Sample Size Comparison with R-Programming between Two ANOM: Type Methods for Testing the Homogeneity of Variances

Abstract

Pran Kumar and Anjaneyulu (Int J Phys Appl Sci 3:34–40, 2016) and Pran Kumar and Anjaneyulu (Bull Math Stat Res 5:54–58, 2017) derived two sample size expressions for two ANOM-type methods developed by Rao and Harikrishna (J Appl Stat 24:279–287, 1997) and Pran Kumar and Rao (Commun Stat Simul Comput 27:459–468, 1998) respectively for testing the homogeneity of several variances. In this article, an empirical comparative study is done with R-software code programming of R Core Team (R: a language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, 2019) for the sample sizes derived by Pran Kumar and Anjaneyulu (2016, 2017) between the two methods developed by Rao and Harikrishna (1997) and Pran Kumar and Rao (1998). The study is carried to detect the significance of one of the population variance among k normal population variances from their grand average by at least a specified amount ‘d’ for fixed level of significance α and fixed power P in the case of equal sample sizes. The specified amount $\Delta$ in the sample size given by Pran Kumar and Anjaneyulu (2016) and the specified amount D in the sample size given by Pran Kumar and Anjaneyulu (2017) are derived in terms of some common amount ‘d’ for comparison. The tables of comparison of sample sizes are given particularly for one of the significant variance taken as unity among k variances from their grand average and for α = 0.01, 0.05, P = 0.8, 0.9, 0.95, 0.99, d = 1, 3, 5, k = 3(1) 20, 30, 60, The comparison reveals that the sample size derived by Pran Kumar and Anjaneyulu (2017) for the method developed by Pran Kumar and Rao (1998) is less than that of the sample size derived by Pran Kumar and Anjaneyulu (2016) for the method developed Rao and Harikrishna (1997).

Appendix

1.1 Execution and Output of R-Program Given in Sect. 4.1

In the R-program given in Sect. 4.1, the script ‘var’ is used for variance value, ‘alfa’ is used for significance level α value, ‘diff’ is used for significant difference d value, ‘k’ is used to indicate number of samples and also the notations

• n10.8, n20.8, and d0.8 indicate the sample sizes n₁, n₂ and their difference n₁- n₂ respectively at the power 0.8. Similarly,

• n10.9, n20.9, and d0.9 indicate the sample sizes n₁, n₂ and their difference n₁- n₂ respectively at the power 0.9.

• n10.95, n20.95, and d0.95 indicate the sample sizes n₁, n₂ and their difference n₁- n₂ respectively at the power 0.95.

• n10.99, n20.99, and d0.99 indicate the sample sizes n₁, n₂ and their difference n₁- n₂ respectively at the power 0.99.

The execution of the R- program given in Sect. 4.1 and its output for the computations of Tables 1, 2, 3, 4, 5 and 6 are as follows.

1.2 R-Program for Generating a Single Table of Comparison of Sample Sizes

The R program for generating a single table (with respect to Table 1) is as follows:

Similarly, remaining Tables 2, 3, 4, 5 and 6 can be generated by substituting their corresponding values of α (alfa) and d (diff) in the above program.

1.3 R-Program for Generating Multiple Tables of Comparison of Sample Sizes

A general R-program for generating multiple tables of comparison of sample sizes simultaneously for various values of variance (var = 0.5, 1, 2, 5) and levels of significance α (alfa = 0.01, 0.05, 0.10) is as follows.

The above R-software code program can be extended for any number of values of variance, values of levels of significance α, values of significant difference d and values of number of samples k.