Relationship Strength
Overview
Statistical relationships between variables can be strong or weak. Imagine, for example, that there is a statistical relationship between whether or not people own pets and how stressed they are; pet owners are less stressed on average than non-owners. (Note that this is a D relationship.) We can still ask whether this is a strong relationship (pet owners are a lot less stressed) or a weak one (pet owners are just a bit less stressed.) Similarly, imagine that there is a relationship between how many hours students studied and the score they got on an exam (an R relationship). Again, we can still ask whether this is a strong relationship (people who studied more tended to do a lot better) or a weak one (people who studied more tended to do just a bit better).
Relationship Strength for D Relationships
There are different ways of measuring the strength of a D relationship, but one commonly used statistic is the standardized mean difference, or d. To get the standardized mean difference, simply find the difference between the two group means and divide this difference by the standard deviation. For example, imagine that pet owners have a mean stress score of 13 and non-owners a mean stress score of 15. Imagine further that the standard deviation in both groups is 4. Then d is the difference between the two means, 2, divided by the standard deviation, 4. So d = 2 / 4 = 0.50.
Of course, in most situations, the standard deviations of the two groups are not exactly the same. So which one should we use? Different researchers have answered this question in different ways. What we will do, however, is use the mean of the two group standard deviations. For example, if the two standard deviations above had been 4 and 3, then we would have divided by a standard deviation of 3.5. Although most researchers use a somewhat more complicated approach to combine the two standard deviations, this approach works pretty well and is quite simple. In fact, we can use it to make quick estimates of d in our heads. For example, imagine that one group of students has a mean exam score of 75.88 and another group has a mean exam score of 82.35. Imagine further that the two standard deviations are 12.35 and 16.22. This means that the difference between the means is about 7 and the mean of the two standard deviations is about 14. So again, d is about 1/2 or 0.50.
One way to think about d is that it is a measure of how far apart the two means are in “standard deviation units.” In the pets-stress example above, the two means are 2 units apart, but one standard deviation is 4 units. So the two means are half a standard deviation apart. If the two group means were 13 and 17, then they would be a full standard deviation apart: d = 1.00.
Relationship Strength | Cohen's d | Pearson's r |
Large | 0.80 | .50 |
Medium | 0.50 | .30 |
Small | 0.20 | .10 |
The psychologist Jacob Cohen popularized this measure of relationship strength for D relationships, which is why it is often called Cohen’s d. But he also suggested some rough guidelines for determining whether a value of d represents a small, medium, or large effect for psychological research in many areas. According to Cohen, d = 0.20 can generally be considered a small effect, d = 0.50 can generally be considered a medium effect, and d = 0.80 can generally be considered a large effect.
Relationship Strength for R Relationships
The most common statistic used to measure the strength of R relationships is Pearson’s r. Pearson’s r is just one example of a correlation coefficient, but when researchers refer to “the correlation coefficient,” they almost always mean Pearson’s r. One thing to remember about Pearson’s r, however, is that it is a measure of the strength of a linear relationship. It is not appropriate for describing relationships that are non-linear. There are methods for describing the strength of non-linear relationships, but they will have to wait until you take intermediate statistics.
Although we need not worry about computing Pearson’s r at this point, we still need to understand it thoroughly. Pearson’s r is always a number that ranges from -1.00 to + 1.00. A negative value of r indicates a negative relationship and a positive value of r indicates a positive relationship. A value of zero indicates no relationship, and as r moves away from zero in either direction it indicates a stronger relationship. When it reaches either -1.00 or +1.00, it indicates a relationship that is as strong as an R relationship can be. Correlations of -1.00 and +1.00 are also called perfect negative and perfect positive relationships. Again, the strength of an R relationship is indicated by how far r is above or below zero. So, for example, r values of -.70 and +.70 represent equally strong relationships. The difference is only that one is a negative relationship and the other is a positive relationship.
Cohen also provided guidelines for determining whether a value of r represents a small, medium, or large effect in psychological research. According to him, r = .10 can be considered a small effect, r = .30 can be considered a medium effect, and r = .50 can be considered a large effect. Be careful not to confuse the guidelines for r with the guidelines for d. It would be useful to commit the information in the table above to memory.
For a better understanding of Pearson's r, check out the following Web sites:
http://stat-www.berkeley.edu/users/stark/SticiGui/Text/ch4.1.htm http://bcs.whfreeman.com/ips4e/cat_010/applets/CorrelationRegression.html The first allows you to specify a sample size and a value of Pearson's r, and then it creates a scatterplot that is consistent with those values. The second allows you to add data points to a generic scatterplot, and it computes Pearson's r for you. You can also drag the points around to see how Pearson's r changes. |