Which correlation we have to use for two dichotomous variables?
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Differences and Relationships
The correlation coefficient is a measure of how two variables are related. t-tests examine how two groups are different. What if I told you these two types of questions are really the same question?
Examine the following histogram. This histogram shows a large difference in the dependent variable, y, between the groups No X and X. The mean is much higher for the X than the No X group.

This is an example in which X has two groups. Now imagine that there are several values (levels or groups) for variable X: No X, a little X, some X, a lot of X, all of X. For instance, what if instead of having a study with no drug vs. drug, there is a study in which several levels of dosage used (e.g., 0mg, 5mg, 10mg, etc). Then there would be several intermittent values of X. The following graph shows one potential outcome of the study in which there are several levels of X.

In this second graph, we see that as the value of X increases, the value of Y increases. That is the definition of a relationship. In other words, there is a correlation between X and Y.
The same holds true for the first figure in which there were only two values of X. As X increases, Y increases. So both graphs demonstrate that there is a relationship between X and Y. At the same time, we can see that both graphs demonstrate that the values of Y differ for the values of X. So, a difference between values of X is the same as a relationship between X and Y.
Point-biserial Correlation
A point-biserial correlation is simply the correlation between one dichotmous variable and one continuous variable. It turns out that this is a special case of the Pearson correlation. So computing the special point-biserial correlation is equivalent to computing the Pearson correlation when one variable is dichotmous and the other is continuous.
Because we know that differences are the same as correlations now, we can show that the t-test for two groups is the same as the correlation between the grouping or independent variable (X) and the dependent variable (Y). Let's go back to our last t-test example and check this out. In this example, we were comparing first and last siblings to see who was more introverted.
First or Last Sibling
Introversion
1
65
1
48
1
63
1
52
1
61
1
53
1
63
1
70
1
65
1
66
2
61
2
42
2
66
2
52
2
47
2
58
2
65
2
62
2
64
2
69
The correlation coefficient is a measure of how two variables are related. t-tests examine how two groups are different. What if I told you these two types of questions are really the same question?
Examine the following histogram. This histogram shows a large difference in the dependent variable, y, between the groups No X and X. The mean is much higher for the X than the No X group.

This is an example in which X has two groups. Now imagine that there are several values (levels or groups) for variable X: No X, a little X, some X, a lot of X, all of X. For instance, what if instead of having a study with no drug vs. drug, there is a study in which several levels of dosage used (e.g., 0mg, 5mg, 10mg, etc). Then there would be several intermittent values of X. The following graph shows one potential outcome of the study in which there are several levels of X.

In this second graph, we see that as the value of X increases, the value of Y increases. That is the definition of a relationship. In other words, there is a correlation between X and Y.
The same holds true for the first figure in which there were only two values of X. As X increases, Y increases. So both graphs demonstrate that there is a relationship between X and Y. At the same time, we can see that both graphs demonstrate that the values of Y differ for the values of X. So, a difference between values of X is the same as a relationship between X and Y.
Point-biserial Correlation
A point-biserial correlation is simply the correlation between one dichotmous variable and one continuous variable. It turns out that this is a special case of the Pearson correlation. So computing the special point-biserial correlation is equivalent to computing the Pearson correlation when one variable is dichotmous and the other is continuous.
Because we know that differences are the same as correlations now, we can show that the t-test for two groups is the same as the correlation between the grouping or independent variable (X) and the dependent variable (Y). Let's go back to our last t-test example and check this out. In this example, we were comparing first and last siblings to see who was more introverted.
First or Last Sibling
Introversion
1
65
1
48
1
63
1
52
1
61
1
53
1
63
1
70
1
65
1
66
2
61
2
42
2
66
2
52
2
47
2
58
2
65
2
62
2
64
2
69
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