6.4.1.2. Hodges-Lehmann Estimator (Unpaired)

If the product of the two sample sizes does not exceed 2 x 10⁹ then an estimate of the difference between the two sample medians and its confidence interval are computed.

First, all n1 x n2 differences between each pair of numbers from the two samples are sorted in increasing order. Then, the median (the Hodges-Lehmann estimator or the shift parameter) is found.

The limits of the asymptotic confidence interval are the K^th smallest and the K^th largest difference:

where K is rounded up to the nearest integer and the mean and standard deviation of the Mann-Whitney statistic are as given in the previous section.

The exact confidence interval is also displayed, which is based on the exact distribution of the Mann-Whitney statistic. To determine the lower bound of the exact interval (the K_lth smallest difference), find k_l such that:

round k_l up to the nearest integer and calculate:

The upper limit is determined likewise, for:

      .

For the paired case of this test see 6.4.2.2. Hodges-Lehmann Estimator (Paired).

Example 1

Example 13.4 on p. 458 from Armitage, P. & G. Berry (1994). Gain in weight of rats receiving diets with high and low protein content are measured. The null hypothesis “there is no difference in median weights” is tested at 95% level.

Open NONPAR12 and select Statistics 1 → Nonparametric Tests (1-2 Samples) → Unpaired Samples. Select High (C1) and Low (C2) as [Variable]s and check the Hodges-Lehmann Estimator (Unpaired) output option to obtain the following results:

Unpaired Samples

Hodges-Lehmann Estimator (Unpaired)

For High and Low

6.4.1.3. Wald-Wolfowitz Runs Test

The null hypothesis “two independent samples have been drawn from the same population” is tested against the alternative hypothesis “they differ in respect of their medians, variability or skewness”. It is assumed that the variable under consideration has a continuous distribution.

All cases from the two samples are sorted together. If the two distributions are similar, then cases belonging to two samples must be scattered randomly. Then the program counts the number of runs (i.e. the number of groups of cases which belong to the same sample). If there are ties between cases belonging to two samples then the minimum and the maximum possible number of runs are reported separately. Two sets of results using the normal approximation are reported.

Asymptotic without Continuity Correction: In this case the Z-statistic is defined as:

      where:

Asymptotic with Continuity Correction: The Z-statistic with continuity correction is defined as:

      In some applications, the test statistic with continuity correction is reported for  and without continuity correction otherwise. The same normal approximation is also used for the Runs Test.

Exact: The exact one- and two-tailed probabilities are reported. Their use is recommended for n £ 30.

Data in one of the three types supported for Two Sample Tests can be used for this test. Missing values are omitted by case.

Example

Table 100 on p. 251 from Cohen, L. & M. Holliday (1983). Aggression scores in 20 nursery school children following violent (Condition 1) and neutral (Condition 0) cartoons are given.

Open NONPAR12 and select Statistics 1 → Nonparametric Tests (1-2 Samples) → Unpaired Samples. Select Score (C13) as [Variable] and Condition (C14) as [Factor]. From the next dialogue uncheck the Run a separate analysis for each option selected box and select only the Wald-Wolfowitz Runs Test option:

Unpaired Samples

Wald-Wolfowitz Runs Test

Data variable: Score

Subsample selected by: Condition

This result is not significant at the 10% level. Hence do not reject the null hypothesis “watching violent cartoons does not cause a significant change in the aggression of nursery school children”.

6.4.1.4. Moses Extreme Reaction Test

This test is used to determine the difference in range between two samples. Cases from the two samples are ranked together. Ranks corresponding to the smallest and largest group 1 cases are determined. The span is the difference between these two ranks plus one.

The program will prompt for the number of cases to be trimmed from either side of the span. The suggested number is either 1 or the integer closest to 5% of the number of cases in group 1, whichever is larger. of cases in two groups as well as the span and the one-tailed probability.

The exact one-tailed probability is computed for n £ 150. This limit can be changed by entering the following line with the appropriate number in the [Options] section of Documents\Unistat60\Unistat60.ini  file:

WMWMaxExactSize=150

Example

Open DEMODATA and select Statistics 1 → Nonparametric Tests (1-2 Samples) → Unpaired Samples. Select Wages (C2) and Energy (C3) as [Variable]s and click on the [Opt] button next to the Moses Extreme Reaction Test option. Accept the default value of 3 from the next dialogue.

Unpaired Samples

Moses Extreme Reaction Test

6.4.1.5. Two Sample Median Test

This test is used to determine whether two samples are drawn from populations with similar medians. The median for the two combined samples is calculated, the two samples are dichotomised and a 2 x 2 table is formed. It is possible to edit the computed median and to enter any values. The without and with a continuity correction and the exact probabilities.

Asymptotic without Continuity Correction: The following chi-square statistic with one degree of freedom is compared with the chi-square distribution:

Asymptotic with Continuity Correction: In this case the numerator is slightly different:

where g1 and g2 are the number of cases greater than the median in samples 1 and 2 respectively.

Exact: Two-tailed and table probabilities are reported using Fisher’s exact probability formula (see 6.4.5.2. Fisher’s Exact Test).

Example

Example 8.18 on p. 156 from Zar, J. H. (1999). The null hypothesis “the medians of the two sampled populations are equal” is tested.

Open NONPAR12 and select Statistics 1 → Nonparametric Tests (1-2 Samples) → Unpaired Samples and select Assistant A (C20) and Assistant B (C21) as [Variable]s. Note that these are the rank data in descending order. Select the Two Sample Median Test output option to obtain the following results:

Unpaired Samples

Two Sample Median Test

Since P > 0.05, do not reject the null hypothesis. Note that in the 5^th edition of Biostatistical Analysis (2010) Example 8.15 on p. 173, Zar employs a different method where observations at the median are omitted. With this approach the total number of valid cases is 23 and the chi-squared statistic with continuity correction is 0.473.