Revisit the Two Sample t-Test with a Known Ratio of Variances

ABSTRACT

Inference for the difference of two independent normal means has been widely studied in staitstical literature. In this paper, we consider the case that the variances are unknown but with a known relationship between them. This situation arises frequently in practice, for example, when two instruments report averaged responses of the same object based on a different number of replicates, the ratio of the variances of the response is then known, and is the ratio of the number of replicates going into each response. A likelihood based method is proposed. Simulation results show that the proposed method is very accurate even when the sample sizes are small. Moreover, the proposed method can be extended to the case that the ratio of the variances is unknown.

Inference for the difference of two independent normal means has been widely studied in staitstical literature. In this paper, we consider the case that the variances are unknown but with a known relationship between them. This situation arises frequently in practice, for example, when two instruments report averaged responses of the same object based on a different number of replicates, the ratio of the variances of the response is then known, and is the ratio of the number of replicates going into each response. A likelihood based method is proposed. Simulation results show that the proposed method is very accurate even when the sample sizes are small. Moreover, the proposed method can be extended to the case that the ratio of the variances is unknown.

KEYWORDS

Behrens-Fisher Problem, Canonical Parameter, Exponential Family Model, Likelihood Based Inference, Modified Signed Log-Likelihood Ratio Statistic, Satterthwaite Method.

Behrens-Fisher Problem, Canonical Parameter, Exponential Family Model, Likelihood Based Inference, Modified Signed Log-Likelihood Ratio Statistic, Satterthwaite Method.

Cite this paper

nullY. She, A. Wong and X. Zhou, "Revisit the Two Sample t-Test with a Known Ratio of Variances,"*Open Journal of Statistics*, Vol. 1 No. 3, 2011, pp. 151-156. doi: 10.4236/ojs.2011.13018.

nullY. She, A. Wong and X. Zhou, "Revisit the Two Sample t-Test with a Known Ratio of Variances,"

References

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[2] A. Wong and Y. Wu, “Likelihood Analysis for the Difference in Means of Two Independent Normal Distributions with One Variance Unknown,” Journal of Statistical Research, Vol. 42, 2008, pp. 17-35.

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[5] O. E. Barndorff-Nielsen, “Inference on Full and Partial Parameters, Based on the Standardized Signed Log-like- lihood Ratio,” Biometrika, Vol. 73, 1986, pp. 307-322.

[6] O. E. Barndorff-Nielsen, “Modified Signed Log-likeli- hood Ratio,” Biometrika, Vol. 78, No. 3, 1991, pp. 557- 563. doi:10.1093/biomet/78.3.557

[7] T. DiCiccio, C. Field and D. A. S. Fraser, “Approximation of Marginal Tail Probabilities and Inference for Scalar Parameters,” Biometrika, Vol. 77, 1990, pp. 77-95. doi:10.1093/biomet/77.1.77

[1] A. Maity and M. Sherman, “The Two Sample t-Test with One Variance Unknown,” The American Statistician, Vol. 60, No. 2, 2006, pp. 163-166. doi:10.1198/000313006X108567

[2] A. Wong and Y. Wu, “Likelihood Analysis for the Difference in Means of Two Independent Normal Distributions with One Variance Unknown,” Journal of Statistical Research, Vol. 42, 2008, pp. 17-35.

[3] E. Schechtman and M. Sherman, “The Two-sample t-Test with a Known Ratio of Variances,” Statistical Methodology, Vol. 4, No. 4, 2007, pp. 508-514. doi:10.1016/j.stamet.2007.03.001

[4] D. A. Sprott and V.T. Farewell, “The Difference between Two Normal Means,” The American Statistician, Vol. 47, No. 2, 1993, pp. 126-128. doi:10.2307/2685194

[5] O. E. Barndorff-Nielsen, “Inference on Full and Partial Parameters, Based on the Standardized Signed Log-like- lihood Ratio,” Biometrika, Vol. 73, 1986, pp. 307-322.

[6] O. E. Barndorff-Nielsen, “Modified Signed Log-likeli- hood Ratio,” Biometrika, Vol. 78, No. 3, 1991, pp. 557- 563. doi:10.1093/biomet/78.3.557

[7] T. DiCiccio, C. Field and D. A. S. Fraser, “Approximation of Marginal Tail Probabilities and Inference for Scalar Parameters,” Biometrika, Vol. 77, 1990, pp. 77-95. doi:10.1093/biomet/77.1.77