Overview Association refers to a wide variety of coefficients which measure strength of relationship, defined various ways. In common usage "association" refers to measures of strength of relationship in which at least one of the variables is a dichotomy, nominal, or ordinal. Correlation, which is a type of association used when both variables are interval, is discussed separately. Reliability, which is a type of association used to establish the consistency of a measure or to assess inter-rater similarity on a variable, is also discussed separately.

Contents Key concepts and terms Assumptions Frequently asked questions Bibliography




Key Concepts and Terms

• Significance versus association. Measures of significance test the null hypothesis that the strength of an observed relationship is not different from what would be expected due to the chance of random sampling. Significance coefficients reflect not only strength of relationship but also sample size and sometimes other parameters. Therefore it is possible to have a relationship which displays strong association but is not significant (ex., all males are Republicans and all females are Democrats, but the sample size is only 8) or a relationship which displays an extremely weak association but is very significant (ex., 50.1% of males are Republicans compared to 50.0% of females, but sample size is 15,000 and the significance level is .001). Because significance and association are not at all equivalent, researchers ordinarily must report both significance and association when discussing their findings. Note also that significance is relevant only when one has a random sample, whereas association is always relevant to research inferences.

• Coefficients of association. Most coefficients of association vary from 0 (indicating no relationship) to 1 (indicating perfect relationship) or -1 (indicating perfect negative relationship). As discussed below, however, there are various types of "perfect relationship" and various types of "no relationship." Which definitions the researcher selects may strongly affect the conclusions to which he or she comes. When particular coefficients are discussed later in this section, their definitions of perfect and null relationships are cited and this is one important criterion used by researchers in selecting among possible measures of association. If you wish to skip the rather long discussion below, just keep in mind that most but not all coefficients of association define "perfect relationship" as strict monotonicity and define a "null relationship" as statistical independence.
  • Types of perfect relationship. There are four definitions of "perfect" linear relationship in association, plus the definition of perfect curvilinear relationships. The linear definitions are those dealing with strict monotonic, ordered monotonic, predictive monotonic, and weak monotonic relationships. All relationships which are perfect by strict monotonicity are also perfect by the others. Likewise, perfect ordered and predictive monotonic relationships will also be perfect by the criterion of weak monotonicity. One cannot have perfect ordered monotonicity and perfect predictive monotonicity at the same time. None of the definitions based on monotonicity are appropriate for curvilinear or discontinuous relationships.
    1. The concept of pairs. Strength of linear relationship is defined in terms of degree of monotonicity, which is based on counting various types of pairs in a relationship shown in a table. A pair is a two cases, each of which is in a different cell in the table representing the joint distribution of two variables. Let x be an independent variable with three values and let y be a dependent with two values, with a, b, ..., f being the cell counts in the resulting table, illustrated below:

       	x
       y		1	2	3
       	1	a	b	c
       	2	d	e	f

       The four types of pairs, how they are counted, and their symbols are shown in the table below.

       Type of Pair	Number of Pairs	Symbol
       Concordant	a(e+f) + b(f)	P
       Discordant	c(d+e) + b(d)	Q
       Tied on x	ad + be +cf	Xo
       Tied on y	a(b+c) + bc + d(e+f) + ef	Yo

       All definitions of "perfect relationship" increase the coefficient of association toward + 1 as concordant pairs increase. However, there is disagreement about how to handle ties, leading to the different definitions below.

    2. Strict monotonicity. Perfect positive strict monotonicity requires that discordant pairs (Q), ties on x (X_o), and ties on y (Y_o) all be zero. For perfect negative monotonicity, concordant pairs (P), ties on x (X_o), and ties on y (Y_o) all must be be zero. That is, by this most common definition, a relationship is "perfect" if (1) when the independent variable x increases, then the dependent variable y increases (or decreases in the case of perfect negative relationships), and (2) if each value of x corresponds to only one y value. Likewise, when y increases, x also increases (or decreases for negative relationships), and each y value corresponds to only one x value. An implication is that there are no ties on x and no ties on y (that is, no cases other than on the diagonal). Examples of perfect strict monotonic association are below:

       x  y   15 0 0 0 15 0 0 0 15

       x  y   15 0 0 0 0 15 0 0 0 0 0   15

    3. Ordered monotonic. Perfect positive ordered monotonicity exists when discordant pairs (Q) and ties on y (Y_o) are zero. Perfect negative ordered monotonicity exists when concordant pairs (P) and ties on y (Y_o) are zero. That is, perfect ordered monotonicity exists when (1) as x increases, y also increases (or decreases for perfect negative relationships) and (2) when every y value corresponds to just one x value. Examples of perfect ordered monotonic association are below:

       x  y   15 0 0 0 15 0 0 15 0

       x  y   15 0 0 0 0 0   0 15 0 0 0   15

    4. Predictive monotonic. Perfect positive predictive monotonicity exists when discordant pairs (Q) and ties on x (X_o) are zero. Perfect negative predictive monotonicity exists when concordant pairs (P) and ties on x (X_o) are zero. That is, perfect predictive monotonicity exists when (1) as x increases, y also increases (or decreases for perfect negative relationships) or remains the same, and (2) when every y value corresponds to just one y value. Examples of perfect ordered monotonic association are below:

       x  y   15 0 0 0 15 15 0 0 0

       x  y   15 15 0 0 0 0   0 0 0 0 15 15

       Note this form of association is called "predictive" because the dependent variable can be predicted uniquely from knowing the value of the independent variable, given that each independent x value corresponds uniquely to one dependent y value.

    5. Weak monotonic. Perfect positive weak monotonicity exists when discordant pairs (Q) are zero. Perfect positive weak monotonicity exists when concordant pairs (P) are zero. Perfect weak monotonicity exists when (1) as x increases, y also increases (or decreases for perfect negative relationships) or remains the same. In 2-by-2 tables this corresponds to having a zero cell in the table. Examples of perfect ordered monotonic association are below:

       x  y   15 0 0 15 0 0 15 15 15

       x  y   15 0 0 0 15 15 0 0 0 0 15 15

    6. Curvilinear. Curvilinear association is perfect when every x value of the independent corresponds to only one y value of the dependent variable. The reverse need not be true, nor need the relationship be continuous. Most investigations of curvilinear relationships involve the use of curve-fitting software, however, which usually do require the distribution be continuous. Some applications also require that the curve be describable as a mathematical function.
       

       Curvilinear association is asymmetric in that its definition depends of which variable is independent and which is dependent. Thus for hypotheses in which y is the independent variable, then curvilinear association is perfect when every y value corresponds to only one x value. Note curvilinear association is never applicable to nominal variables.

  • Types of null relationship. There are four ways to define "no relationship" between two variables. The leading definition, independence, is a symmetric criterion making no assumption about the direction of causation, whereas accord is asymmetric. Both independence and accord are nominal criteria, making no assumption about the level of data. Balance is an ordinal criterion, except for dichotomies, and assumes the values of the two variables are ordered. Cleavage is a sufficient condition for independence and balance, but is a more stringent definition such that independence or balance do not imply cleavage.
    1. Independence. By far the most common definition of null relationship is based on the laws of probability. Two variables are independent when their joint distribution is as would be predicted on the basis of the number of cases in their individual categories. The expected value for any joint category, calculated as in chi-square, is the product of the number of cases in their separate categories divided by n, the sample size. For instance, if in a sample of 100 there are 50 men and 40 Republicans, the expected number of male Republicans is 50*40/100 = 20. If every joint category (each of the cells in a table) is the expected value, then there is a null relationship as defined by the criterion of independence. Note independence makes no assumption about which is the independent and which is the dependent variable (it is symmetric). When a relationship is independent, chi-square will be zero and thus chi-square may be viewed as a test of independence.

    2. Accord. By this criterion, two variables have a null relationship if the largest-count categories of the independent variable all have the same value on the dependent variable. For instance, let the independent be low, medium, and high education and let the dependent be unsatisfactory, satisfactory, and meritorious performance evaluations. There might be a tendency to have more meritorious ratings as one moved from low to medium to high education. However, it might be true at the same time the most low-educated, most medium-education, and most high-educated employees all received satisfactory ratings. By the criterion of accord there would be a null relationship, whereas by the criterion of independence there would be a relationship. Accord is the second most common definition of strength of relationship and is an asymmetric definition

    3. Balance. When the value categories of both variables are ordered and crosstabulated, by this criterion a null relationship is said to exist when the number of cases on the right-sloping diagonal(s) is equal to the number of cases on the left-sloping diagonal(s). Consider the following table:

       Degree/Rating	< BA	BA	> BA	Row total
       Unsatisfactory	8	4	7	19
       Satisfactory	4	4	3	11
       Meritorious	3	3	2	8
       Column Total	15	11	12	38

       left diagonals =28; right diagonals = 28

       In this table there is a tendency for those with less than a BA degree or more than a BA degree to receive low performance ratings, and for those with exactly a BA to do proportionately best. However, since the count on the right- and left-sloping diagonals is 28 in each case, by accord there is a null relationship.

    4. Cleavage. By this criterion, a null relationship exists when the number of cases associated with each category of the independent variable is split evenly among the dependent variable categories. Consider the following table:

       Degree/Rating	< BA	BA	> BA	Row total
       Unsatisfactory	3	5	8	16
       Satisfactory	3	5	8	16
       Meritorious	3	5	8	16
       Column Total	9	15	24	48

       left diagonals =37; right diagonals = 37

       When a null relationship exists by cleavage, as above, there will also be a null relationship by balance and independence. Since there are equal numbers of cases in each dependent category for each independent category, accord cannot be computed but it also approaches null for tables with perfect cleavage. However, note that the reverse is not true: tables with a null relationship by any of the other criteria need not have a null relationship by the cleavage criterion.

• Association with Control Variables ("The Elaboration Model"). In crosstabulation, for an original table of X and Y, the researcher may seek to control for the effects of a third variable, Z. For instance, for a table of religious affiliation with party vote, one may seek to control for gender. This is done by computing measures of association for the original table (X=religion and Y=vote) and for similar X-Y tables for each value of Z (in this case, a male table and a female table of religin and vote). By comparing an appropriate coefficient of association in the original table with its counterparts for the control subtables, one of several effects may be noticed:
  • No effect occurs when the original and subtable measures of association are equivalent in magnitude and sign.

  • Explanation occurs when the control variable is an anteceding cause of the independent and dependent, or when it is an intevening variable on the path from the independent to the dependent, and there is no direct causal path from the independent to the dependent. In this case, the subtable associations approach 0 and for random samples should test as not significant. In such cases the original association is said to be spurious. Note that one cannot differentiate statistically between an anteceding and an intervening control effect but rather must do so on some other basis, such as knowledge of time sequences of related events.

  • Partial explanation occurs when there is a direct path from the independent to the dependent variable, but the control variable is also either an anteceding or intevening cause. In this case subtable association drops only part way to 0 compared to the original bivariate association. Note that if the association drops sufficiently far as no longer to be significant, this is considered indicative of explanation, not partial explanation.

  • Suppression occurs when the control variable has a positive effect on the dependent through one path and a negative effect through another path. For three-variable models, suppression may occur when there is an odd number of negative arrows. In such situations, the control variable acts in one direction by way of the independent and in the opposite direction in terms of direct effect on the dependent, thereby masking some of the correlation which would exist in the absence of the control. When suppression occurs, subtable association will be higher than the original bivariate association.

• Specific Measures of Association. With the exception of eta, when data are mixed by data level, the researcher uses a measure of association for the lower data level. Thus, for nominal-by-ordinal association one would use a measure for nominal-level association.
  • Dichotomous Association (2-by-2 tables): Percent Difference, Yule's Q, Yule's Y, Risk

  • Nominal Association: Phi, Contingency Coefficient, Tschuprow's T, Cramer's V, Lambda, Uncertainty Coefficient

  • Nominal-by-Interval Nonlinear Association: Eta

  • Ordinal Association: Gamma, Kendall's tau-b and tau-c, Somers' d

  • Association for Inter-rater Agreement (rows and columns are the same variable): Kappa

  
  

  Assumptions

  • Assumptions are discussed in the sections for each particular measure of association. Measures of association may assume nominal, ordinal, or interval levels of measurement; symmetry or asymmetry of causal direction; square versus any shape table; and alternative definitions of "perfect relationship" and "null relationship" as described above.

  • Strict monotonicity and the assumption of equal marginals. Measures of association which define perfect relationship in terms of strict monotonicity can reach 1.0 only when the two variables have the same marginal distribution, ignoring null rows and null columns. One such measure, tau b, is used to illustrate this in the four tables below:

    TABLE A Male Female Row Totals Republican 15 10 25 Democrat 5 20 25 Column Totals 20 30 n = 50 tau b = .408	TABLE B Male Female Row Totals Republican 20 5 25 Democrat 0 25 25 Column Totals 20 30 n = 50 tau b = .816
    TABLE C Male Female Row Totals Republican 25 5 30 Democrat 5 15 20 Column Totals 30 20 n = 50 tau b = .583	TABLE D Male Female Row Totals Republican 30 0 30 Democrat 0 20 20 Column Totals 30 20 n = 50 tau b = 1.0

    Table A illustrates a hypothetical relationship between gender and political party, shown to have a level of association by tau b of .408. Table B represents the strongest possible relationship between gender and party if one is forced to keep the marginal totals the same as in Table A. Even though Table B is as strong as possible keeping the same total number of men and women, and Republicans and Democrats, its association is less than 1.0 (it is .817). Table C illustrates a relationship between the same two variables, but where gender and party have equal marginals, with a tau b strength of .583. Table D represents the strongest possible relationship between gender and party, keeping the marginal totals the same as in Table C, and its strength is a perfect 1.0, reflecting strict monotonicity. That is, a monotonic measure of association like tau b can reach 1.0 only when the marginal distributions of the two variables are the same, as they are in Tables C and D. In the 2-by-2 case, ordered and predictive monotonic measures of association exhibit the same behavior, although in larger tables they can reach 1.0 even when row and column marginals are not the same.

  • Monotonicity and table size. In a non-square table with no null rows and no null columns, there will always be ties on the variable with the smaller number of classes. When the row variable has fewer classes there will be ties on the row variable (y), and thus such a table cannot have perfect association by strict or ordered monotonicity, but may be perfect by predictive or weak monotonicity. When the column variable has fewer classes there will be ties on the column variable (x), and thus such a table cannot have perfect association by strict or predictive monotonicity, but may be perfect by ordered or weak monotonicity.

  
  

  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, Yule's Y, or Tschuprow's T. Another SPSS Inc. product, SYSTAT, offers Yule's Q and Yule's Y.

  
  

  Bibliography

  • Garson, G. David (1976). Political Science Methods. Boston, MA: Holbrook Press. Chapter 11 covers most measures of association. See also any standard introductory textbook on statistics associated with survey research.

  • 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 update, 3/24/2008.