I have two binary variables, naturally dichotomous, yes no They present information on whether a college has multi-stream recycling or single-stream recycling. I did Pearson's product-moment correlation as I read in a research paper that naturally dichotomous variables can be used in Pearson's correlation. I also did Phi coefficient correlation to see the comparison. I found that the value of association is the same, but the signs have reversed from negative in Pearson's to positive in phi. I am not able to understand why the signs changed.

Also, additional information, the variables are not exclusive; there are some colleges that have both types of recycling streams. Will that cause a problem in the analysis? Should I re-code these variables?


1 Answer 1


I think this has to do with how the phi ($\phi$) is calculated and reported by the software you use. Let's use an example and assume that we have a small data set with two dichotomous variables (0=no, 1=yes):

ss <- c(0,0,1,0,0,1,1,0,1,0) 
ms <- c(1,1,0,1,0,1,0,1,0,1) 

And we want to calculate the Pearson's correlation ($r$) using this formula:

$$r = \frac{\sum(x_i - \bar x)(y_i - \bar y)}{\sqrt{\sum(x_i-\bar x)^2 \sum(y_i-\bar y)^2}}$$

You can plug the numbers into the formula or simply run the function cor in R:

cor(ss, ms)

As in your case, we have a negative $r$. Now, we can calculate $\phi$ for $2 \times 2$ contingency table using our two variables:

|     | ms |     |
| ss  | no | yes |
| no  | 1  | 5   |
| yes | 3  | 1   |

We can use the formula on this wikipedia page for $\phi$ coefficient. Let's assign letters to make it simple, and then plug the numbers:

|     | ms |     |
| ss  | no | yes |
| no  | A  | B   |
| yes | C  | D   |

$$\phi = \frac{AD-BC}{\sqrt{(A+B)(C+D)(A+C)(B+D)}} = \frac{1-15}{\sqrt{6 \times 4 \times 4 \times 6}} = -0.5833333$$

Exactly the same result. So, why do you have a positive value then? If you google around, you can see a different formula for $\phi$:

$$\phi = \sqrt{\frac{\chi^2}{N}}$$

In fact, this is the formula of Cramér's V (sometimes called Cramér's phi, $\phi_c$) simplified for $2 \times 2$ contingency tables. These two measures are related: "In the case of a $2 \times 2$ contingency table Cramér's V is equal to the absolute value of Phi coefficient." We can check this using our small data, where $\chi^2 \approx 3.403$

$$\phi_c = \sqrt{\frac{3.403}{10}} \approx 0.583 $$

This could explain why you observe a reversal of sign.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.