Monday, July 19, 2010

Measures of Relationships

As educators and as scientists, there are numerous times when we wish to know the relationship between one thing and another; this is often a means for us to gain knowledge about what we do and perhaps how well we do it. We could, for example, look at National Board scores, and perhaps we could assume that it is doing just what we want it to in terms of understanding how well we all teach. Of course, this could be an incorrect assumption as well. But if we look at the test as a means of teaching effectiveness, there are ways to check up on it; we can attempt to correlate how well the scores relate to some criteria we select as indicative of teaching effectiveness. What we need, of course, is some sort of an index of relationship- “a number that when low indicates a low degree and when high a high degree of relationship between two variables.” (1) This is the coefficient of correlation.

Phillips suggests the following thought experiment: Consider that we are interested in two variables, height and weight, of toy soldiers. Imagine that these soldiers are all the same shape, but that they differ in size, and thus also in weight. Imagine that we graph a series of toy soldiers, and we see that the smaller soldiers all weight less than the taller soldiers; the smaller ones are lighter and the larger ones are heavier. But what is the exact relationship here? Is it strong? Is it directly proportional? Is it perfect? If you had to put this relationship on a scale of 0 to 1.0, where would you place it, if 0 equals no relationship and 1.0 indicates a perfectly linear relationship (that is, a solder twice as large as a smaller one weights twice as much as it)?

A coefficient of correlation does just this; it places the relationship on a scale from 0-1.0, but in addition to providing you information about the strength of the relationship, it also gives you information about direction (i.e., positive or negative). There are a number of coefficients that you might see. Phillips states that the Rank-Difference Coefficient (Spearman rho) is the easiest to comprehend, while the Product-Moment Correlation (Pearson’s r) is the most useful and most frequently used. Spearman’s rho is used with ordinal data, while the Product-Moment correlation requires interval data; to put this another way, the former does not allow for the precision of the latter. Spearman’s test requires you to rank the order of the variable from smallest to largest, to then find the differences between ranks and square them, and use that information in the formula. In this test, as in the other, the size of a coefficient is independent of its direction; that is, a correlation of .75 is the same strength as a correlation of -.75. In Pearson’s test, we use the ordinary interval scores that tell us how far apart the subjects are on each variable. The product moment part of the name comes from the way in which it is calculated, by summing up the products of the deviations of the scores from the mean (2).

We do not need to concern ourselves with how these are actually calculated. We need but know that these measures demonstrate the strength of a relationship and its direction, from 0 indicating no real relationship to 1.0 or -1.0 indicating a perfect relationship, in linear fashion. This can help us with interpreting studies which demonstrate the relationships between variables that we are interested in.

1. Phillips JL, Jr. How to think about statistics. New York, NY; WH Freeman & Company;1998;44
2., accessed July 19, 2010

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