The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Minimum Bias Design for Polynomial Regression

다항회귀모형에 대한 최소편의 실험계획

  • Jang, Dae-Heung (Department of Statistics, Pukyong National University) ;
  • Kim, Youngil (School of Business and Economics, ChungAng University)
  • Received : 2015.10.28
  • Accepted : 2015.12.12
  • Published : 2015.12.31
Download PDF
⟨ Previous Next ⟩

Abstract

Traditional criteria for optimum experimental designs depend on the specifications of the model; however, there will be a dilemma when we do not have perfect knowledge about the model. Box and Draper (1959) suggested one direction to minimize bias that may occur in this situation. We will demonstrate some examples with exact solutions that provide a no-bias design for polynomial regression. The most interesting finding is that a design that requires less bias should allocate design points away from the border of the design space.

전통적으로 최적실험을 위한 실험기준들은 기본적으로 가정된 모형에 의존한다. 따라서 모형에 대한 완벽한 정보를 가지지 않는 경우 실험자는 곤란에 빠질 수 밖애 없다. Box와 Draper (1959) 이런 상황에 대비해 적분된 평균제곱오차의 편의부분에 해당하는 적분된 편의를 최소화하는 실험기준을 제안하고 필요충분조건을 명시하였다. 그러나 간단한 예제를 제외하고는 문헌에서는 이러한 필요충분조건을 만족하는 실험에 대한 구채적인 예제는 계산상의 문제로 예상외로 많이 연구가 되어 있지 않다. 비록 수치적인 해이긴 하지만 다항회귀모형을 중심으로 최소편의를 만족하는 실험의 성격을 파악하였는데 결론적으로 양극단에서 안쪽 방향으로 이탈되는 위치에서 받힘점이 형성되는 것을 알 수 있었다.

Keywords

References

  1. Abdelbasit, K. M. and Butler, N. A. (2006). Minimum bias design for generalized linear models, The Indian Journal of Statistics, 68, 587-599.
  2. Box, G. E. P. and Draper, N. R. (1959). A bias for the selection of a response surface design, Journal of the Statistical Association, 54, 622-654. https://doi.org/10.1080/01621459.1959.10501525
  3. Draper, N. R. and Guttman, I. (1992). Treating bias as variance for experimental design purposes, Annals of the Institute of Statistical Mathematics, 44, 659-671. https://doi.org/10.1007/BF00053396
  4. Hunga, K. and Fan, S. (2004). A note on minimum bias estimation in response surfaces, Statistics and Probability Letters, 70, 71-85. https://doi.org/10.1016/j.spl.2004.08.009
  5. Jones, B. and Nachtsheim, C. J. (2011a). A class of three-level designs for definitive screening in the presence of second-order effects, Journal of Quality Technology, 43, 1-15. https://doi.org/10.1080/00224065.2011.11917841
  6. Jones, B. and Nachtsheim, C. J. (2011b). Efficient designs with minimal aliasing, Technometrics, 53, 62-71. https://doi.org/10.1198/TECH.2010.09113
  7. Karson, M. J. (1970). Design criterion for minimum bias estimation of response surfaces, Journal of the Statistical Association, 65, 1565-1572. https://doi.org/10.1080/01621459.1970.10481185
  8. Kim, Y. (1995). A Study for additional characteristics of $I_{\lambda}$-optimal experimental designs, Communications for Statistical Applications and Methods, 2, 55-63.
  9. Montepiedra, G. and Fedorov, V. V. (1997). Minimum bias design with constraints, Journal of Statistical Planning and Inference, 63, 97-111. https://doi.org/10.1016/S0378-3758(96)00199-1
  10. Myer, R. K. and Nachtsheim, C. J. (1995). The coordinate exchange algorithm for constructing exact experimental designs, Technometrics, 37, 60-69. https://doi.org/10.1080/00401706.1995.10485889