^{1}

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This paper, comparison of two sample tests, is motivated by the fact that in the test of significant difference between two independent samples, numerous methods can be adopted; each may lead to significant different results; this implies that wrong choice of test statistic could lead to erroneous conclusion. To prevent misleading information, there is a need for proper investigation of some selected methods for test of significant difference between variables/subjects most especially, independent samples. The paper examines the efficiency and sensitivity of four test statistics to ascertain which test performs better. Based on the results, the relative efficiency favours median test as being more efficient than modified median test for both symmetric and asymmetric distributions. In terms of power of test, median test is more sensitive than Modified Median (MMED) test since it has higher power irrespective of the sample sizes for both symmetric and asymmetric distribution. In terms of relative efficiency for asymmetric distribution Modified Mann-Whitney U test is more efficient than Mann-Whitney U test (MMWU), and then for symmetric distribution, Mann-Whitney U test (MMWU) is more efficient than Modified Mann-Whitney in sample size of 5; but for other sample sizes considered Modified Mann-Whitney U test (MMWU) is better than Mann-Whitney. Using power of test for both symmetric and asymmetric distributions, Mann-Whitney is more sensitive than Modified Mann-Whitney U test (MMWU) because it has higher power.

One of the challenges faced by researchers most especially, statisticians, is to take decisions in the presence of uncertainties. Most often, intelligent guess is made and statistical methods are applied to validate or reject any possible assumptions that might have been made to enable the use of such methods. Numerous methods exist for testing statistical hypotheses in various conditions. In some cases, the probability distribution of the population from which samples are drawn is known. For instance, if the population is assumed to be normal; then, the sample size is assumed to be sufficiently large to justify the assumption of normality. In special cases, the sample sizes are very small and the probability distribution of the populations from which samples are drawn is unknown; hence, the sample is said to be distribution free and only non-parametric methods are applied. Thus, in most cases where the assumption of parametric methods is violated or not met, the non-parametric methods are usually preferred. Non-parametric methods that readily suggest themselves include the Median and the Mann- Whitney U test [

(1) MEDIAN TEST: Median test is a procedure for testing whether two independent groups (samples) differ in central tendencies represented by the population median [

where

The test statistic is

which under

Reject

(2) Modified median test intrinsically adjusted for ties is used for test of equality in population media [

The null hypothesis

The test statistic is

where

m is sample size of variable X.

n is sample size of variable Y.

Reject H_{0} at α-level of significance if

(3) Mann-Whitney U test is used for determination of the likelihood that two samples/groups emanated from the same population/distribution [

The test statistic is

Then,

where

n_{1} is the total number of the first group/observation.

n_{2} is the total number of the first group/observation.

R_{1} is the sum of the ranks for the first group/observation.

Then

is the mean and

is the standard deviation.

This Z-score is, as usual, compared at a given level of significance with an appropriate critical value obtained from a normal distribution table for a rejection or acceptance of the null hypothesis.

(4) Modified Intrinsically Ties Adjusted Mann-Whitney U test is used to check whether two samples could have been drawn from the same population/distribution [

The test statistic is

where

n_{1} is the sample size of variable

n_{2} is the sample size of variable

R_{1} and R_{2} are the respectively sums of the ranks assigned to observations from populations

_{1} is on the average greater than or less than observations or scores by subject from population X_{2}._{ }

The test hypothesis will be

vs

Reject H_{0} at α-level of significance if

Relative Efficiency of two test statistics (R.E) is the ratio of the variances of one of the two test statistics to the other (say:

Between test 1 and test 2, test 2 is relatively more efficient than test 1 if the relative efficiency of the tests,

Power of a statistical test is the probability of rejecting the null hypothesis when it is in fact false and should be rejected (i.e. the probability of not committing a type II error [

In this paper, Monte Carlo’s Simulation techniques was used in the generation of data of different distributions and varying sample sizes ranging from 5 to 100 which was repeated 30 times for each sample size. In the simulation, sample size of 5, 10, 50 and 100 were considered to cover both small and large sample sizes. Monte Carlo simulation is defined as a method to generate random sample data based on some known distribution for numerical experiments. Monte Carlo simulation is an algorithm used to determine performance of an estimator or test statistic under various scenarios [

1) Specify the data generation process.

2) Choose a sample size N for the MC simulation.

3) Choose the number of times to repeat the MC Simulation.

4) Generate a randomsample of size N based on the data generation process.

5) Using random sample generated in 4 above, calculate the test statistic(s).

6) Go backto (4) and (5) until desirable replicate is achieved.

7) Examine parameter estimates, test statistics, etc.

In the paper, for data from a known family of distributions, Gamma (4, 0.3) and Beta (2, 2) were used.

From the simulated data using Monte Carlo simulation approach, the following results were obtained: Tables 1-4 are test statistics value of asymmetric distribution for different sample size while Tables 5-8 are test statistics value of symmetric distribution for different sample size.

Variances were computed from

Median | Modified median | Mann-Whitney U test | MMWU |
---|---|---|---|

0.4 | 2.1267361 | 31 | 5.425347 |

0.4 | 0.0400641 | 27 | 5.008013 |

0.4 | 1.0416667 | 25 | 5.208333 |

0.4 | 0.0400641 | 27 | 5.008013 |

0.4 | 3.7224265 | 32 | 5.744485 |

0.4 | 1.0416667 | 30 | 5.208333 |

3.6 | 3.42E+01 | 37 | 1.18E+01 |

0.4 | 3.7224265 | 23 | 5.744485 |

0.4 | 3.7224265 | 23 | 5.744485 |

0.4 | 2.1267361 | 31 | 5.425347 |

3.6 | 6.0019841 | 33 | 6.200397 |

0.4 | 6.0019841 | 33 | 6.200397 |

3.6 | 9.265350877 | 34 | 6.85307 |

0.4 | 1.41E+01 | 20 | 7.8125 |

0.4 | 0.0400641 | 27 | 5.008013 |

0.4 | 1.41E+01 | 20 | 7.8125 |

3.6 | 2.1267361 | 24 | 5.425347 |

0.4 | 3.7224265 | 23 | 5.744485 |

0.4 | 0.3652597 | 29 | 5.073052 |

3.6 | 6.0019841 | 22 | 6.200397 |

3.6 | 2.15E+01 | 36 | 9.300595 |

0.4 | 0.3652597 | 29 | 5.073052 |

0.4 | 0.0400641 | 27 | 5.008013 |

0.4 | 0.0400641 | 27 | 5.008013 |

3.6 | 2.1267361 | 24 | 5.425347 |

0.4 | 2.1267361 | 24 | 5.425347 |

0.4 | 9.265350877 | 34 | 6.85307 |

0.4 | 3.72E+00 | 23 | 5.744485 |

3.6 | 3.42E+01 | 18 | 1.18E+01 |

0.4 | 1.0416667 | 30 | 5.208333 |

Median | Modified median | Mann-Whitney U test | MMWU |
---|---|---|---|

0 | 1.461039 | 111 | 10.1461 |

3.2 | 4.16666667 | 95 | 10.41667 |

0.8 | 3.34849111 | 96 | 10.33485 |

0.8 | 9.89010989 | 90 | 1.10E+01 |

3.2 | 1.69E+01 | 124 | 1.17E+01 |

0.8 | 0 | 105 | 10 |

0 | 0.1602564 | 107 | 10.01603 |

0.8 | 2.68E+01 | 128 | 1.27E+01 |

0.8 | 1.14E+01 | 121 | 1.11E+01 |

0 | 6.1120543 | 93 | 10.61121 |

0.8 | 1.69E+01 | 86 | 1.17E+01 |

0 | 0 | 105 | 10 |

0 | 1.010101 | 100 | 10.10101 |

0 | 0.1602564 | 103 | 10.01603 |

0 | 0.6441224 | 109 | 10.06441 |

0.8 | 1.010101 | 100 | 10.10101 |

0 | 0.3613007 | 102 | 10.03613 |

0 | 0.3613007 | 108 | 10.03613 |

0 | 1.461039 | 99 | 10.1461 |

0 | 1.010101 | 110 | 10.10101 |

0.8 | 1.010101 | 115 | 10.41667 |

0 | 4.16666667 | 101 | 10.06441 |

0.8 | 0.6441224 | 98 | 10.19992 |

0 | 4.16666667 | 95 | 10.41667 |

0 | 4.16666667 | 95 | 10.41667 |

0 | 0.04001601 | 104 | 10.004 |

3.2 | 0.1602564 | 107 | 10.01603 |

0.8 | 2.40E+01 | 127 | 1.24E+01 |

0 | 2.40E+01 | 127 | 1.24E+01 |

0 | 1.999184 | 98 | 10.19992 |

Median | Modified median | Mann-Whitney U test | MMWU |
---|---|---|---|

1.44 | 1.58E+01 | 2624 | 5.03E+01 |

0.64 | 2.29E+01 | 2406 | 5.05E+01 |

1.44 | 5.77330169 | 2465 | 5.01E+01 |

0.16 | 3.539404 | 2572 | 5.01E+01 |

0.16 | 1.58E+01 | 2426 | 5.03E+01 |

1.44 | 84.48324 | 2751 | 5.17E+01 |

1.44 | 3.41E+01 | 2670 | 5.07E+01 |

1.71 | 1.99E+01 | 2636 | 5.04E+01 |

0.64 | 2.4359713 | 2564 | 5.00E+01 |

1.44 | 2.3125372 | 2563 | 5.00E+01 |

1.44 | 5.23E+01 | 2346 | 5.10E+01 |

0 | 0.9219399 | 2501 | 5.00E+01 |

0.64 | 4.11E+01 | 2684 | 5.08E+01 |

0 | 5.77330169 | 2465 | 5.01E+01 |

0.16 | 2.25E+01 | 2643 | 5.04E+01 |

1.44 | 98.1891 | 2768 | 5.20E+01 |

0.64 | 3.22E+01 | 2666 | 5.06E+01 |

1.44 | 4.49E+01 | 2359 | 5.09E+01 |

0.16 | 1.78E+01 | 2420 | 5.04E+01 |

0 | 1.22E+01 | 2612 | 5.02E+01 |

0.49 | 2.0753214 | 2489 | 5.00E+01 |

0 | 1.85E+01 | 2418 | 5.04E+01 |

1.44 | 5.29E+01 | 2705 | 5.11E+01 |

0.64 | 0.4096671 | 2509 | 5.00E+01 |

0.16 | 10.28211555 | 2445 | 5.02E+01 |

4 | 4.49E+01 | 2691 | 5.09E+01 |

2.56 | 113.9621 | 2786 | 5.23E+01 |

0 | 3.41E+01 | 2670 | 5.07E+01 |

5.76 | 318.1589 | 2105 | 5.64E+01 |

1.44 | 4.54E+01 | 2692 | 5.09E+01 |

Median | Modified median | Mann-Whitney U test | MMWU |
---|---|---|---|

2 | 146.1039 | 9450 | 101.461 |

2.88 | 115.816 | 9515 | 101.158 |

0.32 | 0.2916085 | 10023 | 100.003 |

0.08 | 3.28E+01 | 9764 | 100.328 |

0.08 | 1.78E+01 | 9839 | 100.178 |

0.32 | 1.6902857 | 9985 | 100.017 |

0.32 | 6.97446092 | 10182 | 100.07 |

2 | 2.71E+01 | 10310 | 100.271 |

0.72 | 196.4389 | 9356 | 101.964 |

0 | 3.16940419 | 9961 | 100.032 |

0.32 | 4.24E+01 | 10375 | 100.424 |

0 | 2.41E+01 | 10295 | 100.241 |

0.32 | 76.26727 | 10485 | 100.763 |

1.28 | 125.2026 | 9494 | 101.252 |

2.88 | 124.747 | 10605 | 101.248 |

3.92 | 213.5569 | 9327 | 102.136 |

0.08 | 1.87E+01 | 10266 | 100.187 |

2 | 91.84752 | 9573 | 100.919 |

1.28 | 4.56E+01 | 9713 | 100.456 |

0 | 1.28E+01 | 10229 | 100.128 |

0 | 4.16E+01 | 9728 | 100.417 |

0.72 | 2.59E+01 | 9796 | 100.259 |

0.72 | 2.77E+01 | 10313 | 100.277 |

0.32 | 3.54E+01 | 9753 | 100.354 |

3.92 | 169.7601 | 9404 | 101.698 |

0.32 | 82.76295 | 9597 | 100.828 |

5.12 | 414.0671 | 11047 | 104.141 |

2 | 244.2155 | 10822 | 102.442 |

0 | 0.2916085 | 10077 | 100.003 |

0.72 | 4.35E+01 | 10379 | 100.435 |

Median | Modified median | Mann-Whitney U test | MMWU |
---|---|---|---|

10 | 2.1267361 | 15 | 5.425347 |

0.4 | 0.0400641 | 27 | 5.008013 |

0.4 | 3.7224265 | 32 | 5.744485 |

3.6 | 5.99E+01 | 17 | 1.70E+01 |

0.4 | 2.1267361 | 24 | 5.425347 |

0.4 | 1.0416667 | 30 | 5.208333 |

0.4 | 2.1267361 | 31 | 5.425347 |

0.4 | 0.0400641 | 27 | 5.008013 |

0.4 | 0.0400641 | 28 | 5.008013 |

0.4 | 0.0400641 | 27 | 5.008013 |

0.4 | 0.0400641 | 27 | 5.008013 |

0.4 | 9.265350877 | 34 | 6.85307 |

0.4 | 2.1267361 | 24 | 5.425347 |

0.4 | 1.0416667 | 30 | 5.208333 |

0.4 | 6.0019841 | 22 | 6.200397 |

0.4 | 1.0416667 | 30 | 5.208333 |

3.6 | 3.7224265 | 32 | 5.744485 |

0.4 | 9.265350877 | 21 | 6.85307 |

0.4 | 0.0400641 | 28 | 5.008013 |

0.4 | 0.3652597 | 26 | 5.073052 |

0.4 | 0.0400641 | 28 | 5.008013 |

3.6 | 137.7604 | 39 | 3.26E+01 |

0.4 | 1.0416667 | 30 | 5.208333 |

0.4 | 1.0416667 | 30 | 5.208333 |

0.4 | 3.7224265 | 32 | 5.744485 |

3.6 | 5.99E+01 | 38 | 1.70E+01 |

3.6 | 1.41E+01 | 20 | 7.8125 |

0.4 | 9.265350877 | 34 | 6.85307 |

0.4 | 0.3652597 | 29 | 5.073052 |

0.4 | 6.0019841 | 22 | 6.200397 |

Median | Modified median | Mann-Whitney U test | MMWU |
---|---|---|---|

0 | 0.04001601 | 106 | 10.004 |

3.2 | 2.99E+01 | 129 | 1.30E+01 |

3.2 | 7.25010725 | 118 | 10.72501 |

0 | 6.1120543 | 117 | 10.61121 |

0 | 5.08617066 | 94 | 10.50862 |

0.8 | 4.16666667 | 115 | 10.41667 |

0 | 3.34849111 | 96 | 10.33485 |

0 | 4.16666667 | 95 | 10.41667 |

0 | 2.627258 | 113 | 10.26273 |

0 | 5.08617066 | 94 | 10.50862 |

3.2 | 5.08617066 | 116 | 10.50862 |

0 | 0.04001601 | 106 | 10.004 |

3.2 | 1.49E+01 | 87 | 11.48897 |

0.8 | 0.1602564 | 103 | 10.01603 |

0 | 3.34849111 | 96 | 10.33485 |

0.8 | 3.34849111 | 114 | 10.33485 |

0.8 | 5.08617066 | 116 | 10.50862 |

0.8 | 1.90E+01 | 125 | 1.19E+01 |

0.8 | 3.34849111 | 96 | 10.33485 |

0.8 | 8.506944444 | 119 | 1.09E+01 |

0.8 | 0.04001601 | 104 | 10.004 |

0.8 | 1.010101 | 100 | 10.10101 |

0 | 0.6441224 | 109 | 10.06441 |

0.8 | 5.08617066 | 116 | 10.50862 |

0 | 2.627258 | 97 | 10.26273 |

0 | 0.04001601 | 106 | 10.004 |

0 | 0.3613007 | 102 | 10.03613 |

0.8 | 9.89010989 | 90 | 1.10E+01 |

0.8 | 1.14E+01 | 89 | 1.11E+01 |

0.8 | 1.999184 | 112 | 10.19992 |

Median | Modified median | Mann-Whitney U test | MMWU |
---|---|---|---|

1.96 | 2.02E+01 | 2413 | 5.04E+01 |

0.04 | 2.4359713 | 2457 | 5.01E+01 |

0.36 | 0.77464 | 2547 | 5.00E+01 |

0.36 | 134.0615 | 2807 | 5.27E+01 |

0.04 | 3.10144281 | 2481 | 5.01E+01 |

0.36 | 3.10144281 | 2719 | 5.12E+01 |

1 | 7.86466359 | 2498 | 5.00E+01 |

0.04 | 0 | 2489 | 5.00E+01 |

3.24 | 6.24E+01 | 2720 | 5.12E+01 |

0.36 | 5.02769078 | 2469 | 5.01E+01 |

3.24 | 83.71228 | 2750 | 5.17E+01 |

0.36 | 3.18E+01 | 2385 | 5.06E+01 |

1 | 6.64E+01 | 2324 | 5.13E+01 |

1.96 | 3.96E+01 | 2681 | 5.08E+01 |

0.64 | 1.55E+01 | 2623 | 5.51E+01 |

7.84 | 2.65E+01 | 2906 | 5.51E+01 |

0.04 | 3.88E+01 | 2406 | 5.05E+01 |

0.36 | 0.05760133 | 2519 | 5.00E+01 |

0.36 | 9.03251706 | 2450 | 5.02E+01 |

0.04 | 5.39401309 | 2467 | 5.01E+01 |

0.04 | 0.2704292 | 2538 | 5.00E+01 |

3.24 | 124.258 | 2797 | 5.25E+01 |

3.24 | 149.5787 | 2822 | 5.30E+01 |

0.36 | 1.88E+01 | 2416.5 | 5.04E+01 |

0.04 | 8.555579317 | 2598 | 5.02E+01 |

0.36 | 2.99E+01 | 2661 | 5.06E+01 |

0.36 | 0.1024042 | 2517 | 5.00E+01 |

4.84 | 2.99E+01 | 2661 | 5.06E+01 |

1 | 1.4408299 | 2495 | 5.00E+01 |

0.04 | 2.0753214 | 2489 | 5.00E+01 |

Median | Modified median | Mann-Whitney U test | MMWU |
---|---|---|---|

0 | 2.23E+01 | 10286 | 100.2233 |

0.32 | 2.92E+01 | 10320 | 100.2925 |

0.72 | 5.18E+01 | 9691 | 100.5182 |

0.32 | 7.846151383 | 9910 | 100.0785 |

1.28 | 8.88829316 | 10199 | 100.0889 |

2 | 278.4764 | 9227 | 102.7848 |

5.12 | 193.564 | 9361 | 101.9356 |

1.28 | 5.39E+01 | 9684 | 100.5387 |

0 | 1.7959225 | 10117 | 100.018 |

0.08 | 5.38529858 | 9934 | 100.0539 |

0.32 | 4.32827258 | 9946 | 100.0433 |

0.08 | 2.2505064 | 10125 | 100.0225 |

1.28 | 0 | 10357 | 100.3784 |

0.32 | 3.765017 | 10147 | 100.0377 |

2.88 | 233.3302 | 9295 | 102.3333 |

1.28 | 2.65E+01 | 10307 | 100.2649 |

0 | 3.88E+01 | 9739 | 100.3884 |

0.32 | 1.2101464 | 10105 | 100.0121 |

0.32 | 4.7546596 | 10159 | 100.0475 |

1.28 | 104.7176 | 10559 | 101.0472 |

2 | 294.7239 | 9204 | 102.9472 |

0.72 | 8.301285418 | 10194 | 100.083 |

3.92 | 157.6999 | 9427 | 101.577 |

1.28 | 5.48E+01 | 10418.5 | 96.96971 |

0.32 | 4.24540158 | 9947 | 100.0425 |

0 | 5.29279989 | 10165 | 100.0529 |

0.08 | 5.29279989 | 9935 | 100.0529 |

0.08 | 4.24540158 | 9947 | 100.0425 |

0.08 | 2.90E+01 | 9781 | 100.2903 |

0.72 | 116.2544 | 9514 | 101.1625 |

Type of distr. | Test statistic | 5 | 10 | 50 | 100 |
---|---|---|---|---|---|

Symmetric distribution | Median | 1.695 | 1.081 | 3.219 | 1.487 |

MMED | 791.97 | 41.56 | 1748.91 | 7470.9 | |

Asymmetric distribution | Median | 2.072 | 0.924 | 1.575 | 1.918 |

MMED | 82.56 | 64.37 | 3605.1 | 8664.9 | |

Symmetric distribution | Mann-Whitney | 27.513 | 124.97 | 22424.7 | 137572.1 |

MMWU | 31.68 | 0.416 | 1.978 | 1.18 | |

Asymmetric distribution | Mann-Whitney | 25.082 | 125.18 | 23213.6 | 200042.0 |

MMWU | 3.302 | 0.641 | 1.442 | 0.867 |

As shown in _{1} to M_{4} are the methods considered as M_{1} is the Median test and M_{2} is Modified Median test (MMED), M_{3} is the Mann-Whitney U test and M_{4} is Modified Mann-Whitney U (MMWU) test statistic.

All the ratios for the first and second rows are less than 1.0 which implies the method used as numerator is better and more reliable than the method used as denominator for all the sample sizes considered, i.e. Median Test is better than Modified Median intrinsically Adjusted for Ties (MMED) using Relative Efficiency (

Moreover, considering methods 3 and 4 for asymmetric distribution, M_{4} (Modified Mann-Whitney U test) is better than M_{3} (Mann-Whitney U test) since the values of R.E are all greater than 1.0. Considering symmetric distribution, the efficiency of M_{3} (Mann-Whitney U test) is better/stronger for small sample size (5) and as sample size increases, the strength of M_{4} (Modified Mann-Whitney U test) increases and outweighs M_{3} (Mann- Whitney U test); this implies that the method is inconsistent because its efficiency decreases as sample size increases.

As shown in

As shown in

Power of test were computed from

Power of test is the sensitivity of a test statistic and the greater the value, the more sensitive the test statistic for both symmetric and asymmetric distributions. Median test is more sensitive than MMED because median has higher power than MMED irrespective of the sample sizes.

Considering both Mann-Whitney and MMWU, Mann-Whitney U test is more sensitivity than MMWU for both symmetric and asymmetric distribution. For better understanding of sensitivity of the four test statistics, line chart of power of test is constructed as shown in

As shown in

As shown in

Distribution | 5 | 10 | 50 | 100 |
---|---|---|---|---|

Asymmetric (M_{1}M_{2}) | 0.0251 | 0.0144 | 0.0004 | 0.0002 |

Symmetric (M_{1}M_{2}) | 0.0021 | 0.0260 | 0.0018 | 0.0002 |

Asymmetric (M_{3}M_{4}) | 7.5960 | 195.2286 | 16098.1969 | 230728.9504 |

Symmetric (M_{3}M_{4}) | 0.8685 | 300.4807 | 11337.0576 | 116586.5254 |

Type of distr. | Test statistic | 5 | 10 | 50 | 100 |
---|---|---|---|---|---|

Symmetric distribution | Median | 1 | 1 | 0.9667 | 0.9 |

MMED | 0.6667 | 0.5 | 0.3333 | 0.1667 | |

Asymmetric distribution | Median | 1 | 1 | 0.9333 | 0.9 |

MMED | 0.6667 | 0.6 | 0.2 | 0.1333 | |

Symmetric distribution | Mann-Whitney | 0.9 | 1 | 0.9333 | 0.9333 |

MMWU | 0 | 0 | 0 | 0 | |

Asymmetric distribution | Mann-Whitney | 1 | 1 | 0.9667 | 0.9667 |

MMWU | 0 | 0 | 0 | 0 |

Type of distr. | Test statistic | 5 | 10 | 20 | 30 | 50 | 100 |
---|---|---|---|---|---|---|---|

Symmetric distribution | Median | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0333 | 0.1000 |

MMED | 0.3333 | 0.5000 | 0.7000 | 0.8000 | 0.6667 | 0.8333 | |

Asymmetric distribution | Median | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0667 | 0.1000 |

MMED | 0.3333 | 0.4000 | 0.6000 | 0.6667 | 0.8000 | 0.8667 |

Type of distr. | Test statistic | 5 | 10 | 50 | 100 |
---|---|---|---|---|---|

Symmetric distribution | Mann-Whitney | 0.1000 | 0.0000 | 0.0667 | 0.0667 |

MMWU | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |

Asymmetric distribution | Mann-Whitney | 0.0000 | 0.0000 | 0.0333 | 0.0333 |

MMWU | 1.0000 | 1.0000 | 1.0000 | 1.0000 |

We have in this paper presented a nonparametric statistical method for the analysis of two sample tests. Based on the result of the analysis used, it is observed that for both symmetric and asymmetric distributions, median test is more efficient than Modified Median (MMED) test using relative efficiency as a measure of the efficiency of test statistic since the relative efficiency values are less than 1.0 while in terms of power of test for both symmetric and asymmetric distributions, median test is more sensitive than Modified Median (MMED) test since it has higher power. For Mann-Whitney U test and Modified Mann-Whitney U test (MMWU) using both relative efficiency and power of test, Mann-Whitney U test is more efficient and more sensitive than Modified Mann-Whitney U test (MMWU) since the relative efficiency values are greater than 1 and also it has higher power. In terms of sample size, efficiency of the method is independent of sample sizes except Modified Median Test which has higher power for small sample sizes.

Edith Uzoma Umeh,Nkiru Obioma Eriobu, (2016) Comparison of Two Sample Tests Using Both Relative Efficiency and Power of Test. Open Journal of Statistics,06,331-345. doi: 10.4236/ojs.2016.62029