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Improving the Welch-Satterthwaite Approximation
Mejorando la aproximación de Welch-Satterthwaite
DOI:
https://doi.org/10.15446/rce.v48n2.117675Keywords:
Approximated inference, t-test, Generalized Gama Distribution, Delta method, Maximum Likelihood, Monte Carlo Simulation. (en)Inferencia aproximada, Prueba t, Distribución gamma generalizada, Método delta, Máxima verosimilitud, Simulación Monte Carlo. (es)
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TheWelch-Satterthwaite (WS) methodology is typically used in medicine, biology and economic courses to make inferences about the difference between two population means. Despite his wide-spreading applications, it has been pointing out in many references the multiple limitations of the inferences based on it. In this work, we propose three simple ways to improve the classical WS approach. Under balanced samples scenarios, we give exact inference results of two of the proposed estimators. Additionally, under unbalanced samples scenarios, we offer first-order approximation results and through several Monte Carlo simulations, we assess the mean and variance of the proposed estimators under (very) small and moderate sample sizes. Nonetheless, the simplicity of the proposed approach we obtain a much better performance than the WS proposal. Lastly, one application is presented in which the proposed estimators potentially improve the performance of t-student interval estimation and hypothesis testing procedures.
La metodología de Welch-Satterthwaite (WS) se utiliza típicamente en medicina, biología y economía para realizar inferencias sobre la diferencia entre dos medias poblacionales. A pesar de su amplia aplicación, se ha señalado en numerosas referencias las múltiples limitaciones de las inferencias basadas en esta metodología. En este trabajo, proponemos tres maneras sencillas de mejorar el enfoque clásico de WS. En escenarios de muestras balanceadas, proporcionamos resultados de inferencia exactos de dos de los estimadores propuestos. Además, en escenarios con muestras no balanceadas, ofrecemos resultados de aproximación de primer orden y mediante simulación Monte Carlo, evaluamos la media y la varianza de los estimadores propuestos con tamaños de muestra (muy) pequeños y moderados. No obstante, gracias a la simplicidad del enfoque propuesto, obtenemos un rendimiento mucho mejor que la propuesta de WS. Finalmente, se presenta una aplicación en la que los estimadores propuestos mejoran potencialmente el rendimiento de la estimación del intervalo t-Student y los procedimientos de prueba de hipótesis.
References
Ballico, M. (2000), 'Limitations of the welch-satterthwaite approximation for measurement uncertainty calculations', Metrologia 37(1), 61-64. DOI: https://doi.org/10.1088/0026-1394/37/1/8
Cardozo, C., Paula, G. & Vanegas, L. (2022), 'Generalized log-gamma additive partial linear models with P-spline smoothing', Statistical Papers 63, 1953-1978. DOI: https://doi.org/10.1007/s00362-022-01300-4
Casella, G. & Berger, B. (2003), Statistical Inference, Duxbury-Thomson Learning.
Crowder, S. & Kupferman, S. (2004), 'Use of Welch-Satterthwaite approximation in calibration of voltage standards', Journal of Quality Technology 36(1), 38-52. DOI: https://doi.org/10.1080/00224065.2004.11980251
Hall, B. D. & Willink, R. (2001), 'Does Welch-Satterthwaite make a good uncertainty estimate?', Metrologia 38, 9-15. DOI: https://doi.org/10.1088/0026-1394/38/1/2
Hogg, R., McKean, J. & Craig, A. (2019), Introduction to Mathematical Statistics, Pearson Education.
Lawless, J. (1980), 'Inference in the generalized gamma and log gamma distributions', Technometrics 22, 409-419. DOI: https://doi.org/10.1080/00401706.1980.10486173
Miao, W. & Chiou, P. (2008), 'Confidence intervals for the difference between two means', Computational Statistics & Data Analysis 52, 2238-2248. DOI: https://doi.org/10.1016/j.csda.2007.07.017
R Development Core Team (2024), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria.
Satterthwaite, F. E. (1946), 'An approximate distribution of estimates of variance components', Biometrics Bulletin 2, 110-114. DOI: https://doi.org/10.2307/3002019
Walpole, R. E., Myers, R. H., Myers, S. L. & Ye, K. (2017), Probability and Statistics for Engineers and Scientists, 9 edn, Pearson Education.
Welch, B. L. (1947), 'The generalization of Student's problem when several different population variances are involved', Biometrika 34, 28-35. DOI: https://doi.org/10.1093/biomet/34.1-2.28
Xiao, Y. (2018), 'On the solution of a generalized Behrens-Fisher problem', Far East Journal of Theoretical Statistics 54, 21-140 DOI: https://doi.org/10.17654/TS054010021
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