Objective Bayesian inference for bilateral data. (English) Zbl 1334.62193
Summary: This paper presents three objective Bayesian methods for analyzing bilateral data under Dallal’s model and the saturated model. Three parameters are of interest, namely, the risk difference, the risk ratio, and the odds ratio. We derive Jeffreys’ prior and Bernardo’s reference prior associated with the three parameters that characterize Dallal’s model. We derive the functional forms of the posterior distributions of the risk difference and the risk ratio and discuss how to sample from their posterior distributions. We demonstrate the use of the proposed methodology with two real data examples. We also investigate small, moderate, and large sample properties of the proposed methodology and the frequentist counterpart via simulations.
MSC:
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
| 62F15 | Bayesian inference |
| 62H17 | Contingency tables |
Keywords:
Bayes factor; Dallal’s model; Jeffreys’ prior; odds ratio; product trinomial distribution; reference prior; risk difference; risk ratioReferences:
| [1] | Berger, J. O. and Bernardo, J. M. (1989). “Estimating a Product of Means: Bayesian Analysis with Reference Priors.” Journal of the American of Statistical Assocation , 84: 200-207. · Zbl 0682.62018 · doi:10.2307/2289864 |
| [2] | - (1992a). “Ordered Group Reference Priors with Applications to a Multinomial Problem.” Biometrika , 79: 25-37. · Zbl 0763.62014 · doi:10.1093/biomet/79.1.25 |
| [3] | - (1992b). “Reference Priors in a Variance Components Problem.” In Goel, P. K. and Iyengar, N. (eds.), Bayesian Analysis in Statistics and Econometrics , 323-340. New York: Springer-Verlag. |
| [4] | - (1992c). “On the Development of Reference Priors.” In Bernado, J. M., Berger, J. O., Dawid, A. P., and Smith, A. F. M. (eds.), Bayesian Statistics , volume 4, 35-60. New York: Oxford: University Press. |
| [5] | Bernardo, J. M. (1981). “Reference Posterior Distributions for Bayes Inference.” Journal of the Royal Statistical Society, Series B , 41: 113-147. · Zbl 0428.62004 |
| [6] | Chen, M.-H. and Shao, Q.-M. (1999). “Monte Carlo Estimation of Bayesian Credible and HPD Intervals.” Journal of Computational and Graphical Statistics , 8: 69-92. |
| [7] | Cox, D. R. and Reid, N. (1987). “Parameter Orthogonality and Approximate Conditional Inference.” Journal of the Royal Statistical Society, Series B , 49: 1-39. · Zbl 0616.62006 |
| [8] | Dallal, G. E. (1988). “Paired Bernoulli Trials.” Biometrics , 44: 253-257. · Zbl 0715.62039 · doi:10.2307/2531913 |
| [9] | Datta, G. S. and Ghosh, M. (1996). “On the Invariance of Noninformative Priors.” The Annals of Statistics , 24: 141-159. · Zbl 0906.62024 · doi:10.1214/aos/1033066203 |
| [10] | Ghosh, J. K. and Mukerjee, R. (1992). “Non-informative Priors.” In Bernardo, J. M., Berger, J. O., Dawid, A. P., and Smith, A. F. M. (eds.), Bayesian Statistics , volume 4, 195-210. Oxford: Oxford University Press. |
| [11] | Jeffreys, H. (1946). “An Invariant Form for the Prior Probability in Estimation Problems.” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences , 186: 453-461. · Zbl 0063.03050 · doi:10.1098/rspa.1946.0056 |
| [12] | Mandel, E. M., Bluestone, C. D., Rockette, H. E., Blatter, M. M., Reisinger, K. S., Wucher, F. P., and Harper, J. (1982). “Duration of Effusion after Antibiotic Treatment for Acute Otitis Media: Comparison of Cefaclor and Amoxicillin.” Pediatric Infectious Diseasee , 1: 310-316. |
| [13] | Morris, R. W. (1993). “Bilateral Procedures in Randomised Controlled Trials.” The Journal of Bone and Joint Surgery , 75: 675-6. |
| [14] | Pei, Y.-B., Tang, M.-L., and Guo, J. (2008). “Testing the Equality of Two Proportions for Combined Unilateral and Bilateral Data.” Communications in Statistics - Simulation and Computation , 37: 1515-1529. · Zbl 1152.62393 · doi:10.1080/03610910802140232 |
| [15] | Pei, Y.-B., Tang, M.-L., Wong, W.-K., and Guo, J. (2010). “Confidence Intervals for Correlated Proportion Differences from Paired Data in a Two-Arm Randomized Clinical Trial.” Statistical Methods in Medical Research , 21: 167-187. · doi:10.1177/0962280210365018 |
| [16] | Postlethwaite, A. E., Wong, W. K., Clements, P., Chatterjee, S., Fessler, B. J., Kang, A. H., Korn, J., Mayes, M., Merkel, P. A., Molitor, J. A., Moreland, L., Rothfield, N., Simms, R. W., Smith, E. A., Spiera, R., Steen, V., Warrington, K., White, B., Wigley, F., and Furst, D. E. (2008). “A Multicenter, Randomised, Double-Blind, Placebo-Controlled Trial of Oral Type I Collagen in Patients with Diffuse Cutaneous Systemic Sclerosis: I. Oral Type I Collagen Does not Improve Skin in all Patients, but may Improve Skin in Late-Phase Disease.” Arthritis and Rheumatism , 58: 1810-1822. |
| [17] | Qiu, S.-F., Tang, N.-S., and Tang, M.-L. (2009). “Sample Size for Testing Difference between two Proportions for the Bilateral-Sample Design.” Journal of Biopharmaceutical Statistics , 19: 857-871. · doi:10.1080/10543400903105372 |
| [18] | Rosner, B. (1982). “Statistical Methods in Ophthalmology: An Adjustment for the Intraclass Correlation between Eyes.” Biometrics , 38: 105-114. |
| [19] | Sun, D. and Berger, J. O. (1998). “Reference Priors with Partial Information.” Biometrika , 85: 55-71. · Zbl 1067.62521 · doi:10.1093/biomet/85.1.55 |
| [20] | Tang, M.-L., Pei, Y.-B., Wong, W.-K., and Li, J.-L. (2010). “Goodness-of-fit Tests for Correlated Paired Binary Data.” Statistical Methods in Medical Research , 1-15. |
| [21] | Tang, M.-L., Tang, N.-S., and Rosner, B. (2006). “Statistical Inference for Correlated Data in Ophthalmologic Studies.” Statistics in Medicine , 25: 2771-2783. · doi:10.1002/sim.2425 |
| [22] | Tang, N.-S., Qui, S.-F., Tang, M.-L., and Pei, Y.-B. (2011). “Asymptotic Confidence Interval Construction for Proportion Difference in Medical Studies with Bilateral Data.” Statistical Methods in Medical Research , 20: 233-259. · doi:10.1177/0962280209358135 |
| [23] | Tang, N.-S., Tang, M.-L., and Qiu, S.-F. (2008). “Testing the Equality of Proportions for Correlated Otolaryngologic Data.” Computational Statistics and Data Analysis , 52: 3719-3729. · Zbl 1452.62851 |
| [24] | Yang, R. (1995). “Invariance of the Reference Prior Under Reparametrization.” Test , 4: 83-94. · Zbl 0852.62003 · doi:10.1007/BF02563104 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
