Constrained parameters in applications: review of issues and approaches. (English) Zbl 1268.62022
Summary: This review article provides an introduction to statistical issues that arise when some statistical model parameters are constrained. This often happens in applications, in particular in testing for variance components (e.g., genomics) and construction of one-sided confidence intervals (e.g., environmental risk analysis). Heuristic explanations are provided, and a number of general and recent statistical results that appeared in statistical literature are summarized for use in applications. Simulation results are shown for illustration of consequences of ignoring parameters on the boundary. Special attention is paid to likelihood ratio tests, but other approaches to confidence interval construction, such as Wald, bootstrap, and Bayesian are also briefly discussed. This paper presents examples from the risk assessment field and genomics, but all conclusions apply to whenever one-sided testing is conducted. Recommendations are provided for dealing with parameters on the boundary for a range of situations.
MSC:
| 62F03 | Parametric hypothesis testing |
| 62F25 | Parametric tolerance and confidence regions |
| 65C60 | Computational problems in statistics (MSC2010) |
References:
| [1] | Cox, D. R.; Hinkley, D. V., Theoretical statistics (1974), London, UK: Chapman and Hall, London, UK · Zbl 0334.62003 |
| [2] | Self, S. G.; Liang, K.-Y., Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions, Journal of American Statistical Association, 82, 398, 605-610 (1987) · Zbl 0639.62020 |
| [3] | Eaton, D. L.; Klaassen, C. D.; Doull; Casarett, Principles of Toxicology, Toxicology: The Basic Science of Poisons (2001), New York, NY, USA: McGraw-Hill, New York, NY, USA |
| [4] | Crump, K. S., A new method for determining allowable daily intakes, Fundamental and Applied Toxicology, 4, 5, 854-871 (1984) |
| [5] | Crump, K. S.; Guess, H. A.; Deal, K. L., Confidence intervals and test of hypotheses concerning dose response relations inferred from animal carcinogenicity data, Biometrics, 33, 3, 437-451 (1977) · Zbl 0371.92011 |
| [6] | Molenberghs, G.; Verbeke, G., Likelihood ratio, score, and wald tests in a constrained parameter space, American Statistician, 61, 1, 22-27 (2007) · doi:10.1198/000313007X171322 |
| [7] | Dominicus, A.; Skrondal, A.; Gjessing, H. K.; Pedersen, N. L.; Palmgren, J., Likelihood ratio tests in behavioral genetics: problems and solutions, Behavior Genetics, 36, 2, 331-340 (2006) · doi:10.1007/s10519-005-9034-7 |
| [8] | Chernoff, H., On the distribution of the likelihood ratio, Annals of Mathematical Statistics, 25, 573-578 (1954) · Zbl 0056.37102 |
| [9] | Feder, P. I., On the distribution of the log likelihood ratio test statistic when the true parameter is “near” the boundaries of the hypothesis regions, Annals of Mathematical Statistics, 39, 2044-2055 (1968) · Zbl 0212.23002 |
| [10] | Moran, P. A. P., Maximum-likelihood estimation in non-standard conditions, Proceedings of the Cambridge Philosophical Society, 70, 441-450 (1971) · Zbl 0224.62013 |
| [11] | Chant, D., On asymptotic tests of composite hypotheses in nonstandard conditions, Biometrika, 61, 2, 291-298 (1974) · Zbl 0289.62022 |
| [12] | Sinha, B.; Kopylev, L.; Fox, J., Some new aspects of dose-response multistage models with applications, UMBC technical report (2009) |
| [13] | Kopylev, L.; Sinha, B., On the asymptotic distribution of likelihood ratio test when parameters lie on the boundary, Sankhya B, 73, 1, 20-41 (2011) · Zbl 1230.62021 · doi:10.1007/s13571-011-0022-z |
| [14] | Bailer, A. J.; Smith, R. J., Estimating upper confidence limits for extra risk in quantal multistage models, Risk Analysis, 14, 6, 1001-1010 (1994) · doi:10.1111/j.1539-6924.1994.tb00069.x |
| [15] | Gelfand, A. E.; Smith, A. F. M.; Lee, T.-M., Bayesian analysis of constrained parameter and truncated data problems using Gibbs sampling, Journal of the American Statistical Association, 87, 418, 523-532 (1992) |
| [16] | Dunson, D. B.; Neelon, B., Bayesian inference on order-constrained parameters in generalized linear models, Biometrics, 59, 2, 286-295 (2003) · Zbl 1210.62097 · doi:10.1111/1541-0420.00035 |
| [17] | Hans, C.; Dunson, D. B., Bayesian inferences on umbrella orderings, Biometrics, 61, 4, 1018-1026 (2005) · Zbl 1087.62123 · doi:10.1111/j.1541-0420.2005.00373.x |
| [18] | Visscher, P. M., A note on the asymptotic distribution of likelihood ratio tests to test variance components, Twin Research and Human Genetics, 9, 4, 490-495 (2006) · doi:10.1375/183242706778024928 |
| [19] | Meyer, K., Likelihood calculations to evaluate experimental designs to estimate genetic variances, Heredity, 101, 3, 212-221 (2008) · doi:10.1038/hdy.2008.46 |
| [20] | Stoel, R. D.; Garre, F. G.; Dolan, C.; Van Den Wittenboer, G., On the likelihood ratio test in structural equation modeling when parameters are subject to boundary constraints, Psychological Methods, 11, 4, 439-455 (2006) · doi:10.1037/1082-989X.11.4.439 |
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