Optimal Scaling: Discretization

Cleophas, Ton J.; Zwinderman, Aeilko H.

doi:10.1007/978-94-007-5824-7_3

Ton J. Cleophas³ &
Aeilko H. Zwinderman⁴

3773 Accesses
1 Citations

Abstract

In clinical trials the research question is often measured with multiple variables, and multiple regression is commonly used for analysis. The problem with multiple regression is that consecutive levels of the variables are assumed to be equal, while in practice this is virtually never true. Optimal scaling is a method designed to maximize the relationship between a predictor and an outcome variable by adjusting their scales.

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Author information

Authors and Affiliations

European College Pharmaceutical Medicine, Lyon, France
Ton J. Cleophas
Department of Epidemiology and Biostatistics, Academic Medical Center, Amsterdam, Netherlands
Aeilko H. Zwinderman

Authors

Ton J. Cleophas
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Aeilko H. Zwinderman
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Appendix: Datafile of 250 Subjects Used as Example

G1	G2	G3	G4	G16	G17	G18	G19	G24	G25	G26	G27	O1	O2	O3	O4
8	8	9	5	7	10	5	6	9	9	6	6	6	7	6	7
9	9	10	9	8	8	7	8	8	9	8	8	8	7	8	7
9	8	8	8	8	9	7	8	9	8	9	9	9	8	8	8
8	9	8	9	6	7	6	4	6	6	5	5	7	7	7	6
10	10	8	10	9	10	10	8	8	9	9	9	8	8	8	7
7	8	8	8	8	7	6	5	7	8	8	7	7	6	6	7
5	5	5	5	5	6	4	5	5	6	6	5	6	5	6	4
9	9	9	9	8	8	8	8	9	8	3	8	8	8	8	8
9	8	9	8	9	8	7	7	7	7	5	8	8	7	6	6
10	10	10	10	10	10	10	10	10	8	8	10	10	10	9	10
2	2	8	5	7	8	8	8	9	3	9	8	7	7	7	6
7	8	8	7	8	6	6	7	8	8	8	7	8	7	8	8
8	9	9	8	10	8	8	7	8	8	9	9	7	7	8	8
7	7	8	8	8	9	10	7	9	4	8	8	9	8	7	7
3	4	3	8	4	4	4	3	4	3	4	4	4	4	3	4
7	8	8	5	8	8	7	6	7	7	8	7	10	8	8	7
8	8	8	8	6	8	5	1	9	7	7	8	7	7	8	6
7	8	8	8	8	9	8	7	10	10	9	8	9	9	9	9
8	4	3	8	3	5	5	3	2	10	1	0	5	3	4	3
8	7	6	10	8	8	7	6	4	4	5	5	7	7	7	5
9	9	10	8	8	9	7	7	8	9	8	9	8	7	8	7
6	6	6	6	4	5	4	5	3	9	3	4	4	5	4	3
8	8	8	7	7	7	8	6	8	7	9	4	6	7	8	9
9	9	10	9	10	10	7	10	10	10	10	10	8	8	8	5
8	7	8	8	9	8	9	8	8	8	8	8	8	8	8	9
8	5	5	4	2	1	1	0	0	1	0	0	3	2	4	5
6	6	6	6	5	6	3	5	4	4	4	5	5	6	3	4
7	8	9	8	8	9	9	6	9	8	8	10	9	8	7	7
8	8	8	7	7	7	7	6	7	8	7	8	7	6	6	6
8	8	8	8	9	8	9	8	9	8	9	9	9	8	7	8
7	7	7	6	7	7	9	7	7	7	7	8	8	6	7	7
9	9	9	9	6	9	8	7	8	8	8	9	8	8	8	8
10	10	10	10	9	9	10	5	10	2	9	9	8	10	8	8
9	8	9	9	8	7	7	8	9	9	9	9	8	5	9	7
8	9	9	9	8	7	7	6	7	8	8	8	8	7	8	6
3	4	2	5	4	2	2	4	4	4	3	4	6	2	3	2
8	8	9	9	8	8	8	8	8	8	8	8	8	8	8	8
8	6	7	6	7	7	8	6	7	6	5	5	6	7	7	6
10	10	10	10	7	10	10	8	10	10	10	10	10	8	8	7
8	10	9	8	8	8	7	6	7	7	10	8	9	8	8	7
8	8	8	8	8	8	7	8	7	8	8	8	9	9	8	7
5	7	7	8	5	7	7	3	1	6	3	10	5	6	6	5
10	9	9	10	7	9	9	9	9	9	9	8	8	9	7	7
9	7	7	9	3	6	4	2	1	8	2	1	6	6	6	6
8	8	10	8	9	8	7	8	8	7	8	8	9	6	5	7
6	8	8	8	9	10	10	9	10	9	9	10	9	8	5	5
8	8	8	8	10	8	7	10	8	8	7	10	9	7	8	6
6	5	5	6	6	6	4	6	3	5	0	3	7	5	5	3
9	9	9	8	8	9	8	7	6	7	8	10	8	8	8	6
9	10	8	8	9	10	10	9	7	8	9	7	8	8	7	7
8	8	8	9	6	8	7	6	8	9	8	8	7	7	6	5
8	5	6	7	8	8	7	7	4	6	7	6	8	8	7	6
4	1	4	9	0	0	7	0	0	10	0	10	0	0	0	0
5	5	7	5	7	7	8	5	7	7	5	5	7	7	7	7
5	5	6	5	4	4	4	3	3	2	3	3	3	4	3	3
7	9	9	10	5	9	9	9	9	6	7	6	10	7	10	9
10	10	10	10	8	9	9	6	7	8	8	10	7	7	7	6
8	8	8	8	6	9	8	7	7	6	6	2	7	7	7	5
6	6	7	9	8	8	7	6	1	9	0	4	6	7	7	6
6	7	7	7	6	5	5	5	5	7	3	5	7	6	6	8
9	9	9	9	8	8	9	6	8	7	6	10	8	7	7	8
7	7	7	7	6	8	8	6	7	7	7	8	6	6	5	10
9	7	8	9	8	10	8	9	8	9	7	9	7	7	8	3
8	9	9	8	7	8	7	8	8	6	7	8	7	8	7	6
8	8	8	8	6	8	8	5	8	9	8	7	7	7	6	5
7	7	7	7	4	5	6	6	3	6	7	7	1	5	6	5
9	10	9	9	8	9	8	8	9	8	9	9	8	7	8	8
8	9	9	8	8	8	8	7	8	7	8	9	6	6	5	6
7	8	8	8	6	7	7	6	8	5	7	7	7	6	7	5
4	2	2	6	5	5	4	4	6	4	3	2	4	6	7	2
5	5	7	5	5	5	5	2	2	9	5	5	4	5	5	4
9	9	10	9	7	8	7	8	8	9	8	8	8	8	6	9
8	8	8	8	7	7	7	9	8	9	7	8	7	7	5	6
8	8	9	8	8	9	5	9	8	5	7	6	8	8	8	6
9	9	9	9	6	8	8	4	7	5	6	6	7	7	8	8
9	8	8	8	7	9	9	9	10	10	10	10	10	9	7	10
9	9	9	8	8	8	8	7	7	7	7	7	8	8	8	7
8	5	7	9	2	8	8	2	9	10	1	9	5	5	5	5
7	6	9	8	5	7	7	6	5	7	4	4	6	7	6	7
8	8	9	8	6	7	7	6	8	7	7	10	8	7	8	6
10	10	10	10	8	10	10	7	8	8	7	8	9	9	9	7
9	9	6	6	4	5	5	5	2	3	5	4	2	3	3	3
3	3	3	8	0	7	0	0	0	7	0	10	0	0	8	8
5	4	4	7	4	4	4	2	0	4	2	8	3	3	3	3
8	10	10	10	7	8	7	10	10	9	8	10	10	9	9	8
5	8	8	8	7	8	8	6	7	7	7	10	7	8	6	6
7	4	5	9	5	8	7	5	5	8	0	7	6	6	6	6
5	6	5	8	10	9	0	8	8	8	8	5	8	8	5	4
7	5	7	6	3	6	6	3	5	6	6	5	5	5	5	5
10	8	9	8	8	8	8	6	8	8	6	6	8	7	5	8
10	10	10	10	10	10	8	10	9	10	10	10	10	9	9	8
6	6	4	5	0	5	5	5	5	8	5	9	6	4	5	5
10	3	7	9	0	5	7	7	10	8	10	10	5	5	5	5
5	7	8	7	8	7	8	7	8	6	7	6	8	6	7	6
9	10	9	9	10	6	6	7	9	8	8	8	10	7	7	10
10	10	10	10	9	10	10	9	10	10	10	10	9	9	9	9
10	10	10	10	10	10	10	8	10	10	6	10	7	6	8	8
7	7	7	8	7	8	8	6	8	8	7	7	8	7	8	8
9	5	7	9	6	8	8	4	6	7	4	5	6	5	5	4
9	9	10	8	8	9	8	7	8	8	7	9	8	7	5	7
6	6	5	4	4	4	4	3	4	3	4	5	4	5	5	5
7	8	8	9	7	5	4	7	10	8	8	8	6	4	4	7
8	6	6	5	7	6	0	8	7	9	7	7	7	7	6	7
6	8	8	9	9	9	9	5	9	8	7	9	9	5	5	9
9	5	6	7	10	10	8	7	8	9	10	10	8	8	7	8
8	7	8	5	8	7	4	5	8	5	5	9	3	5	3	5
7	8	7	4	8	8	8	7	7	6	6	7	8	7	7	7
8	7	10	10	10	10	10	10	10	10	10	10	9	6	4	8
5	9	10	5	9	9	6	8	10	10	10	9	8	7	9	9
9	6	6	7	10	10	6	6	9	10	10	9	10	10	10	9
0	4	7	5	10	8	9	9	9	7	8	7	8	9	9	9
4	8	8	6	9	9	7	2	9	9	9	9	8	8	8	7
8	8	10	8	7	7	5	5	5	10	8	3	7	7	6	7
9	10	10	7	5	4	0	7	10	10	10	10	5	4	5	9
10	10	10	10	7	0	0	8	2	8	1	0	4	5	3	3
10	8	8	8	5	5	8	8	10	10	10	10	6	6	5	5
7	10	10	8	10	10	8	8	10	9	10	10	7	8	10	6
10	9	9	6	9	9	0	9	10	8	9	9	8	7	10	7
8	10	8	5	7	6	5	7	10	10	10	10	6	6	7	7
10	8	8	7	8	8	7	5	10	8	8	10	8	8	7	8
8	7	8	8	10	10	2	1	8	10	8	8	9	7	9	10
8	8	8	8	6	7	7	4	8	8	7	7	7	5	6	7
7	9	8	8	9	8	8	7	9	9	9	7	10	9	7	7
8	8	9	9	7	7	8	7	7	8	7	7	7	7	8	8
8	7	8	7	8	8	8	7	8	8	7	8	8	8	8	7
8	7	7	8	7	7	8	7	8	8	7	8	7	7	8	7
8	8	8	8	7	6	8	6	9	8	7	9	8	8	6	6
8	8	8	9	9	6	8	9	8	9	10	10	8	8	8	5
7	8	8	6	8	9	9	6	8	8	8	8	8	8	6	8
7	9	9	8	6	8	8	5	8	7	5	9	7	5	7	4
10	10	10	8	9	8	8	8	10	10	10	10	10	10	9	9
6	8	7	8	9	8	10	8	8	9	9	8	8	7	7	5
8	8	8	8	8	8	8	8	8	8	5	10	8	8	8	7
10	0	0	10	0	7	5	0	0	3	0	10	0	0	0	0
8	5	9	4	6	8	8	5	6	6	4	5	6	5	5	4
9	9	9	9	8	8	8	7	7	3	0	9	7	7	8	8
8	9	8	8	8	8	8	8	8	9	9	8	8	8	9	5
7	7	7	7	7	7	7	5	7	7	7	5	8	7	5	6
9	9	9	9	7	7	8	8	8	7	8	6	8	6	6	7
5	7	4	10	0	10	10	0	5	5	0	10	0	0	0	0
9	9	9	9	9	10	10	9	10	10	10	10	10	10	5	5
8	8	9	7	7	8	8	7	8	7	7	8	8	8	6	8
9	10	10	7	9	9	8	4	9	9	9	8	8	7	9	9
10	10	10	10	10	10	9	7	10	10	10	9	7	7	5	9
8	6	9	9	7	9	8	5	6	6	5	5	6	7	5	4
7	7	8	5	8	8	7	6	5	5	7	4	5	6	6	6
9	10	10	10	9	8	9	8	8	8	8	9	9	8	6	7
7	7	6	6	4	6	6	4	4	6	3	5	4	4	4	4
8	8	8	8	9	8	7	9	10	3	7	10	9	8	7	7
8	8	8	8	7	8	5	8	10	10	7	10	8	7	7	7
10	10	10	10	10	10	10	10	10	10	10	9	10	10	10	10
10	10	10	10	9	10	10	9	10	10	10	10	9	9	9	8
9	10	10	10	8	10	10	8	10	10	10	10	9	8	8	7
4	6	8	8	7	7	7	5	4	7	5	9	6	6	7	5
8	8	8	7	7	8	9	7	7	5	7	4	8	9	9	9
8	8	8	8	6	7	7	4	6	10	6	6	7	7	7	5
8	8	4	8	5	5	5	1	0	5	0	10	2	2	2	2
7	7	7	7	7	8	8	4	7	7	6	6	6	6	6	6
8	7	7	8	10	9	8	9	10	9	8	9	9	8	7	8
9	9	7	8	9	8	8	8	8	8	9	8	9	7	8	6
5	3	4	3	4	5	3	5	2	3	5	4	4	2	4	7
6	8	8	8	9	9	8	7	9	8	9	10	8	8	7	7
9	10	10	10	6	8	9	8	0	10	10	10	10	9	6	9
4	5	5	7	4	4	5	4	2	4	2	7	5	5	3	3
8	8	8	8	10	10	10	10	10	10	10	8	10	7	7	7
9	9	9	9	10	8	8	8	8	8	7	8	9	9	8	8
10	10	10	10	8	8	8	8	8	8	8	9	9	9	8	8
10	10	10	9	10	10	10	10	10	10	10	10	10	10	10	10
10	10	10	10	7	5	5	5	6	8	8	5	8	5	5	10
7	8	8	8	4	5	5	4	5	4	5	8	7	6	8	4
8	8	8	8	5	8	8	5	5	5	5	7	6	6	5	5
8	6	8	5	5	5	5	3	3	9	3	2	5	3	5	3
10	10	10	10	10	10	10	10	9	10	10	10	10	9	10	10
7	7	7	7	7	8	8	5	6	7	7	9	6	7	5	5
8	7	7	8	8	9	5	5	6	7	6	5	7	7	6	6
10	10	10	10	9	10	10	10	9	10	10	10	10	9	10	5
7	9	9	9	8	9	8	8	9	8	8	7	9	10	8	8
9	8	8	8	9	9	8	7	10	8	9	10	9	8	7	8
8	6	6	7	5	7	5	4	5	2	5	5	6	5	5	4
8	9	9	9	6	8	7	6	6	5	5	7	7	6	7	6
7	8	9	9	9	10	10	7	10	5	8	8	10	10	5	9
9	8	8	8	8	9	7	8	0	5	7	10	8	8	9	2
10	10	10	10	6	10	7	8	10	9	2	8	9	9	7	6
10	10	9	10	10	10	10	9	10	10	10	10	10	10	9	10
8	9	9	8	8	8	8	8	8	8	8	8	8	8	9	8
8	10	10	10	8	8	8	8	9	9	9	8	9	8	9	8
8	8	8	5	5	8	8	8	6	8	10	5	7	7	5	7
6	6	7	7	6	7	5	2	5	5	5	0	6	10	6	6
10	10	10	10	5	10	10	10	10	10	10	10	10	10	5	10
8	7	8	8	7	9	9	7	6	8	8	8	7	7	5	6
8	7	8	7	8	8	8	8	9	9	8	9	8	7	7	6
7	7	7	8	8	9	8	7	8	8	8	9	7	7	7	7
10	10	10	10	10	10	10	10	10	10	10	10	10	10	10	10
10	10	10	9	7	9	9	7	8	8	8	7	8	8	8	8
10	10	10	10	10	10	10	5	10	10	10	10	9	10	9	9
10	10	10	10	10	10	10	10	10	10	10	10	10	10	9	9
10	10	10	9	10	10	9	9	10	6	10	10	10	10	7	9
7	9	9	8	9	10	9	8	8	8	8	8	8	7	5	7
9	9	9	9	9	9	8	8	9	9	8	7	9	8	8	8
6	5	5	7	1	5	6	5	5	10	5	10	3	0	5	5
10	10	10	10	7	10	10	10	10	10	10	10	10	10	5	10
8	9	10	9	9	10	9	9	9	10	10	9	10	9	10	9
6	8	8	9	3	8	5	5	5	5	7	6	5	5	6	6
9	9	9	9	5	8	5	6	9	9	8	10	8	8	8	8
8	9	9	8	5	8	8	8	8	8	7	9	7	7	5	7
6	7	7	7	6	6	6	3	3	6	0	6	5	5	5	5
8	8	8	9	7	8	8	8	5	8	7	10	7	7	7	6
8	8	9	6	6	7	5	5	10	5	0	10	7	7	5	5
8	9	9	7	6	7	7	6	9	7	7	7	7	6	7	7
8	4	6	7	3	6	6	6	0	6	0	9	6	5	4	6
9	9	9	9	9	8	8	8	7	8	8	8	8	8	8	8
6	7	7	6	6	6	4	4	5	6	8	5	2	3	3	4
6	7	7	7	4	6	4	4	4	8	4	5	6	7	7	5
8	7	7	9	7	10	5	6	8	8	6	9	6	7	6	7
10	10	10	9	8	7	8	7	8	9	9	8	5	5	5	4
8	7	8	10	8	9	6	7	8	7	8	8	8	8	7	8
8	9	7	8	9	8	8	7	8	7	5	9	6	8	8	8
7	7	5	7	8	8	6	6	9	7	8	8	7	7	6	7
9	9	10	8	8	8	6	5	10	10	10	10	7	7	5	6
8	6	9	9	8	9	8	9	8	7	7	8	9	9	7	8
7	7	8	9	7	7	7	8	7	8	9	7	6	8	7	7
7	7	8	7	8	7	8	7	8	8	6	5	7	8	7	7
9	10	9	9	8	7	9	9	6	6	6	6	7	9	8	8
7	7	7	6	6	6	9	9	8	3	5	8	6	9	9	8
9	10	7	8	7	5	10	10	10	10	7	10	6	8	9	7
4	6	5	7	4	4	4	3	10	9	10	9	4	4	3	4
8	8	8	8	8	8	7	6	10	8	10	10	10	8	8	7
8	6	3	8	6	8	5	1	10	7	10	10	7	7	8	6
7	7	8	7	8	8	8	6	4	6	5	9	7	7	5	6
9	10	9	9	8	7	7	9	7	4	7	4	5	9	8	6
10	10	8	9	7	6	5	8	6	7	6	6	7	9	9	5
4	6	9	8	9	9	7	7	0	1	0	0	8	7	9	4
8	8	7	7	4	6	5	5	4	4	4	5	3	5	4	3
8	8	8	8	7	7	8	6	8	7	8	9	6	7	7	9
8	8	3	4	9	8	7	6	6	8	6	8	7	8	8	5
6	8	8	7	5	7	7	7	8	7	8	8	7	7	7	7
9	10	9	8	7	8	8	8	6	7	6	8	8	8	8	8
4	5	5	7	6	9	6	8	9	9	8	8	8	10	10	7
6	8	8	8	8	8	8	6	5	4	6	6	9	9	9	8
9	9	7	7	4	6	5	3	3	5	3	6	4	3	7	5
8	9	5	5	7	8	7	6	6	7	6	6	7	6	7	4
10	10	8	7	7	8	6	5	6	8	5	7	5	6	8	8
7	9	8	7	7	8	10	8	9	8	7	8	9	8	8	7
9	9	9	9	6	3	4	4	7	8	6	8	6	3	4	4
6	5	5	7	6	7	6	4	9	8	8	9	7	7	7	6
9	9	9	9	9	10	10	8	7	6	8	8	8	8	8	7
8	9	9	7	9	9	8	8	3	5	3	6	7	6	7	8
8	8	8	8	4	5	5	8	8	2	8	7	8	7	8	6
8	9	9	4	8	9	9	7	6	7	8	6	9	7	8	7
10	9	7	7	7	8	8	8	8	7	5	7	7	7	7	6

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Cleophas, T.J., Zwinderman, A.H. (2013). Optimal Scaling: Discretization. In: Machine Learning in Medicine. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5824-7_3

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DOI: https://doi.org/10.1007/978-94-007-5824-7_3
Published: 04 November 2012
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-5823-0
Online ISBN: 978-94-007-5824-7
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Optimal Scaling: Discretization

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Appendix: Datafile of 250 Subjects Used as Example

Appendix: Datafile of 250 Subjects Used as Example

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