Tricky Correlations

Dear Students,

Last night, I received a message from someone in our class who was a bit confused about my comment earlier today that "the meaning of a phi coefficient is dependent on how the 2 variables are coded." I'm now sending everyone the response I prepared and sent to the person who asked about phi, for I suspect many of you did not understand fully why it's a tricky task to interpret someone else's phi coefficient. I hope the following explanation helps clarify the point I was trying to make in class yesterday.

Suppose I want to correlate gender (male/female) with pet ownership (yes/no) for a group of 5 people. Also suppose I code my data in this manner: for the gender variable, 1 = male and 0 = female; for the pet ownership variable, 1 = yes, 0 = no.

Here are the 2 scores for each of my 5 people (with the gender score coming first and the pet ownership score coming second):

  • Tammy: 0,1
  • Teddy: 1,0
  • Terry: 0,1
  • Tippi: 0,1
  • Timmy: 1,0

Note that there is a "high-low, low-high" relationship in the data. Zeros are paired with 1s; 1s are paired with 0s. The phi correlation, for these data, turns out equal to -1.00. This is, of course, a perfect negative correlation. It turns out that way because we have the strongest possible inverse relationship between the gender and pet ownership variables.

But now I want you to go back and imagine that I coded the data differently. To be more specific, I want you to imagine that I decide that gender will be coded such that 1 = female and 0 = male. (I'll keep pet ownership as it was: 1 = yes, 0 = no.)

Now the data for the same 5 people would look like this:

  • Tammy: 1,1
  • Teddy: 0,0
  • Terry: 1,1
  • Tippi: 1,1
  • Timmy: 0,0

Here, the relationship is clearly a "high-high, low-how" relationship. In fact, phi now turns out equal to +1.00, a perfect positive correlation. It turns out like this because we've got the strongest possible direct relationship between gender and pet ownership.

Now, think about what we've just seen. We started with 5 people and measured them in terms of gender and pet ownership. Both times, we wanted to know the relationship between these 2 variables. Both times, we measured the relationship by means of the "phi" procedure (because we had 2 dichotomous variables, with each being a "true" dichotomy). However, the phi coefficient turned out equal to -1.00 the first time but +1.00 the second time. How could this happen, since both times we measured the same 5 people?

As I hope you can see, I ended up with different values for phi because my coding of gender was different. When I computed phi the first time, I arbitrarily decided to code males = 1 and females = 0. I'm free to code gender the other way, and that's exactly what I did the second time I computed phi.

The "moral of the story" is simply this: You can't interpret a phi coefficient unless you know how the researcher coded the 2 variables. For example, if I told you that I used phi to correlate handedness (right/left) with political affiliation (Republican/Democrat) and found phi to be +.80, you would know that there was a fairly strong "high-high, low-low" relationship between my 2 sets of scores. But would that mean Repulicans tended to be right-handed while Democrats tended to be left-handed, OR would the phi of +.80 mean that Republicans tended to be left-handed while Democrats tended to be right-handed? To know which of these interpretations is correct, you'd need to know how I coded my data!

I'm sorry this message is so long. However, I hope it has helped you understand why you must exercise care when interpreting phi coefficients. Of course, the same care must be exercised when dealing with point biserial, biserial, and tetrachoric correlations; that's the case because each of these, like phi, involves an arbitrary coding of at least one of the 2 variables.

Sky Huck

Copyright © 2012

Schuyler W. Huck
All rights reserved.

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