# Kruskal-Wallis – Exact

This tool is used to estimate the exact P-Value using Monte Carlo. Typically this would not be necessary unless the sample sizes were smaller (each sample N <= 5 for Kruskal-Wallis), but this gives continuity on the example.

Computing an exact P-Value for Kruskal-Wallis is very computationally intensive. The Network Model by Mehta and Patel cannot be used for this test (see Appendix Exact and Monte Carlo P-Values for Nonparametric and Contingency Tests). In this example, the total number of permutations are:

(31+42+27)! / (31! * 42! * 27!) = 7.42 E44

(i.e., more than the number of stars in the observable universe). So we will not attempt to compute the exact, but rather use Monte Carlo.

- Open
**Customer Data.xlsx**, click on**Sheet 1**tab (or press**F4**to activate last worksheet). - Click
**SigmaXL > Statistical Tools > Nonparametric Tests - Exact > Kruskal-Wallis - Exact**. If necessary, check**Use Entire Data Table,**click**Next**. - Ensure that
**Stacked Column Format**is checked. Select*Overall Satisfaction*, click**Numeric Data Variable (Y) >>**; select*Customer Type*, click**Group Category (X) >>**. Select**Monte Carlo Exact**with the**Number of Replications**= 1e6 and**Confidence Level for P-Value**= 99%. One million replications are used because the expected P-Value is very small as estimated from the “large sample” Kruskal-Wallis above. This will take up to a minute to run, so if you have a slow computer, use 1e5 replications instead of 1e6.

**Tip**: The Monte Carlo 99% confidence interval for P-Value is**not**the same as a confidence interval on the test statistic due to data sampling error. The confidence level for the hypothesis test statistic is still 95%,**so all reported P-Values less than .05 will be highlighted in red**to indicate significance. The 99% Monte Carlo P-Value confidence interval is due to the uncertainty in Monte Carlo sampling, and it becomes smaller as the number of replications increases (irrespective of the data sample size). The Exact P-Value will lie within the stated Monte Carlo confidence interval 99% of the time. - Click
**OK**.

Click on cell B16 to view the P-Value with more decimal place precision (or change the cell format to scientific notation). The Monte Carlo P-Value here is 0.000009 (9 e-6) with a 99% confidence interval of .000002 (2 e-6) to 0.000016 (1.6 e-5). This will be slightly different every time it is run (the Monte Carlo seed value is derived from the system clock). So we reject H0: at least one pairwise set of medians are not equal.

Note that the large sample (asymptotic) P-Value of 2.3 e-5 lies outside of the Monte Carlo exact confidence interval. -
Now we will consider a small sample problem. Open
**Snore Study.xlsx**. This data is from:

Gibbons, J.D. and Chakraborti, S. (2010). Nonparametric Statistical Inference (5th Edition). New York: Chapman & Hall, (Example 10.2.1 data, page 347; Example 10.4.2 analysis, pp. 360 – 362).

An experiment was conducted to determine which device is the most effective in stopping snoring or at least in reducing it. Fifteen men who are habitual snorers were divided randomly into three groups to test the devices. Each man’s sleep was monitored for one night by a machine that measures amount of snoring on a 100-point scale while using a device. - Select
**Snore Study Data**tab. Click**SigmaXL > Statistical Tools > Nonparametric Tests – Exact > Kruskal-Wallis - Exact**. If necessary, check**Use Entire Data Table**, click**Next**. - With
**Unstacked Column Format**checked, select*Device A, Device B and Device C*, click**Numeric Data Variables (Y) >>**. Select**Exact**with the default**Time Limit for Exact Computation**= 60 seconds.

- Click
**OK**. Results:

With the Exact P-Value = 0.0042 we reject H0, and conclude that there is a significant difference in median snore study scores. This exact P-Value matches that given in the reference textbook using SAS and StatXact.

By way of comparison we will now rerun the analysis using the “large sample” or “asymptotic” Kruskal-Wallis test. - Select
**Snore Study Data**tab (or press**F4**to activate last worksheet). Click**SigmaXL > Statistical Tools > Nonparametric Tests > Kruskal-Wallis**. If necessary, check**Use Entire Data Table**, click**Next.** - With
**Unstacked Column Format**checked, select*Device A, Device B and Device C*, click**Numeric Data Variables (Y) >>**. - Click
**OK**. Results:

With the P-Value = .0118 we reject H0 (using alpha = .05), but note that if we were using alpha = 0.01, we would have incorrectly failed to reject the null hypothesis. This “large sample” P-Value matches that given in the reference textbook using Minitab.

In conclusion, whenever you have a small sample size and are performing a Nonparametric test, always use the Exact option.

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