Trend Analysis validation

Free Mann-Kendall and Seasonal Mann-Kendall trend tests with Theil-Sen slope estimation for groundwater monitoring time series.

Scenarios: 16
Values agreeing: 318
Values differing: 2
Validated against: R EnvStats / nortest, GWSDAT, USEPA ProUCL, pymannkendall

Open Trend Analysis

Configured for this tool but not run in this pass: GSI Mann-Kendall Toolkit (not included in this comparison run)

Four reference implementations are compared: R EnvStats, GWSDAT, USEPA ProUCL and the GSI Mann-Kendall Toolkit. They agree on the Mann-Kendall statistic, its variance, z and p across all scenarios, including tied data, seasonal data and the smallest and largest series the tools accept. The GSI Confidence Factor comparison identified a defect in this toolkit: where repeated results place Kendall's S between two values covered by the exact probability table, it read the adjacent table entry. It now interpolates between the two, matching the GSI Toolkit including on the worked example in GSI's documentation.

Scenarios

Every result identical - S = 0 and var(S) fully tie-correctedBoundary case
Monthly data, twelve seasons over three yearsTypical use
Quarterly data with a clear declineTypical use
Six quarterly results with a clear downward trendTypical use
GSI's own published benzene exampleSpecial case
A strong seasonal cycle with no underlying trendBoundary case
Forty sampling events - the last size the GSI workbook acceptsBoundary case
Forty-one sampling events - one past the GSI workbook's capacityBoundary case
Four points - the smallest series the toolkit will testBoundary case
Heavy ties around a weak rising trendBoundary case
Repeated values exercise the tie correction in var(S)Boundary case
Seasons that trend in opposite directionsBoundary case
The same question at ten resultsBoundary case
Ties inside every seasonBoundary case
Ties producing a Kendall S the untied distribution cannot reachBoundary case
Two seasons, the fewest the test allowsBoundary case

Every result identical - S = 0 and var(S) fully tie-corrected

Boundary case

Input

8 points

x	value
1	5
2	5
3	5
4	5
5	5
6	5
7	5
8	5

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	8	8	0	exact match	Agree
Mann-Kendall S s	0	0	0	exact match	Agree
Variance of S varS	0	0	0	within 1e-9 relative	Agree
Standard normal z z	NaN	0	-	within 0.000001 relative	Differs
Why these differ: With no variation at all the Mann-Kendall variance is zero, so z is 0/0 and the test is formally undefined. The four engines split: this toolkit reports z = 0 and p = 1; GWSDAT's Kendall routine also returns p = 1, while raising its own internal fault code for the degenerate input; R's EnvStats returns NaN; ProUCL returns no statistic at all. So a third package agrees with the number reported here, and the two that decline are declining rather than disagreeing. S and the variance - the quantities that are actually defined for a flat record - agree exactly across all four.
p-value pValue	NaN	1	-	within 1e-9	Differs
Why these differ: Same reason as z. Reporting p = 1 says 'no evidence of a trend', which is the honest reading of a series that never moves, and GWSDAT reports the same. The result carries a note saying the test is undefined here, so the number is a stated convention rather than a silent one.
Theil-Sen slope theilSenSlope	0	0	0	within 0.000001 relative	Agree
Intercept intercept	5	5	0	within 0.000001 relative	Agree

Against GWSDAT (GWSDAT 3.3.0, R 4.6.1)

Value	GWSDAT	EnviroAutomate	Agreement band	Verdict
Sample size (n) n	8	8	exact match	Agree
Mann-Kendall S s	0	0	exact match	Agree
Variance of S varS	0	0	within 0.000001 relative	Agree
p-value pValue	1	1	within 0.00001 relative, or 1e-12 absolute	Agree

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Agreement band	Verdict
Sample size (n) n	8	8	exact match	Agree
Mann-Kendall S s	0	0	exact match	Agree
Variance of S varS	0	0	within 1e-9 relative	Agree

Against pymannkendall (pymannkendall 1.4.3)

Value	pymannkendall	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	8	8	0	exact match	Agree
Mann-Kendall S s	0	0	0	exact match	Agree
Variance of S varS	0	0	0	within 1e-9 relative	Agree
Standard normal z z	0	0	0	within 0.000001 relative	Agree
p-value pValue	1	1	0	within 1e-9	Agree
Theil-Sen slope theilSenSlope	0	0	0	within 0.000001 relative	Agree
Trend direction direction	No trend	No trend	-	exact match	Agree

Not run for this scenario

GSI Mann-Kendall Toolkit - not included in this comparison run

Monthly data, twelve seasons over three years

Typical use

Input

36 points

x	value	season
1	12	0
2	11	1
3	13	2
4	10	3
5	9	4
6	8	5
7	7	6
8	6	7
9	8	8
10	9	9
11	11	10
12	12	11
13	11	0
14	10	1
15	12	2
16	9	3
17	8	4
18	7	5
19	6	6
20	5	7
21	7	8
22	8	9
23	10	10
24	11	11
25	10	0
26	9	1
27	11	2
28	8	3
29	7	4
30	6	5
31	5	6
32	4	7
33	6	8
34	7	9
35	9	10
36	10	11

Settings: seasonPeriod = 12

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	36	36	0	exact match	Agree
Mann-Kendall S s	-36	-36	0	exact match	Agree
Variance of S varS	43.99999999999999	43.99999999999999	0	within 1e-9 relative	Agree
Standard normal z z	-5.276448530110864	-5.276448530110864	0	within 0.000001 relative	Agree
p-value pValue	1.31711483891223e-7	1.31711483941643e-7	5.04e-17	within 1e-9	Agree
season.0.s season.0.s	-3	-3	0	within 0.000001 relative	Agree
season.0.varS season.0.varS	3.6666666666666665	3.6666666666666665	0	within 0.000001 relative	Agree
season.1.s season.1.s	-3	-3	0	within 0.000001 relative	Agree
season.1.varS season.1.varS	3.6666666666666665	3.6666666666666665	0	within 0.000001 relative	Agree
season.2.s season.2.s	-3	-3	0	within 0.000001 relative	Agree
season.2.varS season.2.varS	3.6666666666666665	3.6666666666666665	0	within 0.000001 relative	Agree
season.3.s season.3.s	-3	-3	0	within 0.000001 relative	Agree
season.3.varS season.3.varS	3.6666666666666665	3.6666666666666665	0	within 0.000001 relative	Agree
season.4.s season.4.s	-3	-3	0	within 0.000001 relative	Agree
season.4.varS season.4.varS	3.6666666666666665	3.6666666666666665	0	within 0.000001 relative	Agree
season.5.s season.5.s	-3	-3	0	within 0.000001 relative	Agree
season.5.varS season.5.varS	3.6666666666666665	3.6666666666666665	0	within 0.000001 relative	Agree
season.6.s season.6.s	-3	-3	0	within 0.000001 relative	Agree
season.6.varS season.6.varS	3.6666666666666665	3.6666666666666665	0	within 0.000001 relative	Agree
season.7.s season.7.s	-3	-3	0	within 0.000001 relative	Agree
season.7.varS season.7.varS	3.6666666666666665	3.6666666666666665	0	within 0.000001 relative	Agree
season.8.s season.8.s	-3	-3	0	within 0.000001 relative	Agree
season.8.varS season.8.varS	3.6666666666666665	3.6666666666666665	0	within 0.000001 relative	Agree
season.9.s season.9.s	-3	-3	0	within 0.000001 relative	Agree
season.9.varS season.9.varS	3.6666666666666665	3.6666666666666665	0	within 0.000001 relative	Agree
season.10.s season.10.s	-3	-3	0	within 0.000001 relative	Agree
season.10.varS season.10.varS	3.6666666666666665	3.6666666666666665	0	within 0.000001 relative	Agree
season.11.s season.11.s	-3	-3	0	within 0.000001 relative	Agree
season.11.varS season.11.varS	3.6666666666666665	3.6666666666666665	0	within 0.000001 relative	Agree
Theil-Sen slope theilSenSlope	-0.08333333333333333	-0.08333333333333333	0	within 0.000001 relative	Agree

Not run for this scenario

USEPA ProUCL - ProUCL does not implement the seasonal Mann-Kendall test, so these scenarios are compared against R EnvStats, which does.
GWSDAT - GWSDAT has no seasonal Mann-Kendall; nothing to compare

Quarterly data with a clear decline

Typical use

Input

16 points

x	value	season
1	10	0
2	14	1
3	9	2
4	6	3
5	11	0
6	15	1
7	10	2
8	7	3
9	9	0
10	13	1
11	8	2
12	5	3
13	8	0
14	12	1
15	7	2
16	4	3

Settings: seasonPeriod = 4

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	16	16	0	exact match	Agree
Mann-Kendall S s	-16	-16	0	exact match	Agree
Variance of S varS	34.666666666666664	34.666666666666664	0	within 1e-9 relative	Agree
Standard normal z z	-2.5476233268253408	-2.5476233268253408	0	within 0.000001 relative	Agree
p-value pValue	0.010845950917060448	0.01084595091660101	4.59e-13	within 1e-9	Agree
season.0.s season.0.s	-4	-4	0	within 0.000001 relative	Agree
season.0.varS season.0.varS	8.666666666666666	8.666666666666666	0	within 0.000001 relative	Agree
season.1.s season.1.s	-4	-4	0	within 0.000001 relative	Agree
season.1.varS season.1.varS	8.666666666666666	8.666666666666666	0	within 0.000001 relative	Agree
season.2.s season.2.s	-4	-4	0	within 0.000001 relative	Agree
season.2.varS season.2.varS	8.666666666666666	8.666666666666666	0	within 0.000001 relative	Agree
season.3.s season.3.s	-4	-4	0	within 0.000001 relative	Agree
season.3.varS season.3.varS	8.666666666666666	8.666666666666666	0	within 0.000001 relative	Agree
Theil-Sen slope theilSenSlope	-0.20833333333333331	-0.20833333333333331	0	within 0.000001 relative	Agree

Not run for this scenario

USEPA ProUCL - ProUCL does not implement the seasonal Mann-Kendall test, so these scenarios are compared against R EnvStats, which does.
GWSDAT - GWSDAT has no seasonal Mann-Kendall; nothing to compare

Six quarterly results with a clear downward trend

Typical use

Input

6 points

x	value
1	10
2	9
3	7
4	8
5	5
6	4

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	6	6	0	exact match	Agree
Mann-Kendall S s	-13	-13	0	exact match	Agree
Variance of S varS	28.333333333333332	28.333333333333332	0	within 1e-9 relative	Agree
Standard normal z z	-2.2544074479065386	-2.2544074479065386	0	within 0.000001 relative	Agree
p-value pValue	0.024170547174545247	0.024170547173495427	1.05e-12	within 1e-9	Agree
Theil-Sen slope theilSenSlope	-1.2	-1.2	0	within 0.000001 relative	Agree
Intercept intercept	11.7	11.7	0	within 0.000001 relative	Agree

Against GWSDAT (GWSDAT 3.3.0, R 4.6.1)

Value	GWSDAT	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	6	6	0	exact match	Agree
Mann-Kendall S s	-13	-13	0	exact match	Agree
Variance of S varS	28.33333396911621	28.333333333333332	6.36e-7	within 0.000001 relative	Agree
p-value pValue	0.02417055144906044	0.024170547173495427	4.28e-9	within 0.00001 relative, or 1e-12 absolute	Agree

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	6	6	0	exact match	Agree
Mann-Kendall S s	-13	-13	0	exact match	Agree
Variance of S varS	28.33333333333333	28.333333333333332	3.55e-15	within 1e-9 relative	Agree
Standard normal z z	-2.2544074479065386	-2.2544074479065386	0	within 0.000001 relative	Agree
p-value pValue	0.024170547174454805	0.024170547173495427	9.59e-13	within 1e-9	Agree

Against pymannkendall (pymannkendall 1.4.3)

Value	pymannkendall	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	6	6	0	exact match	Agree
Mann-Kendall S s	-13	-13	0	exact match	Agree
Variance of S varS	28.333333333333332	28.333333333333332	0	within 1e-9 relative	Agree
Standard normal z z	-2.2544074479065386	-2.2544074479065386	0	within 0.000001 relative	Agree
p-value pValue	0.024170547174545254	0.024170547173495427	1.05e-12	within 1e-9	Agree
Theil-Sen slope theilSenSlope	-1.2	-1.2	0	within 0.000001 relative	Agree
Trend direction direction	Decreasing	Decreasing	-	exact match	Agree

Not run for this scenario

GSI Mann-Kendall Toolkit - not included in this comparison run

GSI's own published benzene example

Special case

Input

7 points

x	value
1	2190
2	384
3	357
4	19
5	1
6	1
7	1

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	7	7	0	exact match	Agree
Mann-Kendall S s	-18	-18	0	exact match	Agree
Variance of S varS	40.666666666666664	40.666666666666664	0	within 1e-9 relative	Agree
Standard normal z z	-2.665812704795515	-2.665812704795515	0	within 0.000001 relative	Agree
p-value pValue	0.007680246786159427	0.007680246785836786	3.23e-13	within 1e-9	Agree
Theil-Sen slope theilSenSlope	-118.66666666666667	-118.66666666666667	0	within 0.000001 relative	Agree
Intercept intercept	493.6666666666667	493.6666666666667	0	within 0.000001 relative	Agree

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	7	7	0	exact match	Agree
Mann-Kendall S s	-18	-18	0	exact match	Agree
Variance of S varS	40.666666666666664	40.666666666666664	0	within 1e-9 relative	Agree
Standard normal z z	-2.665812704795515	-2.665812704795515	0	within 0.000001 relative	Agree
p-value pValue	0.007680246785861419	0.007680246785836786	2.46e-14	within 1e-9	Agree

Against GWSDAT (GWSDAT 3.3.0, R 4.6.1)

Value	GWSDAT	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	7	7	0	exact match	Agree
Mann-Kendall S s	-18	-18	0	exact match	Agree
Variance of S varS	40.66666793823242	40.666666666666664	1.27e-6	within 0.000001 relative	Agree
p-value pValue	0.007680247072130442	0.007680246785836786	2.86e-10	within 0.00001 relative, or 1e-12 absolute	Agree

Against pymannkendall (pymannkendall 1.4.3)

Value	pymannkendall	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	7	7	0	exact match	Agree
Mann-Kendall S s	-18	-18	0	exact match	Agree
Variance of S varS	40.666666666666664	40.666666666666664	0	within 1e-9 relative	Agree
Standard normal z z	-2.665812704795515	-2.665812704795515	0	within 0.000001 relative	Agree
p-value pValue	0.007680246786159417	0.007680246785836786	3.23e-13	within 1e-9	Agree
Theil-Sen slope theilSenSlope	-118.66666666666667	-118.66666666666667	0	within 0.000001 relative	Agree
Trend direction direction	Decreasing	Decreasing	-	exact match	Agree

Not run for this scenario

GSI Mann-Kendall Toolkit - not included in this comparison run

A strong seasonal cycle with no underlying trend

Boundary case

Input

16 points

x	value	season
1	5	0
2	20	1
3	12	2
4	3	3
5	6	0
6	19	1
7	13	2
8	2	3
9	4	0
10	21	1
11	11	2
12	4	3
13	6	0
14	19	1
15	12	2
16	3	3

Settings: seasonPeriod = 4

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Agreement band	Verdict
Sample size (n) n	16	16	exact match	Agree
Mann-Kendall S s	0	0	exact match	Agree
Variance of S varS	30.666666666666668	30.666666666666668	within 1e-9 relative	Agree
Standard normal z z	0	0	within 0.000001 relative	Agree
p-value pValue	1	1	within 1e-9	Agree
season.0.s season.0.s	1	1	within 0.000001 relative	Agree
season.0.varS season.0.varS	7.666666666666667	7.666666666666667	within 0.000001 relative	Agree
season.1.s season.1.s	-1	-1	within 0.000001 relative	Agree
season.1.varS season.1.varS	7.666666666666667	7.666666666666667	within 0.000001 relative	Agree
season.2.s season.2.s	-1	-1	within 0.000001 relative	Agree
season.2.varS season.2.varS	7.666666666666667	7.666666666666667	within 0.000001 relative	Agree
season.3.s season.3.s	1	1	within 0.000001 relative	Agree
season.3.varS season.3.varS	7.666666666666667	7.666666666666667	within 0.000001 relative	Agree
Theil-Sen slope theilSenSlope	0	0	within 0.000001 relative	Agree

Not run for this scenario

USEPA ProUCL - ProUCL does not implement the seasonal Mann-Kendall test, so these scenarios are compared against R EnvStats, which does.
GWSDAT - GWSDAT has no seasonal Mann-Kendall; nothing to compare

Forty sampling events - the last size the GSI workbook accepts

Boundary case

Input

40 points

x	value
1	20.8
2	19.3
3	19
4	19.9
5	18.4
6	18.1
7	19
8	17.5
9	17.2
10	18.1
11	16.6
12	16.3
13	17.2
14	15.7
15	15.4
16	16.3
17	14.8
18	14.5
19	15.4
20	13.9
21	13.6
22	14.5
23	13
24	12.7
25	13.6
26	12.1
27	11.8
28	12.7
29	11.2
30	10.9
31	11.8
32	10.3
33	10
34	10.9
35	9.4
36	9.1
37	10
38	8.5
39	8.2
40	9.1

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	40	40	0	exact match	Agree
Mann-Kendall S s	-716	-716	0	exact match	Agree
Variance of S varS	7354.666666666667	7354.666666666667	0	within 1e-9 relative	Agree
Standard normal z z	-8.337283025803837	-8.337283025803837	0	within 0.000001 relative	Agree
p-value pValue	7.60169612913962e-17	0	7.60e-17	within 1e-9	Agree
Theil-Sen slope theilSenSlope	-0.3	-0.3	0	within 0.000001 relative	Agree
Intercept intercept	20.349999999999998	20.349999999999998	0	within 0.000001 relative	Agree

Against GWSDAT (GWSDAT 3.3.0, R 4.6.1)

Value	GWSDAT	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	40	40	0	exact match	Agree
Mann-Kendall S s	-716	-716	0	exact match	Agree
Variance of S varS	7354.66650390625	7354.666666666667	0.000163	within 0.000001 relative	Agree
p-value pValue	7.60168130326596e-17	0	7.60e-17	within 0.00001 relative, or 1e-12 absolute	Agree

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	40	40	0	exact match	Agree
Mann-Kendall S s	-716	-716	0	exact match	Agree
Variance of S varS	7354.666666666668	7354.666666666667	9.09e-13	within 1e-9 relative	Agree
Standard normal z z	-8.337283025803837	-8.337283025803837	0	within 0.000001 relative	Agree
p-value pValue	7.60169612933010e-17	0	7.60e-17	within 1e-9	Agree

Against pymannkendall (pymannkendall 1.4.3)

Value	pymannkendall	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	40	40	0	exact match	Agree
Mann-Kendall S s	-716	-716	0	exact match	Agree
Variance of S varS	7354.666666666667	7354.666666666667	0	within 1e-9 relative	Agree
Standard normal z z	-8.337283025803837	-8.337283025803837	0	within 0.000001 relative	Agree
p-value pValue	0	0	0	within 1e-9	Agree
Theil-Sen slope theilSenSlope	-0.3	-0.3	0	within 0.000001 relative	Agree
Trend direction direction	Decreasing	Decreasing	-	exact match	Agree

Not run for this scenario

GSI Mann-Kendall Toolkit - not included in this comparison run

Forty-one sampling events - one past the GSI workbook's capacity

Boundary case

Input

41 points

x	value
1	20.8
2	19.3
3	19
4	19.9
5	18.4
6	18.1
7	19
8	17.5
9	17.2
10	18.1
11	16.6
12	16.3
13	17.2
14	15.7
15	15.4
16	16.3
17	14.8
18	14.5
19	15.4
20	13.9
21	13.6
22	14.5
23	13
24	12.7
25	13.6
26	12.1
27	11.8
28	12.7
29	11.2
30	10.9
31	11.8
32	10.3
33	10
34	10.9
35	9.4
36	9.1
37	10
38	8.5
39	8.2
40	9.1
41	7.6

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	41	41	0	exact match	Agree
Mann-Kendall S s	-756	-756	0	exact match	Agree
Variance of S varS	7914.666666666667	7914.666666666667	0	within 1e-9 relative	Agree
Standard normal z z	-8.486539504580369	-8.486539504580369	0	within 0.000001 relative	Agree
p-value pValue	2.12881316104394e-17	0	2.13e-17	within 1e-9	Agree
Theil-Sen slope theilSenSlope	-0.3	-0.3	0	within 0.000001 relative	Agree
Intercept intercept	20.2	20.2	0	within 0.000001 relative	Agree

Against GWSDAT (GWSDAT 3.3.0, R 4.6.1)

Value	GWSDAT	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	41	41	0	exact match	Agree
Mann-Kendall S s	-756	-756	0	exact match	Agree
Variance of S varS	7914.66650390625	7914.666666666667	0.000163	within 0.000001 relative	Agree
p-value pValue	2.12881117951504e-17	0	2.13e-17	within 0.00001 relative, or 1e-12 absolute	Agree

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	41	41	0	exact match	Agree
Mann-Kendall S s	-756	-756	0	exact match	Agree
Variance of S varS	7914.666666666667	7914.666666666667	0	within 1e-9 relative	Agree
Standard normal z z	-8.486539504580369	-8.486539504580369	0	within 0.000001 relative	Agree
p-value pValue	2.1288131610894e-17	0	2.13e-17	within 1e-9	Agree

Against pymannkendall (pymannkendall 1.4.3)

Value	pymannkendall	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	41	41	0	exact match	Agree
Mann-Kendall S s	-756	-756	0	exact match	Agree
Variance of S varS	7914.666666666667	7914.666666666667	0	within 1e-9 relative	Agree
Standard normal z z	-8.486539504580369	-8.486539504580369	0	within 0.000001 relative	Agree
p-value pValue	0	0	0	within 1e-9	Agree
Theil-Sen slope theilSenSlope	-0.3	-0.3	0	within 0.000001 relative	Agree
Trend direction direction	Decreasing	Decreasing	-	exact match	Agree

Not run for this scenario

GSI Mann-Kendall Toolkit - not included in this comparison run

Four points - the smallest series the toolkit will test

Boundary case

Input

4 points

x	value
1	12
2	9.5
3	8.1
4	6

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	4	4	0	exact match	Agree
Mann-Kendall S s	-6	-6	0	exact match	Agree
Variance of S varS	8.666666666666666	8.666666666666666	0	within 1e-9 relative	Agree
Standard normal z z	-1.6984155512168937	-1.6984155512168937	0	within 0.000001 relative	Agree
p-value pValue	0.08942935902899354	0.08942935903273863	3.75e-12	within 1e-9	Agree
Theil-Sen slope theilSenSlope	-1.975	-1.975	0	within 0.000001 relative	Agree
Intercept intercept	13.7375	13.7375	0	within 0.000001 relative	Agree

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	4	4	0	exact match	Agree
Mann-Kendall S s	-6	-6	0	exact match	Agree
Variance of S varS	8.666666666666666	8.666666666666666	0	within 1e-9 relative	Agree
Standard normal z z	-1.6984155512168937	-1.6984155512168937	0	within 0.000001 relative	Agree
p-value pValue	0.08942935902767497	0.08942935903273863	5.06e-12	within 1e-9	Agree

Against pymannkendall (pymannkendall 1.4.3)

Value	pymannkendall	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	4	4	0	exact match	Agree
Mann-Kendall S s	-6	-6	0	exact match	Agree
Variance of S varS	8.666666666666666	8.666666666666666	0	within 1e-9 relative	Agree
Standard normal z z	-1.6984155512168937	-1.6984155512168937	0	within 0.000001 relative	Agree
p-value pValue	0.08942935902899363	0.08942935903273863	3.75e-12	within 1e-9	Agree
Theil-Sen slope theilSenSlope	-1.975	-1.975	0	within 0.000001 relative	Agree
Trend direction direction	No trend	No trend	-	exact match	Agree

Not run for this scenario

GSI Mann-Kendall Toolkit - not included in this comparison run

Heavy ties around a weak rising trend

Boundary case

Input

8 points

x	value
1	3
2	3
3	3
4	3
5	2
6	4
7	3
8	5

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	8	8	0	exact match	Agree
Mann-Kendall S s	8	8	0	exact match	Agree
Variance of S varS	48.666666666666664	48.666666666666664	0	within 1e-9 relative	Agree
Standard normal z z	1.0034188133917428	1.0034188133917428	0	within 0.000001 relative	Agree
p-value pValue	0.3156588305734086	0.31565883060042355	2.70e-11	within 1e-9	Agree
Theil-Sen slope theilSenSlope	0	0	0	within 0.000001 relative	Agree
Intercept intercept	3	3	0	within 0.000001 relative	Agree

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	8	8	0	exact match	Agree
Mann-Kendall S s	8	8	0	exact match	Agree
Variance of S varS	48.66666666666667	48.666666666666664	7.11e-15	within 1e-9 relative	Agree
Standard normal z z	1.003418813391743	1.0034188133917428	2.22e-16	within 0.000001 relative	Agree
p-value pValue	0.315658830566905	0.31565883060042355	3.35e-11	within 1e-9	Agree

Against GWSDAT (GWSDAT 3.3.0, R 4.6.1)

Value	GWSDAT	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	8	8	0	exact match	Agree
Mann-Kendall S s	8	8	0	exact match	Agree
Variance of S varS	48.66666793823242	48.666666666666664	1.27e-6	within 0.000001 relative	Agree
p-value pValue	0.3156588077545166	0.31565883060042355	2.28e-8	within 0.00001 relative, or 1e-12 absolute	Agree

Against pymannkendall (pymannkendall 1.4.3)

Value	pymannkendall	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	8	8	0	exact match	Agree
Mann-Kendall S s	8	8	0	exact match	Agree
Variance of S varS	48.666666666666664	48.666666666666664	0	within 1e-9 relative	Agree
Standard normal z z	1.0034188133917428	1.0034188133917428	0	within 0.000001 relative	Agree
p-value pValue	0.3156588305734087	0.31565883060042355	2.70e-11	within 1e-9	Agree
Theil-Sen slope theilSenSlope	0	0	0	within 0.000001 relative	Agree
Trend direction direction	No trend	No trend	-	exact match	Agree

Not run for this scenario

GSI Mann-Kendall Toolkit - not included in this comparison run

Repeated values exercise the tie correction in var(S)

Boundary case

Input

8 points

x	value
1	5
2	5
3	5
4	7
5	7
6	9
7	9
8	12

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	8	8	0	exact match	Agree
Mann-Kendall S s	23	23	0	exact match	Agree
Variance of S varS	59.666666666666664	59.666666666666664	0	within 1e-9 relative	Agree
Standard normal z z	2.8481102232138804	2.8481102232138804	0	within 0.000001 relative	Agree
p-value pValue	0.004397968961524268	0.004397968961341592	1.83e-13	within 1e-9	Agree
Theil-Sen slope theilSenSlope	1	1	0	within 0.000001 relative	Agree
Intercept intercept	2.5	2.5	0	within 0.000001 relative	Agree

Against GWSDAT (GWSDAT 3.3.0, R 4.6.1)

Value	GWSDAT	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	8	8	0	exact match	Agree
Mann-Kendall S s	23	23	0	exact match	Agree
Variance of S varS	59.66666793823242	59.666666666666664	1.27e-6	within 0.000001 relative	Agree
p-value pValue	0.004397988319396973	0.004397968961341592	1.94e-8	within 0.00001 relative, or 1e-12 absolute	Agree

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	8	8	0	exact match	Agree
Mann-Kendall S s	23	23	0	exact match	Agree
Variance of S varS	59.666666666666664	59.666666666666664	0	within 1e-9 relative	Agree
Standard normal z z	2.8481102232138804	2.8481102232138804	0	within 0.000001 relative	Agree
p-value pValue	0.004397968961424414	0.004397968961341592	8.28e-14	within 1e-9	Agree

Against pymannkendall (pymannkendall 1.4.3)

Value	pymannkendall	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	8	8	0	exact match	Agree
Mann-Kendall S s	23	23	0	exact match	Agree
Variance of S varS	59.666666666666664	59.666666666666664	0	within 1e-9 relative	Agree
Standard normal z z	2.8481102232138804	2.8481102232138804	0	within 0.000001 relative	Agree
p-value pValue	0.00439796896152433	0.004397968961341592	1.83e-13	within 1e-9	Agree
Theil-Sen slope theilSenSlope	1	1	0	within 0.000001 relative	Agree
Trend direction direction	Increasing	Increasing	-	exact match	Agree

Not run for this scenario

GSI Mann-Kendall Toolkit - not included in this comparison run

Seasons that trend in opposite directions

Boundary case

Input

16 points

x	value	season
1	10	0
2	10	1
3	10	2
4	10	3
5	11	0
6	9	1
7	11	2
8	9	3
9	12	0
10	8	1
11	12	2
12	8	3
13	13	0
14	7	1
15	13	2
16	7	3

Settings: seasonPeriod = 4

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Agreement band	Verdict
Sample size (n) n	16	16	exact match	Agree
Mann-Kendall S s	0	0	exact match	Agree
Variance of S varS	34.666666666666664	34.666666666666664	within 1e-9 relative	Agree
Standard normal z z	0	0	within 0.000001 relative	Agree
p-value pValue	1	1	within 1e-9	Agree
season.0.s season.0.s	6	6	within 0.000001 relative	Agree
season.0.varS season.0.varS	8.666666666666666	8.666666666666666	within 0.000001 relative	Agree
season.1.s season.1.s	-6	-6	within 0.000001 relative	Agree
season.1.varS season.1.varS	8.666666666666666	8.666666666666666	within 0.000001 relative	Agree
season.2.s season.2.s	6	6	within 0.000001 relative	Agree
season.2.varS season.2.varS	8.666666666666666	8.666666666666666	within 0.000001 relative	Agree
season.3.s season.3.s	-6	-6	within 0.000001 relative	Agree
season.3.varS season.3.varS	8.666666666666666	8.666666666666666	within 0.000001 relative	Agree
Theil-Sen slope theilSenSlope	0	0	within 0.000001 relative	Agree

Not run for this scenario

USEPA ProUCL - ProUCL does not implement the seasonal Mann-Kendall test, so these scenarios are compared against R EnvStats, which does.
GWSDAT - GWSDAT has no seasonal Mann-Kendall; nothing to compare

The same question at ten results

Boundary case

Input

10 points

x	value
1	1
2	5
3	2
4	5
5	5
6	5
7	1
8	5
9	5
10	5

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	10	10	0	exact match	Agree
Mann-Kendall S s	11	11	0	exact match	Agree
Variance of S varS	79.66666666666667	79.66666666666667	0	within 1e-9 relative	Agree
Standard normal z z	1.1203705304599068	1.1203705304599068	0	within 0.000001 relative	Agree
p-value pValue	0.26255589762985626	0.2625558976426787	1.28e-11	within 1e-9	Agree
Theil-Sen slope theilSenSlope	0	0	0	within 0.000001 relative	Agree
Intercept intercept	5	5	0	within 0.000001 relative	Agree

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	10	10	0	exact match	Agree
Mann-Kendall S s	11	11	0	exact match	Agree
Variance of S varS	79.66666666666666	79.66666666666667	1.42e-14	within 1e-9 relative	Agree
Standard normal z z	1.1203705304599068	1.1203705304599068	0	within 0.000001 relative	Agree
p-value pValue	0.2625558976241922	0.2625558976426787	1.85e-11	within 1e-9	Agree

Against GWSDAT (GWSDAT 3.3.0, R 4.6.1)

Value	GWSDAT	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	10	10	0	exact match	Agree
Mann-Kendall S s	11	11	0	exact match	Agree
Variance of S varS	79.66666412353516	79.66666666666667	2.54e-6	within 0.000001 relative	Agree
p-value pValue	0.2625558376312256	0.2625558976426787	6.00e-8	within 0.00001 relative, or 1e-12 absolute	Agree

Against pymannkendall (pymannkendall 1.4.3)

Value	pymannkendall	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	10	10	0	exact match	Agree
Mann-Kendall S s	11	11	0	exact match	Agree
Variance of S varS	79.66666666666667	79.66666666666667	0	within 1e-9 relative	Agree
Standard normal z z	1.1203705304599068	1.1203705304599068	0	within 0.000001 relative	Agree
p-value pValue	0.2625558976298561	0.2625558976426787	1.28e-11	within 1e-9	Agree
Theil-Sen slope theilSenSlope	0	0	0	within 0.000001 relative	Agree
Trend direction direction	No trend	No trend	-	exact match	Agree

Not run for this scenario

GSI Mann-Kendall Toolkit - not included in this comparison run

Ties inside every season

Boundary case

Input

16 points

x	value	season
1	9	0
2	9	1
3	5	2
4	5	3
5	9	0
6	7	1
7	5	2
8	3	3
9	7	0
10	7	1
11	3	2
12	3	3
13	7	0
14	5	1
15	3	2
16	1	3

Settings: seasonPeriod = 4

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	16	16	0	exact match	Agree
Mann-Kendall S s	-18	-18	0	exact match	Agree
Variance of S varS	28.666666666666668	28.666666666666668	0	within 1e-9 relative	Agree
Standard normal z z	-3.175121312459921	-3.175121312459921	0	within 0.000001 relative	Agree
p-value pValue	0.00149773871335081	0.00149773871328951	6.13e-14	within 1e-9	Agree
season.0.s season.0.s	-4	-4	0	within 0.000001 relative	Agree
season.0.varS season.0.varS	6.666666666666667	6.666666666666667	0	within 0.000001 relative	Agree
season.1.s season.1.s	-5	-5	0	within 0.000001 relative	Agree
season.1.varS season.1.varS	7.666666666666667	7.666666666666667	0	within 0.000001 relative	Agree
season.2.s season.2.s	-4	-4	0	within 0.000001 relative	Agree
season.2.varS season.2.varS	6.666666666666667	6.666666666666667	0	within 0.000001 relative	Agree
season.3.s season.3.s	-5	-5	0	within 0.000001 relative	Agree
season.3.varS season.3.varS	7.666666666666667	7.666666666666667	0	within 0.000001 relative	Agree
Theil-Sen slope theilSenSlope	-0.25	-0.25	0	within 0.000001 relative	Agree

Not run for this scenario

USEPA ProUCL - ProUCL does not implement the seasonal Mann-Kendall test, so these scenarios are compared against R EnvStats, which does.
GWSDAT - GWSDAT has no seasonal Mann-Kendall; nothing to compare

Ties producing a Kendall S the untied distribution cannot reach

Boundary case

Input

8 points

x	value
1	2
2	4
3	3
4	3
5	3
6	3
7	3
8	3

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Agreement band	Verdict
Sample size (n) n	8	8	exact match	Agree
Mann-Kendall S s	1	1	exact match	Agree
Variance of S varS	37	37	within 1e-9 relative	Agree
Standard normal z z	0	0	within 0.000001 relative	Agree
p-value pValue	1	1	within 1e-9	Agree
Theil-Sen slope theilSenSlope	0	0	within 0.000001 relative	Agree
Intercept intercept	3	3	within 0.000001 relative	Agree

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	8	8	0	exact match	Agree
Mann-Kendall S s	1	1	0	exact match	Agree
Variance of S varS	36.99999999999999	37	7.11e-15	within 1e-9 relative	Agree
Standard normal z z	0	0	0	within 0.000001 relative	Agree
p-value pValue	1	1	0	within 1e-9	Agree

Against GWSDAT (GWSDAT 3.3.0, R 4.6.1)

Value	GWSDAT	EnviroAutomate	Agreement band	Verdict
Sample size (n) n	8	8	exact match	Agree
Mann-Kendall S s	1	1	exact match	Agree
Variance of S varS	37	37	within 0.000001 relative	Agree
p-value pValue	1	1	within 0.00001 relative, or 1e-12 absolute	Agree

Against pymannkendall (pymannkendall 1.4.3)

Value	pymannkendall	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	8	8	0	exact match	Agree
Mann-Kendall S s	1	1	0	exact match	Agree
Variance of S varS	37	37	0	within 1e-9 relative	Agree
Standard normal z z	0	0	0	within 0.000001 relative	Agree
p-value pValue	1	1	0	within 1e-9	Agree
Theil-Sen slope theilSenSlope	0	0	0	within 0.000001 relative	Agree
Trend direction direction	No trend	No trend	-	exact match	Agree

Not run for this scenario

GSI Mann-Kendall Toolkit - not included in this comparison run

Two seasons, the fewest the test allows

Boundary case

Input

10 points

x	value	season
1	8	0
2	14	1
3	7	0
4	13	1
5	6	0
6	12	1
7	5	0
8	11	1
9	4	0
10	10	1

Settings: seasonPeriod = 2

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	10	10	0	exact match	Agree
Mann-Kendall S s	-20	-20	0	exact match	Agree
Variance of S varS	33.333333333333336	33.333333333333336	0	within 1e-9 relative	Agree
Standard normal z z	-3.2908965343808667	-3.2908965343808667	0	within 0.000001 relative	Agree
p-value pValue	0.000998686379180259	0.00099868637926237	8.21e-14	within 1e-9	Agree
season.0.s season.0.s	-10	-10	0	within 0.000001 relative	Agree
season.0.varS season.0.varS	16.666666666666668	16.666666666666668	0	within 0.000001 relative	Agree
season.1.s season.1.s	-10	-10	0	within 0.000001 relative	Agree
season.1.varS season.1.varS	16.666666666666668	16.666666666666668	0	within 0.000001 relative	Agree
Theil-Sen slope theilSenSlope	-0.5	-0.5	0	within 0.000001 relative	Agree

Not run for this scenario

USEPA ProUCL - ProUCL does not implement the seasonal Mann-Kendall test, so these scenarios are compared against R EnvStats, which does.
GWSDAT - GWSDAT has no seasonal Mann-Kendall; nothing to compare

Evidence generated 2026-08-26 from a full comparison run against the versions named above.

x	value	season
1	12	0
2	11	1
3	13	2
4	10	3
5	9	4
6	8	5
7	7	6
8	6	7
9	8	8
10	9	9
11	11	10
12	12	11
13	11	0
14	10	1
15	12	2
16	9	3
17	8	4
18	7	5
19	6	6
20	5	7
21	7	8
22	8	9
23	10	10
24	11	11
25	10	0
26	9	1
27	11	2
28	8	3
29	7	4
30	6	5
31	5	6
32	4	7
33	6	8
34	7	9
35	9	10
36	10	11

x	value	season
1	10	0
2	14	1
3	9	2
4	6	3
5	11	0
6	15	1
7	10	2
8	7	3
9	9	0
10	13	1
11	8	2
12	5	3
13	8	0
14	12	1
15	7	2
16	4	3

x	value	season
1	5	0
2	20	1
3	12	2
4	3	3
5	6	0
6	19	1
7	13	2
8	2	3
9	4	0
10	21	1
11	11	2
12	4	3
13	6	0
14	19	1
15	12	2
16	3	3

x	value	season
1	10	0
2	10	1
3	10	2
4	10	3
5	11	0
6	9	1
7	11	2
8	9	3
9	12	0
10	8	1
11	12	2
12	8	3
13	13	0
14	7	1
15	13	2
16	7	3

x	value	season
1	12	0
2	11	1
3	13	2
4	10	3
5	9	4
6	8	5
7	7	6
8	6	7
9	8	8
10	9	9
11	11	10
12	12	11
13	11	0
14	10	1
15	12	2
16	9	3
17	8	4
18	7	5
19	6	6
20	5	7
21	7	8
22	8	9
23	10	10
24	11	11
25	10	0
26	9	1
27	11	2
28	8	3
29	7	4
30	6	5
31	5	6
32	4	7
33	6	8
34	7	9
35	9	10
36	10	11

x	value	season
1	10	0
2	14	1
3	9	2
4	6	3
5	11	0
6	15	1
7	10	2
8	7	3
9	9	0
10	13	1
11	8	2
12	5	3
13	8	0
14	12	1
15	7	2
16	4	3

x	value	season
1	5	0
2	20	1
3	12	2
4	3	3
5	6	0
6	19	1
7	13	2
8	2	3
9	4	0
10	21	1
11	11	2
12	4	3
13	6	0
14	19	1
15	12	2
16	3	3

x	value	season
1	10	0
2	10	1
3	10	2
4	10	3
5	11	0
6	9	1
7	11	2
8	9	3
9	12	0
10	8	1
11	12	2
12	8	3
13	13	0
14	7	1
15	13	2
16	7	3

x	value	season
1	12	0
2	11	1
3	13	2
4	10	3
5	9	4
6	8	5
7	7	6
8	6	7
9	8	8
10	9	9
11	11	10
12	12	11
13	11	0
14	10	1
15	12	2
16	9	3
17	8	4
18	7	5
19	6	6
20	5	7
21	7	8
22	8	9
23	10	10
24	11	11
25	10	0
26	9	1
27	11	2
28	8	3
29	7	4
30	6	5
31	5	6
32	4	7
33	6	8
34	7	9
35	9	10
36	10	11

x	value	season
1	10	0
2	14	1
3	9	2
4	6	3
5	11	0
6	15	1
7	10	2
8	7	3
9	9	0
10	13	1
11	8	2
12	5	3
13	8	0
14	12	1
15	7	2
16	4	3

x	value	season
1	5	0
2	20	1
3	12	2
4	3	3
5	6	0
6	19	1
7	13	2
8	2	3
9	4	0
10	21	1
11	11	2
12	4	3
13	6	0
14	19	1
15	12	2
16	3	3

x	value	season
1	10	0
2	10	1
3	10	2
4	10	3
5	11	0
6	9	1
7	11	2
8	9	3
9	12	0
10	8	1
11	12	2
12	8	3
13	13	0
14	7	1
15	13	2
16	7	3