1 1 2
1 1 2 10
1 2 1 10
1 2 2 2
2 1 1 10
2 1 2 2
2 2 1 2
2 2 2 10
END DATA.
WEIGHT by n.
CROSSTABS r by c by table / STATISTICS=PHI.
This will probably not align very nicely, but the output shows:
Value p
Table 1:
Phi -.667 .001
Cramer's V .667 .001
Table 2
Phi .667 .001
Cramer's V .667 .001
EDIT: That output did not align at all well. Here's another try:
TABLE 1: phi = -.667, Cramer's V = .667
TABLE 2: phi = .667, Cramer's V = .667
All p-values = .001
Remember that Phi = Pearson's r on a pair of dichotomies. So negative values must be possible. Add this to the code above for a demonstration:
SPLIT FILE BY table.
CORRELATIONS r WITH c.
SPLIT FILE OFF.
HTH.
Thank you Salvatore S. Mangiafico. I just checked my data set it's actually a 2*2 table hence the negative data readings are Phi values and not Cramer v. please can I have some more references related to your explanation on possible negative value for phi. Once again thank you
Philip Govule , I don't have a great source for this distinction.
In some textbook and online descriptions, you will find that phi is treated as not directional. For example, J. Cohen, 1988, Statistical Power Analysis for the Behavioral Sciences, defines phi, in the 2 x 2 case, as sqrt(chi-square / N). ( https://www.clintools.com/products/org/WebHelp/images/ebx_-695398319.gif ). This would always be positive. But he does note that phi is equal to the product-moment correlation for the 2 x 2. [And so, could be positive or negative]. The sqrt(chi-square/N) definition is one you will find most commonly, but isn't the one most software uses.
Another definition you will find is (a - (a + b) * (a + c))/sqrt((a + b) * (c + d) * (a + c) * (b + d)) . Where a, b, c, and d, are the cells of the 2 x 2 table ( https://methods.sagepub.com/images/virtual/the-sage-encyclopedia-of-communication-research-methods/10.4135_9781483381411-table54.jpg ). This definition is directional (positive or negative).
You can see a note I've written on the topic at the following.
https://rcompanion.org/handbook/H_10.html
Hello all,
I concur with Salvatore S. Mangiafico 's note that the phi coefficient can be negative in sign (and the usual formula for phi in textbooks is just a simplified version of the Pearson product-moment correlation as is the point-biserial; both likely developed to speed hand calculation before the days of statistical software).
My intrigue is, that spss algorithm documentation for the crosstabs subprogram clearly identifies phi as the square root of (chi-square / N), which can't be negative. So, RG readers, that's just another case of a sloppy job of IBM SPSS documentation of what goes on in their packages. As I've never seen a Cramer's V formula other than square root of: chi-square / (N * MD), where MD is minimum table dimension minus 1, it can't be negative.
Good luck to all during these challenging times.
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