Description

This function inverts pairwise Wilcoxon tests, comparing a common control sample with each of several treatment samples to provide simultaneous confidence bounds for the respective shift parameters by which the sampled treatment populations may differ from the control population. It is assumed that all samples are independent and that the sampled distributions are continuous to avoid ties. The joint coverage probability for all bounds/intervals is calculated, estimated, or approximated, see Details. For treatment of ties also see Details.

Usage

Arguments

Details

The first sample is treated as control sample with sample size n_1. The remaining s=k-1 samples are treatment samples. Let W_{1i}, i=2,…,k denote the respective Wilcoxon statistics comparing the common control sample (index 1) with each of the s treatment samples (indexed by i). For each comparison of control and treatment i sample only the observations of the two samples involved are ranked. By W_i=W_{1i}-n_i(n_i+1)/2 we denote the corresponding Mann-Whitney test statistic. Furthermore, let D_{i(j)} denote the j-th ordered value (ascending order) of the n_1n_i paired differences between the observations in treatment sample i and those of the control sample. By simple extension of results in Lehmann (2006), pages 87 and 92, the following equations hold, relating the null distribution of the Mann-Whitney statistics and the joint coverage probabilities of the D_{i(j_i)} for any set of j_1,…,j_s with 1≤ j_i ≤ n_1 n_i.

P_Δ(Δ_i ≤ D_{i(j_i)}, i=2,…,k)=P_0(W_i≤ j_i -1, i=2,…,k)

and

P_Δ(Δ_i ≥ D_{i(j_i)}, i=2,…,s)=P_0(W_{i}≤ n_1 n_i -j_i, i=2,…,k)

where P_Δ refers to the distribution under Δ=(Δ_2,…,Δ_k) and P_0 refers to the joint null distribution of the W_i when all sampled distributions are the same and continuous. There are k-1 indices j_i that can be manipulated to affect the achieved confidence level. To limit the computational complexity standardized versions of the W_i, i.e., (W_i-μ_i)/τ_i with μ_i and τ_i representing mean and standard deviation of W_i, are used to choose a common value for (j_i -1-μ_i)/τ_i (satisfying the γ level) from the multivariate normal approximation for the W_i (see Miller (1981) and Scholz (2016)), and reduce that to integer values for j_i, rounding up, rounding down, and rounding to the nearest integer. These integers j_i are then used in approximating the actual joint probabilities P_0(W_i≤ j_i -1, i=2,…,k), and from these three coverage probabilities the one that is closest to the nominal confidence level γ and ≥ γ and also also the one that is closest without the restriction ≥ γ are chosen.

When method = "exact" or = "simulated" is specified, the same process is used, using either the fully enumerated exact distribution of W_i, i=2,…,k (based on a recursive version of Chase's sequence as presented in Knuth (2011)) for all sample splits, or the simulated distribution of W_i, i=2,…,k. However, since these distributions are discrete the starting point before rounding up is the smallest quantile such that the proportion of distribution values less or equal to it is at least γ. The starting point before rounding down is the highest quantile such that the proportion of distribution values less or equal to it is at most γ. The third option of rounding to the closest integer is performed using the average of the first two.

Confidence intervals are constructed by using upper and lower confidence bounds, each with same confidence level of (1+γ)/2.

When the original sample data appear to be rounded, and especially when there are ties, one should widen the computed intervals or bounds by the rounding ε, as illustrated in Lehmann (2006), pages 85 and 94. For example, when all sample values appear to end in one of .0, .2, .4, .6, .8, the rounding ε would be .2. Ultimately, this is a judgment call for the user. Such widening of intervals will make the actually achieved confidence level ≥ the stated achieved level.

Value

A list of class kSamples with components

warning

method = "exact" should only be used with caution. Computation time is proportional to the number of enumerations. Experiment with system.time and trial values for Nsim to get a sense of the required computing time.

References

Knuth, D.E. (2011), The Art of Computer Programming, Volume 4A Combinatorial Algorithms Part 1, Addison-Wesley

Lehmann, E.L. (2006), Nonparametrics, Statistical Methods Based on Ranks, Revised First Edition, Springer Verlag.

Miller, Rupert G., Jr. (1981), Simultaneous Statistical Inference, Second Edition, Springer Verlag, New York.

Scholz, F.W. (2016), "On Steel's Test with Ties", submitted to Journal of Nonparametric Statistics.

Examples