ADS-79D-HDBK Downloaded from https://www.everyspec.com A two-sample t test is commonly used to test for the equality of means from two distributions. However, this test assumes that both distributions are normal (also, it usually assumes that both distributions have equal variances, in which case the means are equal if both distributions are subsets of the same normally distributed population). 54 Based on the histogram date, the CI distribution does not appear to follow a normal distribution. The Lilliefors normality test was applied to all the CI data plotted in Figure I-13, before removal and after replacement, and found that neither distribution passed the normality test.
55 For this reason, the non-parametric Wilcoxon Rank Sum test was used, which tests the null hypothesis that the two distributions are identical against the alternative hypothesis that the two distributions differ only with respect to the median. 56 Per this test, the null hypothesis was rejected at a significance level of .05. CI values for the damage and no damage data sets differ significantly. The significance level of .05 or Type I error indicates at 5% chance of rejecting the null hypothesis when it is true. Type I and Type II error rates are described in more details in Table I-III.
For this example, Figure I-14 provides a block diagram of the analysis steps required prior to defining a sample size. Typical statistical analyses for determining sample size are based on the assumption that data follows a normal distribution. One method uses the t statistic to test the hypothesis. Sample size for a hypothesis test using a t statistic can be defined determining the minimum sample size that provides a t statistic that is at the minimum critical region of the distribution 57 . Can the t statistic be used to determine a sample size if the data is non-parametric? Lehman 58 determined the efficiency of the Wilcoxon rank sum test when compared to the two sample t test.
Conover 59 also defined the asymptotic relative efficiency. Both found the maximum loss of efficiency 54 Montgomery, Douglas C. Design and Analysis of Experiments. New York: Wiley, 1991. 55 Conover, W.J Practical Nonparametric Statistics. New York, NY: J. Wiley and Sons, 1999. 56 Pappas, P. and DePuy, V., ―An Overview of Non Parametric Tests in SAS®: When, Why, and How,‖ Paper TU04, Duke Clinical Research Institute, Durham, North Carolina, USA, 2004 57 NIST/SEMATECH e, National Institute of Standards and Technology, ―Handbook of Statistical Methods,‖ Engineering Statistics Handbook, 2011. 58 Lehmann, E. and H. D'Abrera, ―Nonparametrics:
Statistical Methods Based on Ranks,‖ Upper Saddle River, NJ: Prentice Hall, 1998. 59 Conover, Practical Nonparametric Statistics , New York, NY: J. Wiley and Sons, 1999. 162 TABLE I-III: State of system—health of component H is True o CI = CI damaged undamaged No Damage H is False o CI ≠ CI damaged undamaged Damage Decision Reject H o Indicate Damage False Positive (FP) (α = Type I Error = significance = p-value) False Alarms True Positive (TP) (1-β or power) Hits Fail to reject H o Indicate No Damage True Negative (TN) (1-α) Correct Indication False Negative (FN) (β = Type II Error) Missed Hits