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# Statistics reference pages: Measures of association for bivariate data

The following table is your starting point for deciding which measure of association to use. After the table, you'll find one- or two-paragraph descriptions of each of these measures.

If one variable is... |
And the other variable is... |
Use... |

Nominal | Nominal |
Contingency coefficient Cramer's V Lambda |

Nominal | Ordinal |
Contingency coefficient Cramer's V Lambda |

Nominal | Numeric |
Contingency coefficient Cramer's V Lambda NOTE: If you have a nominal response variable and a numeric explanatory variable, you can use Eta and Eta |

Ordinal | Ordinal |
Kendall's tau-b Kendall's tau-c Spearman's rank correlation coefficient |

Ordinal | Numeric |
Kendall's tau-b Kendall's tau-c Spearman's rank correlation coefficient |

Numeric | Numeric |
Pearson's linear correlation coefficient Coefficient of determination Spearman's rank correlation coefficient |

Note: Lambda, eta, and eta^{2} are the only measures in this table whose values change if the row variable is interchanged with the column variable. So they're the only measures in this table for which you have to decide which variable is explanatory (make this one your column variable in SPSS) and which is response (make this one your row variable in SPSS).

### Coefficient of determination *r*^{2}

This coefficient is literally the percent of variation in responses explained by the variation in explanatory values, via the linear association between the two variables. THIS INTERPRETATION IS VALID ONLY FOR LINEAR ASSOCIATIONS. Because *r*^{2} is a percent, it is always between 0 and 1 (inclusive). It can be calculated by squaring Pearson's *r*, hence the name "*r*^{2}."

### Contingency coefficient

The contingency coefficient measures the strength of associations between nominal variables. (Sadly, the exact meaning of the word "strength" here is not clear.) It ranges from 0 to a maximum value less than 1. This maximum value depends on the number of rows and the number of columns in the table. So you don't have a way to know whether a given value of the contingency coefficient is high or very high or as high as possible. (But see the next paragraph.) On the other hand, if the contingency coefficient is 0, there is no association between your two variables, and the higher the contingency coefficient is, the stronger the association. It's usally best to use the contingency coefficient together with another measure of association.

NOTE: It turns out that you can fix the contingency coefficient so that it ranges from 0 to 1. As it happens, the maximum value of the contingency coefficient is the square root of (k - 1) / k, where k is the number of rows or of columns, whichever is smaller. So if you divide the contingency coefficient by the square root of (k - 1) / k, you get an adjusted contingency coefficient that ranges from 0 to 1.

### Cramer's V

This coefficient is used with nominal variables. Its minimum value is 0 (no association) and its maximum value is 1 (perfect association), no matter how many rows or columns your two-way table has. Cramer's V suffers from the problem that there is no good way to interpret values between 0 and 1. If one table has a higher V than another table, you can say that the association in the first table is stronger than the association in the second table. But there's no way to know how much stronger. It's usually best to use Cramer's V together with another measure of association.

### Eta (η)

Eta was invented specifically for the situation in which you have a nominal explanatory (column) variable and a numeric response (row) variable. Eta has the same kind of interpretation as Pearson's *r*, but Eta does not assume the association is linear. (It can't be, when one of the variables is not numeric!) Unlike *r*, eta is always between 0 and 1. Eta requires the counts to be "large enough" for its interpretation as the strength of an association to be reliable. It also performs better when the number of categories of the nomial variable is "large." Eta close to 0 means no association; eta close to 1 means any association there may be is strong.

### Eta^{2} (η^{2})

Eta^{2} was invented specifically for the situation in which you have a nominal explanatory (column) variable and a numeric response (row) variable. Eta^{2} has the same kind of interpretation as the coefficient of determination *r ^{2}*, but Eta

^{2}does not assume the association is linear. (It can't be, when one of the variables is not numeric!) Like

*r*, eta

^{2}^{2}is always between 0 and 1, and represents the percentage of variation in the numeric variable that is explained by the association between the two variables. Eta

^{2}requires the counts to be "large enough" for its interpretation as the strength of an association to be reliable. It also performs better when the number of categories of the nomial variable is "large." Eta

^{2}close to 0 means no association; eta

^{2}close to 1 means any association there is may be is strong.

### Gamma (Γ)

This outdated coefficient literally measures the surplus of "concordant" data points over "discordant" data points, as a percentage of all points except ties. (Note to matrix-savvy students: Think of the two-way table as a matrix *T*. Then gamma is the ratio of the determinant of *T* , to the permanent of *T*.) Gamma ranges from -1 to 1. As with other measures of association, if the association present in the data is very weak, gamma wil be close to 0. Likewise, when gamma is close to -1 or 1, the association is strong and negative or positive (respectively). However, if cell counts are low in either the upper or the lower triangle of the table, gamma will be close to 0 whether there is an association or not, no matter how strong the association may actually be. Because gamma ignores ties, it is not as sensitive as other measures of association. On the other hand, large numbers of ties can make measures like Kendall's tau-b and tau-c to be lower than they should. So some authors recommend using gamma when there are many ties.

### Kendall's tau-b (τ_{b})

Kendall's tau-b was originally used only with 2 x 2 two-way tables of ordinal data or binomial data. Nowadays, it is used with larger tables, though Kendall's tau-c is more appropriate for larger tables. Tau-b measures the extent to which "concordant" data points outweigh "discordant" data points, taking ties into account. "Concordant" means basically that the value of the row variable is higher than the value of the column response variable (for the same individual). Oddly enough, if you make the row variable change places with the column variable, you get the same number of concordant points. "Discordant" means roughly that the value of the row variable is lower than the value of the column variable. If there is no association between the variables, tau-b is (approximately) 0. If the two-way table is "square," that is, if the number of rows equals the number of columns, then tau-b can be as high as 1 (for perfect, positive associations) and as low as -1 (for perfect, negative associations). But if the table is not square, the highest and lowest possible values of tau-b are lower than 1 and higher than -1, respectively.

(Note: Some authors say that tau-b is actually a measure of the strength of linear associations. Unfortunately, I haven't been able to figure out what that means. The data are, after all, ordinal and not numeric...?)

### Kendall's tau-c (τ_{c})

Tau-c is just like tau-b, except it includes an adjustment for the size of the table ("size" meaning "number of row-by-number of columns"). This adjustment makes it more suitable for non-square tables (and for tables larger than 2 x 2) than tau-b.

### Lambda (*Λ*)

Lambda is for use with nominal variables. There are two versions of lambda, depending on whether the row variable is response or explanatory. Lambda is like the coefficient of determination. It tells you how well you can forecast responses. More precisely, it tells you how much better you can predict the value of the response variable (row variable) if you know the value of the explanatory variable (column variable). (E.g. lambda = .6 means your forecasts are 60% more likely to be right if you know the value of the explanatory variable than if you don't.) However, due to a fluke in the formula for lambda, if one of the row totals is much larger than the others, lambda can be 0 even if there is an association between the variables. So if the row totals are "very" different, it's less risky to use Cramer's V or the contingency coeffiecent.

### Pearson's linear correlation coefficient *r*

This is the number most people mean when they use the word "correlation" in conversations about the strength of associations. It is a much misunderstood and abused coefficient. Interpreting Pearson's *r* as a measure of the strength of an association is valid ONLY for linear associations. It is very sensitive to outliers, skewing, and nonlinearity (but not in any usefully systematic way). Therefore, Pearson's *r* should only be used with well-behaved (preferrably "jointly normal") data that exhibit strictly linear associations. If *r* is -1 or 1, then the linear association is perfect (and either negative or positive, respectively). If r is (close to) 0 then any linear association that may be present is very weak, indeed.

Pearson's *r* is, strictly speaking, for numeric data only. Some people code ordinal data with numbers, calculate Pearsons's *r* ,and interpret it as a measure of the strength of association. (That an association might be linear in such a case is absolutely out of the question.) This practice is to be discouraged, as better measures of association for ordinal data exist.

### Spearman's rank correlation coefficient rho (ρ)

Spearman's rho is Pearson's *r* applied not to the data themselves, but to their ranks. In this way, Spearman's rho can be applied to nonlinear associations. Unfortunately, in such a case, it is not valid to interpret *ρ*^{2} as a percentage of anything, in contrast to the interpretation of *r*^{2}. Spearman's *ρ* tends to be a bit lower than Pearson's *r*, which is a good thing in my opinion, especially when any amount of nonlinearity is present in the data.

### Other measures of association

The notion of "association" is rather vague. Consequently, the variety of measures of association is very wide, indeed. Other measures than those listed above include the following: odds ratio, relative risk, sensitivity, specificity, prediction error rates (including so-called "false positive" and "false negative" rates), and so on. The Internet has much fascinating information on these measures. Just make sure you read about any given measure at three or four reputable web sites, because any one web site--including this one!--may give incomplete (or even mistaken) information on any given topic.