Alternative Correlational Techniques

11/8/01

Announcements:

• Hand back any papers
• Today is really more of a lecture than a lab, but we have a lot of stuff that I think we should cover, and this is the most efficient way to do so.

Alternative Techniques:

• Much of what I am talking about today is just an extension of what students already know from Pearson correlation.
• Part of the idea is that we have a whole set of coefficients that are just Pearson coefficients in unusual places.
• Most of what I will say applies to correlation much more than to regression, although I will discuss a bit about regression.

Point-biserial correlation

• Biserial correlations are correlations when one of the variables (either the dependent variable or the independent variable) is a dichotomy.
• When we call it a "point"-biserial we are speaking about a case where we treat the dichotomous variable as a true dichotomy.
  • By this I mean that we don't imagine some underlying continuity along that variable that we want to take into account.
  • Even when there is some underlying continuity, we still ignore it in most situations.
• To calculate the correlation, we just apply the standard Pearson formula. There is nothing new here.
  
• Example--I'm going back to the study by Jo Epping on cancer patients.
  

Jo Epping’s practicum presentation looked at cancer data. There is an argument in the literature that psychological factors play a role in survival from cancer. One important variable seems to be "avoidance" Patients with a high incidence of reported avoidance is believed to have a poorer outcome. The data are available at Jo.sav and Jo.dat.

She divided her subjects into two groups. Group 1 was classed as a success because those subjects were cancer free at follow–up. Group 2 were classed as Failed because they were either not cancer free or had died at follow–up.

The dependent variable is the reported level of avoidance at Time 1

Have them run this test using SPSS

Data

Group	1 (Success)	2 (Fail)
Mean	14.41	17.00
s.d.	21.247	18.706
ni	49	18

Pooled estimates





Here we will reject the null hypothesis that the two groups show equal levels of avoidance—The exact probability under H₀ =is .0434.

A good psychologist now would go out and compute a measure of the magnitude of the difference. I might suggest Cohen's d.

But suppose that we treated this as a problem in correlation. 

Correlational Approach

• The Outcome variable is coded 1 and 2, and we could correlate that with Avoidance.
• I'm going to run this with the regression procedure rather than the correlation procedure, just because it gives me more information to play with.
• That means I have to decide which is the dependent variable and which is the independent variable. Since survival comes later in time than avoidance, it makes sense to make that the dependent variable.
• Have them do this using SPSS
• SPSS printout:

• Notice that the correlation is .248. The test on the slope is also a test on the correlation (with one predictor), and it shows that the correlation has a t of 2.061 and a p = .043, which is exactly what we found with the t test.
• First of all, this shows us that we are really asking the same kind of question.
• Second, note the slope = .0237. It tells us that a one unit change in avoidance leads to a .0237 unit change in survival.
  • The beta translates this to standard deviation units, which is more meaningful.

Magnitude of effect

• When we talked about correlation, we talked about r² as the percentage of variability in one variable accounted for by variability in another.
• Our r² here is .06, which says that about 6% of the variability in survivability is accounted for by variability in avoidance.
  • that's not huge, but it's something.
• But this really tells us that any 2-sample t can be converted to an r² type measure.
  • We could either do this by rerunning as a correlation, or by a simple formula that does the same thing.
  • 
  • If we just want to have a whole matrix of correlations (i.e. our interest isn't really in predicting), then the correlation we want is the point-biserial when we have a dichotomous predictor.
  • But keep in mind that the way we calculate the point-biserial is just to apply the plain old Pearson formula.

Logistic Regression

• I'll talk about this next semester, but not now.
• I just want to say that logistic regression is a better way of predicting a dichotomy, though it is not much better unless the odds of survival get very low or very high for some values of the predictor.

Phi Coefficient

• We have just worked with a single dichotomous variable. What happens when we have two dichotomous variables?
• If we don't assume (or want to work with) any underlying continuity, the coefficient we want is the phi coefficient.
• This coefficient is also calculated by using the standard Pearson formula.
  
• Example
• I'm taking an example from the chi-square lecture.
  • Friedman, Katcher, Lynch, and Thomas (1980) did an interesting study on the effect of having a pet for people recovering from heart attacks. I don't recall whether they supplied a pet or just found people who did, and did not, have pets.
    • We looked at this example when we looked at chi-square.
    • What difference would this make from a methodological perspective?
  • They found 92 people who had recently had a heart attack, and classified them in terms of whether or not they had a pet. They then determined whether these people were alive one year later.
    • Here we have two variables of classification:
      • Pet (yes/no)
      • Alive/Dead
      • We want to test the null hypothesis that Pet and Survival are independent.
        

• But, these data could be entered into a standard data file as two variables, each coded 1 and 2 (or 0 and 1, or whatever).
• We could then use the correlation (or regression) procedure to obtain a correlation and a test on that correlation.
  • Here we would call it phi (f) just to indicate that it is calculated on dichotomous data.
• SPSS printout
  

• If I ran this as a chisquare, I'd have

• Notice that the probability levels are exactly the same for r and chi-square.
• We can go from r to chi-square by

• But this tells us that we can also get a "percentage of variance" measure from chi-square, just by converting to phi and squaring.
  • In this case, phi-squared = .31² = .09.

   

Biserial and Tetrachoric coefficients

• These are coefficients that we don't seem to use anymore, so I'm not going to give formulae.
• If we believed in an underlying continuity, and if we wanted to make use of it, we could "adjust" the coefficients to what they would most likely be if each variable were normally distributed instead of a dichotomy.
• Students should know what these are, but that's all.

Rank Correlations

• There are two general kinds of correlations calculated with ranked data.
• Spearman's rho
  • Just rank each set of data from lowest to highest and apply Pearson's formula to the ranks.
  • Discuss why you might want to do this.
    • Particularly useful with outliers or where you have absolutely no faith in the underlying metric.
• Kendall's tau
  • This statistic is not a Pearson r.
  • It is generally thought of as a better statistic for ranked data than Spearman's.
  • It works on "concordances," which refer to the ordering of data.
    • If you take the numbers 8, 12, and 9, 12 is greater than 8,  and 9 is greater than 8, so those two comparisons are "concordant." but 12 is less than 9, so that pair is "discordant."
    • The simplest idea is to rank that data with respect to one of the variables (.e.g. X) and then count the number of concordances in the other variable (Y).
    • If the data line up perfectly, then the number of concordant pairs will equal the number of pairs (n(n-1)/2)
    • Kendall gives a formula for tau that you see in the book. It is set up to give values between +1 and -1, just as for standard correlations. That formula uses "inversions", which are just the opposite of concordances. That simply makes the sign come out right.
    • I won't spend more time on it here, but I do recommend tau whenever your data has some heavy-duty outliers that can screw up a standard correlation.

Kendall's Coefficient of Concordance (W)

• I have never actually used this statistic, but I have recommended it to people.
• It is a good way of obtaining a measure of the degree to which several judges agree on their ratings.
• I give the formula in the text, and say that W comes out as an average Spearman correlation between pairs of judge's ratings. I also give a significance test, should you need that.
• This is really all I want to say on that topic.

Ordinal chi-square

In the book I give a pretty limp suggestion of what to do when you have a chi-square contingency table and actually have an ordinal prediction. I do a much better job at the following web site.

• http://www.uvm.edu/~dhowell/StatPages/More_Stuff/OrdinalChisq/OrdinalChiSq.html

Last revised: 11/05/01