Runs Up/Down Test

Basic Concepts

For a sequence of numeric data, the runs test can be used to test for randomness based on the sign of the data elements. The runs up/down test, however, uses + when the sequence increases and – when the sequence decreases. In this case, for sequences that are large enough the number of runs follows a normal distribution N(μ, σ2) with

Normal approximation

The normal approximation is valid for a two-tailed test for samples of size n at least as large as the value in the following table (divide the alpha value by 2 for a one-tailed test):

Sample size requirements

Figure 1: Normal approximation sample size requirements

Example

Example 1: Repeat Example 1 of One-Sample Runs Test using the runs up/down test.

The result is shown in Figure 2. We can’t reject the null hypothesis that the data sequence is random with p-value = 1 (cell G11).

Runs Up/Down test 1

Figure 2 – Runs Up/Down Test

Columns B, C, and D are used to calculate the number of runs (in cell G8). We do this by inserting 0 in cells B2 and C2, and then the formulas =IF(A3>A2,1,IF(A3=A2,0,-1)) in cell B3, =IF(B3=0,C2,B3) in cell C3, and =IF(C3<>C2,1,0) in cell D3. Finally, you highlight the range B3:D16, and press Ctrl-D.

Example 2: Repeat Example 1 with the modified data in Figure 3. Note that this time there are 3 ties.

Once again, we can’t reject the null hypothesis that the data sequence is random with p-value = .433944 (cell G11).

Test with ties

Figure 3 – Runs Up/Down Test with ties

Worksheet Function

Real Statistics Function: The Real Statistics Resource Pack provides the following worksheet function.

RunsUpDn(R1, lab, tails): returns a column array with the results of the runs up/down test for the data in column array R1.

lab and tails are as for RUNSTEST. Also, the output has the same format as the output from RUNSTEST.

You can obtain the output shown in F4:G11 in either Figure 2 or 3 by using the array formula =RunsUpDn(A2:A16,TRUE).

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

Becketti, S. (2023) Runtest. Stata
https://www.stata.com/manuals/rruntest.pdf

Zar. J. H. (2010) Biostatistical analysis 5th Ed. Pearson
https://bayesmath.com/wp-content/uploads/2021/05/Jerrold-H.-Zar-Biostatistical-Analysis-5th-Edition-Prentice-Hall-2009.pdf

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