A modified Reed-Muench method of estimation in dose-effect relationship

M. S. Tikhov   ; K. N. Shkileva

Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 4 (2019), pp. 5-26

Résumé

This paper revises the estimators for a distribution function in the dose-effect relationship as Reed and Muench (1938) by proposing a new statistics to estimate the distribution function when the main variable is not necessarily normally distributed. We prove the consistency and asymptotic normality of this estimates without assuming the form of a parametric family. The initial statistics of Reed and Mench designed to estimate the median dose of ED$_{50}$ are modified in such a way that it is possible to construct stable estimates of the distribution function and effective doses of ED$_{100\lambda}$ in a wide range of $\lambda$: from 0.05 to 0.95. A stochastic approximation algorithm for estimating of the distribution function is designed. The convergence theorem of this algorithm is proved. To illustrate the practical utility of our approach, the techniques developed in the paper are used for computing the mean age of eruption of the premolars in boys. The statistical data are from the Hayes&Mantel (1958) work. Finally, a Monte Carlo exercise is performed based on the on simulated data. The results show that the nonparametric estimates of the distribution function considered in this paper work well in practice, in some cases even for relatively small sample sizes.

Détail

Keywords: dose-effect relationship, method of Reed and Muench, stochastic approximation, asymptotic normality, simulation modeling.

Bibliographie

[1] Tikhov M. S., “Statistical Estimation on the Basis of Interval-Censored Data”, Interuniversity Transactions on Statistical Method of Estimation and Testing Hypotheses, Perm University, Perm, 2000, 49–70 (in Russian)

[2] Tikhov M. S., “Statistical estimation based on interval censored data”, Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis, and Quality of Life, Statistics for Industry and Technology, eds. N. Balakrishnan, M.S. Nikulin, M. Mesbah, N. Limnios, Birkhauser, Boston, MA, 2004, 211–218 | DOI | MR

[3] Tikhov M. S., Yaroshchuk M. V., “Asymptotic normality in dose-effect relationship”, Bulletin of Lobachevsky State University of Nizhniy Novgorod. Series: Mathematics, 2006, no. 1(4), 129–137 (in Russian)

[4] Tikhov M., Ivkin M., “Multivariate k-Nearest Neighbors Distribution Function Estimates in Dose-effect Relationship”, Proceedings of the 2014 International Conference on Mathematical Models and Methods in Applied Sciences, MMAS'2014, 325–329 | Zbl

[5] Tikhov M., Ivkin M., “A new Yang-type estimator of Distribution Function at Quantal Response over Indirect Data”, Wseas Transactions on Mathematics, 13 (2014), 684–693

[6] Borodina T. S., Tikhov M. S., “Asymptotic normality of nonparametric estimates of quantiles”, Bulletin of Lobachevsky State University of Nizhniy Novgorod. Series: Mathematics, 2013, no. 1, 196–207 (in Russian)

[7] Tikhov M. S., Borodina T. S., “Kernel Quantile Estimators in the Dose-Effect Dependence”, Automatic Control and Computer Sciences, 47:2 (2013), 75–86 | DOI | MR

[8] Dette H., Neumeyer N., Pilz K., “A Note on Nonparametric Estimation of the Effective Dose in Quantal Bioassay”, Journal of the American Statistical Association, 100 (2005), 503–510 (in Russian) | DOI | MR | Zbl

[9] Tikhov M. S., “Neparametricheskoe otsenivanie effektivnykh doz po dannym binarnykh otklikov”, Ufa Mathematical Journal, 5:2 (2013), 94–108 (in Russian) | MR

[10] Hengartner N. W., Matzner-Lober E., “Asymptotic unbiased density estimators”, ESAIM: Probability and Statistics, 13 (2009), 1–14 | DOI | MR | Zbl

[11] Cornillon P. A., Hengartner N., Matzner-Lober E., “Recursive Bias Estimation for multivariate regression smoothers”, ESAIM: Probability and Statistics, 18 (2014), 483–502 | DOI | MR | Zbl

[12] Tikhov M. S., Dolgih I. S., “Asymptotically unbiased estimates of a distribution function in dose-effect relationships”, Journal Mathematical Sciences, v. 205, 2015, 113–120 | DOI | MR | Zbl

[13] Finney D. J., Probit analysis, Cambridge University Press, 2009, 256 pp. https://dspace.gipe.ac.in | MR | Zbl

[14] Urbakh V. Yu., Statistical analysis in biological and medical research, Medicina Publ., Moscow, 1975, 297 pp. (in Russian)

[15] Reed L. J., Muench H., “A simple method of estimating fifty per cent endpoints”, The American Journal of Hygiene, 27:3 (1938), 493–497

[16] Gurevich K. G., “Evaluation of Parameters of the Dose-Effect Curve by Soline-Interpolation Method”, Moscow University Chemistry Bulletin, 41:1 (2000), 69–70 (in Russian) | MR

[17] Gnedenko B. V., The Theory of Probability, AMS Chelsea Publishing, Providence, Rhode Island, 2005, 529 pp. | MR | MR

[18] Tikhov M. S., “Fourier methods for recursive estimating of distribution function in dose-effect relationship”, Herald of Tver State University. Series: Applied Mathematics, 2018, no. 4, 31–49 (in Russian) | DOI

[19] Gantmakher F. R., The Theory of Matrices, Nauka Publ., Moscow, 2010, 560 pp. (in Russian) | MR

[20] Vazan M., Stochastic Approximation, Mir Publ., Moscow, 1972, 296 pp. (in Russian)

[21] Robbins H., Monro S., “Stochastic Approximation Method”, The Annals of Mathematical Statistics, 22:3 (1951), 400–407 | DOI | MR | Zbl

[22] Galambos J., Seneta E., “Regularly varying sequences”, Proceeding of the American Mathematical Society, 41:1 (1973), 110–116 | DOI | MR | Zbl

[23] Seneta E., “Variants Karamata's Iteration Theorem”, Publications de l'Institute Mathématique, 80(94) (2006), 241–251 | DOI | MR | Zbl

[24] Gradshtejn I. S., Ryzhik I. M., Table of Integrals, Series and Products, GIFML Publ., Moscow, 1963, 1100 pp. (in Russian) | MR

[25] Chen C. P., “Inequalities for the Euler-Mascheroni constant”, Applied Mathematics Letters, 23 (2010), 161–164 | DOI | MR

[26] Seneta E., Regularly Varying Functions, Nauka Publ., Moscow, 1985, 142 pp. (in Russian)

[27] Hayes R. L., Mantel N., “Procedures for computing the mean age of eruption of human teeth”, Journal of Dental Research, 35:5 (1958), 938–947 | DOI