Thursday, October 02, 2008

Hypothesis Testing with Correlations

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

The correlation statistic presents the first instance in which we'll be examining statistical significance (here and here). The question is whether we can reject the null hypothesis (Ho) that the correlation between a given pair of variables in the full population is zero (RHO = 0).

We, of course, obtain correlations (r) for our sample, and then see if our sample correlation is sufficiently different from zero (in either a positive or negative direction) so that it would have been sufficiently unlikely to have arisen from pure chance when the population RHO was truly zero. That's what we mean by statistical significance. When we achieve statistical significance, we can reject the Ho of zero RHO.

In order to have a statistically significant correlation, the correlation (r) itself should be appreciably different from zero, either above zero (a positive correlation) or below zero (a negative correlation).

Also, in order for the correlation to be significant, the significance (or probability) level displayed for a given correlation in your SPSS output must be very small (p < .05, or if the probability is even smaller, you can use one of the other conventional cut-off points, p < .01 or p < .001). Any time the probability p is larger than .05, the correlation is nonsignificant (in my opinion, if you get a correlation with a p level of .06 or .07, it's OK to note in your report that the correlation narrowly missed being significant under conventional standards).

Suppose you find that the correlation between two variables is r = .30, p < .01. This is telling us that, if the null hypothesis (Ho) is true -- that is, there truly is no correlation in the population from which the sample was drawn (rho = 0, where rho looks like a curvy capital P) -- then it would be extremely unlikely (p < .01) for a correlation of .30 to crop up purely by chance when the correlation throughout the society is truly zero.

[Here's a figure I've added in October 2007, to convey the idea of there truly being no correlation in a large population, but a correlation occurring in one's sample purely by random sampling error:

This web document is also helpful. The opposite problem, where the full population truly has a correlation, but you draw a sample that fails to show it, will be discussed later in the course.]

A significant correlation in our sample thus allows us to reject the null hypothesis and assert, based on an inference from our sample, that there is a correlation in the population. Again, note the inference from sample to population.

The essence of scientific hypothesis testing can thus be distilled to three steps:

1. State the null hypothesis (Ho) that there is no correlation between your two variables in the population (rho = 0). The investigator probably doesn't believe Ho (generally, we seek to uncover significant relationships), but Ho is part of the scientific protocol.

2. Obtain the sample correlation (r) between your two variables and the associated significance (p) level.

3. If the correlation is statistically significant (r is well above or well below zero, and p < .05), reject Ho. If the correlation is nonsignificant (r is close to zero and p is larger than .05), then the null hypothesis that there is no correlation between your two variables in the population must be maintained. We never accept the truth of the null hypothesis for certain; we just say it cannot be rejected.

I've stated above that, in order to be significant, a correlation (r) needed to be well above or well below zero. That's not always true, however. As we saw in some of our SPSS illustrations, with a very large sample size (n = 1,000 or more), a correlation does not necessarily need to be that far from zero to be significant. If a correlation appears to be small, yet is listed in the output as being significant (probably only due to the large sample size), you can say the correlation was "significant, though weak." A Wikipedia document on correlation (which I've just added to the links section on the right) displays guidelines developed by the late statistician Jacob Cohen for labeling correlations as "small, medium, and large."

Two concepts we will be taking up later in the course, statistical power and confidence intervals, will elaborate upon the issue of small correlations sometimes being significant and, conversely, relatively large correlations not being significant.