# Week 38: The Tetrachoric Correlation Coefficient

Let’s talk about another measure of association today: the **tetrachoric correlation coefficient**!

**When Would You Use It?
**The tetrachoric correlation coefficient is a parametric test used to determine, in the population, if the correlation between values on two variables is some value other than zero. More specifically, it is used to determine if there is a significant linear relationship between the two variables.

**What Type of Data?
**The tetrachoric correlation coefficient requires both variables to be interval or ratio data, but also that both of them have been transformed into dichotomous nominal or ordinal scale variables.

**Test Assumptions**

- The sample has been randomly selected from the population it represents.
- The underlying distributions for both the variables involved is assumed to be continuous and normal.

**Test Process**

Step 1: Formulate the null and alternative hypotheses. The null hypothesis claims that in the population, the correlation between the scores on variable X and variable Y is equal to zero. The alternative hypothesis claims otherwise (that the correlation is less than, greater than, or simply not equal to zero).

Step 2: Compute the test statistic, a z-value. To do so, the actual correlation coefficient, r_{tet}, must be calculated first. This calculation requires the information on the variables X and Y to be displayed in a table such as the following:

Where “0” and “1” are the coded values of the dichotomous responses for X and Y, and the values a, b, c, and d represent the number of points in the sample that belong to the different combinations of 0 and 1 for the two variables.

Once the table is constructed, r_{tet} is computed as follows:

To compute the z-statistic, the following equation is used:

To obtain h for each variable, first find the z-value that delineates the point on the normal curve for which the proportion of cases corresponding to the smaller of p_{0} and p_{1} falls above that point and the larger of the two proportions p_{0} and p_{1} falls below. This table lists the ordinates for specific z-scores.

Step 3: Obtain the p-value associated with the calculated z-score. The p-value indicates the probability of observing a correlation as extreme or more extreme than the observed sample correlation, under the assumption that the null hypothesis is true.

Step 4: Determine the conclusion. If the p-value is larger than the prespecified α-level, fail to reject the null hypothesis (that is, retain the claim that the correlation in the population is zero). If the p-value is smaller than the prespecified α-level, reject the null hypothesis in favor of the alternative.

**Example
**Let’s look at the exam grades for one of the old STAT 213 classes. I want to see if there is a significant correlation between the grades on midterm 1 and midterm 2 as far as whether they got a grade higher than a C+. I will code a grade higher than a C+ as 1 and a grade equal to or lower than a C+ as a 0. Let the X variable be the grade on the first midterm, and the Y variable be the grade on the second midterm. I suspect a negative correlation between X and Y, since a lot of students who did poorly on the first midterm either dropped the class or worked really hard to do well on the second one. Here, n = 105 and let α = 0.05.

H0: ρ_{tet} = 0

Ha: ρ_{tet} > 0

The following table shows the distribution of the 0’s and 1’s for these two variables:

**Computations:
**First, let’s find the h values. For midterm 1, p

_{0}= 0.18 and p

_{1}= 0.82. The z-score for which 0.82 of the distribution falls below and 0.21 of the distribution falls above is 0.92. The ordinate, h, of this value is 0.2613 according to the table. For midterm 2, p

_{0}= 0.31 and p

_{1}= 0.69. The z-score for which 0.69 of the distribution falls below and 0.31 of the distribution falls above is 0.5. The ordinate, h, of this value is 0.3521 according to the table. So,

Since our calculated p-value is smaller than our α-level, we reject H_{0} and conclude that the correlation in the population is significantly greater than zero.