Table 1 Statistics for power transformation parameter and statistical test for structural effects based on Monte Carlo simulations

Simulations with non negative estimate (λ) Sample size Transformation parameter and its empirical estimate ± Standard deviation Test for residual normality (proportion with P > 0.05)
N n λ Mean (λ 1 ) ± Std Mean (λ 0 ) ± Std Y λ ${\left({Y}^{\lambda }\right)}^{\left({\lambda }_{0\right)}\right)}$ ${\left({Y}^{\lambda }\right)}^{\left({\lambda }_{1}\right)}$
973 36 0.4 0.397 ± 0.160 0.296 ± 0.123 0.698 0.962 0.968
996 54 0.4 0.386 ± 0.125 0.289 ± 0.107 0.522 0.950 0.971
999 72 0.4 0.393 ± 0.104 0.295 ± 0.010 0.387 0.963 0.978
1000 90 0.4 0.395 ± 0.089 0.295 ± 0.087 0.251 0.941 0.977
1000 108 0.4 0.393 ± 0.085 0.295 ± 0.083 0.179 0.929 0.975
1000 126 0.4 0.393 ± 0.075 0.296 ± 0.074 0.120 0.920 0.976
1000 144 0.4 0.395 ± 0.068 0.299 ± 0.067 0.082 0.912 0.979
1000 162 0.4 0.395 ± 0.067 0.398 ± 0.068 0.056 0.891 0.972
1000 180 0.4 0.395 ± 0.063 0.030 ± 0.063 0.037 0.889 0.968
1000 198 0.4 0.396 ± 0.059 0.301 ± 0.059 0.018 0.880 0.971
1000 216 0.4 0.396 ± 0.057 0.302 ± 0.057 0.010 0.867 0.977
1. Notation: λ is the preset value for generating the data; λ 0 represents the estimated value for the transformation parameter with no factorial treatment effect in the model; λ 1 stands for estimated value of transformation parameter with factorial treatment effect in the model. P-value is obtained by Shapiro-Wilk test with SAS Univariate procedure. A total of 1000 datasets were generated for each fixed sample size and transformation parameter. 