Mann-Kendall Tests

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Mann-Kendall Tests

EQuIS Mann-Kendall Calculations

oMann-Kendall S Value

oMann-Kendall Z Value

EQuIS Mann-Kendall Tests

 

The Mann-Kendall test determines the trend of varying data (whether it increases or decreases over time). This page describes the methods used in EQuIS reporting to evaluate the Mann-Kendall test.

 

EQuIS Mann-Kendall Calculations

 

The Mann-Kendall test uses the following equation:

30444_Mann-Kendall

The calculation compares each value to each of the preceding values in the data set using the sign() function. This function outputs either 1, -1, or 0 depending on whether yj - yi is either positive, negative, or zero respectively. The sum of these calculations within the iteration signifies an upward, downward, or stagnate step.

 

Mann-Kendall S Value

 

If the sample size is less than 10, then EQuIS compares the Mann-Kendall S value to the following values (Small Sample Method) to determine the confidence percentage:

 

Sample #

90% Confidence

95% Confidence

99% Confidence

4

6

6

5

7

8

10

6

8

11

13

7

10

13

17

8

11

16

20

9

14

18

24

10

16

21

27

Compare calculated value of S to values [source].

 

Note: Sample size must be greater than 3 for Mann-Kendall S to be calculated.

 

Should the sample size be greater than 10, then the variance of the Mann-Kendall S value is calculated by Normal Approximation:

30444_M-K_napprox

where g is the number of tied groups (where a number has repeated) and wp is the number of data points (tied values) in the pth group

For example, a dataset of {17, 14, 39, 16, 39, 14, 14, 39, 17, 39} contains the following:

Three tied groups (g = 3) for 14, 17, and 39

Tied group 1 (p = 1) for the three values of 14 (w1 = 3)

Tied group 2 (p = 2) for the two values of 17 (w2 = 2)

Tied group 3 (p = 3) for the three values of 39 (w3 = 4)

 

Mann-Kendall Z Value

 

The following conditions for the Mann-Kendall S value determine the next step in calculation:

1.If S > 0:
30444_positiveS

2.If S < 0:
30444_negativeS

3.If S = 0, then Z = 0.
 

The Z value is then checked with the following conditions (see table below) to test the null hypothesis – that there is no monotonic trend, i.e. no increasing or decreasing between two subsequent steps. The corresponding confidence level of rejecting that hypothesis is then assigned.

Values

Confidence Level

30444_Z99CL

99

30444_Z95CL

95

30444_Z90CL

90

 

Note that α = 1 - (confidence percentage).

 

EQuIS Mann-Kendall Tests

 

The following EQuIS reports perform Mann-Kendall calculations:

Analytical Results - Statistics

Analytical Results II - Statistics

Statistics: Analytical Statistics (by Location)

 

EnviroInsite uses a separate method of evaluating the Mann-Kendall test. For more information, see Mann Kendall Trend Evaluation.

 

In EQuIS, the sign difference between the detect or non-detect (Process multiple non-detect limits of an analyte at a location parameter) entries are compared for each summation step, and these iterations are added through a double summation to determine the total count as the Mann-Kendall S value.

 

Note: The “Process multiple non-detect limits of an analyte at a location” parameter can be set at Max, Min, Average, or Last, defaulting to Last if there is no selection.

 

Source – Table A‐11 of Guidance for Data Quality Assessment: Practical Methods for Data Analysis, EPA QA/G‐9, US EPA Office of Environmental Information EPA/600/R‐96/084, July 2000