The Annals of Statistics

Optimal crossover designs for the proportional model

Wei Zheng

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Abstract

In crossover design experiments, the proportional model, where the carryover effects are proportional to their direct treatment effects, has draw attentions in recent years. We discover that the universally optimal design under the traditional model is E-optimal design under the proportional model. Moreover, we establish equivalence theorems of Kiefer–Wolfowitz’s type for four popular optimality criteria, namely A, D, E and T (trace).

Article information

Source
Ann. Statist., Volume 41, Number 4 (2013), 2218-2235.

Dates
First available in Project Euclid: 23 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1382547519

Digital Object Identifier
doi:10.1214/13-AOS1148

Mathematical Reviews number (MathSciNet)
MR3127864

Zentralblatt MATH identifier
1277.62191

Subjects
Primary: 62K05: Optimal designs
Secondary: 62J05: Linear regression

Keywords
Crossover designs A-optimality D-optimality E-optimality equivalence theorem proportional model pseudo symmetric designs

Citation

Zheng, Wei. Optimal crossover designs for the proportional model. Ann. Statist. 41 (2013), no. 4, 2218--2235. doi:10.1214/13-AOS1148. https://projecteuclid.org/euclid.aos/1382547519


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References

  • Bailey, R. A. and Kunert, J. (2006). On optimal crossover designs when carryover effects are proportional to direct effects. Biometrika 93 613–625.
  • Bose, M. and Stufken, J. (2007). Optimal crossover designs when carryover effects are proportional to direct effects. J. Statist. Plann. Inference 137 3291–3302.
  • Chêng, C. S. and Wu, C.-F. (1980). Balanced repeated measurements designs. Ann. Statist. 8 1272–1283.
  • Fedorov, V. V. and Hackl, P. (1997). Model-oriented Design of Experiments. Lecture Notes in Statistics 125. Springer, New York.
  • Hedayat, A. S. and Afarinejad, K. (1975). Repeated measurements designs. In A Survey of Statistical Design and Linear Models (N. Srivastava, ed.) 229–240. North-Holland, Amsterdam.
  • Hedayat, A. and Afsarinejad, K. (1978). Repeated measurements designs. II. Ann. Statist. 6 619–628.
  • Hedayat, A. S. and Yang, M. (2003). Universal optimality of balanced uniform crossover designs. Ann. Statist. 31 978–983.
  • Hedayat, A. S. and Yang, M. (2004). Universal optimality for selected crossover designs. J. Amer. Statist. Assoc. 99 461–466.
  • Hedayat, A. S. and Zheng, W. (2010). Optimal and efficient crossover designs for test-control study when subject effects are random. J. Amer. Statist. Assoc. 105 1581–1592.
  • Kempton, R. A., Ferris, S. J. and David, O. (2001). Optimal change-over designs when carry-over effects are proportional to direct effects of treatments. Biometrika 88 391–399.
  • Kiefer, J. (1975). Construction and optimality of generalized Youden designs. In A Survey of Statistical Design and Linear Models (Proc. Internat. Sympos., Colorado State Univ., Ft. Collins, Colo., 1973) (J. N. Srivastava, ed.) 333–353. North-Holland, Amsterdam.
  • Kunert, J. (1984). Optimality of balanced uniform repeated measurements designs. Ann. Statist. 12 1006–1017.
  • Kunert, J. and Martin, R. J. (2000). On the determination of optimal designs for an interference model. Ann. Statist. 28 1728–1742.
  • Kunert, J. and Stufken, J. (2002). Optimal crossover designs in a model with self and mixed carryover effects. J. Amer. Statist. Assoc. 97 898–906.
  • Kushner, H. B. (1997). Optimal repeated measurements designs: The linear optimality equations. Ann. Statist. 25 2328–2344.
  • Kushner, H. B. (1998). Optimal and efficient repeated-measurements designs for uncorrelated observations. J. Amer. Statist. Assoc. 93 1176–1187.
  • Park, D. K., Bose, M., Notz, W. I. and Dean, A. M. (2011). Efficient crossover designs in the presence of interactions between direct and carry-over treatment effects. J. Statist. Plann. Inference 141 846–860.
  • Stufken, J. (1991). Some families of optimal and efficient repeated measurements designs. J. Statist. Plann. Inference 27 75–83.
  • Zheng, W. (2013a). Universally optimal crossover designs under subject dropout. Ann. Statist. 41 63–90.
  • Zheng, W. (2013b). Supplement to “Optimal crossover designs for the proportional model.” DOI:10.1214/13-AOS1148SUPP.

Supplemental materials

  • Supplementary material: Appendix for optimal crossover designs for the proportional model. This document is to provide a general algorithm to derive optimal $P_{\langle d\rangle}$ for arbitrary values of $\lambda_{0}$ and $\Sigma$ based on the equivalence theorems.