CHAPTER 14: SUPPLEMENT

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1 CHAPTER 4: SUPPLEMENT OTHER MEASURES OF ASSOCIATION FOR ORDINAL LEVEL VARIABLES: TAU STATISTICS AND SOMERS D. Introduction Gamma ignores all tied pairs of cases. It therefore may exaggerate the actual strength of association between two ordinal variables, especially for small tables where there are many ties. Kendall's tau-b (τ b ), Kendall's tau-c (τ c ), and Somers d were designed to correct for this problem of ties by including them in their formulas. (Note, there is third Kendall tau measure, tau-a, but we will not consider it here since it has limited application in social research). The Tau statistics and Somers d are simply modifications of gamma. Each contains exactly the same quantity in the numerator as gamma: N s - N d, where N s is the number of same pairs and N d the number of different pairs. Unlike gamma, the Tau statistics and Somers' d include tied pairs in one way or another: tau-b and tau-c include pairs tied on both the dependent variable and the independent variable while Somers' d includes pairs tied on the dependent variable but not the independent variable.. Kendall's tau b and tau-c Kendall's tau-b (named after British statistician, Maurice Kendall) is a symmetrical (the same value is produced regardless of which variable is considered the dependent) measure of association used for square tables. Tau-b can only reach a maximum of +.0 (or a minimum of -.0 for negative relationships) in square tables. Tau-c, a variant of taub, was designed (adjusts for table size) to reach these limits for non-square tables. Tau-b and tau-c have the same interpretation: they range from -.0, negative relationship to +.0, positive relationship, where a value of zero indicates no relationship. Note that unlike gamma, tau-b and tau-c do NOT have a PRE interpretation, except in a very special case of tau-b. The formula for computing tau-b is N + + T N + N τ b = ( s Nd x)( s d y) + T where, N s is the number of same pairs N d is the number of different pairs T x is the number of pairs tied on the independent variable, X T y is the number of pairs tied on the dependent variable, Y

2 The formula for computing tau-c is τ c = N [( m - ) /m] where, N the total number of cases m is the minimum value, whichever is less, of the number of rows or the number of columns. 3. Somers' d Somers' d, developed by the sociologist Robert H. Somers, is a modified version of tau-b. However, Somers' d was designed specifically as an asymmetric measure of association and assumes that you can identify one of the variables as the dependent variable. Thus it differs from tau-b in that it corrects only for pairs tied on the dependent variable. The formula for computing Somers' d is Somers' d = + Nd +T y where, N s is the number of same pairs N d is the number of different pairs T y is the number of pairs tied on the dependent variable, Y Somers' d ranges fro.0 to -.0 (for negative relationships). It does NOT have a PRE interpretation. 4. An Example Continuing with the example of the relationship between length of service and burnout for a sample of 00 teachers (see Table 4.), we will compute each statistic. To compute the tau statistics and Somers' d two sets of numbers need to be calculated. First we need to find the number of pairs of cases that are ranked the same on both variables, N s, and the number of pairs of cases ranked differently on the variables, N d. Recall that these values were calculated in computing gamma in Chapter 4: N s =,83 and N d = 499. That is, there are,83 pairs of cases are ranked the same on length of service and burnout, and 499 pairs of cases ranked differently on both variables.

3 Second we need to find the number of pairs tied on the dependent variable, T y, and the number of pairs tied on the independent variable, T x. To find T y, start with the cell containing the cases ranked the lowest on both variables, then multiply the frequency in that cell by the total of all frequencies immediately to the right of that cell. Repeat this procedure for each cell and add the products together. The total of these products is T y. Figure, which reproduces the data from Table 4., shows the direction of multiplication for each of the six cells in a 3x3 table. The frequency in the darker-shaded cell is multiplied by the product of the frequencies in the lighter-shaded cells. Figure : Calculating T y =00 =00 =56 =4 =75 =3 Hence, T y = = 986. There are 986 pairs tied on the dependent variable burnout. 3

4 To find T x, we follow the same logic, but in the opposite direction as shown in Figure. Figure : Calculating T x =360 =56 =04 =80 =65 =05 Hence, T x = = 970. There are 970 pairs tied on the independent variable length of service τ b = ( )( ) 33 = = τ c = = 00 [( 3 -)/3] = = = [.667] [.667] Somers' d = = = Gamma, as calculated in Chapter 4, equals Hence the Tau and Somers' d statistics are lower than gamma. This will always be the case (except when there are no ties) because the denominators in these statistics include ties and will therefore be larger than the denominator of gamma. Using Table 3.3 (as opposed to Table 4. which is designed to reflect the higher value produced by gamma ) as a guide to interpret the Tau statistics and Somers' d, these measures all indicate a moderately strong and positive association between length of service and burnout. 4

5 5. Which Measure to Use? There is no hard and fast rule on which ordinal measure of association to use. When there are relatively few ties, gamma should be used because it has a PRE interpretation and the others (tau statistics and Somers d) do not. When there are many ties (which becomes more likely as table size decreases), gamma tends to exaggerate the real strength of relationship, so an adjusted measure (one that takes ties into account) should used; that is, either tau-b when tables are square (e.g., Table 4.) or tau-c when tables are nonsquare, or alternatively, Somers d when one of the variables is specifically identified as the dependent. 5