Combined nonparametric tests for equality of location and scale under ordered alternatives

Introduction

Ordered alternatives tests are sometimes used in life-testing experiments and drug-screening studies. An ordered alternative test is used to gain power if the researcher thinks parameters will be ordered in a certain way if they are different. Let 𝑋 𝑖1 , 𝑋 𝑖2 , … , 𝑋 𝑖𝑛 𝑖 , 𝑖 = 1,2, … , 𝑘 (𝑘 ≥ 3)be random independent samples each of size 𝑛 𝑖 from 𝑘 populations, where -∞ < 𝜇 𝑖 < +∞ and 𝜎 𝑖 > 0 are location and scale parameters, respectively. We are interested in testing the hypotheses in (1) Versus 𝐻 𝑎: 𝜇 1 ≤ 𝜇 2 ≤ ⋯ ≤ 𝜇 𝑘 and 𝜎 1 ≤ 𝜎 2 ≤ ⋯ ≤ 𝜎 𝑘 with at least one inequality strict.

Background

Lepage [1] proposed a two-sample nonparametric test for location and scale. The Lepage test combines the Wilcoxon test [2] for location and the Ansari-Bradley test [3] for scale alternatives. Several extensions of the Lepage test have been proposed, and Marozzi [4] concluded that no single test performs best under all situations. Alsubie and Magel (2020a) [5] proposed two nonparametric tests, L1 and L2, for the simple tree alternative. Both tests combine the Fligner-Wolfe test [6] for location with a modified Ansari-Bradley test for scale.

The test L1 standardizes the two components separately before combining them, whereas L2 combines the components first and then standardizes the sum. Their study found that L2 generally has higher power when only location parameters differ, while L1 performs better when scale differences are present, or when both are different. Alsubie and Magel (2020b) [7] proposed three additional tests, M1, M2, and M3, combining the Fligner-Wolfe and Moses [8] statistics. Test M1 standardizes the components separately, M2 standardizes the combined statistic, and M3 places additional weight on the Moses component. Their results showed that M2 performs well under location alternatives, whereas M1 is often preferred when scale differences are present or when both differences are present. M3 does perform well when just the scale parameters are different. Zhu and Magel (2022) [9] proposed six nonparametric tests for the nondecreasing ordered alternative by combining the Jonckheere-Terpstra, Modified Jonckheere-Terpstra, and Shan statistics with the Moses technique. Their simulation study evaluated significance levels and power for three, four, and five populations under Normal, t, and Exponential distributions using both equal and unequal sample sizes. The present study extends the work of Zhu and Magel (2022) by focusing on JT-and MJT-based procedures, introducing weighted combinations that place greater emphasis on the Moses scale component, comparing mean-based and medianbased centering approaches, and evaluating performance under additional distributions including Logistic, Laplace, Cauchy, and Chi-square distributions. The purpose of this paper is to compare four nonparametric tests, T 1 , T 2 , T 3 , and T 4 , for the nondecreasing ordered alternative when testing for differences in location, scale, or both. All tests will be compared using both the sample mean and sample median of the subgroups when calculating the sum of squared deviations for the Moses technique. T 1 and T 2 using the sample mean were introduced by Zhu and Magel (2022) T 3 and T 4 are introduced in this research as well as using the sample median in the Moses technique. A Monte Carlo simulation study is conducted to evaluate Type I error control and power under six distributions for k = 4 populations.

Materials and Methods

We compared four tests to test the hypothesis in (1). All four tests use the Moses technique (Moses, 1963) [8] to transform data so that tests for scale parameters can be done by using tests for location parameters on the transformed data.

Jonckheere-Terpstra (JT) Test + Moses Test

The tests, 𝑇 1 and 𝑇 3 , use the Jonckheere-Terpstra (JT) test (Jonckheere, 1954; Terpstra, 1952) [9,10] on both the original data and the data transformed by the Moses technique. For 𝑘 (𝑘 ≥ 3) populations, the JT test is defined by

Modified Jonckheere-Terpstra (MJT) Test + Moses Test

The second and fourth tests, 𝑇 2 and 𝑇 4 , use the Modified Jonckheere-Terpstra (MJT) test [12] on both the original data and the data transformed by the Moses technique. For 𝑘 greater than 3 populations, the MJT test is defined by

The MJT test is applied to the original data and standardized to obtain 𝑀𝐽𝑇 * . The MJT test is applied to the transformed values using the Moses technique, MJS, and standardized to obtain 𝑀𝐽𝑆 * . The expected value and variance of 𝑀𝐽𝑇are:

Simulation Study

A simulation study was conducted to evaluate the two versions of the four proposed tests and was implemented in SAS version 9.4. The test statistics were evaluated assuming random samples from six distributions: Normal, Logistic, Laplace, Cauchy, Exponential, and Chi-square. The RAND function was used to generate random samples from each distribution. Both mean-based and median-based centering approaches were implemented within the Moses-type subgroup framework. The simulation was conducted in two steps. The first step was to estimate the Type I error rates of the test statistics. Replications of 10,000 samples were used for all simulations. The stated significance level was 0.05. Type I error rates were estimated by counting the total number of times the null hypothesis was rejected when the null hypothesis was true, divided by 10,000. A test was considered to maintain nominal size if the estimated Type I error rate fell within the range 0.045 to 0.055. The second step was to compare the powers of the test statistics under various conditions. Powers were estimated by counting the total number of times each proposed test rejected the null hypothesis under a given condition, divided by 10,000. Three and four populations were considered, with only the results for four populations given in this paper since the results for three populations were found to be similar. Equal and unequal sample sizes were used. For equal sample sizes, 𝑛 𝑖 = 18 and 𝑛 𝑖 = 36 for all populations were considered. For unequal sample sizes, configurations with a larger control group, 𝑛 1 =36, and the remaining samples each of size 18 and configurations with larger treatment groups with the sample sizes of the treatment populations (considered the 2 nd through 4 th populations) being each 36 and the sample size from the control population being 18 were examined. Two subgroup sizes (𝑚 = 3 and 𝑚 = 6) were used for each scenario when applying the Moses technique. Three alternative structures were examined. First, the location parameters were different and the scale parameters were equal. Second, the location parameters were equal and the scale parameters were different. Third, both the location and scale parameters were different simultaneously. The null hypothesis was rejected for large values of the test statistics, using the critical value 𝑧 0.95 = 1.645for the one-sided ordered alternative at 𝛼 = 0.05.

Results and Discussion

The results were similar between the equal and unequal sample sizes considered in this study. Therefore, only the results are given for equal sample sizes of 36. Results are given for each test statistic when the sample mean was used in calculating the sum of squared deviations for each subsample for the Moses technique and when the sample median was used. Under all symmetric distributions, T₁ and T₂ showed substantially higher power than T₃ and T₄. For the Normal distribution with location parameters (0, 0, 1, 1), T₁ and T₂ achieved power around 0.97, while T₃ and T₄ achieved only around 0.45. Similar patterns were observed for the Logistic, Laplace, and Cauchy distributions, with T₁ and T₂ consistently outperforming T₃ and T₄. There was little difference in powers between mean-based and median-based procedures across all distributions.

Table 3 presents power results for pure scale alternatives. The pattern was reversed: T₃ and T₄ showed substantially higher powers than T₁ and T₂ across all symmetric distributions. For the Normal distribution with scale parameters (1, 1, 2, 2), T₃ and T₄ achieved power around 0.94, while T₁ and T₂ achieved only around 0.75. This pattern held consistently across the Logistic, Laplace, and Cauchy distributions. These results confirm that the weighted scale component in T₃ and T₄ provides an advantage when scale differences are present without location shifts. It is noted that the chi-square distribution with 10 degrees of freedom is less skewed. This behavior is consistent with the variance inflation observed in Table 5 and helps explain the elevated Type 1 error rates under skewed distributions.

Conclusion

Four combined nonparametric test statistics with two variations each were compared for detecting nondecreasing ordered alternatives in both location and scale. Two of these were proposed in this research, T₃ and T₄, and two were existing tests. The sample median variation was also investigated in this research for all four test statistics. A Monte Carlo simulation study was conducted to evaluate Type 1 error control and power under six distributions for 3 and 4 populations with the results for 4 populations given in this paper. The simulation results support the following recommendations. When the researcher suspects that location parameters differ and scale parameters are equal, either T₁ or T₂ is recommended. When scale parameters differ and location parameters are equal, either T₃ or T₄ is recommended. When both location and scale parameters differ, the choice depends on which component is expected to dominate; T₃ or T₄ are recommended when scale differences are expected to be larger, with T₄ having slightly higher estimated powers. There was little practical difference between mean-based and median-based procedures, so either may be used. The tests perform well under symmetric distributions including the normal, logistic, laplace, and Cauchy. However, Type 1 error rates were substantially inflated under strongly skewed distributions such as the exponential and chi-square with small degrees of freedom. The tests should therefore not be used when the underlying distributions are strongly skewed. Future work will address the correction of Type 1 error inflation under skewed distributions by incorporating the covariance between the location and scale components.

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