Gamma
ORDINAL DATA: Association
Gamma Measure of Association(ordinal
data): 
Where: fa = frequency of agreements in rank order
fi = frequency of inversions in rank order.
A. Procedure for data without ties:
1. Put one variable in order from largest to smallest (or
highest to lowest).
2. Rearrange other variables to correspond.
3. Count fa and fi on second variable:
a) fa for each score is the number of scores located above
it which are larger than it is. Sum these.
b) fi for each score is the number of scores located above
it which are smaller than it is. Sum these.
Example:
Q: What is the relationship between I.Q. and income?
A: Rearrange one of the variables from largest to smallest
(choose I.Q.), then rearrange the other variable.
| Person | I.Q. | Income |
| 001 | 138 | 950 |
| 002 | 101 | 750 |
| 003 | 108 | 600 |
| 004 | 115 | 800 |
| 005 | 140 | 1000 |
transform to a table in order of one variable (highest to lowest
here)
| Person | I.Q. | Income | fa | fi | |
| E | 140 | 1000 | 0 | 0 | = (9-1)/(9+1) |
| Q | 138 | 950 | 1 | 0 | = |
| D | 115 | 800 | 2 | 0 | 8/10 = +.80 |
| C | 108 | 600 | 3 | 0 | |
| B | 101 | 750 | 3 | 1 | |
| fa = 9 | 1=fi |
Interpretation:
Use above scale or Convert to a percent and include in the statements, "There is a ________ association between (variable 1) and (variable 2)." for a positive relationship - "There is ____% more agreement than disagreement in the rank order of (variable 1) and (variable 2)." For a negative relationship - "There is _____% more disagreement than agreement in the rank order of (variable 1) and (variable 2)."
Example: G = +.80 = large positive association between I.Q. and income. 80% more agreement than disagreement in the rank order of I.Q. and income.
B. Procedure for data with ties:
1. Put data in a table (in standard form).
2. To get fa, multiply the number in each cell by the sum
of the numbers in the cells located both below and to the right
of it (the numbers not in the same row or column). Start in the
left upper corner and work across the first row. Then go to the
next row and start at the left side working again across the row.
Sum these products.
3. To get fi, multiply the number in each cell by the sum
of the numbers in the cells located both below and to the left
of it. Continue as with fa but start with the right upper
corner and work to the left. Sum these products.
Example:
# OF CLASS ABSENCES
| HI | MED | LO | ||
| HI | 9 | 26 | 13 | |
| SUNNY DAYS | MED | 19 | 75 | 83 |
| LOW | 16 | 56 | 110 |
Q: What is the association between sunny days and class
absences?
A: Notice there are many ties in the data, but only one
tie is needed to qualify for procedure for data with ties. The
data is already in a (standard form) table.
# OF CLASS ABSENCES
| HI | MED | LO | ||
| HI | 9 | 26 | 13 | |
| SUNNY DAYS | MED | 19 | 75 | 83 |
| LOW | 16 | 56 | 110 |
fa = below and to the right = 9(75 + 83 + 56 + 110)+26(83
+ 110)+13(0) +19(56 + 110)+75(110)+83(0)
+16(0)+56(0)+ 110(0) = 19,338
* Right hand column and bottom row cells
both yield zeros so it is not necessary to include them
# OF CLASS ABSENCES
| HI | MED | LO | ||
| HI | 9 | 26 | 13 | |
| SUNNY DAYS | MED | 19 | 75 | 83 |
| LOW | 16 | 56 | 110 |
fi = below and to the left = 13(19 + 75 + 16 + 56) + 26(19 + 16) +9(0+83(16 + 56) +75(16) +19(0) + 110(0) +56(0)+ 16(0) = 10,244
*The left hand column and
bottom row cells both yield zeros so it is not necessary to include
them.
G = 19338 - 10244 = .31 Interpretation: Moderately
small positive relationship and 31%
19338 + 10244 more agreement than disagreement in rank order of
sunny days and class absences.