Overview Association refers to coefficients which gauge the strength of a relationship. Coefficients in this section are designed for use with 2-by-2 tables. Note that measures for larger tables, discussed separately for nominal and ordinal data, may also be used with 2-by-2 tables. Of the measures in this set, percent difference is the simplest and most common. Yule's Q is usually reported as gamma, to which it is equivalent in the 2-by-2 case. Yule's Y is rarely used. The relative risk coefficient is common in medical research but less so in social science, but it provides a 2-by-2 measure defining null association as balance, whereas percent difference and Yule's Q/gamma use the more conventional definition of statistical independence, as discussed in the section on association. The odds ratio, another common measure of association for 2-by-2 tables, is also discussed further separately.

Contents Key concepts and terms Assumptions Frequently asked questions Bibliography




Key Concepts and Terms

• Percent Difference. Percent difference (%d) is perhaps the simplest of all measures of association. It is a simple, traditional measure of strength of relationship for 2-by-2 tables. It is computed by subtracting the difference, measured in percent, between the first and second columns in either row, as in the example below. The same percent difference will be computed whichever row is selected. In this case, %d = 25%.

  City Size/Arenas	Small	Large
  No Arena	20 (25%)	0 (0%)
  Have Arena	60 (75%)	80 (100%)

  • Interpretation: Given %d = 25%, we may state, "City size makes a 25% difference in having an arena, or not having an arena."
  • Meaning of association: As discussed in the separate section on association, %d defines "perfect association" as strict monotonicity and "null relation" as statistical independence.
  • Symmetricalness: Note percent difference is asymmetric. The independent variable forms the columns, while the dependent variable is the row variable. Percentages add to 100% in the columns. Reversing the independent and the dependent will lead to a different result (in this case, 57% rather than 25%).
  • Data level: Percent difference handles any level of data, including nominal variables such as gender.
  • Other features: Percent difference is identical to Somers' d for the 2-by-2 case.

• Yule's Q is a symmetric measure based on the difference between concordant (P) and discordant (Q) data pairs, as discussed separately in the section on association.. It equals this difference (P - Q) as a percentage of all non-tied pairs (P + Q). For a table whose cells are labeled as below, Q equals (ad - bc)/(ad + bc), which, for the data below means Q = (10*80 - 5*60)/(10*80 + 5*60) = 500/1100 = .455.

  City Size/Arenas	Small	Large
  No Arena	a = 10	b = 5
  Have Arena	c = 60	d = 80

  • Interpretation: In the example above, we may say that, "The surplus of consistent data pairs over inconsistent pairs is 45.5% of all non-tied data pairs." Typically, "consistent" means consistent with the researcher's hypothesis.
  • Meaning of association: Q reaches 1.0 under under conditions of weak monotonicity, which it defines as perfect association. Q defines null relationship as statistical independence.
  • Symmetricalness: Q is symmetrical. It makes no absolute difference which is the independent or column variable.
  • Data level: Q requires dichotomous data, which may be nominal or higher in level.
  • Other features: Yule's Q is gamma for the case of 2-by-2 tables. Q will often be higher than gamma for the same data in dichotomized and non-dichotomized forms respectively, since unnecessary dichotomization tends to wash out many of the small differences which lead to inconsistent pairs in gamma. Such forced dichotomization is not recommended.

• Yule's Y, also called Yule's coefficient of colligation, is a variant on Yule's Q, using the geometric mean of diagonal and off-diagonal pairs rather than the number of pairs, as does Q. That is, Y = (SQRT(P) - SQRT(Q))/(SQRT(P) + SQRT(Q)). Thus for the table above (in the section on Yule's Q), Y = (SQRT(800) - SQRT(300))/(SQRT(800) + SQRT(300)) = (28.3 - 17.3)/(28.3 + 17.3) = 11/45.6 = .24.
  • Interpretation: Yule's Y has no easily expressible interpretation. In this case, the geometric mean of the surplus of consistent over inconsistent data pairs as a percentage of all non-tied pairs is .24. Y can also be interpreted as the difference between the probabilities in the diagonal and off-diagonal where row and column probabilities have been standardized to .5.
  • Meaning of association: Y reaches 1.0 under under conditions of weak monotonicity, which it defines as perfect association. Y defines null relationship as statistical independence.
  • Symmetricalness: Y is symmetrical. It makes no absolute difference which is the independent or column variable.
  • Data level: Y requires dichotomous data, which may be nominal or higher in level.
  • Other features: Under conditions of less than perfect weak monotonicity, Yule's Y will be lower than Yule's Q. That is, Yule's Y penalizes for near-weak monotonic relationships, similar to Somers' d, which is used far more commonly due to its more readily intuited meaning. Yule's Y is less sensitive than Q to differences in the marginal distributions of the two variables.

• The Relative Risk Coefficient, RR. Risk is a common measure of association for dichotomous variables. It is widely used in medical studies of risk factors, such as the relation of a treatment to heart attacks. While common in health-related research, risk is applicable to any situation in which the independent (column) variable is a treatment or cause and the dependent (row) variable is an outcome or effect.

  Selecting "Risk" in SPSS gives the relative risk estimates and the odds ratio.. Note that SPSS will set up the ratio such that the risk factor (treatment factor) is the rows and the control (effect) is the columns. In the table below, the ratio will be treatment:control. Consider the table below, where Attack=1 is heart attack, 2 is no heart attack; and Group=1 is treatment, 2 is control:

  

  Risk of a treated person getting an attack = a/(a+b) = 80/500 = .16

  Risk of a control person getting an attack = c/(c+d) = 100/500 = .20

  The relative risk is the ratio of the two ratios = .16/.20 = .80. That is, the relative risk of a treated person getting a heart attack compared to a control person is .8. Put another way, the relative risk of a heart attack after treatment is 80%. Relative risk is shown in SPSS output in the Risk Estimate table under "For cohort Attack = 1.0":

  

  In SPSS. select Analyze, Descriptive Statistics, Crosstabs; select the row and column variables, letting the independent be the rows; click Cells and ask for observed counts and row percentages; click Statistics and select Risk. Risk ratios appear in the "Value" column. SPSS will compute a risk ratio for each value of the dependent variable (ex., for heart attack and for no heart attack). If the relative risk ratio value for the heart attack cohort is 0.80, for instance, then the probability of a heart attack is 80% in the treatment group compared to the control group. If the relative risk for the no heart attack cohort is 1.05, then the probability of no heart attack is 5% greater for the treatment group compared to the control group.

  • Interpretation: RR for these data means that the treatment group has 80% of the risk of the control group.
  • Meaning of association: RR is 0 and a null relationship exists when a/c = b/d, which corresponds to the conditions of statistical independence and balance. RR is maximum when the treatment/effect cell approaches 0 (cell a) and others are non-zero, which corresponds to a condition of weak monotonicity. RRR has the same attributes but is 1 - RR.
  • Symmetricalness: Risk is an asymmetric measures. Different coefficients will result depending on which is the independent (column) variable.
  • Data level: Dichotomous data are assumed, of nominal level or higher.
  • Other features: Risk does not vary from 0 to plus or minus 1 as do most measures of association. As risk in the treatment group approaches 0 for > 0 risk in the control group, RR approaches 0; as risk in the treatment group approaches 0, RR approaches infinity for >0 risk in the treatment group. Since RRR (see below) is 1 - RR, RRR displays the opposite characteristics.

• Relative risk reduction (RRR) is 1 minus risk = 1 - RR. RRR signifies for these data that treatment reduces risk by 20%.

• The odds ratio is also related to relative risk. It is also shown in the Risk Estimate table illustrated above. The odds of being a treatment group heart attack victim are a/b = 80/420 = .19. The odds of being a control group heart attack victim are c/d = 100/400 = .25. The odds ratio is the ratio of these two odds. Odds ratio = (a/b)/(c/d) = .19/.25 = .76. For further discussion of the odds ratio as a measure of association, see the section on log-linear analysis.
  • Interpretation: The odds ratio of .76 signifies that the odds of getting a heart attack after treatment is 76% of the odds for the control group.
  • Meaning of association: An odds ratio of 1.0 means the two variables are independent. The larger the odds ratio is above 1.0, the more effect the independent has on the dependent. The further below 1.0 the odds ratio is, the more the negative association between the two variables. In this example the odds ratio (.76) is below 1.0 and there is a negative relationship: being in the lower group (1=treatment) is correlated with being in the higher attack variable (2=no attack).
  • Significance: One can test the null hypothesis that the odds ratio = 1.0 (independence) using Cochran's test or Mantel-Haenszel continuity-corrected chi-square, as illustrated in output below. If either has "Asymp. Sig. (2-tailed)" probability less than or equal to .05, then the researcher rejects the null hypotheses and concludes the two variables are not independent. In SPSS, select Analyze, Descriptive Statistics, Crosstabs; enter the row and column variables; click Cells and ask for observed counts and row percentages; click Statistics and select Risk, Cochran's and Mantel-Haenszel.

    

  • Testing for equality of odds ratios. If one has entered a Layer (grouping) variable, then the question can be raised whether the odds ratio is homogenous across categories of the layer variable: The Breslow-Day and Tarone's statistics test for homogeneity. If "Asymp. Sig. (2-sided)" by either test is equal to or less than .05, then the researcher rejects the null hypothesis that the odds ratios are equal across groups. By a different rule of thumb, used in the SPSS tutorial, if for either test p > .10, the researcher concludes the odds ratios are homogenous across layers. In SPSS, when specifying row and column variables, also add the grouping variable (ex., race) as Layer 1. Breslow-Day and Tarone's tests will be included in the output. Note that Tarone's test, which is a modification of Breslow-Day to cover more conditions, is generally preferred.
  • Symmetry: The odds ratio is an asymmetric measures. Different coefficients will result depending on which is the independent (column) variable.
  • Data level: Dichotomous data are assumed, of nominal level or higher.




Assumptions

• Assumptions for each coefficient are discussed above.
• Note measures of association, unlike significance, do not assume randomly sampled data.




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. SPSS does not offer Yule's Q or Yule's Y but another SPSS Inc. product, SYSTAT, does. Also, SPSS does offer gamma, which is identical to Yule's Q for 2-by-2 tables.

  For the heart attack treatment example above, here are measures of significance and association output by SPSS, in addition to risk and odds ratio statistics:

  

  

  




Bibliography

• Bishop, Y. M. M., S. E. Feinberg, and P. W. Holland (1975). Discrete multivariate analysis: Theory and practice. Cambridge, MA: MIT Press. Discusses relative risk.
• Davis, James A. (1971). Elementary survey analysis. Englewood Cliffs, NJ: Prentice-Hall. Davis discusses other measures in this section more extensively, including setting confidence limits on Q (pp. 51-58).
• Liebetrau, Albert M. (1983). Measures of association. Newbury Park, CA: Sage Publications. Quantitative Applications in the Social Sciences Series No. 32.

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