A Geometric Analysis of When Fixed Weighting Schemes Will Outperform Ordinary Least Squares

Psychometrika

Volumn 76, Issue 4, 2011, Pages 650-669

  Davis Stober, Clintin P ^a  

^a UNIVERSITY OF MISSOURI   (United States)

Author keywords

alternative weights; equal weights; fixed weights; hyper cylinders; improper models; mean squared error

Indexed keywords

EID: 80355144456     PISSN: 00333123     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11336-011-9229-1     Document Type: Article

References (55)

1. Azen, R., & Budescu, D. V. (2009). Applications of multiple regression in psychological research. In Millsap, R., & Olivares, A. M. (Eds.), Handbook of quantitative methods in psychology (pp. 285-310). Thousand Oaks: Sage.
2. Barron, F. H., & Barrett, B. E. (1996). Decision quality using ranked attribute weights. Management Science, 42, 1515-1523.
3. Baucells, M., Carrasco, J. A., & Hogarth, R. M. (2008). Cumulative dominance and heuristic performance in binary multiattribute choice. Operations Research, 56, 1289-1304.
4. Ben-Haim, Z., & Eldar, Y. C. (2005). Minimax estimators dominating the least-squares estimator. In ICASSP'05: Proceedings of the 30th IEEE international conference on acoustics, speech, and signal processing (Vol. 4, pp. 53-56). Washington: IEEE Computer Society.
5. Bickel, P. J., & Doksum, K. A. (2001). Mathematical statistics: basic ideas and selected topics (Vol. 1, 2nd ed.). Upper Saddle River: Prentice Hall.
6. Bobko, P., Roth, P. L., & Buster, M. A. (2007). The usefulness of unit weights in creating composite scores-a literature review, application to content validity, and meta-analysis. Organizational Research Methods, 10, 689-709.
7. Boothby, W. M. (2003). An introduction to differentiable manifolds and Riemannian geometry. San Diego: Academic Press.
8. Borda, J. C. (1781). Mémoire sur les élections au scrutin. Paris: Histoire de l'Académie Royale des Sciences.
9. Czerlinski, J., Gigerenzer, G., & Goldstein, D. G. (1999). How good are simple heuristics? In G. Gigerenzer, P. M. Todd, & the ABC Research Group (Eds.), Simple heuristics that make us smart (pp. 97-118). New York: Oxford University Press.
10. Dana, J. (2008). What makes improper linear models tick? In J. Krueger (Ed.), Rationality and social responsibility: essays in honor of Robyn Mason Dawes Mahwah: Lawrence Erlbaum Associates.
11. Dana, J., & Dawes, R. M. (2004). The superiority of simple alternatives to regression for social science predictions. Journal of Educational and Behavioral Statistics, 3, 317-331.
12. Davis-Stober, C. P., Dana, J., & Budescu, D. V. (2010a). Why recognition is rational: Optimality results on single-variable decision rules. Judgment and Decision Making, 5, 216-229.
13. Davis-Stober, C. P., Dana, J., & Budescu, D. V. (2010b). A constrained linear estimator for multiple regression. Psychometrika, 75, 521-541.
14. Dawes, R. M. (1971). A case study of graduate admissions: Applications of three principles of human decision making. American Psychologist, 26, 180-188.
15. Dawes, R. M. (1979). The robust beauty of improper linear models. American Psychologist, 34, 571-582.
16. Dawes, R. M., & Corrigan, B. (1974). Linear models in decision making. Psychological Bulletin, 81, 95-106.
17. Dorans, N., & Drasgow, F. (1978). Alternative weighting schemes for linear prediction. Organizational Behavior and Human Performance, 21, 316-345.
18. Einhorn, H. J., & Hogarth, R. M. (1975). Unit weighting schemes for decision making. Organizational Behavior and Human Performance, 13, 171-192.
19. Eldar, Y. C., Ben-Tal, A., & Nemirovski, A. (2005). Robust mean-squared error estimation in the presence of model uncertainties. IEEE Transactions on Signal Processing, 53, 168-181.
20. Fasolo, B., McClelland, G. H., & Todd, P. M. (2007). Escaping the tyranny of choice: When fewer attributes make choice easier. Marketing Theory, 7, 13-26.
21. Gigerenzer, G., & Goldstein, D. G. (1996). Reasoning the fast and frugal way: Models of bounded rationality. Psychological Review, 103, 650-669.
22. Gigerenzer, G., Todd, P. M., & the ABC Research Group (1999). Simple heuristics that make us smart. New York: Oxford University Press.
23. Goldstein, D. G., & Gigerenzer, G. (2002). Models of ecological rationality: The recognition heuristic. Psychological Review, 109, 75-90.
24. Goldstein, D. G., & Gigerenzer, G. (2009). Fast and frugal forecasting. International Journal of Forecasting, 25, 760-772.
25. Guttman, L. (1984). A corollary of least-squares: For every unbiased estimator there is generally a better biased estimator. Unpublished manuscript.
26. Hoerl, A. E., & Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12, 55-67.
27. Hogarth, R. M., & Karelaia, N. (2005a). Ignoring information in binary choice with continuous variables: When is less "more"? Journal of Mathematical Psychology, 49, 115-124.
28. Hogarth, R. M., & Karelaia, N. (2005b). Simple models for multiattribute choice with many alternatives: When it does and does not pay to face tradeoffs with binary attributes? Management Science, 51, 1860-1872.
29. Hogarth, R. M., & Karelaia, N. (2006a). "Take-The-Best" and other simple strategies: Why and when they work "well" with binary cues. Theory and Decision, 61, 205-249.
30. Hogarth, R. M., & Karelaia, N. (2006b). Regions of rationality: Maps for bounded agents. Decision Analysis, 3, 124-144.
31. Hogarth, R. M., & Karelaia, N. (2007). Heuristic and linear models of judgment: Matching rules and environments. Psychological Review, 114, 733-758.
32. James, W., & Stein, C. (1961). Estimation with quadratic loss. In Proceedings of the 4th Berkeley symposium on mathematical statistics and probability (Vol. 1, pp. 361-379). Berkeley: University of California Press.
33. Katsikopoulos, K. V., & Martignon, L. (2006). Naïve heuristics for paired comparisons: Some results on their relative accuracy. Journal of Mathematical Psychology, 50, 488-494.
34. Katsikopoulos, K. V., Schooler, L. J., & Hertwig, R. (2010). The robust beauty of ordinary information. Psychological Review, 117, 1259-1266.
35. Keren, G., & Newman, J. R. (1978). Additional considerations with regard to multiple regression and equal weighting. Organizational Behavior and Human Performance, 22, 143-164.
36. Koopman, R. F. (1988). On the sensitivity of a composite to its weights. Psychometrika, 53, 547-552.
37. Lehmann, E. L., & Casella, G. (1998). Theory of point estimation (2nd ed.). New York: Springer.
38. Li, S. (2011). Concise formulas for the area and volume of a hyperspherical cap. Asian Journal of Mathematics and Statistics, 4, 66-70.
39. Manning, H. P. (1914). Geometry of four dimensions. New York: The MacMillan.
40. Martignon, L., & Hoffrage, U. (1999). Why does one-reason decision making work? A case study in ecological rationality. In G. Gigerenzer, P. M. Todd, & the ABC Research Group (Eds.), Simple heuristics that make us smart (pp. 119-140). New York: Oxford University Press.
41. O'Neill, B. (1997). Elementary differential geometry. San Diego: Academic Press.
42. R Development Core Team (2010). R: a language and environment for statistical computing. Vienna: R Foundation for Statistical Computing. ISBN 3-900051-07-0. URL www. R-project. org.
43. Payne, J. W., Bettman, J. R., & Johnson, E. J. (1993). The adaptive decision maker. New York: Cambridge University Press.
44. Regenwetter, M., Grofman, B., Marley, A. A. J., & Tsetlin, I. M. (2006). Behavioral social choice. New York: Cambridge University Press.
45. Schmidt, F. L. (1971). The relative efficiency of regression and simple unit predictor weights in applied differential psychology. Educational and Psychological Measurement, 31, 699-714.
46. Stillwell, W. G., Seaver, D. A., & Edwards, W. (1981). A comparison of weight approximation techniques in multiattribute utility decision making. Organizational Behavior and Human Performance, 28, 62-77.
47. The Mathworks (2010). Matlab software, version 2009b. Natick: USA.
48. Wainer, H. (1976). Estimating coefficients in linear models: It don't make no nevermind. Psychological Bulletin, 83, 213-217.
49. Wainer, H. (1978). On the sensitivity of regression and regressors. Psychological Bulletin, 85, 267-273.
50. Wainer, H., & Thissen, D. (1976). Three steps toward robust regression. Psychometrika, 41, 9-34.
51. Waller, N. G. (2008). Fungible weights in multiple regression. Psychometrika, 73, 691-703.
52. Waller, N. G., & Jones, J. A. (2009). Locating the extrema of fungible regression weights. Psychometrika, 74, 589-602.
53. Waller, N. G., & Jones, J. A. (2010). Correlation weights in multiple regression. Psychometrika, 75, 58-69.
54. Waller, N. G., & Jones, J. A. (2011). Investigating the performance of alternate regression weights by studying all possible criteria in regression models with a fixed set of predictors. Psychometrika, 76, 410-439.
55. Wilks, S. S. (1938). Weighting systems for linear functions of correlated variables when there is no dependent variable. Psychometrika, 3, 23-40.