Two statistical tests for trends are used for stability monitoring in Phase 5, a parametric trend test based on the linear regression model and the nonparametric Mann-Kendall trend test. A choice of one these statistical tests should be made before stability monitoring is begun. The selected test procedure is then applied for all monitored wells. Selection of a single testing approach is necessary for comparability of results across wells and to avoid the possibility of running both tests, then selecting the test with most favorable results.

Linear regression and the nonparametric Mann-Kendall trend test are recommended for trend detection in EPA 2006a and EPA 2009. The linear regression trend test relies on a variety of assumptions (e.g., normality) that must be verified. The Mann-Kendall trend test may be used with any series of four or more independent samples to test for trends in well parameters. Trend tests are used in Phase 1 to check for unexpected trends in baseline samples, and in Phase 5 to establish long-term stability.

Test Form 1 is used for the regression t-test and the Mann-Kendall trend test. In regression, a trend is measured by the slope of the regression line θ (theta). Here θ represents the true (and unknowable) value of the slope of the trend line. In the Mann-Kendall test, a trend is charactered by the parameter τ (tau), which is a nonparametric measure of the correlation of the sample values with time. In each case, hypothesis tests are used to determine if there is a significant trend over time. The null hypothesis for both tests is that there is no trend. The time series is analyzed for evidence of a significant trend. If such evidence is found, then the null hypothesis is rejected.

Testing for a trend:

   * The null hypothesis (H0): There is no significant trend in the series. Symbolically, the null hypothesis for the trend test is H0: θ=0.
   * The alternative hypothesis (HA): There is a significant trend in the series. (HA: θ != 0).

The null and alternative hypotheses for the Mann-Kendall test are similar, with θ being replaced by τ.

Unlike the tests for a comparison with baseline conditions, the trend tests require only one DQO parameter, the Type 1 error rate α. This parameter provides a control on the frequency of false positives, i.e., incorrectly deciding there is a trend when, in fact, the series is stationary. The power (1-β) of the trend tests to detect a trend depends on the variability of the series, the type of trend, duration of the sampling program and the frequency of sampling. The power of these tests may be estimated using Monte Carlo simulation. The power of the rank-based Mann-Kendal test and the linear regression t-test for slope are compared by Yue and Pilon (2004), who report that the power of the t-test is slightly higher than that of the Mann-Kendall test for normally distributed data, and for nonnormally distributed series, such as time series with the Weibull, Gumbel, Pearson Type III, or extreme value Type II distributions, the power of the Mann-Kendall test is higher than that of the t-test. The t-test usually performs better with a distribution that is relatively symmetric. In contrast, the Mann-Kendall is a more powerful test for data sets that present skewness. Attachment G contains a detailed discussion of the simulations performed for this study to determine a sufficient number of samples to provide adequate power to detect trends.

Adopting hypothesis tests and a DQO approach described in EPA QA/G9S (EPA 2006a, Section 3.4) can help control the probability of making decision errors. However, incorrect use of hypothesis tests can lead to incorrect decisions. Each type of hypothesis test is based on a set of assumptions that should be verified to confirm proper use of the test. The tests recommended in this document for verifying stability and determining when the site has met the remedial goals include both parametric and nonparametric tests. Nonparametric tests generally have fewer assumptions to verify. The following section provides additional information on parametric versus nonparametric tests.

Selecting the Statistical Approach  -  Parametric Versus Nonparametric Methods

Statistical methods are based on a set of assumptions about the data. The difference between parametric and nonparametric statistical methods is based on the form and details of these assumptions. Many statistical tests and models are appropriate only for data that follow a particular probability distribution. These distributions typically are characterized by one or more parameters, like the mean or standard deviation, and are called parametric statistical methods. Parametric methods use the actual data values and assume that the data values follow a specific probability distribution. Two of the most important distributions for analysis of environmental data are the normal distribution and the lognormal distribution. If two samples are to be compared, then parametric methods often require that both sets of data follow the same type of distribution.

Parametric statistical tests have several distinct advantages over nonparametric tests when the data follow the assumed distribution. They are often easier to apply and communicate, as they are based on well-known statistics, such as the mean and standard deviation. On the other hand, many nonparametric methods rely on the ranks of the sample values or computations involving all possible pairs of sample values. These calculations become more difficult to implement when there are a large number of samples to consider. If the sample size is 30 or more, parametric tests often lead to the same results as their corresponding nonparametric tests unless large outliers are present.

Parametric tests will have more power than a nonparametric counterpart if the assumptions for the test are met. However, the distributional assumptions are often strict, and deviation from these assumptions can lead to misleading results. Parametric tests also have difficulty dealing with outliers and nondetects. If either is found in the data, then a nonparametric statistical method may be the preferred approach. In general, nonparametric methods handle outliers and nondetects better than parametric methods. Nonparametric tests typically use the ranks of the data, and do not assume that the data follow a specific probability distribution. Because of reliance on fewer or weaker assumptions about the distribution of the data, nonparametric methods are often more generally applicable.

This document describes both parametric methods and nonparametric methods. The choice of the statistical method depends on how the data are distributed. If the distributions of the data appear to be normal (bell-shaped), then parametric methods are appropriate and have the advantage of greater power with minimal computations. If the distributions have long tails to the right, than taking the logarithms of the data is recommended. If the logged values are normally distributed, then parametric methods may be applied to the logged data. If both data sets cannot be determined to have normal (or lognormal) distributions, then nonparametric methods might be appropriate.

Determining If Data Have a Normal Distribution

Many statistical tests and models are appropriate only for data that follow a particular distribution. Methods in this section apply to the normal distribution. The tests for normality also may be used with lognormal data by applying the prescribed test to the logarithms of the data. If the data appear to have a distribution other than the normal distribution or the lognormal distribution, a nonparametric test such as the Wilcoxon Rank Sum test (see Section 7.9.2) should be considered.

The Shapiro-Wilk W Test

The Shapiro-Wilk W test is considered one of the most powerful tests for normality. This test is similar to computing a correlation between the quantiles of the standard normal distribution and the ordered values of a data set. Several EPA guidance documents and many statistical texts recommend the W test. Tables of critical values for sample sizes up to 50 have been developed for determining the significance of the test statistic. However, many software packages can perform the W test for data sets with sample sizes as large as 5,000. This test is difficult to compute by hand as it requires many summations and multiplications. Therefore, this document does not give directions for implementing the W test. The Studentized Range test described below is another test for normality that is much easier to compute, but is not as powerful as the W test.

The Studentized Range Test

The Studentized Range test for normality is based on the fact that nearly 100% of the area of a normal curve lies within +-5 standard deviations from the mean. This test uses a ratio of the sample range to the sample standard deviation. Very large and very small values of the ratio imply that the data are not well modeled by a normal distribution

This test for normality compares the sample range (maximum value minus minimum value) to the sample standard deviation. The test statistic is the ratio of the sample range to the standard deviation. Table 7-17 shows the critical values for determining whether the absolute value of the ratio for a sample of size N is significantly too small or too large. If the calculated ratio is less than the lower tabulated value (a) or exceeds the upper value (b), then the data are not normally distributed at the specified level of confidence. The Studentized Range test does not perform well if the data are asymmetric or if the tails of the data are heavier than the normal distribution. In addition, this test may be sensitive to a few extreme values. The test for outliers described in the following section may be used to screen the data for outliers before the Studentized Range test is conducted.
Many environmental data sets are positively skewed with a long but narrow tail of high values and are similar to a lognormal distribution. If the data appear to be lognormally distributed, then the range test may be applied to the logged data values.
Table 7-17.	Critical Values for the Studentized Range Test

                               Confidence Level

                                      90%
                                      95%
                                      99%
                                       N
                                       a
                                       b
                                       a
                                       b
                                       a
                                       b
                                       3
                                                                           1.78
                                                                           2.00
                                                                           1.76
                                                                           2.00
                                                                           1.74
                                                                           2.00
                                       4
                                                                           2.04
                                                                           2.41
                                                                           1.98
                                                                           2.43
                                                                           1.87
                                                                           2.45
                                       5
                                                                           2.22
                                                                           2.71
                                                                           2.15
                                                                           2.75
                                                                           2.02
                                                                           2.80
                                       6
                                                                           2.37
                                                                           2.95
                                                                           2.28
                                                                           3.01
                                                                           2.15
                                                                           3.10
                                       7
                                                                           2.49
                                                                           3.14
                                                                           2.40
                                                                           3.22
                                                                           2.26
                                                                           3.34
                                       8
                                                                           2.59
                                                                           3.31
                                                                           2.50
                                                                           3.40
                                                                           2.35
                                                                           3.54
                                       9
                                                                           2.68
                                                                           3.45
                                                                           2.59
                                                                           3.55
                                                                           2.44
                                                                           3.72
                                      10
                                                                           2.76
                                                                           3.57
                                                                           2.67
                                                                           3.69
                                                                           2.51
                                                                           3.88
                                      11
                                                                           2.84
                                                                           3.68
                                                                           2.74
                                                                           3.80
                                                                           2.58
                                                                           4.01
                                      12
                                                                           2.90
                                                                           3.78
                                                                           2.80
                                                                           3.91
                                                                           2.64
                                                                           4.13
                                      13
                                                                           2.96
                                                                           3.87
                                                                           2.86
                                                                           4.00
                                                                           2.70
                                                                           4.24
                                      14
                                                                           3.02
                                                                           3.95
                                                                           2.92
                                                                           4.09
                                                                           2.75
                                                                           4.34
                                      15
                                                                           3.07
                                                                           4.02
                                                                           2.97
                                                                           4.17
                                                                           2.80
                                                                           4.44
                                      16
                                                                           3.12
                                                                           4.09
                                                                           3.01
                                                                           4.24
                                                                           2.84
                                                                           4.52
                                      17
                                                                           3.17
                                                                           4.15
                                                                           3.06
                                                                           4.31
                                                                           2.88
                                                                           4.60
                                      18
                                                                           3.21
                                                                           4.21
                                                                           3.10
                                                                           4.37
                                                                           2.92
                                                                           4.67
                                      19
                                                                           3.25
                                                                           4.27
                                                                           3.14
                                                                           4.43
                                                                           2.96
                                                                           4.74
                                      20
                                                                           3.29
                                                                           4.32
                                                                           3.18
                                                                           4.49
                                                                           2.99
                                                                           4.80
                                      25
                                                                           3.45
                                                                           4.53
                                                                           3.34
                                                                           4.71
                                                                           3.15
                                                                           5.06
                                      30
                                                                           3.59
                                                                           4.70
                                                                           3.47
                                                                           4.89
                                                                           3.27
                                                                           5.26
                                      35
                                                                           3.70
                                                                           4.84
                                                                           3.58
                                                                           5.04
                                                                           3.38
                                                                           5.42
                                      40
                                                                           3.79
                                                                           4.96
                                                                           3.67
                                                                           5.16
                                                                           3.47
                                                                           5.56
                                      45
                                                                           3.88
                                                                           5.06
                                                                           3.75
                                                                           5.26
                                                                           3.55
                                                                           5.67
                                      50
                                                                           3.95
                                                                           5.14
                                                                           3.83
                                                                           5.35
                                                                           3.62
                                                                           5.77
                                      55
                                                                           4.02
                                                                           5.22
                                                                           3.90
                                                                           5.43
                                                                           3.69
                                                                           5.86
                                      60
                                                                           4.08
                                                                           5.29
                                                                           3.96
                                                                           5.51
                                                                           3.75
                                                                           5.94
                                      65
                                                                           4.14
                                                                           5.35
                                                                           4.01
                                                                           5.57
                                                                           3.80
                                                                           6.01
                                      70
                                                                           4.19
                                                                           5.41
                                                                           4.06
                                                                           5.63
                                                                           3.85
                                                                           6.07
                                      75
                                                                           4.24
                                                                           5.46
                                                                           4.11
                                                                           5.68
                                                                           3.90
                                                                           6.13
                                      80
                                                                           4.28
                                                                           5.51
                                                                           4.16
                                                                           5.73
                                                                           3.94
                                                                           6.18
                                      85
                                                                           4.33
                                                                           5.56
                                                                           4.20
                                                                           5.78
                                                                           3.99
                                                                           6.23
                                      90
                                                                           4.36
                                                                           5.60
                                                                           4.24
                                                                           5.82
                                                                           4.02
                                                                           6.27
                                      95
                                                                           4.40
                                                                           5.64
                                                                           4.27
                                                                           5.86
                                                                           4.06
                                                                           6.32
                                      100
                                                                           4.44
                                                                           5.68
                                                                           4.31
                                                                           5.90
                                                                           4.10
                                                                           6.36
                                      150
                                                                           4.72
                                                                           5.96
                                                                           4.59
                                                                           6.18
                                                                           4.38
                                                                           6.64
                                      200
                                                                           4.90
                                                                           6.15
                                                                           4.78
                                                                           6.39
                                                                           4.59
                                                                           6.84
                                      500
                                                                           5.49
                                                                           6.72
                                                                           5.47
                                                                           6.94
                                                                           5.13
                                                                           7.42
                                     1000
                                                                           5.92
                                                                           7.11
                                                                           5.79
                                                                           7.33
                                                                           5.57
                                                                           7.80

Outlier Detection

Potential outliers are measurements that are extremely large or small relative to the rest of the data and may not be representative of the population from which they were collected. Outliers may result from transcription errors, data-coding errors, or measurement system problems such as errors in chemical analyses. However, it is also possible that outliers may represent true extreme values of a distribution (hot spots, for instance) and indicate a high degree of variability in the population. Failure to remove true outliers, or the removal of false outliers, both lead to distorted estimates of the population parameters. As noted previously, it is essential to remove true outliers before performing the Studentized Range test for normality. This section discusses methods to test outliers.

Outlier tests provide statistical evidence that an extreme value does not "fit" the distribution of the other data and is therefore a potential outlier. Outlier tests should be used only to identify data points that require further investigation. An outlier test alone cannot determine whether a statistical outlier should be discarded from a data set. This decision also should be based on judgmental or scientific considerations such as replicate sampling.

Potential outliers may be identified using graphical methods. Graphs such as histograms, box and whisker plots, and normal probability plots can all be used to identify observations that are much larger or smaller than the rest of the data. If potential outliers are identified, the next step is to apply one of the statistical tests for outliers.

If an outlier is identified, the next step depends on the reason for the outlier. Data points containing transcription errors should be corrected, while data points collected while an instrument was malfunctioning may be discarded. Discarding an outlier from a data set should be done with extreme caution, as environmental data sets often contain legitimate extreme values. If any data points are found to be outliers through the use of a statistical test, this information should be documented when the results of the analysis of the data are reported. This information is critical for subsequent review of the analyses.

Parametric Tolerance Limits for Outliers

A tolerance interval is a range of concentrations containing a pre-specified proportion of the population of all possible sample values. As the interval is constructed from random samples, a tolerance interval is expected to contain the specified proportion with only a certain level of statistical confidence. Two coefficients are required to specify a tolerance interval:

  (1) The population proportion that the interval is expected to contain, called the coverage (g).
  (2) The level of confidence with which the interval reaches the specified coverage.

A tolerance interval with 95% coverage and a confidence level of 90% is expected to contain, on average, 95% of the distribution of all possible samples with a probability of 90%.

A "tolerance limit" is a one-sided tolerance interval. Tolerance limits are a parametric statistical method used for outlier detection. The one-sided upper tolerance limit (UTL) is of most interest in ground water monitoring. A UTL is designed to be exceeded only in a small percentage of the measurements. The UTL gauges whether a sample is too extreme relative to the other sample values and thus should be identified as an outlier.

Tolerance limits may be used to identify outliers in the baseline and stabilization periods. For example, to test the null hypothesis (H0) that a baseline sample is not an outlier, a UTL is constructed using the baseline data from all wells. Each sample value is then compared to the UTL to determine if it is an outlier. Specifying a UTL with 95% (or 99%) coverage and a confidence level of at least 95% is recommended for baseline data sets. A UTL with coverage of 95% (or 99%) would be exceeded, on average by less than 5% (or 1%) of the samples. A confidence level of 95% indicates that the overall false positive rate for the test is set to approximately 5%.
Parametric tolerance limits assume normality of the baseline data used to construct the limit. If the background sample is non-normal, a logarithmic transformation should be applied before using the tolerance limit. If the log-transformed values appear to be normal, the UTL should be constructed using the logarithms of the sample values; then, each sample value is compared with e[UTL] to determine if it is an outlier. Measurements below the limit of detection may be set equal to the detection limit, or alternatively, may be omitted from the calculations, and the sample size is adjusted accordingly.

Mathematically, a tolerance limit can be computed with as few as three samples. However, a sample size of at least eight measurements will be needed to generate an adequate tolerance limit for outlier detection. Pooling baseline data from multiple wells is recommended to increase the sample size.

Calculating an Upper Tolerance Limit

--------------------------------------------------------------------------------
      Step 1. Calculate the mean M and the standard deviation S of the N baseline samples.
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
      Step 2. Construct the one-sided UTL as
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
      where kN(g, 1 - α) is the one-sided normal tolerance factor found in Table 7-18 with a sample size of N, coverage coefficient of g, and a confidence level of (1 - α). For other values of N, use the next lower tabulated value, or calculate approximate values for kN using linear interpolation.
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
      Step 3. Compare each baseline sample with the UTL. If any sample exceeds the UTL, there is statistically significant evidence that the sample is an outlier.
Table 7-18.	One-Sided Upper Tolerance Limit Factors with g% Coverage for Selected Values of N

                                95% Confidence
                                99% Confidence
                                       N
                                     g=90%
                                      95%
                                      99%
                                     g=90%
                                      95%
                                      99%
                                                                              4
                                                                          4.162
                                                                          5.144
                                                                          7.042
                                                                          7.380
                                                                          9.083
                                                                         12.387
                                                                              5
                                                                          3.407
                                                                          4.203
                                                                          5.741
                                                                          5.362
                                                                          6.578
                                                                          8.939
                                                                              6
                                                                          3.006
                                                                          3.708
                                                                          5.062
                                                                          4.411
                                                                          5.406
                                                                          7.335
                                                                              7
                                                                          2.755
                                                                          3.399
                                                                          4.642
                                                                          3.859
                                                                          4.728
                                                                          6.412
                                                                              8
                                                                          2.582
                                                                          3.187
                                                                          4.354
                                                                          3.497
                                                                          4.285
                                                                          5.812
                                                                              9
                                                                          2.454
                                                                          3.031
                                                                          4.143
                                                                          3.240
                                                                          3.972
                                                                          5.389
                                                                             10
                                                                          2.355
                                                                          2.911
                                                                          3.981
                                                                          3.048
                                                                          3.738
                                                                          5.074
                                                                             11
                                                                          2.275
                                                                          2.815
                                                                          3.852
                                                                          2.898
                                                                          3.556
                                                                          4.829
                                                                             12
                                                                          2.210
                                                                          2.736
                                                                          3.747
                                                                          2.777
                                                                          3.410
                                                                          4.633
                                                                             13
                                                                          2.155
                                                                          2.671
                                                                          3.659
                                                                          2.677
                                                                          3.290
                                                                          4.472
                                                                             14
                                                                          2.109
                                                                          2.614
                                                                          3.585
                                                                          2.593
                                                                          3.189
                                                                          4.337
                                                                             15
                                                                          2.068
                                                                          2.566
                                                                          3.520
                                                                          2.521
                                                                          3.102
                                                                          4.222
                                                                             16
                                                                          2.033
                                                                          2.524
                                                                          3.464
                                                                          2.459
                                                                          3.028
                                                                          4.123
                                                                             17
                                                                          2.002
                                                                          2.486
                                                                          3.414
                                                                          2.405
                                                                          2.963
                                                                          4.037
                                                                             18
                                                                          1.974
                                                                          2.453
                                                                          3.370
                                                                          2.357
                                                                          2.905
                                                                          3.960
                                                                             19
                                                                          1.949
                                                                          2.423
                                                                          3.331
                                                                          2.314
                                                                          2.854
                                                                          3.892
                                                                             20
                                                                          1.926
                                                                          2.396
                                                                          3.295
                                                                          2.276
                                                                          2.808
                                                                          3.832
                                                                             21
                                                                          1.905
                                                                          2.371
                                                                          3.263
                                                                          2.241
                                                                          2.766
                                                                          3.777
                                                                             22
                                                                          1.886
                                                                          2.349
                                                                          3.233
                                                                          2.209
                                                                          2.729
                                                                          3.727
                                                                             23
                                                                          1.869
                                                                          2.328
                                                                          3.206
                                                                          2.180
                                                                          2.694
                                                                          3.681
                                                                             24
                                                                          1.853
                                                                          2.309
                                                                          3.181
                                                                          2.154
                                                                          2.662
                                                                          3.640
                                                                             25
                                                                          1.838
                                                                          2.292
                                                                          3.158
                                                                          2.129
                                                                          2.633
                                                                          3.601
                                                                             26
                                                                          1.824
                                                                          2.275
                                                                          3.136
                                                                          2.106
                                                                          2.606
                                                                          3.566
                                                                             27
                                                                          1.811
                                                                          2.260
                                                                          3.116
                                                                          2.085
                                                                          2.581
                                                                          3.533
                                                                             28
                                                                          1.799
                                                                          2.246
                                                                          3.098
                                                                          2.065
                                                                          2.558
                                                                          3.502
                                                                             29
                                                                          1.788
                                                                          2.232
                                                                          3.080
                                                                          2.047
                                                                          2.536
                                                                          3.473
                                                                             30
                                                                          1.777
                                                                          2.220
                                                                          3.064
                                                                          2.030
                                                                          2.515
                                                                          3.447
                                                                             35
                                                                          1.732
                                                                          2.167
                                                                          2.995
                                                                          1.957
                                                                          2.430
                                                                          3.334
                                                                             40
                                                                          1.697
                                                                          2.125
                                                                          2.941
                                                                          1.902
                                                                          2.364
                                                                          3.249
                                                                             45
                                                                          1.669
                                                                          2.092
                                                                          2.898
                                                                          1.857
                                                                          2.312
                                                                          3.180
                                                                             50
                                                                          1.646
                                                                          2.065
                                                                          2.862
                                                                          1.821
                                                                          2.269
                                                                          3.125
                                                                             55
                                                                          1.626
                                                                          2.042
                                                                          2.833
                                                                          1.790
                                                                          2.233
                                                                          3.078
                                                                             60
                                                                          1.609
                                                                          2.022
                                                                          2.807
                                                                          1.764
                                                                          2.202
                                                                          3.038
                                                                             65
                                                                          1.594
                                                                          2.005
                                                                          2.785
                                                                          1.741
                                                                          2.176
                                                                          3.004
                                                                             70
                                                                          1.581
                                                                          1.990
                                                                          2.765
                                                                          1.722
                                                                          2.153
                                                                          2.974
                                                                             75
                                                                          1.570
                                                                          1.976
                                                                          2.748
                                                                          1.704
                                                                          2.132
                                                                          2.947
                                                                             80
                                                                          1.559
                                                                          1.964
                                                                          2.733
                                                                          1.688
                                                                          2.114
                                                                          2.924
                                                                             85
                                                                          1.550
                                                                          1.954
                                                                          2.719
                                                                          1.674
                                                                          2.097
                                                                          2.902
                                                                             90
                                                                          1.542
                                                                          1.944
                                                                          2.706
                                                                          1.661
                                                                          2.082
                                                                          2.883
                                                                             95
                                                                          1.534
                                                                          1.935
                                                                          2.695
                                                                          1.650
                                                                          2.069
                                                                          2.866
                                                                            100
                                                                          1.527
                                                                          1.927
                                                                          2.684
                                                                          1.639
                                                                          2.056
                                                                          2.850
      Source: Odeh and Owen (1980)
Determining the Number of Samples per Well

The procedure for calculating the number of post-restoration samples for determining compliance must address three kinds of comparisons with baseline conditions:

  (1) Comparison of post-restoration samples from an individual monitoring well with the baseline samples from that well
  (2) Comparison of post-restoration samples from an individual monitoring well with the pooled samples from all baseline wells
  (3) Comparison of pooled post-restoration samples from all monitoring wells with the pooled samples all baseline wells

The number of samples may be determined using several approaches. The first, and most basic, approach is based on the assumptions that the number of samples collected post-restoration is the same as the number of samples collected in the baseline period, and that the standard deviations are the same. Let m and σ1 denote the baseline sample size and standard deviation, and n and σ2 denote the post-restoration sample size and standard deviation. The first approach is based on the assumptions that the sample size is the same in both periods (n=m) and that the standard deviations are the same (σ2=σ1). These constraints preclude most practical applications in comparisons with different sample sizes in the baseline and post-restoration periods. For each of the three kinds of comparisons, the sample size and standard deviation will be the same only in unusual cases.

Generalizing to an approach which permits different sample sizes offers greater flexibility than is available in the first approach, but at a cost. The cost is additional complexity in the form of a larger number of tables for determining sample size. This is necessary in order to span the possible ranges of baseline versus post-restoration sample sizes, as well as several combinations of the DQO parameters α and β, and a range of possible values for the resolution of the test MDD/σ1.

Further generalization to a third approach which permits both different sample sizes and different standard deviations offers additional flexibility than is available in the second approach, but with additional complexity. This is necessary in order to span the possible ranges of baseline versus post-operational sample variability, in addition to the parameters noted for the second approach.

Power and sample size calculations tend to be much more difficult for nonparametric procedures than for parametric procedures. Nonparametric procedures usually have less statistical power than parametric tests when the data follow a known distribution. An adjustment factor of 1.16 is usually applied to account for the possible loss of efficiency when the nonparametric WRS test is used. In addition, MARRSIM (EPA 2000a, Section <IP_ADDRESS>) recommends increasing the number of samples per well by 20% to account for possible underestimation of σ and to prepare for unplanned events that result in missing or unusable data.

Same Sample Sizes (n=m) and Same Standard Deviation (σ1=σ2)

Let nw denote the number of samples to be collected from the post-restoration monitoring wells. In the first two kinds of comparisons, nw denotes the number of samples to be collected from an individual post-restoration monitoring well. In the third type of comparison, nw denotes the number of pooled samples to be collected from all post-restoration monitoring wells.

The number of samples from the post-restoration wells will depend on the selected DQO values of α and β, and the number of baseline samples (m) that were collected. For the first kind of comparison, m denotes the number of baseline samples collected from the individual monitoring well in question. For the second and third kinds of comparisons, m denotes the pooled number of samples collected from all baseline wells. In this section, it is assumed that the sample sizes are equal (m=n).

The theory of hypothesis testing provides methods to control the frequency of decision errors when determining compliance. The decision error rate is reduced by increasing the number of samples during the post-operational phases. When the sample sizes are equal, the minimum number of samples per well (nw) to achieve a confidence level of 1  α and power of 1  β using the t-test is obtained using the approximate formula (Campbell et al. 1995):

Here, zp is the p[th] percentile of the standard normal distribution.

The number of measurements required to achieve the desired decision error rates has a strong inverse relationship with MDD/σ. Smaller values of α and β (leading to larger values for the z terms) magnify the strength of this inverse relationship. Hence, a tradeoff exists between cost (number of samples required) and benefit (better power of resolution of the test). This document does not recommend a specific sample size, as each site will have different variability (σ) and DQO parameters (α and β). A complete set of sample size estimates for nw for α = 0.01/0.025/0.05/0.10/0.20 and for β = 0.01/0.025/0.05/0.10/0.20 is tabulated for a range of the MDD/σ ratios in Table E-4 in Attachment E. Table E-4 is based on an assumption that the sample size for the baseline is the same as the sample size for the post-restoration period (m=n) and that the standard deviations are the same (σ2=σ1).
Different Sample Sizes (n!=m), Same Standard Deviations (σ1=σ2)

The formula for the number of samples when the sample sizes are unequal is provided by Campbell et al. (1995). This formula uses the number of samples when the sample sizes are equal (nw = m = n) given in Section 7.6.1 as the starting point for determining unequal sample sizes. When m is known in advance and is larger than nw/2, the number of samples per well in the post-restoration period is given by:

When the number of baseline samples m is smaller than nw/2, there is no value of n that will achieve the DQO parameters α, β, and MDD/σ1 used to calculate the value of nw.

When the number of samples in the baseline data set is larger than nw, the number of samples required in the post-restoration period will be smaller than nw, and vice versa. The total number of samples (n+m) will be greater than 2nw when the sample sizes are unequal. This occurs because the use of equal sample sizes is the optimal allocation for achieving the DQO parameters with a minimum number of samples.

The values of  are provided for 9 alternative values for α and β in Tables E-5 through E-13. The  combinations of the DQO parameters α and β are provided for (α, β) =

                   (0.05, 0.05), (0.05, 0.10), (0.05, 0.20);
                   (0.10, 0.05), (0.10, 0.10), (0.10, 0.20),
                   (0.20, 0.05), (0.20, 0.10), (0.20, 0.20);

The rows of these tables show the number of samples that were collected in the baseline (m). For the indicated value of m, the columns of the table are indexed by the desired level of resolution MDD/σ.

Different Sample Sizes (n!=m) and Different Standard Deviations (σ1!=σ2)

Generalizing to a model which permits different sample sizes offers greater flexibility than is available in the approach underlying Table E-4, but at a cost. The cost is additional complexity in the form of a larger number of tables for determining sample size. This is necessary to span the wide possible ranges of baseline versus post-restoration sample sizes, baseline versus post-operational sample variability, as well as several combinations of the DQO parameters α and β and the test resolution of MDD/σ1.

If there are m baseline samples with standard deviation σ1, the desired degree of resolution for the test is MDD/σ1, and the post-restoration samples are estimated to have standard deviation σ2, then the number of samples nw is the smallest value of n satisfying the inequality:

where  is the ratio of the standard deviations. Here z1-α and z1-β have the standard definitions.

When the number of baseline samples is known, the equation may be solved for the number of post-restoration samples required to achieve the DQO parameters. An iterative solution of the equation is found by using a series of trial values of n starting at 1 and continuing until the inequality is achieved. The expression on the left of the inequality decreases asymptotically to as n increases. Hence, if , there is no n sufficiently large to achieve the desired test resolution of MDD/σ1 given the specified values of m, α, and β.

Table E-14 shows the number of post-restoration samples nw for the DQO parameters (α,β) = (0.05,0.05). The rows in this table are segmented by the number of baseline samples m. For the selected value of m, the table includes rows for all achievable values of MDD/σ1. The columns of the table indicate the ratio of the post-operational to baseline standard deviation: . Large sample sizes may be encountered in pooled comparisons in large fields. The table shows post-restoration sample size estimates up to 1,000. Estimates exceeding this level are indicated by an asterisk.

Only Table E-14 is included in the current Attachment E. Eight other tables showing the number of samples nw for  other combinations of the DQO parameters (α, β) =

                          (0.05, 0.10), (0.05, 0.20);
                   (0.10, 0.05), (0.10, 0.10), (0.10, 0.20),
                   (0.20, 0.05), (0.20, 0.10), (0.20, 0.20),

have been prepared. As these tables are voluminous, they have not been included in Attachment E, but can be provided on request.

Statistical Methods for Trends and Seasonality

The existence of seasonality may complicate trend analysis and comparisons with baseline conditions. Sufficient data must be collected to estimate seasonal trends and account for seasonality in the statistical analysis. This requires at least two years of data (a minimum of one year during the pre-operational phase plus one year during the post-operational phase) under the assumption the seasonal pattern is not affected by the mining and restoration activities.

Sampling locations should be monitored for at least a full year prior to operations, as well as after restoration, with consistent timing of observations within these years. A carefully designed monitoring plan, in which each well has equivalently timed baseline and post-restoration measurements (quarterly or monthly measurements taken at the same periods in each quarter or month), will largely eliminate the need for seasonal adjustment when statistical tests of post/pre-operational differences are performed. Use of consistent timing in both periods has the effect of "subtracting out" any seasonal component when post-restoration data are compared to the initial conditions.

If seasonal patterns are either predicted by hydrological models or observed to be highly variable from year to year, then a single year of pre-operational information may not be adequate. A design in which pre-operational data are collected for more than 1 year may be preferable for ISR operations within shallow aquifers. This is especially true in unusual years of extreme drought or flooding, for example, that may bias the baseline measurements one way or another. One approach for extending the time seasonality is monitored is the use of additional monitoring data collected outside the ore zone. Up gradient wells may be monitored for seasonal effects throughout operational period in lieu of additional pre-operational monitoring.

The first step in analyzing measurements in one or more wells is to plot the data as a time series. Such plots are shown in the example discussed in Attachment D. Plots of the data may reveal patterns such as seasonality and/or the existence of outliers or blunders in the data. Outliers are values that appear to be unusually high or low when compared to the other values. Outliers may be valid data or may arise from unusual circumstances unrelated to the process being measured. Blunders are outright errors made in recording the data, transcription, or calculations. A common blunder is a mistake in the units of measure. Plotting is used to detect these situations, but does not provide for an explanation or resolution for the unusual value. Blunders are sometimes outliers and thus a given value could be both. But blunders need not be outliers -- data may have been keyed in wrong but not be far from the main body of observations. So plotting will not always reveal blunders in the data.
If a value is identified as erroneous, it should be removed from the data set. In cases of doubt, the value should be retained. The nonparametric statistical tests discussed in this section were selected because of their robustness. The statistical term "robust" is loosely defined as resistant to the effects of outliers and blunders in the data.
Adjusting for Seasonality

Seasonality may occur in baseline samples in Phase 1, while the site is reaching steady state in Phase 4 and/or in Phase 5, where seasonality may affect decisions concerning long-term stability and whether target remediation values are attained.

Seasonality is a pattern that repeats periodically in a cycle. An annual seasonal pattern has a cycle that spans 12 months or 4 quarters. A seasonal index measures how far the average for a particular period is above (or below) the average for all periods. The unified guidance for RCRA (EPA 2009) recommends the following concerning seasonality:
      Seasonal fluctuations in intrawell background can be treated in one of two ways. A seasonal Mann-Kendall trend test built to accommodate such fluctuations can be employed (Section 14.3.4). Otherwise, the seasonal pattern can be estimated and removed from the background data, leaving a set of seasonally-adjusted data to be analyzed with either a prediction limit or control chart. In this latter approach, the same seasonal pattern needs to be extrapolated beyond the current background to more recent measurements from the compliance well being tested. These later observations also need to be seasonally-adjusted prior to comparison against the adjusted background, even if there is not enough compliance data yet collected to observe the same seasonal cycles.
However, the guidance adds the following caveat:

      Corrections for seasonality should be used cautiously, as they represent extrapolation into the future. There should be a good physical explanation for the seasonal fluctuation as well as good empirical evidence for seasonality before corrections are made. Higher than average rainfall for two or three Augusts in a row does not justify the belief that there will never be a drought in August, and this idea extends directly to groundwater quality. At least three complete cycles of the seasonal pattern should be observed on a time series plot before attempting the adjustment below. If seasonality is suspected but the pattern is complicated, the user should seek the help of a professional statistician.

The seasonal Mann-Kendall test is a variation of the Mann-Kendall test for trends described below in Section <IP_ADDRESS>. The test is described in detail in EPA 2009 in Section 14.3.4.

Seasonal adjustment procedures are commonly applied to ecological and economic data to account for seasonal patterns. The process of "deseasonalizing" the data removes these periodic seasonal variations to reveal the underlying longer term pattern. The i[th] seasonal component (Qi) is defined as the deviation of the seasonal mean (Yi) from the overall mean (YM): Qi = Yi - YM. The deseasonalized time series (X) is obtained by subtracting the seasonal means from the original data series: Xt,i = Yt,i - Qi (EPA 2009, Eq. 14.23). The deseasonalized data series has the short-term seasonal variations removed; longer-term trends remain in the data. Plots of the seasonally adjusted data series are useful for determining when suspected outliers in sample values reflect the normal variability of monitored parameters after adjusting for the seasonal variations.
When there are four quarterly measurements in each year, the data may be seasonally adjusted by the procedure described in Section D.1 in Attachment D. Appropriate modifications must be made for periodic variations based on other time frames. Some parameters may require seasonal adjustment and others not. Formal tests for the presence of seasonality across several wells are based on an analysis of variance. This procedure is described in EPA 2002b (Sections 14.2.2 and 14.3.3).

The seasonal adjustment procedures are applicable to data that are approximately symmetric and normally distributed. For highly skewed lognormal data series, the calculations above would be applied to the logarithms of the measurements. This is equivalent to using the ratio of the quarterly mean to the overall mean (Q[*]i=Yi/YM) as the seasonality index in place of the additive index above. If this index is 1.2, this means that, on average, the period (season) is 20% higher than average. In this case, the seasonally adjusted data series is obtained by dividing the original data series by the seasonal index: Xt,I = Yt,i/Q[*]i.

It is assumed that there is a complete set of quarterly measurements for 3 years with no missing or nondetect values. If one or two nondetects occur in the data series, one should replace those values with the limit of detection. If there is at most one missing data value, these methods may be applied using the averages of the available data to compute the seasonal index. If more than one value is missing, the appropriateness of adjusting for seasonal variation should be discussed with a statistician familiar with environmental sampling.
--------------------------------------------------------------------------------
      Unless otherwise noted, in the remaining sections of this chapter, the term "data" refers to the seasonally adjusted data series Xt,i.

Using Trend Tests to Determine Stability

Mining by ISR causes a major perturbation of the physical and chemical environment of the ore zone. When mining operations end and restoration activities are completed, the ground water and the minerals it contacts begin to shift toward a new geochemical steady state. It should be recognized that the system may never return to steady-state conditions exactly comparable to pre-operational baseline conditions.

In the short run, post-restoration samples are used to provide statistical evidence of trends or lack thereof. However, the long-term stability of the site is best understood using a conceptual model that considers these data along with the hydrology, contaminant transport, and geochemical reactions to provide a qualitative estimate of the evolution of the system after post-restoration monitoring is completed. The NRC (2007) has emphasized that development of a justifiable conceptual model capturing the major chemical and physical phenomena at each site is required. This approach allows for site-specific flexibility.

In addition to water quality data, mineralogical data can be used to evaluate the long-term stability of the system. Hydro-geochemical modeling can be a valuable tool for modeling the rate of the return to stability and for predicting how long a system will take to return to baseline conditions. Although complete mineralogical characterization may require additional resources, a fully developed quantitative model supported by site post-restoration measurements can provide additional confidence that the restoration goal of site stability after closure has been met.

Statistical tests for trends are used to demonstrate stability within the specified monitoring period. These tests may be used with any time series of four or more independent samples to test for trends in well parameters. Trend tests are employed in Phase 1 to check for unexpected trends in baseline samples, and particularly in Phase 5 to affirm long-term stability. Trends may be detected using parametric and/or nonparametric statistical tests. The power of a trend test to detect a trend depends on several factors, including the underlying variability of the series, the magnitude of slope we wish to detect, level of confidence for the test and the length of time the series is observed.

A Monte Carlo simulation was conducted to examine the time it takes to detect a trend. Two popular methods of trend detection are considered: the linear regression t-test and the Mann-Kendall test. The regression t-test is a parametric statistical test, and the Mann-Kendall test is a nonparametric test.

Detecting Trends Using Hypothesis Tests

Given a series of T sample values collected from a well at times , the objective is to decide if there is a significant trend in the series. In general, time may be measured in any units, such as months, quarters, or years. However, the interpretation of the slope of the trend line will depend on the units selected for the time axis as well as the units of measurement for the sample values. In this discussion, quarterly sampling is assumed as quarterly measurements are commonly used. Sampling frequencies other than quarterly sampling are discussed in Section <IP_ADDRESS>. The time variable is measured in number of years from the first observation. The slope estimates are expressed as a percentage change per year. The simulations assume that the sample collection times are equally spaced, one per quarter. In general, this is not a necessary assumption. Both methods of trend detection may be applied with irregularly spaced sampling times.

The linear trend model provides a parametric statistical test for a significant trend in the percentage deviations. A linear trend is described by a trend line with the equation:

Here the coefficient a is the initial value of y at time t=0, often described as the intercept of the trend line. The coefficient b is the slope of the trend line. If time is increased by one unit, the value of y is increased (or decreased) by b units when b is positive (or negative).

The regression t-test is used to test if an observed positive or negative slope in the series is statistically significant at a specified level of confidence. The t-statistic for the test is the slope estimate  divided by its standard error of estimation: . If the absolute value of the t-statistic for the slope coefficient is larger than the critical value for the test, then the trend has a significantly positive (or negative) slope. If the slope of the trend line is not zero, all statistical tests will detect the trend eventually. In the short term, the test may or may not detect a trend. The power to detect a trend will depend on the magnitude of the trend, the degree of temporal variability, the level of confidence for the test, and how long we are willing to wait to detect the trend. A complete discussion of the use of linear and nonlinear regression models for assessing trends is provided in Chapter 6 of EPA 1992.

Simulation of Trend Detection

A Monte Carlo simulation study was conducted to estimate the time required to detect a linear trend using regression t-test and the Mann-Kendall trend test. In the simulation study, a time series of T quarterly samples is generated for each combination of slope and variability. Quarterly samples were generated at times tj, j=1,..., T where the time t is measured in years from the first sample. For example, with five quarters of data (T=5), the five values of tj are t1=0, t2=0.25, t3=0.5, t4=0.75, and t5=1.0 years. This section presents a summary of the simulation results. Details of the simulation analysis are provided in Attachment G.

Regression analysis provides an estimate of the magnitude of the trend, and the regression t-test is used to determine if the trend is statistically significant. The Mann-Kendall test is a nonparametric test for detecting trends in the data series. The Mann-Kendall test may be used with any series of four or more independent samples to test for trends. The Mann-Kendall test does not provide an estimate of the slope, however; only a test of whether the trend is significant.

The Mann-Kendall test and the regression t-test have very similar characteristics when the observation errors are normally distributed. Regression analysis is sensitive to outliers and requires careful inspection of the data before the results of the t-test can be validated. Regression also requires numerical values for all samples, and pseudovalues must be assigned for nondetects. The Mann-Kendall test is not unduly influenced by outliers and is therefore more robust than regression for detecting trends. The Mann-Kendall test also may be used with nondetects.

Table 7-19 through Table 7-21 show the results of the simulations. The tables contain results for 11 selected values of the slope that is to be detected, ranging from 1% per year to 100% per year. For each slope, 9 levels of variability are considered ranging from 5% to 150%. Each table shows the minimum number of quarterly samples required, such that the probability of detecting the slope is at least 90%, 95% or 99%, respectively.

In the upper right corner of the Mann-Kendall tables, an asterisk is used when the number of samples required exceeds 100. In these regions of the tables, the variability is too large to detect the specified slope with less than 25 years of quarterly samples. The regression tables do not have this constraint, and the series were permitted to be as long as necessary to detect the slope.

Using Table 7-20 for a 95% chance of detection using regression, 21 samples are required when the slope is 10% and the variability is 20%. The same number of samples (21) is required for a slope of 15% and a variability of 30%. The corresponding number of samples using the Mann-Kendall test is 22 for both cases. These two cases are similar in that the ratio of the slope to the variability is 0.5.

The simulation results suggest that linear trends will be detected using the Mann-Kendall test within 15 quarters in 95% or more of the cases if the standardized slope ratio is greater than or equal to 1. When using the regression t-test under the most favorable assumptions, linear trends may be detected within 14 quarters in 95% or more of the cases if the standardized slope ratio is greater than or equal to 1. Doubling of the number of samples to 30 (or 28 for regression) enables detection of slopes equal approximately to 1/3 of the variability.

Table 7-19.	Number of Quarterly Samples Required for 90% Probability of Detecting Slope Using a Mann-Kendall or Regression Trend Test
                             Regression Trend Test
                            Mann-Kendall Trend Test
                                Slope (%/year)
                                Variability (%)
                                Slope (%/year)
                                Variability (%)

                                       5
                                      10
                                      15
                                      20
                                      30
                                      50
                                      75
                                      100
                                      150

                                       5
                                      10
                                      15
                                      20
                                      30
                                      50
                                      75
                                      100
                                      150
                                       1
                                      36
                                      56
                                      73
                                      88
                                      114
                                      161
                                      210
                                      254
                                      333
                                       1
                                      37
                                      57
                                      74
                                      90
                                       *
                                       *
                                       *
                                       *
                                       *
                                       3
                                      18
                                      27
                                      35
                                      43
                                      56
                                      78
                                      101
                                      122
                                      159
                                       3
                                      18
                                      28
                                      37
                                      44
                                      57
                                      78
                                       *
                                       *
                                       *
                                       5
                                      13
                                      20
                                      26
                                      31
                                      40
                                      56
                                      72
                                      87
                                      114
                                       5
                                      13
                                      20
                                      26
                                      32
                                      41
                                      57
                                      75
                                      89
                                       *
                                      10
                                       9
                                      13
                                      17
                                      20
                                      26
                                      36
                                      46
                                      56
                                      73
                                      10
                                       9
                                      13
                                      17
                                      20
                                      27
                                      36
                                      47
                                      57
                                      74
                                      15
                                       7
                                      10
                                      13
                                      15
                                      20
                                      27
                                      35
                                      43
                                      56
                                      15
                                       8
                                      11
                                      13
                                      16
                                      21
                                      28
                                      36
                                      44
                                      57
                                      20
                                       6
                                       9
                                      11
                                      13
                                      16
                                      23
                                      30
                                      35
                                      46
                                      20
                                       7
                                       9
                                      12
                                      13
                                      17
                                      24
                                      30
                                      37
                                      47
                                      30
                                       5
                                       7
                                       9
                                      10
                                      13
                                      18
                                      23
                                      27
                                      35
                                      30
                                       6
                                       8
                                       9
                                      11
                                      14
                                      18
                                      24
                                      28
                                      37
                                      40
                                       5
                                       6
                                       7
                                       9
                                      11
                                      15
                                      19
                                      23
                                      30
                                      40
                                       5
                                       7
                                       8
                                       9
                                      11
                                      15
                                      20
                                      24
                                      31
                                      50
                                       4
                                       6
                                       7
                                       8
                                      10
                                      13
                                      16
                                      20
                                      26
                                      50
                                       5
                                       6
                                       7
                                       9
                                      10
                                      13
                                      17
                                      20
                                      27
                                      75
                                       4
                                       5
                                       5
                                       6
                                       8
                                      10
                                      13
                                      15
                                      20
                                      75
                                       4
                                       5
                                       6
                                       7
                                       9
                                      11
                                      13
                                      16
                                      20
                                      100
                                       4
                                       4
                                       5
                                       5
                                       7
                                       9
                                      11
                                      13
                                      17
                                      100
                                       4
                                       5
                                       5
                                       6
                                       7
                                       9
                                      12
                                      13
                                      17
        * Over 100 quarters. Mann-Kendall test simulation is limited to 25 years.
Table 7-20.	Number of Quarterly Samples Required for 95% Probability of Detecting Slope Using a Mann-Kendall or Regression Trend Test
                             Regression Trend Test
                            Mann-Kendall Trend Test
                                Slope (%/year)
                                Variability (%)
                                Slope (%/year)
                                Variability (%)

                                       5
                                      10
                                      15
                                      20
                                      30
                                      50
                                      75
                                      100
                                      150

                                       5
                                      10
                                      15
                                      20
                                      30
                                      50
                                      75
                                      100
                                      150
                                       1
                                                                             38
                                                                             61
                                                                             79
                                                                             95
                                                                            124
                                                                            175
                                                                            225
                                                                            271
                                                                            358
                                       1
                                                                             39
                                                                             62
                                                                             80
                                                                             97
                                                                              *
                                                                              *
                                                                              *
                                                                              *
                                                                              *
                                       3
                                                                             19
                                                                             30
                                                                             38
                                                                             46
                                                                             60
                                                                             84
                                                                            109
                                                                            133
                                                                            175
                                       3
                                                                             20
                                                                             30
                                                                             39
                                                                             47
                                                                             62
                                                                             85
                                                                              *
                                                                              *
                                                                              *
                                       5
                                                                             14
                                                                             21
                                                                             27
                                                                             33
                                                                             44
                                                                             60
                                                                             79
                                                                             96
                                                                            123
                                       5
                                                                             14
                                                                             22
                                                                             29
                                                                             34
                                                                             44
                                                                             62
                                                                             80
                                                                             97
                                                                              *
                                      10
                                                                              9
                                                                             14
                                                                             18
                                                                             21
                                                                             28
                                                                             39
                                                                             50
                                                                             60
                                                                             79
                                      10
                                                                             10
                                                                             14
                                                                             18
                                                                             22
                                                                             29
                                                                             40
                                                                             51
                                                                             62
                                                                             80
                                      15
                                                                              7
                                                                             11
                                                                             14
                                                                             17
                                                                             21
                                                                             30
                                                                             38
                                                                             46
                                                                             60
                                      15
                                                                              8
                                                                             12
                                                                             14
                                                                             17
                                                                             22
                                                                             31
                                                                             39
                                                                             47
                                                                             61
                                      20
                                                                              6
                                                                              9
                                                                             12
                                                                             14
                                                                             18
                                                                             25
                                                                             32
                                                                             39
                                                                             50
                                      20
                                                                              7
                                                                             10
                                                                             12
                                                                             14
                                                                             19
                                                                             25
                                                                             32
                                                                             39
                                                                             52
                                      30
                                                                              5
                                                                              7
                                                                              9
                                                                             11
                                                                             14
                                                                             19
                                                                             24
                                                                             30
                                                                             38
                                      30
                                                                              6
                                                                              8
                                                                             10
                                                                             12
                                                                             15
                                                                             20
                                                                             25
                                                                             31
                                                                             39
                                      40
                                                                              5
                                                                              6
                                                                              8
                                                                              9
                                                                             11
                                                                             16
                                                                             20
                                                                             25
                                                                             32
                                      40
                                                                              5
                                                                              7
                                                                              9
                                                                             10
                                                                             12
                                                                             16
                                                                             21
                                                                             25
                                                                             33
                                      50
                                                                              5
                                                                              6
                                                                              7
                                                                              8
                                                                             10
                                                                             14
                                                                             18
                                                                             21
                                                                             27
                                      50
                                                                              5
                                                                              7
                                                                              8
                                                                              9
                                                                             11
                                                                             14
                                                                             18
                                                                             22
                                                                             28
                                      75
                                                                              4
                                                                              5
                                                                              6
                                                                              7
                                                                              8
                                                                             11
                                                                             14
                                                                             17
                                                                             21
                                      75
                                                                              4
                                                                              5
                                                                              7
                                                                              7
                                                                              9
                                                                             12
                                                                             14
                                                                             17
                                                                             22
                                      100
                                                                              4
                                                                              4
                                                                              5
                                                                              6
                                                                              7
                                                                              9
                                                                             12
                                                                             14
                                                                             18
                                      100
                                                                              4
                                                                              5
                                                                              6
                                                                              7
                                                                              8
                                                                             10
                                                                             12
                                                                             14
                                                                             18
        * Over 100 quarters. Mann-Kendall test simulation is limited to 25 years.
Table 7-21.	Number of Quarterly Samples Required for 99% Probability of Detecting Slope Using a Mann-Kendall or Regression Trend Test
                             Regression Trend Test
                            Mann-Kendall Trend Test
                                Slope (%/year)
                                Variability (%)
                                Slope (%/year)
                                Variability (%)

                                       5
                                      10
                                      15
                                      20
                                      30
                                      50
                                      75
                                      100
                                      150

                                       5
                                      10
                                      15
                                      20
                                      30
                                      50
                                      75
                                      100
                                      150
                                       1
                                                                             43
                                                                             68
                                                                             89
                                                                            106
                                                                            139
                                                                            197
                                                                            258
                                                                            306
                                                                            405
                                       1
                                                                             45
                                                                             70
                                                                             91
                                                                              *
                                                                              *
                                                                              *
                                                                              *
                                                                              *
                                                                              *
                                       3
                                                                             21
                                                                             34
                                                                             44
                                                                             52
                                                                             68
                                                                             95
                                                                            123
                                                                            150
                                                                            198
                                       3
                                                                             22
                                                                             34
                                                                             44
                                                                             54
                                                                             70
                                                                             96
                                                                              *
                                                                              *
                                                                              *
                                       5
                                                                             16
                                                                             24
                                                                             31
                                                                             37
                                                                             49
                                                                             68
                                                                             88
                                                                            108
                                                                            136
                                       5
                                                                             16
                                                                             25
                                                                             32
                                                                             39
                                                                             50
                                                                             70
                                                                             90
                                                                              *
                                                                              *
                                      10
                                                                             10
                                                                             15
                                                                             20
                                                                             24
                                                                             32
                                                                             43
                                                                             56
                                                                             69
                                                                             89
                                      10
                                                                             11
                                                                             16
                                                                             21
                                                                             25
                                                                             32
                                                                             45
                                                                             58
                                                                             69
                                                                             89
                                      15
                                                                              8
                                                                             12
                                                                             15
                                                                             19
                                                                             24
                                                                             33
                                                                             43
                                                                             52
                                                                             68
                                      15
                                                                              9
                                                                             13
                                                                             16
                                                                             20
                                                                             25
                                                                             34
                                                                             45
                                                                             53
                                                                             69
                                      20
                                                                              7
                                                                             10
                                                                             13
                                                                             15
                                                                             20
                                                                             28
                                                                             36
                                                                             43
                                                                             56
                                      20
                                                                              8
                                                                             11
                                                                             14
                                                                             16
                                                                             21
                                                                             29
                                                                             37
                                                                             44
                                                                             58
                                      30
                                                                              6
                                                                              8
                                                                             10
                                                                             12
                                                                             15
                                                                             21
                                                                             28
                                                                             33
                                                                             43
                                      30
                                                                              7
                                                                              9
                                                                             11
                                                                             13
                                                                             16
                                                                             22
                                                                             29
                                                                             34
                                                                             45
                                      40
                                                                              5
                                                                              7
                                                                              9
                                                                             10
                                                                             13
                                                                             18
                                                                             23
                                                                             28