The binary correlation coefficient formula
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Suppose we want to test whether correlations computed on two different samples NOT 2 correlations from the same sample are equal. We want to know whether these are significantly different, that is, H 0: Also note, we would probably be testing for differences in regression slopes if this were for real. How to do so will be covered later. And, yeah, it's phrased as accepting the null hypothesis, which would cause some people to quibble with the diction.
There will be times, however, when you want to test for differences in independent correlations rather than regressions, and this is how to do it.
It is also possible to test whether multiple independent r s are equal. How to do so is described in Hays and other statistics texts. More than two independent correlations. Suppose we want to test the hypothesis that three or more correlations from independent samples share the same population value. First we have to estimate r, the common population value. To do this, we will compute the average across the studies. But first we will use Fisther's r to z transformation. If the studies differ in their sample sizes, then we will also need to compute a weighed average so that the studies with the larger samples get more weight in the average than do the smaller studies.
We use n -3 instead of n because of the z transformation. Now we need to know how far the individual studies are from the average. To get a handle on this, we compute: Q is just the sum of squared deviations from the mean where the squared deviations are weighted by their sample sizes less 3.
It turns out that when the null hypothesis that all k studies share a single underlying value of r is true , Q is distributed as chi-square with k -1 degrees of freedom. So we can compute Q and compare the result to a tabled value of chi-square to test for significance. If Q is large compared to the tabled value of chi-square, then we can reject the null hypothesis that the study correlations were drawn from a common population with a single value of r.
For example, suppose we are doing test validation work. We have computed the correlation between bank tellers' performance on a video teller test with customer service ratings furnished by actors in our employ who portray customers with banking problems to the tellers and then evaluate the unsuspecting tellers on their customer service skills.
Suppose we have completed studies at three different banks, and we want to test whether the correlation between the test and customer service ratings is the same or different across banks. Suppose we found the following results: The three studies' outcomes or sample r s were. The study sample sizes are shown under the column labeled n. We transform r to z in the next column, which is labeled z. To find the weighted average, we first multiply each z by n -3 , the study sample size less three.
When we add the weighted z s, we find the sum is Our weighted average, zbar, is To find the value of Q , we subtract zbar from each z and square z-zbar 2. Finally, we weight each squared term by its sample size less three. The weighted, squared entries are shown in the final column of the table. The sum of the entries in the rightmost column of the table is our value of Q , It appears that the correlation between test scores and customer services is larger at some banks than at others.
Testing for equality of dependent correlations. Suppose you want to test whether two correlations taken from the same sample are equal, that is, you want to test the equality of two elements from the same correlation matrix. There is a literature on this, and there are several available tests. Note that in the first case, there is one variable in common, and that in the second case, there is no variable in common.
An example of the first type would be a test of whether the correlation between cognitive ability scores and performance in a class on U. An example of the latter would be whether the correlation between scores on a cognitive ability test and performance on a class test in psychology is equal to the correlation between personality test scores and scores on an interview used to determine a person's suitability for personal counseling as a therapist whew, what a long sentence.
Both types of test are described by Steiger , Psychological Bulletin. For the following hypothesis,. Bobko recommends the "Hotelling-Williams test. For example, suppose we have administered a final examination in a statistics class 1. In addition, we have two other measures. One of the measures is the SAT Quantitative score 2 , and the other is self-efficacy questionnaire 3 that asks each student how proficient they think they are at various kinds of stats problems.
We want to know whether the final exam score is more highly correlated with the SAT-Q or the self-efficacy scale. That is, we want to know whether. Suppose that we have students in our class. Further suppose that r 12 the correlation between the exam and the SAT is. If we find the critical value for t with 98 degrees of freedom we find the value 1. We can therefore reject the null that the correlations are equal in the population. Self efficacy is more highly correlated with our exam score than was the SAT-Q.
I just finished a study in which we administered a self-efficacy questionnaire how good are you at problems involving the mean, median or mode? We would expect both self-efficacy and exam performance to increase with instruction. We also expect that students will become increasingly well calibrated over the course of instruction. That is, we expect them to get a better handle on how good they actually are during the semester as they receive feedback on their work. Suppose we have students, and our results look like this:.
We want to test whether the correlations. Here z 12 and z 34 refer to the Fisher transformed values of r , the two correlations. The value in the denominator requires some explanation. In our example, rbar is. We will substitute the value of. Carrying out a bit of arithmetic, we find that. The transformations of our correlations to z give us.
Our test, then, is. Therefore, we can reject the null and conclude that the correlations are different. The students became better calibrated over the course of instruction.
The graph above shows unrestricted data in which both variables are continuous and approximately normally distributed. Here we have deleted the observations between 1 and 3. Note how the resulting r is larger than the original. This says that the correlation between observed measured variables is equal to the correlation between their true scores, that is, the scores that would be observed with no error times the square root of the product of the two reliabilities, one reliability for each variable.
For example, suppose we have a correlation between true scores of. The reliability of X is. Then the observed correlation will be. It sometimes happens that we have an observed correlation, estimates of the reliabilities of x and y, and we want to show what the correlation would be if we had measures without error.
This is sometimes called a correlation that is disattenuated for reliability. The formula to do this is a simple transformation of the formula above:. For example, if we had an observed correlation of. The formula for r is based on the assumption of a bivariate normal distribution. This means that both distributions are normal, and that if you choose any interval of X, the conditional values of Y will still be normal, and if you choose any values of Y, the conditional values of X will be normal.
As long as both variables are fairly continuous and more or less Normal, r will provide a good summary of the linear relations in the data. However, when one or both variables have a bad shape or have only 2 or 3 values, the range of r becomes restricted. For example, if one variable is binary and the other is continuous, r can virtually never reach a value of 1. Its maximum value occurs when about half of the binary values are zero and half are 1, which is also where the maximum variance for the binary measure occurs.
The maximum value of r decreases as the proportion of 1s in the binary variable departs from. For example, let's look at a few numbers. Note that Y2 and Y3 are simple transformations of Y1 that make it binary. Y2 transforms Y1 so that values of 5 or less are zero, while values greater than 5 are 1. Y3 transforms Y1 so that values 8 or less are zero. Note also that when two things correlate 1.
Also note that when you correlate two binary variable Y2 and Y3 , the correlation is even lower. There are variants of the correlation coefficient that try to correct for this problem the biserial and tetrachoric , but there are problems with these.
We often talk about experimental ANOVA and correlational results as if we can infer cause with one and not with the other.
Actually, the statistical analysis has little or nothing to do with inferring causality. The major thing of importance in inferring causality is the way in which data are collected, that is, what matters is the design and not the analysis.