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Symmetric Measures

Symmetric measures of ASSOCIATION describe the relationship between two variables X and Y without differentiating if either variable is an antecedent (or independent variable) or a consequent (or dependent variable). Examples of symmetric measures include PEARSON CORRELATION COEFFICIENT, SPEARMAN CORRELATION, Goodman and Kruskal's GAMMA COEFFICIENT, and point biserial correlation coefficient, in addition to the five measures to be presented below.

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PHI Coefficient

Phi coefficient (φ) is a special case of the Pearson correlation coefficient. When both variables, X and Y, are naturally dichotomous (e.g., male/female, success/failure, survival/death), the Pearson correlation is conventionally labeled as the phi coefficient. Because of its direct relationship with the Pearson correlation coefficient, the null hypothesis test and computation formula for the Pearson correlation coefficient can be applied to the phi coefficient.

Given the dichotomous nature of both variables (e.g., gender and successfulness), the distribution of gender and successfulness can be arranged in a table with four cells, often referred to as a CONTINGENCY TABLE, as shown in Exhibit A of Table 1. As seen, there are 20 successful females, 5 females who fail, 10 successful males, and 15 males who fail. Based on a simplified Pearson correlation formula,

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[assuming (a + c)(b + d)(a + b)(c + d) =/ 0], the correlation between gender (X) and successfulness (Y) is –.41, if females are coded as 1 and males are coded as 0 for variable X, and success is coded as 1 and failure is coded as 0 for variable Y. Alphabetical letters a, b, c, and d represent the frequencies in each cell of the contingency table. We can also calculate the phi coefficient based on the conventional Pearson correlation formula. As demonstrated in Exhibit B of Table 1, the correlation between gender and successfulness is –.41. This negative correlation suggests that the higher the “gender” (from 0 to 1), the lower the “successfulness” (from 1 to 0). In other words, fewer females succeed than do males.

Tetrachoric Correlation

If the phi coefficient is calculated based on two artificially dichotomized variables, actually CONTINUOUS VARIABLES with bivariate NORMAL DISTRIBUTIONS, the phi coefficient will underestimate the relationship between the variables. To estimate what the actual relationship would be if both variables were not artificially dichotomized, we could apply the tetrachoric correlation formula. For example, two variables, sleep(X) and school performance (Y), are artificially dichotomized so that sleep is classified as either short sleep (if a person sleeps less than 7 hours) or long sleep (if a person sleeps equal to or more than 7 hours), and school performance is classified as either poor (if a grade point average is lower than 2.5) or good (if a grade point average is equal to or greater than 2.5). The frequencies for these two variables are summarized in the contingency table depicted in Table 2.

Table 2 Distributions of Sleep (X) and School Performance (Y)
Poor Performance (Y=0)Good Performance (Y=1)Totals
Long Sleep (X = 1)60 = a40 = b100 = (a + b)
Short Sleep (X = 0)40 = c60 = d100 = (c + d)
Totals100 = (a + c)100 = (b + d)200 = a + b + c + d

The tetrachoric correlation formula is defined

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