Overview Association refers to coefficients which gauge the strength of a relationship. Coefficients in this section are designed for use with nominal data. Though nominal-level coefficients may be computed for ordinal data or higher, measures designed for higher levels have greater power and are preferred. Note that measures for dichotomous nominal data are discussed separately. Also, Cohen's Kappa, a nominal measure of agreement, is dicussed in a separate section also. Of the measures in this section, phi, the contingency coefficient, Tschuprow's T, and Cramer's V are based on adjusting chi-square significance to factor out sample size. These measures do not lend themselves to easily expressable interpretation. The adjusted contingency coefficient C* and Cramer's V vary between 0 and 1 regardless of table size, whereas phi, C, and T do not. V is by far the most used measure of association for this subset. Lambda is a popular measure because of its easily-understood interpretation in terms of proportionate reduction in error (PRE) varying from 0 to 1, but it defines perfect association as predictive monotonicity and null relationship as accord, unlike most measures, which use the criteria of strict monotonicity and statistical independence, as discussed in the section on association. The Uncertainty Coefficient also has a PRE meaning but its formula takes into account the entire distribution rather than just the mode (which lambda uses) and therefore may be preferred to lambda..

Contents Key concepts and terms Assumptions Frequently asked questions Bibliography




Key Concepts and Terms

• Phi. Phi is a chi-square based measure of association. Recall that the size of the chi-square coefficient depends on the strength of the relationship and sample size. Phi eliminates sample size by dividing chi-square by n, the sample size, and taking the square root. Phi is sometimes called Pearson's coefficient of mean-square contingency, though sometimes this term is applied to Pearson's contingency coefficient, discussed below, which is a modification of phi.

  Phi can handle larger tables but is often used as a measure of association in 2-by-2 tables formed by true dichotomies. With dichotomized continuous data, tetrachoric correlation is preferred. Phi is the geometric mean of percent difference across rows and percent difference across columns when used with 2-by-2 tables. That is, it can be interpreted as a symmetric version of percent difference when used with 2-by-2 tables. (Recall the geometric mean is a measure of central tendency computed by multiplying n values and taking the nth root of their product.). For larger tables, the maximum value of phi depends on table size and can exceed 1.0, making phi a not particularly suitable measure of association.

  Computationally, phi is the square root of chi-square divided by n, the sample size: phi = SQRT(X²/n). When computing phi, note that Yates' correction to chi-square is not used. Phi thus measures the strength of the relationship defined as the number of cases on one diagonal minus the number on the other diagonal, adjusting for the marginal distribution of the variables.

  • Example. For the table below, chi-square equals 8.33. Given n = 50, therefore phi = SQRT(8.33/50) = .41.

    Party/Vote	Democrat	Republican
    Voted	15	10
    Didn't Vote	5	20

  • Interpretation: In 2-b-2 tables, phi can be interpreted as symmetric percent difference, measuring the percent of concentration of cases on the diagonal. Also in 2-by-2 tables, phi is identical to the correlation coefficient. In larger tables, where phi may be greater than 1.0, there is no simple intuitive interpretation, which is a reason why phi is often used only for 2-by-2 tables. For the example above, phi is .41. The percent difference with party as independent (column) is 42%, and with vote as independent is 40%. Phi is the mean percent difference between party and vote with either considered as causing the other.
  • Meaning of association: Phi defines perfect association as predictive monotonicity (see discussion in the section on association), and defines null relationship as statistical independence.
  • Significance. The significance of phi is the same as for chi-square. SPSS and other major packages report the significance level of phi.
  • Symmetricalness: Phi is a symmetrical measure. It makes no difference which is the independent (column) variable. Phi tends to understate asymmetric relationships. Note, however, that some computer packages, such as Systat, use special formulas for phi in 2-by-2 tables so that phi varies from -1 to +1, allowing it to indicate negative relationships when used with dichotmous ordinal data. However, phi is still a symmetric measure and the sign can be ignored when using dichotomous nominal data.
  • Data level: Phi is used with nominal data, though for 2-by-2 tables, data may be ordinal.
  • Other features: Phi is very sensitive to shifts in marginal distributions. Phi does not vary from 0 to 1. For tables larger than 2-by-2, the maximum value of phi is the square root of (k - 1), where k is the number or rows or the number of columns, whichever is smaller. This means phi can be greater than 1.0 for larger tables, with a theoretical maximum of infinity, and differs depending on table size.
  • In SPSS, select Analyze, Descriptive Statistics, Crosstabs; select the row and column variables; click Statistics; select phi.

• The Contingency Coefficient, Pearson's C. The contingency coefficient is an adjustment to phi, intended to adapt it to tables larger than 2-by-2. C is equal to the square root of chi-square divided by chi-square plus n, the sample size: C = SQRT[X²/(X² + n)]. C has a maximum approaching but never totally reaching 1.0 only for large tables and some researchers recommend it only for 5-by-5 tables or larger. For smaller tables it will underestimated the level of association even when all observations are on the diagonal of a table. For the table above, in the section on phi, C = SQRT[8.33/(8.33 + 50)] = .38.
  • Sakoda's adjusted Pearson's C, C*. By dividing C by C_max, one gets adjusted C (C*). C* varies from 0 to 1 regardless of table size, making for a much more useful measure. Unfortunately, C* is not supported by major statistical packages though given C it is easy to compute manually. Its formula is C/SQRT[(k-1)/k], where k = the number of rows or columns, whichever is smaller. For the table above, in the section on phi, C* = observed C/maximum C = .38/SQRT[(2-1)/2] = .38/.71 = .53.

  • Interpretation: There is no easily intuited interpretation of C or C*, though C* may be viewed as the association between two variables as a percentage of their maximum possible variation. Pearson viewed C as a nominal approximation of Pearsonian correlation, r.
  • Significance. The significance of C is the same as for chi-square. SPSS and other major packages report the significance level of C.
  • Meaning of association: C and C* define a perfect relationship as weak monotonic, and define a null relationship as statistical independence, as discussed in the section on association. However, the smaller the table, the more C (but not C*) will be less than 1.0 regardless of monotonicity.
  • Symmetricalness: C and C* are symmetrical measures. It does not matter which is the independent (column) variable.
  • Data level: C and C* may be used with nominal data or higher.
  • Other features: The upper limit of C is .71 for 2-by-2 tables, but it approaches an upper limit of 1 as the number of rows and columns increases. C_max = SQRT[(k-1)/k], where k = the number of rows or columns, whichever is smaller. C and C* can reach 1.0 only when the two variables have equal marginals.
  • In SPSS, select Analyze, Descriptive Statistics, Crosstabs; enter the row and column variables; click Statistics; select Contingency Coefficient.

• Tshuprow's T. T is a chi-square-based measure of association which reaches 1.0 in square tables where row marginals are identical to column marginals. T is equal to the square root of chi-square divided by sample size n times the square root of the number of degrees of freedom in the table: T = SQRT[X²/(n*SQRT((r-1)(c-1)))], where r is the number of rows and c is the number of columns. Because T is less than 1.0 for non-square tables, it is little used and is not supported by major statistical packages.
  • Interpretation: There is no easily intuited interpretation of T. For 2-by-2 tables, T = phi.
  • Meaning of association: T defines a perfect relationship as one which is weak monotonic, and defines a null relationship as statistical independence, as discussed in the section on association. However, the less square the table and the more unequal the marginals, the more T will be less than 1.0.
  • Significance: The significance of T is the same as for chi-square.
  • Symmetricalness: T is a symmetrical measure. It does not matter which is the independent (column) variable.
  • Data level: T may be used with nominal data or higher.
  • Other features: T reaches 1.0 under strict monotonicity only in square tables. Let k = the smaller of r or c, defined above. Let j = (r-1) or (c-1), whichever is larger. Then T_max = QUARTICROOT[((k-1)/j)⁴]. For three rows and ten columns, for instance, T_max = .69. T can reach 1.0 only when the two variables have equal marginals.
  • SPSS does not support Tshuprow's T.

• Cramer's V. Cramer's V is the most popular of the chi-square-based measures of nominal association because it gives good norming from 0 to 1 regardless of table size, when row marginals equal column marginals. V equals the square root of chi-square divided by sample size, n, times m, which is the smaller of (rows - 1) or (columns - 1): V = SQRT(X²/nm). Cramer's V tends to be lower than phi or C.
  • Interpretation: V may be viewed as the association between two variables as a percentage of their maximum possible variation.V² is the mean square canonical correlation between the variables. For 2-by-2 tables, V = phi (hence some packages like Systat print V only for larger tables).
  • Meaning of association: V defines a perfect relationship as one which is predictive or ordered monotonic, and defines a null relationship as statistical independence, as discussed in the section on association. However, the more unequal the marginals, the more V will be less than 1.0.
  • Significance: The significance of C is the same as for chi-square. SPSS and other major packages report the significance level of the computed V value. The formula for the variance of Cramer's V is given in Liebetrau (1983: 15-16).
  • Symmetricalness: V is a symmetrical measure. It does not matter which is the independent (column) variable.
  • Data level: V may be used with nominal data or higher.
  • Other features: V can reach 1.0 only when the two variables have equal marginals.
  • In SPSS, select Analyze, Descriptive Statistics, Crosstabs; enter the row and column variables; click Statistics; select Cramer's V.

• Lambda. Lambda, also known as Goodman-Kruskal lambda, is a PRE (proportionate reduction in error) measure, which means that its value reflects the percentage reduction in errors in predicting the dependent given knowledge of the independent. This probability is defined as the chance that an observation is in a category other than the most common (modal) one. That is, with no knowlege of the independent, a blind forecaster would guess that each observation of the dependent would have the value of its modal category. Thus the marginal of the modal category is the number of correct guesses one would expect by chance. This forms the denominator of the equation for lambda. The numerator reflects the number of correct guesses knowing the independent variable.
  Computation. In its customary, asymmetric form, lambda = [(SUM(f_i) - F_d]/(n - f_d), where f is the largest frequency for each class of the i classes of the independent variable, and F_d is the largest marginal value of the dependent variable, and n is sample size.

  	City Size 1	City Size 2	City Size 3	Row Totals
  City Manager	80	9	1	90
  Mayor-Council	40	1	9	50
  Column Totals	120	10	10	n = 140

  lambda = [(80 + 9 + 9) - 90]/(140-90) = .16

  In the table above, knowing city size reduces errors in guessing form of government by 16%. Errors made not knowing is based on subtracting the modal category of the dependent, 90, from n, which is 140: 140 - 90 = 50 errors when not knowing the independent. That is, if one did not know the distribution of the independent variable, then one would guess all cities were of the city manager type, and one would be right 90 times and wrong 50 times. The proportionate reduction in error (PRE) made when knowing the distribution of the independent is the sum of the modal categories of each column of the dependent: 80 + 9+ 9 = 98, minus the 90 correct guesses one would have made anyway (98 - 90 = 8), divided by the number of errors one would have made anyway: 8/50 = .16.

  Since lambda has a known sampling distribution (it is asymptotically normal) it is possible to compute its standard error and significance. SPSS, SAS, and other major packages report the ASE (asymptotic standard error). Lambda divided by its ASE gives the coefficient used to compute p_lambda, the probability or significance level of the computed lambda value. The formula for the variance of lambda is given in SAS (1988: 533-534) and in Liebetrau (1983: 20-23).

  • Interpretation: Lambda is the percent one reduces errors in guessing the value of the dependent variable when one knows the value of the independent variable. Specifically, lambda is the surplus of errors made when the marginals of the dependent variable are known, minus the number of errors made when the frequencies of the dependent variable are known for each class (value) of the independent variable, as a percentage of errors of the first type. A lambda of 0 indicates that knowing the distribution of the independent variable is of no help in estimating the value of the dependent variable, compared to estimating the dependent variable solely on the basis of its own frequency distribution.
  • Meaning of association: Lambda defines "perfect relationship" as predictive monotonic or higher, and defines "null relationship" as accord, as described in the section on association. This differs from most measures of association, which use the criteria of strict monotonicity and statistical independence, respectively.
  • Significance: SPSS computes the significance of lambda based on a complex formula for lambda's standard error.
  • Symmetricalness: The usual version of lambda is asymmetric. However, SPSS and other packages print out three versions: a symmetric version, and two asymmetric versions, one with each of the two variables considered as dependent. Lambda symmetric is simply the average of the two asymmetric lambdas.
  • Data level: Lambda may be used with nominal level data or higher. That is, lambda is unaffected by the ordering of the columns of the independent variable.
  • Other features: Note lambda cannot be computed if all entries lie in just one column of the independent variable, but otherwise it varies from 0 to 1. Lambda is sensitive to inequalities in column marginals. For instance, if in the example above there were as many cities of sizes 2 and 3 as there are of size 1, and if the frequency breakdown of each size were the same, then lambda would be .60 rather than .16, leading us to think the relationship was strong rather than weak. This is because knowing the distribution of the independent only helps for city size = 3, and there are only 10 such cities out of 140 -- were there more such cities, then the independent would be of greater predictive help. Sensitivity analysis of this point is obtained by computing lambda by the formula above (observed lambda) and then substituting column percentages as if they were frequencies and recomputing lambda (column-equalized lambda).
  • In SPSS, select Analyze, Descriptive Statistics, Crosstabs; enter row and column variables; click Statistics; select Lambda.

• The Uncertainty Coefficient, UC or Theil's U. The uncertainty coefficient, also called the entropy coefficient, varies from 0 to 1 and has a proportionate reduction in error interpretation based on information theory (see Theil, 1972). UC is the percent reduction in error in accounting for the variance in the dependent variable, where variance is defined in terms of entropy (see formulas below). Though lambda and UC are both PRE measures of nominal association, UC is differentiated in that its formula takes into account the entire distribution rather than just the modal distribution upon which lambda focuses and is therefore sometimes preferred for this reason.

  The formula for UC(R|C), which is the uncertainty coefficient for predicting the row variable on the basis of the column variable, is given below, with its variants:

  UC(R|C) = [H(X) + H(Y) - H(XY)]/H(Y)
  UC(C|R) = [H(Y) + H(X) - H(XY)]/H(X)
  UC_symmetric = 2[H(X) + H(Y) - H(XY)]/[H(X) + H(Y)]

  where

  • X is the column variable
  • Y is the row variable
  • n is sample size
  • r_j are the row totals (marginals) for rows 1...j
  • c_k are the column totals (marginals) for rows 1...k
  • n_jk is the cell count for row j, column k
  • ln is the symbol for the natural log function
  • H(X) = - SUM_j[(r_j/n)*ln(r_j/n)] = entropy for UC(C|R)
  • H(Y) = - SUM_k[(c_k/n)*ln(c_k/n)] = entropy for UC(R|C)
  • H(XY) = - SUM_jSUM_k[(n_jk/n)*ln(n_jk/n)]

• Interpretation: The uncertainty coefficient is the percent reduction in uncertainty in predicting the dependent variable based on knowing the independent variable. When UC is 0, the independent variable is of no help in predicting the dependent variable.
• Meaning of association: The uncertainty coefficient defines "perfect relationship" as predictive monotonic or higher, and defines "null relationship" as statistical independence, as described in the section on association.
• Significance: SPSS and other major statistical packages report the significance of the uncertainty coefficient. The significance is the uncertainty coefficient divided by its asymptotic standard error (ASE). The formula for the variance of UC is given in SAS(1988: 534), along with computational formulas.
• Symmetricalness: The uncertainty coefficient is an asymmetric measure. It will have a different value depending on which is the independent and which the dependent variable. However, major statistical packages like SPSS and SAS output three versions of UC: one with the column as independent, one with the row as independent, and a symmetric version averaging the first two. Nonetheless, whereas asymmetric UC has a clear PRE meaning, symmetric UC does not, other than being an average of the two asymmetric UC's.
• Data level: Data may be nominal level or higher.
• Other features: For historical reasons, the uncertainty coefficient was often computed in terms of predicting the column variable from the row variable -- the opposite of the social science convention of using the column variable as the independent. Thus the default uncertainty coefficient is often UC(C|R): the coefficient for the column variable predicted on the basis of the row variable. Most statistical packages also output UC(R|C) as well as a symmetric version of UC, but the researcher should be careful on this point.




Assumptions

• Assumptions for each coefficient are discussed above.
• Note measures of association, unlike significance, do not assume randomly sampled data.
• Equal marginals. Cramer's V and all measures which define a perfect relationship in terms of strict monotonicity require that the marginal distribution of the two variables be equal for the coefficient to reach 1.0. This is discussed in the section on association.




Frequently Asked Questions

• Where does one find these measures of association in SPSS?
  Most are found in the SPSS CROSSTABS module. From the menu, select Statistics, Summarize, Crosstabs. In the "Crosstabs" dialog box, click the "Statistics" button, then in the "Crosstabs: Statistics" dialog box, check the measures you want. However, Tschuprow's T is not supported in SPSS (or SAS).




Bibliography

• Agresti, Alan (1996). Introduction to categorical data analysis. NY: John Wiley and Sons.
• Goodman, Leo A. and W. H. Kruskal (1954, 1959, 1963, 1972). Measures for association for cross-classification, I, II, III and IV. Journal of the American Statistical Association. 49: 732-764, 54: 123-163, 58: 310-364, and 67: 415-421 respectively. The 1972 installment discusses the uncertainty coefficient.
• Liebetrau, Albert M. (1983). Measures of association. Newbury Park, CA: Sage Publications. Quantitative Applications in the Social Sciences Series No. 32.
• Rosenberg, M. (1968). The logic of survey analysis. NY: Basic Books.
• SAS Institute (1988). SAS/STAT user's guide. Cary, NC: Sas Institute.
• Theil, H. (1972). Statistical decomposition analysis. Amsterdam: North Holland. Discusses the uncertainty coefficient.

Copyright 1998, 2008 by G. David Garson.
Last updated 3/24/2008.