StressStrengthR

Nonparametric estimate of the stress-strength reliability parameter R = P(X<Y)

Function StressStrengthR calculates the nonparametric estimate of the reliability parameter R = P(X<Y) in stress-strength models, with continuous and/or discrete distributions of the random variables X and Y, based on the random samples (X1,...,Xn) from the distribution of X, and the random sample (Y1,...,Ym) from the distribution of Y, computed by using the Wilcoxon-Man-Whitney statistic W = sum_{i=1}^n sum_{j=1}^m [I(X(i)<Y(j)) + I(X(i)==Y(j))/2]. The reliability parameter R is then estimated by R = W/(n*m).

Moreover, the algorithm generates B realizations of the reliability parameter R from the bootstrapped samples, which are used to calculate the estimated confidence interval for R. Under null hypothesis (that the distribution of X is equivalent with the distribution of Y) the true probability value is P(X<=Y) = P(Y>=X) = 0.5 should be covered by the confidence interval with stated probability (1-alpha).

In medicine, important example of applications of the parameter R = P(X<Y) is given by treatment comparisons. Here X is the response (the diagnostic test value) for a control group, and Y refers to a treatment group, P(X<Y) measures the effect of the treatment, which is close to 1 only in exceptional cases. This index can be interpreted as the probability that in a randomly selected pair of healthy and diseased individuals the diagnostic test value is higher for the diseased subject.

For more details and alternative methods for estimation the parameter R = P(X<Y) see e.g. Kotz, Lumelskii and Pensky (2003) and Zhou (2008).