Minimum MSE Weights of Adjusted Summary Estimator of Risk Difference in Multi-Center Studies

ABSTRACT

The simple adjusted estimator of risk difference in each center is easy constructed by adding a value c on the number of successes and on the number of failures in each arm of the proportion estimator. Assessing a treatment effect in multi-center studies, we propose minimum MSE (mean square error) weights of an adjusted summary estimate of risk difference under the assumption of a constant of common risk difference over all centers. To evaluate the performance of the proposed weights, we compare not only in terms of estimation based on bias, variance, and MSE with two other conventional weights, such as the Cochran-Mantel-Haenszel weights and the inverse variance (weighted least square) weights, but also we compare the potential tests based on the type I error probability and the power of test in a variety of situations. The results illustrate that the proposed weights in terms of point estimation and hypothesis testing perform well and should be recommended to use as an alternative choice. Finally, two applications are illustrated for the practical use.

The simple adjusted estimator of risk difference in each center is easy constructed by adding a value c on the number of successes and on the number of failures in each arm of the proportion estimator. Assessing a treatment effect in multi-center studies, we propose minimum MSE (mean square error) weights of an adjusted summary estimate of risk difference under the assumption of a constant of common risk difference over all centers. To evaluate the performance of the proposed weights, we compare not only in terms of estimation based on bias, variance, and MSE with two other conventional weights, such as the Cochran-Mantel-Haenszel weights and the inverse variance (weighted least square) weights, but also we compare the potential tests based on the type I error probability and the power of test in a variety of situations. The results illustrate that the proposed weights in terms of point estimation and hypothesis testing perform well and should be recommended to use as an alternative choice. Finally, two applications are illustrated for the practical use.

Cite this paper

C. Viwatwongkasem, J. Jitthavech, D. Bohning and V. Lorchirachoonkul, "Minimum MSE Weights of Adjusted Summary Estimator of Risk Difference in Multi-Center Studies,"*Open Journal of Statistics*, Vol. 2 No. 1, 2012, pp. 48-59. doi: 10.4236/ojs.2012.21006.

C. Viwatwongkasem, J. Jitthavech, D. Bohning and V. Lorchirachoonkul, "Minimum MSE Weights of Adjusted Summary Estimator of Risk Difference in Multi-Center Studies,"

References

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[2] A. Agresti and B. Caffo, “Simple and Effective Confidence Intervals for Proportions and Differences of Proportions Result from Adding Two Successes and Two Failures,” The American Statistician, Vol. 54, No. 4, 2000, pp. 280-288. doi:10.2307/2685779

[3] B. K. Ghosh, “A Comparison of Some Approximate Confidence Interval for the Binomial Parameter,” Journal of the American Statistical Association, Vol. 74, No. 368, 1979, pp. 894-900. doi:10.2307/2286420

[4] R. G. Newcombe, “Two-Sided Confidence Intervals for the Single Proportion: Comparison of Seven Methods,” Statistics in Medicine, Vol. 17, No. 8, 1998, pp. 857-872. doi:10.1002/(SICI)1097-0258(19980430)17:8<857::AID-SIM777>3.0.CO;2-E

[5] R. G. Newcombe, “Interval Estimation for the Difference between Independent Proportions: Comparison of Eleven Methods,” Statistics in Medicine, Vol. 17, No. 8, 1998, pp. 873-890. doi:10.1002/(SICI)1097-0258(19980430)17:8<873::AID-SIM779>3.0.CO;2-I

[6] D. B?hning and C. Viwatwongkasem, “Revisiting Proportion Estimators,” Statistical Methods in Medical Research, Vol. 14, No. 2, 2005, pp. 147-169. doi:10.1191/0962280205sm393oa

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[9] K. J. Lui and K. C. Chang, “Testing Homogeneity of Risk Difference in Stratified Randomized Trials with Noncompliance,” Computational Statistics and Data Analysis, Vol. 53, No. 1, 2008, pp. 209-221. doi:10.1016/j.csda.2008.07.016

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[11] W. G. Cochran, “Some Methods for Strengthening the Common Chi-Square Test,” Biometrics, Vol. 10, No. 4, 1954, pp. 417-451. doi:10.2307/3001616

[12] N. Mantel and W. Haenszel, “Statistical Aspects of the Analysis of Data from Retrospective Studies of Disease,” Journal of the National Cancer Institute, Vol. 22, 1959, pp. 719-748.

[13] J. Sanchez-Meca and F. Marin-Martinez, “Testing the Significance of a Common Risk Difference in Meta- Analysis,” Computational Statistics & Data Analysis, Vol. 33, No. 3, 2000, pp. 299-313. doi:10.1016/S0167-9473(99)00055-9

[14] J. L. Fleiss, “Statistical Methods for Rates and Proportions,” John Wiley & Sons Inc., New York, 1981.

[15] S. R. Lipsitz, K. B. G. Dear, N. M. Laird and G. Molenberghs, “Tests for Homogeneity of the Risk Difference When Data Are Sparse,” Biometrics, Vol. 54, No. 1, 1998, pp. 148-160. doi:10.2307/2534003

[16] D. G. Kleinbaum, L. L. Kupper and H. Morgenstern, “Epidemiologic Research: Principles and Quantitative Methods,” Lifetime Learning Publications, Belmont, 1982.

[17] D. B. Petitti, “Meta-Analysis, Decision Analysis and Cost-Effectiveness Analysis: Methods for Quantitative Synthesis in Medicine,” Oxford University Press, Oxford, 1994.

[18] W. G. Cochran, “The Chi-Square Test of Goodness of Fit,” Annals of Mathematical Statistics, Vol. 23, No. 3, 1952, pp. 315-345. doi:10.1214/aoms/1177729380

[19] K. J. Lui and C. Kelly, “Tests for Homogeneity of the Risk Ratio in a Series of 2 × 2 Tables,” Statistics in Medicine, Vol. 19, No. 21, 2000, pp. 2919-2932. doi:10.1002/1097-0258(20001115)19:21<2919::AID-SIM561>3.0.CO;2-D

[20] S. J. Pocock, “Clinical Trials: A Practical Approach,” Wiley Publication, New York, 1997.

[21] R. Tuner, R. Omar, M. Yang, H. Goldstein and S. Thompson, “A Multilevel Model Framework for Meta- Analysis of Clinical Trials with Binary Outcome,” Statistics in Medicine, Vol. 19, No. 24, 2000, pp. 3417-3432. doi:10.1002/1097-0258(20001230)19:24<3417::AID-SIM614>3.0.CO;2-L

[22] F. Yates, “Contingency Tables Involving Small Numbers and the Chi-Squared Test,” Journal of the Royal Statistical Society (Supplement), Vol. 1, 1934, pp. 217-235.

[23] K. J. Lui, “A Simple Test of the Homogeneity of Risk Difference in Sparse Data: An Application to a Multicenter Study,” Biometrical Journal, Vol. 47, No. 5, 2008, pp. 654-661. doi:10.1002/bimj.200410150

[24] C. A. Rencher, “Linear Models in Statistics,” Wiley Series in Probability and Mathematical Statistics, New York, 2000.

[25] A. Sen and M. Srivastava, “Regression Analysis: Theory, Methods, and Applications,” Springer-Verlag, New York, 1990.

[1] A. Agresti and B. A. Coull, “Approximate Is Better than Exact for Interval Estimation of Binomial Proportions,” American Statistical Association, Vol. 52, 1998, pp. 119- 126.

[2] A. Agresti and B. Caffo, “Simple and Effective Confidence Intervals for Proportions and Differences of Proportions Result from Adding Two Successes and Two Failures,” The American Statistician, Vol. 54, No. 4, 2000, pp. 280-288. doi:10.2307/2685779

[3] B. K. Ghosh, “A Comparison of Some Approximate Confidence Interval for the Binomial Parameter,” Journal of the American Statistical Association, Vol. 74, No. 368, 1979, pp. 894-900. doi:10.2307/2286420

[4] R. G. Newcombe, “Two-Sided Confidence Intervals for the Single Proportion: Comparison of Seven Methods,” Statistics in Medicine, Vol. 17, No. 8, 1998, pp. 857-872. doi:10.1002/(SICI)1097-0258(19980430)17:8<857::AID-SIM777>3.0.CO;2-E

[5] R. G. Newcombe, “Interval Estimation for the Difference between Independent Proportions: Comparison of Eleven Methods,” Statistics in Medicine, Vol. 17, No. 8, 1998, pp. 873-890. doi:10.1002/(SICI)1097-0258(19980430)17:8<873::AID-SIM779>3.0.CO;2-I

[6] D. B?hning and C. Viwatwongkasem, “Revisiting Proportion Estimators,” Statistical Methods in Medical Research, Vol. 14, No. 2, 2005, pp. 147-169. doi:10.1191/0962280205sm393oa

[7] G. Casella and R. L. Berger, “Statistical Inference,” Duxbury Press, Belmont, 1990.

[8] A. E. Taylor and W. R. Mann, “Advanced Calculus,” John Wiley & Sons, New York, 1972.

[9] K. J. Lui and K. C. Chang, “Testing Homogeneity of Risk Difference in Stratified Randomized Trials with Noncompliance,” Computational Statistics and Data Analysis, Vol. 53, No. 1, 2008, pp. 209-221. doi:10.1016/j.csda.2008.07.016

[10] W. G. Cochran, “The Combination of Estimates from Different Experiments,” Biometrics, Vol. 10, No. 1, 1954, pp. 101-129. doi:10.2307/3001666

[11] W. G. Cochran, “Some Methods for Strengthening the Common Chi-Square Test,” Biometrics, Vol. 10, No. 4, 1954, pp. 417-451. doi:10.2307/3001616

[12] N. Mantel and W. Haenszel, “Statistical Aspects of the Analysis of Data from Retrospective Studies of Disease,” Journal of the National Cancer Institute, Vol. 22, 1959, pp. 719-748.

[13] J. Sanchez-Meca and F. Marin-Martinez, “Testing the Significance of a Common Risk Difference in Meta- Analysis,” Computational Statistics & Data Analysis, Vol. 33, No. 3, 2000, pp. 299-313. doi:10.1016/S0167-9473(99)00055-9

[14] J. L. Fleiss, “Statistical Methods for Rates and Proportions,” John Wiley & Sons Inc., New York, 1981.

[15] S. R. Lipsitz, K. B. G. Dear, N. M. Laird and G. Molenberghs, “Tests for Homogeneity of the Risk Difference When Data Are Sparse,” Biometrics, Vol. 54, No. 1, 1998, pp. 148-160. doi:10.2307/2534003

[16] D. G. Kleinbaum, L. L. Kupper and H. Morgenstern, “Epidemiologic Research: Principles and Quantitative Methods,” Lifetime Learning Publications, Belmont, 1982.

[17] D. B. Petitti, “Meta-Analysis, Decision Analysis and Cost-Effectiveness Analysis: Methods for Quantitative Synthesis in Medicine,” Oxford University Press, Oxford, 1994.

[18] W. G. Cochran, “The Chi-Square Test of Goodness of Fit,” Annals of Mathematical Statistics, Vol. 23, No. 3, 1952, pp. 315-345. doi:10.1214/aoms/1177729380

[19] K. J. Lui and C. Kelly, “Tests for Homogeneity of the Risk Ratio in a Series of 2 × 2 Tables,” Statistics in Medicine, Vol. 19, No. 21, 2000, pp. 2919-2932. doi:10.1002/1097-0258(20001115)19:21<2919::AID-SIM561>3.0.CO;2-D

[20] S. J. Pocock, “Clinical Trials: A Practical Approach,” Wiley Publication, New York, 1997.

[21] R. Tuner, R. Omar, M. Yang, H. Goldstein and S. Thompson, “A Multilevel Model Framework for Meta- Analysis of Clinical Trials with Binary Outcome,” Statistics in Medicine, Vol. 19, No. 24, 2000, pp. 3417-3432. doi:10.1002/1097-0258(20001230)19:24<3417::AID-SIM614>3.0.CO;2-L

[22] F. Yates, “Contingency Tables Involving Small Numbers and the Chi-Squared Test,” Journal of the Royal Statistical Society (Supplement), Vol. 1, 1934, pp. 217-235.

[23] K. J. Lui, “A Simple Test of the Homogeneity of Risk Difference in Sparse Data: An Application to a Multicenter Study,” Biometrical Journal, Vol. 47, No. 5, 2008, pp. 654-661. doi:10.1002/bimj.200410150

[24] C. A. Rencher, “Linear Models in Statistics,” Wiley Series in Probability and Mathematical Statistics, New York, 2000.

[25] A. Sen and M. Srivastava, “Regression Analysis: Theory, Methods, and Applications,” Springer-Verlag, New York, 1990.