Renáta Németh, Dávid Simon
ELTE
Supplemental topic not covered in the exam. We discuss it in order to assign a measure of variability to each level of measurement.
With nominal or ordinal variables.
Takes values from 0 to 1.
If all the cases are in the same category, there is no variability and IQV is 0.
In contrast, when the cases are distributed uniformly across the categories, there is maximum variability and IQV is 1.
Example (ISSP, 1998, Hungary). Education by employment status.
Education | ||||
Below high school |
High school |
College or university |
Total | |
Self employed |
27 35.5% |
32 42.1% |
17 22.4% |
76 100.0% |
Employee |
516 62.6% |
195 23.7% |
113 13.7% |
824 100.0% |
Level of education is more homogeneous within employees: two third of them do not have a high school degree. Compute the IQV within the two groups.
IQV = number of observed differences / maximum possible differences
Calculating the number of observed differences
Consider the sample below:
János without high school degree István university degree Károly university degree Ildikó high school degree
the pairs below differ in their level of education:
János-István János-Károly János-Ildikó István-Ildikó Károly-Ildikó
So there are 5 “differences”. A simpler way is to follow the formula:
without high school degree 1 person university degree 2 persons high school degree 1 person
Different pairs: without high school degree vs. university degree – 2 pairs, without high school degree vs. high school degree – 1 pair, university degree vs. high school degree – 2 pairs, a total of 5 pairs.
If we have K categories, and f_{i} denotes the frequency in the ith category, the following formula gives the result:
Σ_{i=1..K, j=(i+1)..K}f_{i}f_{j}
Following the formula for self-employeds, the number of observed differences is:
27*32+27*17+32*17=1,867
For employees:
516*195+516*113+195*113=180,963
Education | ||||
Below high school |
High school |
College/university |
Total | |
Self employed |
27 |
32 |
17 |
76 |
Employee |
516 |
195 |
113 |
824 |
Calculating the maximum number of possible differences
Follow this formula:
(K(K-1)/2)*(N/K)^{2}
where K is the number of categories of the variable, and N is the sample size.
For self-employeds:
(3*2/2)*(76/3)^{2} = 1,925
For employees:
(3*2/2)*(824/3)^{2} = 226,325
Calculating IQV
IQV = number of observed differences / maximum possible differences
For self-employeds: 1867/1925 = 0.97
For employees: 180,963/226,325 = 0.8
That is, the values of IQV support our earlier observation: level of education has a lower level of variability within employees.
In the previous example IQV was calculated for an ordinal variable. However, the IQV is not sensitive to the ordering of the categories. Its application with ordinal variables causes some information deficit.
Remark:
IQV can be calculated from percentages as well.
In the previous example, IQV for self-employeds is (35.5*42.1+35.5*22.4+42.1*22.4)/((3*2/2)*(100/3)^{2}) = 0.97
Example
Racial diversity in eight states of the USA in 2006 (categories are white/black/Asian/Latino/other (Native American etc)). Interpret the data.
State |
IQV |
Hawaii |
0.89 |
California |
0.84 |
New York |
0.63 |
Alaska |
0.54 |
Washington |
0.33 |
Florida |
0.48 |
Maine |
0.05 |
Vermont |
0.05 |
Source: Frankfort-Nachmias