3. Measures of association - Gamma
Suppose you are involved in a research study investigating the possible relationship between household income and public high school graduation rates in your state. Using current yearly income and public school district data, you find the median household income and high school graduation rate for 1,834 public high schools in your state. You construct the following table showing the results of your research:
Median Income (X) Graduation Rate (Y)
Low (<$50k) High (>$50k) TOTALS
Low (<80%) 534 396 930
High (>80%) 420 484 904
TOTALS 954 880 1,834
*Fictitious Data
Since these are collapsed-ordinal variables (that is, ordinal variables with only a few scores), you decide to use gamma (G) to assess the strength and direction of the relationship. Recall that gamma compares the number of pairs of cases ranked in the same order (Ns) and the number of pairs ranked differently (Nd). Use the following steps to calculate gamma for this table:
Step 1: To compute Ns, multiply the number of cases in the upper-left cell by the number of cases in the bottom-right cell (ignore the Totals column and row). Ns = POSSIBLE ANSWERS (211,464, 191,664, 166,320, 258,456).
Step 2: To compute Nd, multiply the number of cases in the upper-right cell by the number of cases in the bottom-left cell (ignore the Totals column and row). Nd = (211,464, 191,664, 166,320, 258,456).
Step 3: Subtract Nd from Ns. The result is Ns – Nd = (424,776, -256,622, 92,136, -164,486).
Step 4: Add Nd and Ns. The result is Ns + Nd = (424,776, -256,622, 92,136, -164,486).
Step 5: Divide the value you found in Step 3 by the value you found in Step 4. The result is gamma = (4.6103, 0.2169, -4.6103, -0.2169).
This result indicates a (MODERATE, WEAK, STRONG) relationship between median income and high school graduation rate.
According to the proportional reduction in error, or PRE, interpretation of this analysis, taking income into account would improve predictions of graduation rates by (21.7, 461.0, 32.5, 10.8).
As median income increases, graduation rates tend to (STAY THE SAME, DECREASE, INCREASE). This indicates (A NEGATIVE, NO, A POSITIVE) relationship between the variables.