Association Between Variables
Measured at the Ordinal Level
Using the Statistic Gamma and
Conducting a Z-test for Significance

 Gamma is the preferred measure to test strength and direction of two ordinal-level variables that have been arrayed in a bivariate table.


Before computing and interpreting Gamma, it is always useful to find and interpret the column percentages.
Gamma can answer the questions:

1. Is there an association?


2. How strong is the association?
3. What direction (because level is ordinal) is it?

 Gamma can also be tested for significance using a Z or t-test to see if the association (relationship) between two ordinal level variables is significant.
 In this case, you would use the 5 step method, as for χ 2 and conduct a hypothesis test.

 Like Lambda, Gamma is a PRE (Proportional
Reduction in Error) measure: it tells us how much our error in predicting y is reduced when we take x into account.
 With Gamma, we try to predict the order of pairs of cases (predict whether one case will have a higher or lower score than another)
 For example, if case A scores High on
Variable1 and High on Variable 2, will case B also score High-High on both variables?

 To compute Gamma, two quantities must be found:

N s is the number of pairs of cases ranked in the same order on both variables.

N d is the number of pairs of cases ranked differently on the variables.
 Gamma is calculated by finding the ratio of cases that are ranked the same on both variables minus the cases that are not ranked the same (N cases (N s s
+ N
– N d
).
d
) to the total number of

 This ratio can vary from +1.00 for a perfect positive relationship to -1.00 for a perfect negative relationship. Gamma = 0.00 means no association or no relationship between two variables.
 Note that when N s ratio with be positive, and when N than N d is greater than N d
, the is less the ratio will be negative.
s

 Formula for Gamma:
G
 n s n s
 n d
 n d

A Simple Example for Gamma using
Healey #12.1 in 1e or #11.1 in 2/3 e
 As previously seen, this table shows the relationship between authoritarianism of bosses (X) and the efficiency of workers (Y) for 44 workplaces. Since the variables are at the ordinal level, we can measure the association using the statistic Gamma, which is a better statistic to use for ordinal level variables.
Authoritarianism (x)
Efficiency (y) Low High
Low
High
Total
10
17
27
12
5
17
22
22
44

Simple Example (cont.)
For Ns, start with the Low-Low cell (upper left) and multiply the cell frequency by the cell frequency below and to the right .
 N s
= 10(5) = 50
Authoritarianism (x)
Efficiency (y) Low
Low 10
High
12 22
High 17 5 22
Total 27 17 44

Simple Example (cont.)
For N d
, start with the High-Low cell (upper right) and multiply each cell frequency by the cell frequency below and to the left .
 N d
= 12(17) = 204
Authoritarianism (x)
Efficiency (y) Low
Low 10
High
12 22
High 17 5 22
Total 27 17 44

Simple Example (cont.)
G
 n s n s
 n d
 n d

50

204
50

204


154
254
 
0 .
61
Using the table, we can see that G =-0.61 is a strong negative association.
Value
Between 0.0 and 0.30
Between 0.30 and 0.60
Greater than
0.60
Strength
Weak
Moderate
Strong

Simple Example (cont.)
 In addition to strength, gamma also identifies the direction of the relationship. We can look at the sign of Gamma (+ or -). In this case, the sign is negative (G = - 0.61).
 This is a negative relationship: as
Authoritarianism increases, Efficiency decreases.
 In a negative relationship, the variables change in opposite directions.

Example: Healey #14.7 (1e), #12.7 in 2/3e)
 This question involves a more complicated calculation for Gamma. The question asks if aptitude test scores are related to job performance rating for
75 city employees.
 Part a.



Are the two variables, Aptitude, (measured as Low,
Medium and High) and Job Performance (Low,
Medium, and High) associated?
How strong is this association?
What direction is the association?
 Part b.

Is the association significant?



For Ns , start with the Low-Low cell (upper left) and multiply each cell frequency by total of all cell frequencies below and to the right and add together.
For this table, N s is 11(10+9+9+9) + 6(9+9) + 9(9+9) + 10 (9) = 767
Efficiency (y)
Low
Low
Test Scores (x)
Moderate High
11 6 7
Moderate
High
Totals
9
5
25
10
9
25
9
9
25
Total
24
28
23
75



For N d
, start with High-Low cell (upper right) and multiply each cell frequency by total of all cell frequencies below and to the left and add together.
For this table, N d
= 7 (10+9+9+5) + 6 (9+5) + 9(9+5) + 10 (5) = 491
Efficiency (y)
Test Scores (x)
Low Moderate High Total
Low 11 6 7 24
Moderate
High
Totals
9
5
25
10
9
25
9
9
25
28
23
75

G
 n s n s
 n d
 n d

767
767

491

491

267
1258

0 .
22
Using the previous table to interpret the strength of gamma, we see that G =+0.22 is a weak positive association.

 As noted before, gamma also identifies the direction of the relationship. We can look at the sign of Gamma (+ or -). In this case, the sign is positive (G = + 0.22).


This is a positive relationship: as Aptitude
Test Scores increase, Job Performance increases.
Next, we test the association for significance, using the 5 step method.

 The test for significance of Gamma is a hypothesis test, and the 5 step model should be used.
 Step 1: Assumptions

Random sample, ordinal, Sampling Dist. is normal


Step 2: Null and Alternate hypotheses

H o
: γ=0, H
1
: γ≠0 (Note: γ is the population value of G)
Step 3: Sampling Distribution and Critical Region

Z-distribution, α = .05, z = +/-1.96

Part B: Testing Gamma for Significance (cont.)
 Part 4: Calculating Test Statistic:
 Formula : z

G
N n s
 n
1

G d 
2

 Calculate: z

.
22
767
75 1



491
.
22
2
 
.
22 ( 4 .
19 )

0 .
92

Part B: Testing Gamma for Significance (cont.)
 Step 5: Make Decision and Interpret


Z obs
=.92 < Z crit
= +/-1.96
Fail to reject H o


The association between aptitude tests and job performance is not significant.
*Part C: No, the aptitude test should not be continued, because there is no significant association.

st
 The statistic Tau-b is the preferred measure of strength to report when a bivariate table has many “tied pairs” (when cases are scored the same on both variables in a table)
 In this case, gamma will tend to overestimate the strength of the association.


Rule of thumb: when the value of gamma is double that of Tau-b, report Tau-b instead, because it will be a better measure of strength.
* omit Tauc, Somer’s D and Spearman’s rho

 Go to Analyze>Descriptives>Crosstabs (as with Chi-square). Click on Cells for column % and on Statistics, asking for both Gamma and
Tau b.
 Note that SPSS uses a t-test rather than a Ztest to test Gamma for significance. Compare the significance of Gamma (this is the pvalue) to your alpha value. If your p-value is less than your alpha, then the association is significant.

 Healey #14.8 (1e) or #12.8 (2/3e)
 (The solution to this question can be found in the Final Review powerpoint.)
 Also try: Lambda #11.4 (2/3e) or 13.4 (1e)
 And Gamma #12.4 (2/3e) or 14.4 (1e)
 (The answers to these questions can be found in the Lecture list.)