Chapter 14 in 1eCh. 12 in 2/3 Can. Ed.

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

Introduction to Gamma 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?

Introduction to Gamma (cont.)

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 χ2 and conduct a hypothesis test.

Introduction to Gamma (cont.) 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?

Introduction to Gamma (cont.) To compute Gamma, two quantities must be

found: Ns is the number of pairs of cases ranked in the

same order on both variables. Nd 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 (Ns – Nd) to the total number of cases (Ns + Nd).

Computing Gamma 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 Ns is greater than Nd, the ratio with be positive, and when Ns is less than Nd the ratio will be negative.

Formula for Gamma

Formula for Gamma:

ds

ds

nn

nnG

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.

Efficiency (y) Low High

Low 10 12 22

High 17 5 22

Total 27 17 44

Authoritarianism (x)

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.

Ns= 10(5) = 50

Efficiency (y) Low High

Low 10 12 22

High 17 5 22

Total 27 17 44

Authoritarianism (x)

Simple Example (cont.)For Nd, start with the High-Low cell (upper right) and multiply each cell frequency by the cell frequency below and to the left. Nd= 12(17) = 204

Efficiency (y) Low High

Low 10 12 22

High 17 5 22

Total 27 17 44

Authoritarianism (x)

Simple Example (cont.)

Using the table, we can see that G =-0.61 is a strong negativeassociation.

61.0254

154

20450

20450

ds

ds

nn

nnG

Value Strength

Between 0.0 and 0.30

Weak

Between 0.30 and 0.60

Moderate

Greater than 0.60

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?

Part A: Calculating Gamma 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, Ns is 11(10+9+9+9) + 6(9+9) + 9(9+9) + 10 (9) = 767

Efficiency (y) Low Moderate High Total

Low 11 6 7 24

Moderate 9 10 9 28

High 5 9 9 23

Totals 25 25 25 75

Test Scores (x)

Part A: Calculating Gamma (cont.) For Nd, 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, Nd = 7 (10+9+9+5) + 6 (9+5) + 9(9+5) + 10 (5) = 491

Efficiency (y) Low Moderate High Total

Low 11 6 7 24

Moderate 9 10 9 28

High 5 9 9 23

Totals 25 25 25 75

Test Scores (x)

Part A: Calculating Gamma (cont.)

Using the previous table to interpret the strength of gamma, we see that G =+0.22 is a weak positive association.

22.01258

267

491767

491767

ds

ds

nn

nnG

Part A: Calculating Gamma (cont.) 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.

Part B: Testing Gamma for Significance 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 Ho: γ=0, H1: 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 :

Calculate:

21 GN

nnGz ds

92.0)19.4(22.22.175

49176722.

2

z

Part B: Testing Gamma for Significance (cont.) Step 5: Make Decision and Interpret

Zobs=.92 < Zcrit= +/-1.96 Fail to reject Ho

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.

Kendall’s Tau-b* (not in 1st Can. Ed.)*do not need to calculate – for SPSS only 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 Tau-c, Somer’s D and Spearman’s rho

Using SPSS to Calculate Gamma 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 Z-test to test Gamma for significance. Compare the significance of Gamma (this is the p-value) to your alpha value. If your p-value is less than your alpha, then the association is significant.

Practice Questions:

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.)