Measures of Association and Correlation
Proportionate Reduction of Error (PRE) is the logical foundation of determining measures of association. For example, suppose that you were told that there were a 100 people in a room and each person would leave individually. You are asked to guess whether the person is Jewish. How would you make your decision? Logically, you would think about the proportion of Jews to the population of people in the community. If you know that Jews are a minority subgroup in the community, what would make the best guess for each and every person leaving that room? You would probably choose “not Jewish”. You will have some errors, but most of the time you would be correct. Now, is there additional information that would help you improve your prediction? What would happen if you knew that the room was in the temple? Would you change your prediction? If this information improves your predictions (and correspondingly reduces your mistakes, or “proportionately reduces your error”), that is information that you want to know. This is the logic behind the measurements of association.
How do you measure association?
Lambda is a measure of association for nominal variables. Lambda ranges from 0.00 to 1.00. A lambda of 0.00 reflects no association between variables (perhaps you wondered if there is a relationship between a respondent having a dog as a child and his/her grade point average). A Lambda of 1.00 is a perfect association (perhaps you questioned the relationship between gender and pregnancy). Lambda does not give you a direction of association: it simply suggests an association between two variables and its strength.
Gamma is a measure of association for ordinal variables. Gamma ranges from -1.00 to 1.00. Again, a Gamma of 0.00 reflects no association; a Gamma of 1.00 reflects a positive perfect relationship between variables; a Gamma of -1.00 reflects a negative perfect relationship between those variables.
Pearson’s r is a measure of association for continuous variables. Like Gamma, Pearson’s r ranges from -1.00 to 1.00.
If you have differing levels of measures, always use the measure of association of the lowest level of measurement. For example, if you are analyzing a nominal and ordinal variable, use lambda. If you are examining an ordinal and scale pair, use gamma.
It is easy to calculate lambda and gamma using SPSS. Go to Analyze, Descriptive Statistics, Crosstabs. Enter your dependent variable in the “row “and the independent variable in the “column” box. Using the GSS 2008 (1500 cases) database, we can test for the association of the independent variable “SEX” and the dependent variable “Happy”.
Click Statistics. Since we are looking at a nominal and an ordinal variable, we will use lambda. Your screen should look like this:
Click Continue, then OK. Your output will look like this:
Look under the Value column to the dependent variable “General Happiness”. The lambda value is .000, suggesting that there is no association between the variable “SEX” and the dependent variable “Happy”.
To calculate gamma, use the same procedure as you did for lambda, but click “Gamma” instead.
Let’s examine the association between the dependent ordinal variable “Happy” and “Health”, the independent ordinal variable that measures the condition of health (excellent, good, fair poor). In addition to your crosstabs, you will have output that looks like this:
Look at the Gamma in the “Value” column: .385. Note that this positive number tells you that people are generally happier when they are healthier. The value of .385 also suggests that there is a strong association between these two variables.
To calculate Pearson’s r, go to Analyze, Correlate, Bivariate. Enter your two variables. For example, we can examine the correlation between two continuous variables, “Age” and “TVhours” (the number of tv viewing hours per day). Your screen should look like this:
Click OK. Look in your output for the following:
Note that the Pearson’s r value for comparing age to age is 1, suggesting perfect correlation. If you think about this, that makes logical sense. What you are truly interested in examining is the Pearson’s r value of the 2 different variables (in this case, the value is .139). This suggests that someone ages, they watch more television.
Here are guidelines for interpreting the strength of association for Lambda, Gamma, and Pearson’s r (remember, lambda can only have a positive value):
| Strength of Association | Value of Lambda, Gamma, Pearson’s r |
| None | 0.00 |
| Weak association | + .01- .09 |
| Moderate Association | + .10 – .29 |
| Evident of strong association | + .30 – .99 |
| Perfect association, strongest possible | + 1.00 |
Please note: If you need to request accommodations with content linked to on this guide or with your SPSS Software for your coursework, on the basis of a disability, please contact Accessibility Resources and Services by emailing them. Requests for accommodations should be submitted as early as possible to allow for sufficient planning. If you have questions, please visit the Accessibility Resources and Services website.
SPSS eTutor by Dee Britton is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
