The Controlled rounds (Engl. Controlled rounding ) in the, market research and random allocation called, is a method in two-layered populations disproportionately stratified sample of relatively small extent in as undistorted winning ways. Imagine, for example, that through 1000 telephone interviews in the population of at least 18-year-old citizens of the FRG you want to inquire about the attitude towards electromobility, whereby the population is stratified twice, once according to the 16 federal states and the other according to 10 different types of municipalities. There are then 16 x 10 = 160 different layer cells. Mostly proportionally stratified samples are used and as a rule non-integer sample sizes are obtained per stratified cell. For example, 2.3 people in federal state 3 and municipality type 5 are to be interviewed. Simply rounding up or down in the classic way quickly leads to contradictions, as the greatly simplified example below shows. The term controlled rounding goes back to Goodman / Kish (1950).
Greatly simplified example
Both layers each have only two characteristics, A and B or a and b. Assume that a proportionally stratified sample of size n = 10 results in the following sample sizes in the 2 x 2 = 4 stratified cells:
layers
A.
B.
Line total
a
3.1
0.2
3.3
b
0.4
6.3
6.7
Column total
3.5
6.5
10
The total sample size n = 10 should be divided so that 3.1 attempts on cell (a, A), 0.2 on cell (a, B), 0.4 on (b, A) and 6.3 omitted from (b, B). If you now round all the entries in the above table in the classic way, the result is (row and column headings are now omitted):
3
0
3
0
6th
7th
4th
7th
10
You can see many contradictions. The sum of the layer cell circumferences is not 10, the row and column sums are incorrect. The fact that, in classic rounding, stratified cells with values less than 0.5 have no chance of entering the sample also has a distorting effect. The idea of Goodman / Kish is now as follows: Find a random tableau with integer cell entries that correctly complies with the boundary conditions (n = 10, row and column sums) and that is undistorted in the sense that the expected value is equal to the starting table. In the example it is easy to find a random tableau with four possible realizations that meets the requirements mentioned. In the following table only the 2 x 2 = 4 rounded layer cell values are given:
probability
0.1
0.2
0.4
0.3
realization
4th
0
0
6th
3
1
0
6th
3
0
1
6th
3
0
0
7th
In practice it is now the case that one randomly selects one of the four possible tableaus according to the given probabilities and carries out the sample survey accordingly.
General case
In practice you have to deal with much larger panels. The great disadvantage of the Goodman / Kish method is that it is strongly heuristic and, in particular, hardly programmable. L. Cox (1987) chose another, much more successful approach. In an iterative process, he randomly rounds at least one cell of the tableau in each step. Again, the result is a random allocation that is unbiased; H. their expected value agrees with the non-rounded output table. In Germany, K. Rappl (1993) was the first to deal with controlled rounding. Today, for example, the Cox algorithm is used in the ADM sampling system for telephone interviews.
Individual evidence
↑ Goodman, R. and Kish, L .: Controlled Selection - A Technique in Probability Sampling , Journal of the American Statistical Association Vol. 45, 350-372 (1950) [1]
↑ Cox, LH: A Constructive Procedure for unbiased Controlled Rounding , Journal of the American Statistical Association, Vol. 82, 520-525 (1987) [2]
↑ Rappl, K .: Controlled random selection in market research , dissertation, Uni Erlangen-Nürnberg, 1993
