## Wednesday, October 06, 2010

### Correlation

NOTE: I have edited and reorganized some of my writings on correlation to present the information more coherently (10/11/2012).

Our next topic is correlational analysis. There are four major areas to address:

1. A general introduction to correlation, which is available here amidst my Research Methods lecture notes.

2. Running correlations in SPSS. This graphic of SPSS output tries to make clear that a sample correlation and its significance/probability level are two different things (although related to each other).

Second, in graphing the data points and best-fitting line, you start in "Graphs," go to "Legacy Dialogs," and select "Scatter/Dot." Then, select "Simple Scatter" and click on "Define." You will then insert the variables you want to display on the X and Y axes, and say "OK." When the scatter plot first appears, you can click on it to do more editing. To add the best-fit line, under "Elements," choose "Fit Line at Total."

Initially the dots will all look the same throughout the scatter plot. To make each dot represent the number of cases at that point (either by thickness of the dot or through color-coding), click on the "Binning" icon (circled below in red). Thanks to Xiaohui for finding this!

3. Statistical significance and testing the null hypothesis, as applied to correlation. Subthemes within this topic include how sample size affects the ease of getting a statistically significant result (i.e., rejecting the null hypothesis of zero correlation in the full population), and one- vs. two-tailed significance.

4. Partial correlation (i.e., the correlation between two variables, holding constant one or more "lurking" variables).

5. In evaluating the meaning of a correlation that appears as positive or negative in the SPSS output, you must know how each of the variables is keyed (i.e., does a high score reflect more of the behavior or less of the behavior?).

6. Statistical significance is not necessarily indicative of social importance. With really large sample sizes (such as we have available in the GSS), even a correlation that seems only modestly different from zero may be statistically significant. To remedy this situation, the late statistician Jacob Cohen devised criteria for "small," "medium," and "large" correlations.

7. Correlations should also be interpreted in the context of range restriction (see links section on the right). Here's a song to reinforce the ideas:

Restriction in the Range
Lyrics by Alan Reifman
(May be sung to the tune of “Laughter in the Rain,” Sedaka/Cody)

Why do you get such a small correlation,
With variables you think should be related?
Seems you’re not studying the full human spectrum,
Just looking at part of bivariate space,
All kinds of thoughts start to race, through your mind…

Ooh, there’s restriction in the range,
Dampening the slope of the best-fit line,
Ooh, I can correct r for this,
Put a better rho estimate in its place...