Testing for a single mean with transformed data. (English) Zbl 07552584
Summary: In several occasions, before conducting a t-test on a single population mean, we first apply to the data a power transformation. We know that substituting the original null into the transformation is a poor approximation of the null at the transformed scale. By a simulation study, we investigate methods to evaluate the transformed null. A Taylor expansion and an average of the power transformed observations, adjusted for a mean dependent variance, give satisfactory results both in terms of the level of significance and the power of the corresponding test. The importance of the correction is illustrated by two real data examples.
MSC:
| 62-XX | Statistics |
References:
| [1] | Abramowitz, M.; Stegun, I. A., Handbook of mathematical functions: With formulas, graphs, and mathematical tables (1972), Washington, DC: USA Department of Commerce, Washington, DC · Zbl 0543.33001 |
| [2] | Estimating Box-Cox power transformation parameter via goodness-of-fit tests, Communications in Statistics Simulation and Computation, 46, 1, 91-105 (2017) · Zbl 1359.62069 · doi:10.1080/03610918.2014.957839 |
| [3] | Bickel, P. J.; Doksum, K. A., An analysis of transformations revisited, Journal of the American Statistical Association, 76, 374, 296-311 (1981) · Zbl 0464.62058 · doi:10.1080/01621459.1981.10477649 |
| [4] | Box, G. E. P.; Cox, D. R., An analysis of transformations (with discussion, Journal of the Royal Statistical Society Series B, 26, 211-52 (1964) · Zbl 0156.40104 |
| [5] | Chen, H.; Loh, W. Y., Bounds on AREs of tests following Box-Cox transformations, The Annals of Statistics, 20, 3, 1485-1500 (1992) · Zbl 0781.62055 · doi:10.1214/aos/1176348780 |
| [6] | Davison, A. C.; Hinkley, D. V., Bootstrap methods and their application (1997), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0886.62001 |
| [7] | Doksum, K. A.; Wong, C. W., Statistical tests based on transformed data, Journal of the American Statistical Association, 78, 382, 411-417 (1983) · Zbl 0514.62055 · doi:10.1080/01621459.1983.10477986 |
| [8] | Draper, N. R.; Cox, D. R., On distributions and their transformation to normality, Journal of the Royal Statistical Society Series B, 31, 472-447 (1969) · Zbl 0186.53102 |
| [9] | Duan, N., Smearing estimate: A nonparametric retransformation method, Journal of the American Statistical Association, 78, 383, 605-610 (1983) · Zbl 0534.62021 · doi:10.1080/01621459.1983.10478017 |
| [10] | Feng, C.; Wang, H.; Lu, N.; Chen, T.; He, H.; Lu, Y.; Tu, X. M., Log-transformation and its implications for data analysis, Shanghai Archives of Psychiatry, 26, 2, 105-109 (2014) · doi:10.3969/j.issn.1002-0829.2014.02.009 |
| [11] | Freeman, J.; Modarres, R., Inverse Box-Cox: The power-normal distribution, Statistics & Probability Letters, 76, 8, 764-772 (2006) · Zbl 1089.62008 · doi:10.1016/j.spl.2005.10.036 |
| [12] | Freeman, J.; Modarres, R., Efficiency of t-test and Hotelling’s T-test after Box-Cox transformation, Communications in Statistics Theory and Methods, 35, 6, 1109-1122 (2006) · Zbl 1102.62063 · doi:10.1080/03610920600672203 |
| [13] | Hall, P., On the removal of skewness by transformation, Journal of the Royal Statistical Society Series B, 54, 221-228 (1992) · doi:10.2307/2345958 |
| [14] | Heiberger, R. M.; Holland, B., Statistical analysis and data display, an intermediate course with examples in S-plus, R, and SAS (2004), New York, NY: Springer, New York, NY · Zbl 1055.62004 |
| [15] | Hernandez, F.; Johnson, R. A., The large-sample behavior of transformations to normality, Journal of the American Statistical Association, 75, 372, 855-861 (1980) · Zbl 0459.62028 · doi:10.1080/01621459.1980.10477563 |
| [16] | Hinkley, D. V., On power transformations to symmetry, Biometrika, 62, 1, 101-111 (1975) · Zbl 0308.62007 · doi:10.1093/biomet/62.1.101 |
| [17] | Hossain, M. Z., The use of Box-Cox transformation technique in economic and statistical analyses, Journal of Emerging Trends in Economics and Management Sciences, 2, 32-39 (2011) |
| [18] | Lim, W. K.; Lim, A. W., A comparison of usual t-test statistic and modified t-test statistics on skewed distribution functions, Journal of Modern Applied Statistical Methods, 15, 2, 67-89 (2016) · doi:10.22237/jmasm/1478001960 |
| [19] | McCullagh, P.; Nelder, J. A., Generalized linear models (1989), New York, NY: Chapman & Hall/CRC, New York, NY · Zbl 0744.62098 |
| [20] | Niaki, S. T. A.; Abbasi, B., Monitoring multi-attribute processes based on NORTA inverse transformed vectors, Communications in Statistics Theory and Methods, 38, 7, 964-979 (2009) · Zbl 1162.62098 · doi:10.1080/03610920802346119 |
| [21] | Proietti, T.; Lutkepohl, H., Does the Box-Cox transformation help in forecasting macroeconomic time series?, International Journal of Forecasting, 29, 1, 88-99 (2013) · doi:10.1016/j.ijforecast.2012.06.001 |
| [22] | R Core Team. (2016) |
| [23] | Sakia, R. M., The Box-Cox transformation technique: A review, The Statistician, 41, 2, 169-178 (1992) · doi:10.2307/2348250 |
| [24] | Taylor, J. M. G., Power transformations to symmetry, Biometrika, 72, 1, 145-152 (1985) · Zbl 0563.62044 · doi:10.1093/biomet/72.1.145 |
| [25] | Taylor, J. M. G., The retransformed mean after a fitted power transformation, Journal of the American Statistical Association, 81, 393, 114-118 (1986) · doi:10.1080/01621459.1986.10478246 |
| [26] | Verzani, J., Using R for introductory statistics (2014), New York, NY: CRC Press, New York, NY · Zbl 1316.62001 |
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