36
                                      40
                                                                              6
                                                                              8
                                                                             10
                                                                             11
                                                                             14
                                                                             18
                                                                             24
                                                                             28
                                                                             38
                                      50
                                                                              5
                                                                              6
                                                                              8
                                                                              9
                                                                             11
                                                                             16
                                                                             20
                                                                             24
                                                                             31
                                      50
                                                                              5
                                                                              7
                                                                              9
                                                                             10
                                                                             12
                                                                             16
                                                                             21
                                                                             25
                                                                             32
                                      75
                                                                              4
                                                                              5
                                                                              6
                                                                              7
                                                                              9
                                                                             12
                                                                             15
                                                                             19
                                                                             24
                                      75
                                                                              5
                                                                              6
                                                                              7
                                                                              8
                                                                             10
                                                                             13
                                                                             16
                                                                             19
                                                                             25
                                      100
                                                                              4
                                                                              5
                                                                              6
                                                                              6
                                                                              8
                                                                             10
                                                                             13
                                                                             16
                                                                             20
                                      100
                                                                              4
                                                                              5
                                                                              7
                                                                              7
                                                                              9
                                                                             11
                                                                             14
                                                                             16
                                                                             20
        * Over 100 quarters. Mann-Kendall test simulation is limited to 25 years.
Duration of Sampling

Table 7-20 shows the required number of samples to achieve a 95% probability of detection using the regression t-test or the Mann-Kendall test for selected slope and variability. The simulation results also provide estimates of the probability of detecting a trend using a fixed number of quarterly samples. This section presents a comparison of three alternatives: quarterly sampling for 12 quarters (3 years); 20 quarters (5 years); or 32 quarters (8 years). The comparison uses the same set of slopes and variability as in Table 7-20.

Table 7-22 shows the probability of detecting a trend (expressed as a percentage) using the Mann-Kendall trend test or the regression t-test with 12 quarterly samples. The probability of detection was simulated for the combinations of slope and variability shown on the borders of the tables. The slope is expressed as the percentage change per year, and the variability is expressed in terms of the standard deviation from the trend line. The outlined cells on the diagonal of the table indicate where the slope is equal to the variability.

Using 12 quarterly samples, there is approximately an 80% chance of detecting the trend using the Mann-Kendall test when the slope is equal to the variability. When the regression t-test is used with normally distributed observation errors, the chance of detection increases to approximately 87% when the slope is equal to the variability. When the slope is less than the variability, the probability of detection is always smaller than these values, falling quickly to below 50%. The apparently better performance of the regression t-test over the Mann-Kendall test is largely due to the fact that the simulation uses normally distributed errors.

Table 7-23 shows simulation results for 20 quarterly samples spread over 5 years. There is approximately a 100% chance of detecting the trend when the slope is equal to the variability. Table 7-24 shows simulation results for 32 quarterly samples (8 years). In this case, the chance of detecting slopes as small as one-half of the variability is approximately 100%.

Table 7-22.	Probability of Detecting a Trend with 12 Quarterly Samples
                            Mann-Kendall Trend Test
                          Regression t-test for Trend
                                 Slope (%/yr)
                                Variability (%)
                                 Slope (%/yr)
                                Variability (%)

                                       5
                                      10
                                      15
                                      20
                                      30
                                      50
                                      75
                                      100
                                      150

                                       5
                                      10
                                      15
                                      20
                                      30
                                      50
                                      75
                                      100
                                      150
                                       1
                                      12
                                       7
                                       6
                                       6
                                       6
                                       5
                                       5
                                       5
                                       4
                                       1
                                      14
                                       9
                                       7
                                       7
                                       6
                                       6
                                       6
                                       5
                                       5
                                       3
                                      45
                                      18
                                      12
                                       9
                                       8
                                       6
                                       6
                                       5
                                       5
                                       3
                                      51
                                      21
                                      14
                                      11
                                       9
                                       7
                                       7
                                       6
                                       6
                                       5
                                      80
                                      35
                                      20
                                      15
                                      10
                                       7
                                       6
                                       6
                                       5
                                       5
                                      87
                                      40
                                      24
                                      17
                                      12
                                       8
                                       7
                                       7
                                       6
                                      10
                                      100
                                      80
                                      50
                                      33
                                      21
                                      12
                                       9
                                       7
                                       7
                                      10
                                      100
                                      87
                                      58
                                      39
                                      24
                                      15
                                      11
                                       8
                                       8
                                      15
                                      100
                                      98
                                      80
                                      60
                                      35
                                      17
                                      12
                                       9
                                       8
                                      15
                                      100
                                      99
                                      87
                                      67
                                      40
                                      20
                                      14
                                      11
                                       9
                                      20
                                      100
                                      100
                                      96
                                      82
                                      51
                                      25
                                      16
                                      12
                                       9
                                      20
                                      100
                                      100
                                      98
                                      87
                                      59
                                      29
                                      19
                                      14
                                      10
                                      30
                                      100
                                      100
                                      100
                                      98
                                      81
                                      45
                                      25
                                      17
                                      12
                                      30
                                      100
                                      100
                                      100
                                      99
                                      87
                                      51
                                      29
                                      21
                                      14
                                      40
                                      100
                                      100
                                      100
                                      100
                                      96
                                      64
                                      38
                                      25
                                      16
                                      40
                                      100
                                      100
                                      100
                                      100
                                      98
                                      72
                                      44
                                      30
                                      19
                                      50
                                      100
                                      100
                                      100
                                      100
                                      99
                                      81
                                      52
                                      33
                                      20
                                      50
                                      100
                                      100
                                      100
                                      100
                                      100
                                      87
                                      59
                                      39
                                      24
                                      75
                                      100
                                      100
                                      100
                                      100
                                      100
                                      98
                                      81
                                      61
                                      34
                                      75
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      87
                                      68
                                      40
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      96
                                      81
                                      51
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      98
                                      89
                                      58

Key:

Less than 50% chance of detecting trend.

Between 50% and 95% chance of detecting trend.
                                                                       Unshaded
Greater than 95% chance of detecting trend.
                                   Outlined
Outlined cells have slope equal to variability.

Table 7-23.	Probability of Detecting a Trend with 20 Quarterly Samples
                            Mann-Kendall Trend Test
                          Regression t-test for Trend
                                 Slope (%/yr)
                                Variability (%)
                                 Slope (%/yr)
                                Variability (%)

                                       5
                                      10
                                      15
                                      20
                                      30
                                      50
                                      75
                                      100
                                      150

                                       5
                                      10
                                      15
                                      20
                                      30
                                      50
                                      75
                                      100
                                      150
                                       1
                                      31
                                      14
                                      11
                                       9
                                       7
                                       6
                                       6
                                       5
                                       6
                                       1
                                      34
                                      16
                                      11
                                       9
                                       7
                                       6
                                       6
                                       5
                                       6
                                       3
                                      97
                                      56
                                      33
                                      22
                                      14
                                      10
                                       8
                                       7
                                       6
                                       3
                                      98
                                      59
                                      34
                                      23
                                      15
                                      10
                                       8
                                       7
                                       6
                                       5
                                      100
                                      91
                                      63
                                      43
                                      26
                                      15
                                      11
                                       9
                                       6
                                       5
                                      100
                                      93
                                      66
                                      46
                                      27
                                      15
                                      12
                                      10
                                       7
                                      10
                                      100
                                      100
                                      99
                                      91
                                      62
                                      32
                                      19
                                      15
                                      11
                                      10
                                      100
                                      100
                                      99
                                      93
                                      66
                                      33
                                      20
                                      15
                                      12
                                      15
                                      100
                                      100
                                      100
                                      100
                                      90
                                      55
                                      32
                                      22
                                      15
                                      15
                                      100
                                      100
                                      100
                                      100
                                      92
                                      58
                                      34
                                      23
                                      16
                                      20
                                      100
                                      100
                                      100
                                      100
                                      99
                                      76
                                      47
                                      32
                                      19
                                      20
                                      100
                                      100
                                      100
                                      100
                                      99
                                      80
                                      50
                                      35
                                      20
                                      30
                                      100
                                      100
                                      100
                                      100
                                      100
                                      97
                                      77
                                      55
                                      33
                                      30
                                      100
                                      100
                                      100
                                      100
                                      100
                                      98
                                      81
                                      58
                                      34
                                      40
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      94
                                      76
                                      47
                                      40
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      95
                                      79
                                      51
                                      50
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      99
                                      91
                                      63
                                      50
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      99
                                      93
                                      67
                                      75
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      91
                                      75
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      93
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      99
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100

Key:

Less than 50% chance of detecting trend.

Between 50% and 95% chance of detecting trend.
                                                                       Unshaded
Greater than 95% chance of detecting trend.
                                   Outlined
Outlined cells have slope equal to variability.

Table 7-24.	Probability of Detecting a Trend with 32 Quarterly Samples
                            Mann-Kendall Trend Test
                          Regression t-test for Trend
                                 Slope (%/yr)
                                Variability (%)
                                 Slope (%/yr)
                                Variability (%)

                                       5
                                      10
                                      15
                                      20
                                      30
                                      50
                                      75
                                      100
                                      150

                                       5
                                      10
                                      15
                                      20
                                      30
                                      50
                                      75
                                      100
                                      150
                                       1
                                      78
                                      33
                                      19
                                      14
                                      10
                                       8
                                       7
                                       6
                                       5
                                       1
                                      81
                                      36
                                      21
                                      15
                                      10
                                       9
                                       7
                                       7
                                       6
                                       3
                                      100
                                      98
                                      78
                                      58
                                      33
                                      19
                                      12
                                      10
                                       8
                                       3
                                      100
                                      99
                                      82
                                      61
                                      35
                                      20
                                      13
                                      10
                                       8
                                       5
                                      100
                                      100
                                      99
                                      92
                                      66
                                      33
                                      19
                                      14
                                       9
                                       5
                                      100
                                      100
                                      100
                                      94
                                      70
                                      35
                                      21
                                      16
                                      10
                                      10
                                      100
                                      100
                                      100
                                      100
                                      99
                                      78
                                      49
                                      33
                                      20
                                      10
                                      100
                                      100
                                      100
                                      100
                                      99
                                      81
                                      52
                                      35
                                      22
                                      15
                                      100
                                      100
                                      100
                                      100
                                      100
                                      98
                                      78
                                      58
                                      33
                                      15
                                      100
                                      100
                                      100
                                      100
                                      100
                                      99
                                      81
                                      61
                                      35
                                      20
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      95
                                      78
                                      48
                                      20
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      97
                                      82
                                      53
                                      30
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      98
                                      80
                                      30
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      98
                                      83
                                      40
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      95
                                      40
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      96
                                      50
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      99
                                      50
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      75
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      75
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100
                                      100

Key:

Less than 50% chance of detecting trend.

Between 50% and 95% chance of detecting trend.
                                                                       Unshaded
Greater than 95% chance of detecting trend.
                                   Outlined
Outlined cells have slope equal to variability.

Sampling Frequency

When the sampling frequency changes from the base case of one sample per quarter to a higher or lower frequency (say bi-annual or monthly), there are several important factors to consider. Higher sampling frequencies increase the possibility that there will be correlations between successive samples. The simulation model described above does not consider correlations in the series of samples. If there are possible correlations, the required number of samples is larger than the simulation predicts. In the unlikely case of negative correlations, the required number of samples would be smaller than the simulation predicts. In the following discussion, correlation is ignored.

In the absence of correlation, higher sampling frequencies affect the required number of samples in two opposite ways. A higher sampling frequency provides a greater number of samples, but in a shorter period of time. The effect of a greater number of samples is clear: it will increase the power of detection. However, a shorter period of observation means that the time series does not have as long to change and the change that has occurred is more difficult to detect.

To explore the implications of higher or lower sampling frequencies in the absence of correlation, one case in Table 7-20 was selected for a "one-off" analysis. Table 7-20 was generated using 4 quarterly samples per year. The case in Table 7-20 for a 95% chance of detecting a slope of 10%/year and a variability of 20% was selected for this analysis. This case requires 21 samples for a 95% chance of detection using regression, and 22 samples using the Mann-Kendall test when sampling is conducted quarterly.

Sampling frequencies of 2 per year (semiannual), 4 per year (quarterly), 8 per year (approximately 6-week intervals), and 12 per year (monthly) were selected for the simulation analysis. Table 7-25 shows results of the simulation. The selected case from Table 7-20 is highlighted in this table at a sampling frequency of 4 samples per year.

When the sampling frequency is reduced to semiannual (2 per year), the required number of samples for detecting a trend using the regression t-test is reduced from 21 quarterly samples to 14 semiannual samples. It will take 7 years to collect the 14 semiannual samples, compared with 5.25 years to collect 21 quarterly samples. Results for the regression t-test and the Mann-Kendall test are almost identical. In the absence of correlation, less frequent sampling requires fewer sampling events, but the samples are spread over a longer period of time. It is more unlikely to encounter correlations using semiannual data than using quarterly data, so this result of the one-off analysis is reasonably robust with respect to the assumed lack of correlation.

If sampling is conducted more frequently than quarterly, more samples are required. If a 6-week sampling interval is used (approximately 8 per year), the required number of samples for detecting a trend using regression is increased from 21 quarterly samples to 33 samples. It will take slightly over 4 years to collect the 33 six-week samples, compared with 5.25 years to collect 21 quarterly samples. In the absence of correlation, more frequent sampling requires more sampling events, but the samples are spread over a shorter period of time. If a monthly sampling interval is used (12 per year), the required number of samples for detecting a trend using regression is increased to 43 samples. It will take approximately 3.5 years to collect the 43 monthly samples. It should be noted that monthly sampling is the most likely to encounter autocorrelation, which would increase the required number of samples and number of years above the predictions made by the current simulation model.

Figure 7-14 contains a plot comparing the number of samples and number of years required to have a 95% chance of detection using the regression t-test with the sampling frequencies shown in Table 7-25. The point where the two lines cross is the case highlighted in the table. This plot shows the trade-off encountered when the sampling frequency is increased by a factor of 6 from semiannual to monthly. The required number of samples increases by a factor of approximately 3, while the duration of the sampling is decreased by less than half.

Table 7-25.	Number of Samples and Number of Years Required for 95% Chance of Detection Using Regression or Mann-Kendall Test
                (Case: Slope = 10%/year and Variability = 20%)

                              Sampling Frequency
                                  Regression
                                 Mann-Kendall
                               Number of Samples
                                  2 per year
                                      14
                                      14

                                  4 per year
                                      21*
                                      22*

                                  8 per year
                                      33
                                      34

                                  12 per year
                                      43
                                      44
                                Number of Years
                                  2 per year
                                       7
                                       7

                                  4 per year
                                     5.25*
                                     5.5*

                                  8 per year
                                     4.13
                                     4.25

                                  12 per year
                                     3.58
                                     3.67
           *Case in Table G-2 for slope of 10%/year and variability of 20%.

Figure 7-14.	Plot of Number of Samples and Number of Quarters Required for 95% Chance of Detection Using Regression versus Sampling Frequency
                (Case: Slope = 10%/year and Variability = 20%)

Caveats Concerning Regression Analysis of Trends

The linear regression model provides estimates of the slope of the trend line and its standard error based on least squares. The ratio of these estimates is the t-statistic used in the regression t-test for a significant slope. Many parametric tests like the linear regression t-test are based on the normality assumption or equivalent. These tests perform well when the observation errors are normally distributed. When the observation errors are not normally distributed and include random outliers, the least squares estimates of the slope of the regression line and its standard error may be unduly influenced by these outliers.

When regression analysis is used, it is important to verify the assumptions, including:

   * Normality of residuals
   * Equal variances
   * Independence
   * Sensitivity to outliers

The Mann-Kendall test is a nonparametric test for trends. Both tests have very similar characteristics when the observation errors are normally distributed. This fact is confirmed by the similarity of simulation results for these tests in Table 7-19 through Table 7-24. Unlike the regression t-test, the Mann-Kendall test also is expected to perform well when the errors of observation are not normally distributed.

The two tests also differ in the treatment of nondetect values. The Mann-Kendall test is based on counts of how many observations are higher, lower, or the same as the ones before. When the time series contains nondetect values, the counts are made by treating all nondetects as equal and assuming all other values in the series are greater than the nondetects. The counts may be made without assuming a specific value for the nondetects. Such is not the case with parametric models. The most elementary parametric estimate is to use the arithmetic average as an estimate of the mean. This simple estimate cannot be calculated without assuming values for the nondetects. Estimation using linear regression and least squares is only possible if specific values are used for the nondetects, usually one-half of the detection level.

Despite the advantages of the Mann-Kendall test for detecting trends, the test does not provide an estimate of the magnitude of the slope. Other nonparametric methods are available for estimating the slope, including the Theil-Sen trend line estimator.

The following text was excerpted from EPA QA/G-9S (EPA 2006a):

      1..3.1 Estimating a Trend Using the Slope of the Regression Line

      The classic procedures for assessing linear trends involve regression. Linear regression is a commonly used procedure in which calculations are performed on a data set containing pairs of observations (Xi, Yi), so as to obtain the slope and intercept of a line that best fits the data. For temporal data, the Xi values represent time and the Yi values represent the observations. An estimate of the magnitude of trend can be obtained by performing a regression of the data versus time and using the slope of the regression line as the measure of the strength of the trend.

      Regression procedures are easy to apply. All statistical software packages and spreadsheet programs will calculate the slope and intercept of the best fitting line, as well as the correlation coefficient r (see Section 2.2.4). However, regression entails several limitations and assumptions. First of all, simple linear regression (the most commonly used method) is designed to detect linear relationships between two variables; other types of regression models are generally needed to detect non-linear relationships such as cyclical or non-monotonic trends. Regression is very sensitive to outliers and presents difficulties in handling data below the detection limit, which are commonly encountered in environmental studies. Hypothesis testing for linear regression also relies on two key assumptions: normally distributed errors, and constant variance. It may be difficult or burdensome to verify these assumptions in practice, so the accuracy of the slope estimate may be suspect. Moreover, the analyst must ensure that time plots of the data show no cyclical pattern; outlier tests show no extreme data values; and data validation reports indicate that nearly all the measurements were above detection limits. Due to these drawbacks, linear regression is not recommended as a general tool for estimating and detecting trends, although it may be useful as an informal and quick screening tool for identifying strong linear trends. [Emphasis added.]

The nonparametric Mann-Kendall test for trends is recommended in this document for detecting trends. However, the Mann-Kendall test does not provide an estimate of the magnitude of the trend. Once a trend has been detected using the Mann-Kendall test, an estimate of the magnitude of the trend may be required. In this regard, linear regression may be used to estimate the trend, provided that the assumptions required for linear regression are met. Despite the noted drawbacks to using regression for trend analysis, software for regression is readily available and the technique is widely known. Regression also provides traditional confidence intervals and hypothesis tests for evaluating the significance of a trend. Confidence intervals are also useful for comparing trends of different analytes and different wells.

The assumptions concerning outliers and nondetects may preclude the use of linear regression for estimating the magnitude of the trend. If there are outliers and/or nondetects in the data set, a nonparametric method (the Theil-Sen trend line estimator) may be used to estimate the magnitude of the trend.

Testing Multiple Wells for Trends

The Mann-Kendall test is useful for analyzing the trend in data from a single well. If the data were collected systematically across the site at approximately the same sampling times, the Mann-Kendall test statistics Sk for all wells may be combined to make an overall summary for the entire set of wells. In this approach, the statistics Sk are used as a summary measure of the trend in each well. There must be consistency in the data series across wells to make a determination of trend that is valid across all wells.

A single statement applicable to trends across all wells is valid if the wells exhibit approximately steady trends in the same direction (upward or downward), with roughly comparable slopes. Formal statistical tests for the comparability of the data series across wells and for a common trend are described in EPA QA/G9S (EPA 2006a) in the text below. Both tests are based on the chi-squared distribution. The two tests are designed to be implemented sequentially, first testing for comparability of slopes, then for a significant common trend across wells.

The hypothesis tests described in EPA 2006a are:

      Comparability of stations. H0:  Similar dynamics affect all K stations vs. HA:  At least two stations exhibit different dynamics.

      Testing for overall monotonic trend. H0*:  Contaminant levels do not change over time vs. HA*:  There is an increasing (or decreasing) trend consistent across all stations.

      Therefore, the analyst must first test for homogeneity of stations, and then, if homogeneity is confirmed, test for an overall monotonic trend. Directions for the test are contained in Box 4-11 and ideally, the stations in Box 4-11 should have equal sample sizes. However, the numbers of observations at the stations can differ slightly, because of isolated missing values, but the overall time periods spanned must be similar. This guidance recommends that for less than 3 time periods, an equal number of observations (a balanced design) are required. For 4 or more time periods, up to 1 missing value per sampling location may be tolerated.
Plots of the measurements from all wells using a different symbol for each well are examined to assess the consistency across wells. Examples of these plots are shown in Attachment D. Detailed instructions for performing the Mann-Kendall test for multiple wells are shown in Attachment D in Boxes D-4, D-5 and D-6.

Multiple Observations per Time Period for Multiple Wells

If multiple measurements are taken at various times and stations, then previous approaches, with some modifications, are still applicable. Details are provided in Sections <IP_ADDRESS> and <IP_ADDRESS> of EPA 2006a.

Analysis of Post-restoration Trends at ISR Sites

Trend Analysis by Well

Introduction and Examples

Figure 7-15 shows a time plot of the uranium concentrations measured in Well PR-15 at the Crow Butte ISR site over a period of approximately 3(1/2) years. This plot shows the measured concentrations at 16 different times. This is one of the longest sets of measurements at a restored ISR site available for analysis. The mean uranium concentration is also shown on the graph. Although the mean concentration is useful for comparing the post-restoration samples with the baseline, the mean is not essential nor relevant for evaluating trends.

Figure 7-16 shows a similar plot with modified scales. The vertical axis measures the percent deviation of each sample from the mean. The average of the percent deviations is always at 0. The horizontal axis measures time in the number of years since the first measurement. Comparing the graphs in Figure 7-15 and Figure 7-16, it is clear that the linear transformations used to modify the axes do not affect the trend. But the choice of scales does affect the units for the slope. This choice of scales provides estimates of the slope that are easily interpreted, expressed as a percentage of the mean per year. Using these scales, the magnitude and direction of the trends are comparable across production units, wells, and analytes.

The blue line in Figure 7-16 is the best-fitting linear regression trend line estimated using a statistical procedure known as least squares. The equation of the line is shown on the graph, where y represents the vertical axis, and x represents the horizontal axis which in this case is the time axis. The coefficient of x in this equation gives the slope of the trend line, which is equal to 5.9784 or approximately 6%. On average, the uranium concentration in Well PR-15 is rising at about 6% of the mean per year.

Although the symbol x is used to represent the horizontal axis in the figure, the symbol T will be used for the horizontal time axis in the following discussion. The linear trend equation is expressed as, where yT denotes the percent deviation from the mean value at time T, and T is the number of years from the first measurement. The percent deviation from the mean is calculated as , where XT denotes the sample concentration at time T and  is the average concentration in the well over the period in question. The symbol β is slope of the line. In Figure 7-16, β = 5.9784% of the mean per year. The symbol α is called the y-intercept, which is the value of the trend line at time T=0.

The dashed red lines in Figure 7-16 show the temporal variability of the samples around the trend line. These lines are drawn at plus and minus one standard deviation from the trend line. The positive slope of 6% per year is small relative to the temporal variability of approximately +-60%. The question then arises whether the slope is statistically significant. To test for significance of the slope, it is necessary to compute the standard error of the estimated slope, written as SE(β). Table 7-26 shows the regression results for uranium and radium in well PR-15, the estimates of the slope and intercept, standard errors, R-square, t-statistics and degrees of freedom. The slope estimate β=5.98 is in the upper left corner of the table, and the standard error of the slope estimate SE(β)=10.53 is shown in the row below. The t-statistic used to test for a significant slope is equal to the ratio of the slope to its standard error: . When the absolute value of the t-statistic is larger than the critical value for the regression t-test, the slope estimate is statistically significant at a specified level of confidence. Usually a 95% level of confidence is required. A table of the critical values for the t-test is provided in Table 7-31. The smallest critical value in the table for 14 degrees of freedom is 1.345 for a 90% confidence level. The t-statistic of  uranium is much smaller than the critical value; hence the positive slope observed in Figure 7-16 is not statistically significant. The uranium concentrations are considered stable because there is no significant trend.

Table 7-26.	Regression Statistics for Example in Figure 7-16
Uranium (mg/l)
                                     Slope
                                   Intercept

                                                                  Estimate->
                                                                           5.98
                                                                         -10.07

                                                            Standard Error->
                                                                          10.53
                                                                          22.85

                                                                  R-square->
                                                                           0.02

                                                                             14
<-Degrees of Freedom
                                                               t-statistic->
                                                                           0.57
                                                                          -0.44

Radium (pCi/L)
                                     Slope
                                   Intercept

                                                                  Estimate->
                                                                         -10.06
                                                                          15.27

                                                            Standard Error->
                                                                           5.24
                                                                          11.63

                                                                  R-square->
                                                                           0.32

                                                                              8
<-Degrees of Freedom
                                                               t-statistic->
                                                                          -1.92
                                                                           1.31

           Note: An automated tool, such as the linest() function in Excel, can be used to calculate the regression statistics.
Figure 7-17 shows the radium concentrations in well PR-15 over the same time period. After a sudden initial rise, the trend is down for the remaining time. The estimated slope is negative 10% per year, indicating that radium concentrations are decreasing in this well. The standard error for the slope in Table 7-26 is 5.24 and the t-statistic is -1.92. In this case, the downward trend is statistically significant at the 95% level of confidence.

One advantage of the Bayesian approach is it allows for straightforward probability comparisons of the slope parameters. Figure 7-18 shows a Bayesian interpretation of the slope estimates obtained using non-informative prior distributions for uranium and radium in well PR-15 at Crow Butte. In this case, a non-informative prior distribution means a prior opinion that all values of the slope are possible, both negative and positive, and of any magnitude. The figure shows the posterior t-distributions for the uranium and radium slope parameters. The uranium posterior t-distribution is very broad, extending well below the 0 value. Although the least squares slope estimate for uranium is positive, the estimated slope is not statistically significant because a large portion of the posterior distribution lies below 0. The radium posterior t-distribution is much narrower, and almost all of the distribution lies below 0. The negative slope estimate for radium is statistically significant.

Figure 7-19 shows the complementary cumulative distribution functions (CCDFs) for the posterior distributions in Figure 7-18. These curves show the probability that the slope (expressed as the percent of the mean per year) is greater than the value shown on the horizontal axis. The probability that the trend in radium is greater than 0 is less than 0.05 (5%), while the probability that the trend for uranium is greater than 0 is approximately 0.75 (75%).

Figure 7-20 shows a Bayesian comparison of the trends for radium in wells PR-8 and PR-15 at Crow Butte. Although radium was found to have a significant downward trend in well PR-15, radium has a significant upward trend in well PR-8.

Figure 7-15.	Uranium Concentrations in Crow Butte Well PR-15

Figure 7-16.	Deviation of Uranium Concentration from Mean with Variability Bounds (+-1σ)

Figure 7-17.	Radium Concentrations in Crow Butte Well PR-15

Figure 7-18.	Comparison of Bayesian Posterior Distributions for Regression Slope Parameter for Uranium and Radium in Crow Butte Well PR-15

Figure 7-19.	Comparison of Complementary Cumulative Distribution Functions (CCDF) for Regression Slope Parameter for Uranium and Radium in Crow Butte Well PR-15

Figure 7-20.	Comparison of Bayesian Posterior Distributions for Regression Slope Parameter for Radium in Crow Butte Wells PR-8 and PR-15

Well-by-Well Trend Analysis at Four ISR Sites

The regression trend analysis was conducted for seven selected analytes measured in 72 wells in six restored production units at four selected ISR sites with post-stabilization data. Table 7-27 provides a summary of the trend analysis for the seven analytes selected for the study. The analytes include chloride, iron, pH, radium, selenium, total dissolved solids (TDS), and uranium. The table provides information for six regression statistics: the mean sample concentration; the number of quarters covered by the sampling program; the number of samples collected in each well; the slope of the trend line; the t-statistic for the slope; and the variability around the trend line. These results are summaries of the regressions for the number of wells shown. The table shows the minimum, maximum, mean and standard deviation for each regression statistic.

Additional details of the trend analysis are shown in Tables F-1 and F-2 in Attachment F. Table F-1 shows a summary by analyte, while Table F-2 provides details on the regressions for each analyte in each of the six production units. Table F-3 contains the trend analysis results for each well. This table contains the data that are summarized in Tables 7-21, F-1, and F-2. The table identifies the production unit, analyte, well, starting date, ending date, number of  tests, R-square, variability, intercept, slope, standard error of the slope, the lower and upper bounds of the 95% confidence interval for the slope (LCL and UCL), the t-statistic for the slope, the t-test result, and the Mann-Kendall test result. The test results are recorded as a +1 if there is a significantly positive trend, -1 if there is a significantly negative trend, and 0 if the slope is not significant.

A summary of the estimated mean slope and variability is presented in Table 7-28. Figure 7-21 contains a bar plot of the mean slopes shown in Table 7-28. On average, radium, uranium and iron had positive trends, while chloride and selenium showed decreasing trends. Only chloride and uranium have a 95% confidence interval which does not include 0, indicating that only these two mean slopes are significantly different from 0.

The mean values of the variability in Table 7-28 are plotted in Figure 7-22. Iron and selenium had the highest variability, while chloride, TDS and pH show less variability. Radium and uranium are in the mid-range of variability. Figure 7-23 shows the 95% confidence interval for the mean variability of each analyte. In general, the analytes with high variability also have the widest confidence intervals. The mean variability for iron and selenium ranges from 35% to 55%, while radium and uranium have mean variability of 20% to 35%. Figure 7-24 shows the full range of variability encountered in the regression analysis. Although the confidence intervals for the mean variability are all below 55%, variability as high as 100% to 200% was encountered in individual wells.

Figure 7-25 shows a scatter plot of the standard error of the slope versus the number of samples in the regression. Most of the wells available for analysis have only 3 to 5 samples covering a period of approximately 1 year. The standard error of the slope estimates range over more than three orders of magnitude for these wells. This indicates that well-by-well trend analysis is very difficult with only 4 or 5 samples in the data set. Fewer wells were available with 7 to 10 samples, and the standard error is reduced by an order of magnitude in these wells. Only a few wells at Crow Butte were available with more than 15 samples for uranium and radium. These are the wells discussed in the introduction to this section of the report.
Table 7-27.	Summary of Trend Analysis at Four* ISR Sites
                                   Statistic
                                   Parameter
                                Number of Wells
                                    Minimum
                                    Maximum
                                     Mean
                              Standard Deviation
Mean Concentration
Chloride (mg/l)
                                      63
                                                                           2.00
                                                                          97.00
                                                                          16.47
                                                                          16.99

Iron (mg/l)
                                      61
                                                                           0.00
                                                                         163.00
                                                                           3.58
                                                                          20.76

pH (units)
                                      65
                                                                           7.00
                                                                          14.00
                                                                           7.54
                                                                           0.86

Radium (pCi/l)
                                      69
                                                                          23.00
                                                                        3422.00
                                                                         404.85
                                                                         513.01

Selenium (mg/l)
                                      49
                                                                           0.00
                                                                           0.00
                                                                           0.02
                                                                           0.05

TDS (mg/l)
                                      72
                                                                         280.00
                                                                        1183.00
                                                                         536.30
                                                                         211.42

Uranium (mg/l)
                                      70
                                                                           0.00
                                                                           9.00
                                                                           1.50
                                                                           1.87
Number of Quarters
Chloride (mg/l)
                                      63
                                                                           2.00
                                                                           5.00
                                                                           2.69
                                                                           0.62

Iron (mg/l)
                                      61
                                                                           1.00
                                                                          14.30
                                                                           3.43
                                                                           3.74

pH (units)
                                      65
                                                                           1.00
                                                                           5.00
                                                                           2.39
                                                                           1.02

Radium (pCi/l)
                                      69
                                                                           1.00
                                                                          14.30
                                                                           3.47
                                                                           3.51

Selenium (mg/l)
                                      49
                                                                           0.00
                                                                          14.30
                                                                           3.84
                                                                           4.05

TDS (mg/l)
                                      72
                                                                           1.30
                                                                          14.00
                                                                           3.61
                                                                           3.22

Uranium (mg/l)
                                      70
                                                                           2.00
                                                                          14.30
                                                                           3.69
                                                                           3.33
Number of Tests
Chloride (mg/l)
                                      63
                                                                           3.00
                                                                           5.00
                                                                           4.24
                                                                           0.59

Iron (mg/l)
                                      61
                                                                           3.00
                                                                          10.00
                                                                           4.18
                                                                           2.00

pH (units)
                                      65
                                                                           3.00
                                                                           4.00
                                                                           3.62
                                                                           0.49

Radium (pCi/l)
                                      69
                                                                           3.00
                                                                          10.00
                                                                           4.19
                                                                           1.87

Selenium (mg/l)
                                      49
                                                                           3.00
                                                                          10.00
                                                                           5.06
                                                                           1.96

TDS (mg/l)
                                      72
                                                                           3.00
                                                                           9.00
                                                                           4.64
                                                                           1.44

Uranium (mg/l)
                                      70
                                                                           3.00
                                                                          17.00
                                                                           5.00
                                                                           2.75
Slope of Trendline (%/y)
Chloride (mg/l)
                                      63
                                                                        -139.27
                                                                         315.36
                                                                         -18.85
                                                                          73.25

Iron (mg/l)
                                      61
                                                                        -422.10
                                                                         600.91
                                                                          10.10
                                                                         189.99

pH (units)
                                      65
                                                                         -11.48
                                                                         264.25
                                                                           4.66
                                                                          33.08

Radium (pCi/l)
                                      69
                                                                        -160.01
                                                                         363.03
                                                                          17.50
                                                                          92.69

Selenium (mg/l)
                                      49
                                                                        -447.43
                                                                         654.25
                                                                         -12.04
                                                                         187.09

TDS (mg/l)
                                      72
                                                                        -197.67
                                                                         128.42
                                                                           1.89
                                                                          39.70

Uranium (mg/l)
                                      70
                                                                        -240.54
                                                                         242.26
                                                                          27.39
                                                                         106.59
t-statistic
Chloride (mg/l)
                                      63
                                                                         -24.38
                                                                           7.64
                                                                          -0.93
                                                                           4.19

Iron (mg/l)
                                      61
                                                                          -7.44
                                                                          25.96
                                                                           0.42
                                                                           4.76

pH (units)
                                      65
                                                                          -4.87
                                                                           4.43
                                                                           0.14
                                                                           1.39

Radium (pCi/l)
                                      69
                                                                          -5.18
                                                                          50.07
                                                                           2.31
                                                                           8.54

Selenium (mg/l)
                                      49
                                                                          -4.72
                                                                           4.32
                                                                          -0.34
                                                                           2.19

TDS (mg/l)
                                      72
                                                                          -9.96
                                                                         148.78
                                                                           2.51
                                                                          17.84

Uranium (mg/l)
                                      70
                                                                         -19.10
                                                                          15.85
                                                                           1.05
                                                                           3.99
Variability (%)
Chloride (mg/l)
                                      63
                                                                           1.71
                                                                          74.21
                                                                          15.11
                                                                          11.95

Iron (mg/l)
                                      61
                                                                           3.10
                                                                         211.76
                                                                          44.54
                                                                          40.28

pH (units)
                                      65
                                                                           0.18
                                                                          99.55
                                                                           4.53
                                                                          12.18

Radium (pCi/l)
                                      69
                                                                           0.90
                                                                         137.99
                                                                          24.67
                                                                          26.92

Selenium (mg/l)
                                      49
                                                                           2.50
                                                                         185.67
                                                                          43.97
                                                                          37.08

TDS (mg/l)
                                      72
                                                                           0.07
                                                                          48.07
                                                                           9.53
                                                                           8.83

Uranium (mg/l)
                                      70
                                                                           1.06
                                                                         125.06
                                                                          27.44
                                                                          22.45
*	Christensen, MU2 & MU3 (COGEMA 2008a), Highland A (Kearney 2004) and B (Power Resources 2004)), Crow Butte, MU1 (Crow Butte 2002) and Irigaray, 13 wells in 9 units (Irigaray 2004).
Table 7-28.	Mean Slope and Variability Estimates with 95% Confidence Interval for the Mean (LCL to UCL)
Slope
                                      LCL
                                     Mean
                                      UCL
Chloride (mg/l)
                                                                           -37%
                                                                           -19%
                                                                            -1%
Iron (mg/l)
                                                                           -38%
                                                                            10%
                                                                            58%
pH (units)
                                                                            -3%
                                                                             5%
                                                                            13%
Radium (pCi/l)
                                                                            -4%
                                                                            17%
                                                                            39%
Selenium (mg/l)
                                                                           -64%
                                                                           -12%
                                                                            40%
TDS (mg/l)
                                                                            -7%
                                                                             2%
                                                                            11%
Uranium (mg/l)
                                                                             2%
                                                                            27%
                                                                            52%
Variability
                                      LCL
                                     Mean
                                      UCL
Chloride (mg/l)
                                                                            12%
                                                                            15%
                                                                            18%
Iron (mg/l)
                                                                            34%
                                                                            45%
                                                                            55%
pH (units)
                                                                             2%
                                                                             5%
                                                                             7%
Radium (pCi/l)
                                                                            18%
                                                                            25%
                                                                            31%
Selenium (mg/l)
                                                                            34%
                                                                            44%
                                                                            54%
TDS (mg/l)
                                                                             7%
                                                                            10%
                                                                            12%
Uranium (mg/l)
                                                                            22%
                                                                            27%
                                                                            33%

Figure 7-21.	Slope of Trend Line Averaged over All Wells

Figure 7-22.	Temporal Variability (Averaged Over All Wells)

Figure 7-23.	95% Confidence Interval for Mean Temporal Variability

Figure 7-24.	Full Range of Temporal Variability

Figure 7-25.	Scatter Plot of the Standard Error versus the Number of Samples

Pooled Trend Analysis

The trend analyses in the previous section examined over 400 time series of samples collected from a single well. It is also possible to use regression analysis to determine if samples collected from a number of wells have a significant trend. Figure 7-26 shows a scatter plot of the chloride samples collected from eight wells at Christensen Mine Unit 2. This plot uses the same scales as in the previous section, with the percentage deviation of each sample from the well mean plotted on the vertical axis and time on the horizontal axis. With data "pooled" from eight wells, this plot is an example of a pooled regression trend analysis.

In a pooled analysis, data for a single analyte collected from a set of wells at all times are included in a single regression. The regression model for the pooled trend analysis is , where yi,T denotes the (transformed) sample value from well i at time T. Here T=0 at the time of the earliest sample. As before, each value yi,T is a percentage deviation from the mean for that well, and time is expressed in years from the first sample. The model uses only a single coefficient for the intercept, as the mean of the percentage deviations for each well is zero,

An example of a pooled regression trend line is shown in Figure 7-26. The trend line has a positive slope of 47% per year. Due to the larger number of samples available in pooled regression, the standard error of the slope estimate is greatly reduced. In this example the standard error of the slope is 11% and the t-statistic is 3.9 with 30 degrees of freedom. The t-statistic is larger than the critical value in Table 7-31 for 30 degrees of freedom. This confirms that there is a significant upward trend in the chloride measurements.

The plot in Figure 7-27 shows the same data used in Figure 7-26 plotted as circles with a size proportional to the standard error of the slope estimate obtained in the trend analysis of the eight individual wells. The wells with the smallest circles had more precise estimates of their individual trends than wells with larger circles. The trends for the wells with smaller circles have trends which are near the pooled slope estimate.

Figure 7-28 shows an example of the drawbacks of relying on a linear regression model to determine the trend. This plot shows the TDS samples in the same eight wells. The trend line has a positive slope of 27% with a standard error of 9.9% and a t-statistic of 2.73. Although the plot shows that concentrations increased only in the beginning and have trended down since then, the linear trend analysis concludes that there is a significant positive trend for TDS in these wells. The trend in the last 3 samples is downward. In this case, it may be necessary to redo the analysis using only the last 3 sample sets to establish the most recent trend.

The standard errors of the regressions for individual wells were shown in Figure 7-29 of the previous section. The standard error ranged over several orders of magnitude in the analysis of individual wells. A similar plot of the standard errors for the pooled slope estimates is shown in Figure 7-29. The pooled sample sizes ranged from 15 to 100 depending on the production unit and analyte. The smallest pooled sample size is approximately the same as the largest individual well sample size shown in Figure 7-29. Using a pooled trend analysis, the standard errors are reduced to less than 30% when the pooled sample size is 40 or larger.

Figure 7-30 shows the results of the t-test for a significant slope in the pooled trend analysis. A value of +1 on the vertical axis denotes a significantly positive slope, while a value of -1 represents a significantly negative slope. If there is no significant slope, the value is plotted at 0. This plot shows that the likelihood of finding a significantly positive or negative test result is not related to the pooled sample size.

A count of the significant positive and negative trends in Figure 7-30 is shown in Table 7-29. Of the 40 pooled sets of data analyzed, 10 had a significant positive trend and 6 a significant negative trend. The remaining 24 pooled data sets showed no significant trend. The 16 data sets with significant trends are listed in Table 7-29. The first two rows are for chloride and TDS at Christensen MU2, the data used in Figure 7-26 through Figure 7-28. Each of the seven analytes was found to have a significant trend in at least one production unit. When only positive trends are considered, 6 analytes (all but selenium) showed 1 or more positive trends.

Figure 7-26.	Christensen MU2 Chloride Samples over Time with Trend Line

Figure 7-27.	Christensen MU2 Chloride Samples over Time with Trend Line
(Size of circle is equal to the standard error of the slope estimate for each individual well; hence, samples with smaller circles are from wells with more significant trends.)

Figure 7-28.	Christensen MU2 TDS Samples over Time with Trend Line

Figure 7-29.	Scatter Plot of Standard Error of Slope versus Pooled Number of Samples

Figure 7-30.	Scatter Plot of Pooled t-test Results
(+1 = significant positive trend; -1 = significant negative trend; 0 = no significant trend)
Table 7-29.	Summary of Significant Trends in Pooled Trend Analysis

                                     Count
                                    Percent
Positive Trend
                                                                             10
                                                                            25%
Negative Trend
                                                                              6
                                                                            15%
No Trend
                                                                             24
                                                                            60%
Total
                                                                             40
                                                                           100%

Table 7-30.	Significant Positive and Negative Trends Identified using Pooled Trend Analysis
                                     Unit
                                    Analyte
                                   Direction
                                 Trend (%/yr)
Christensen MU2
Chloride
Positive
                                      43%

TDS
Positive
                                      27%

pH (units)
Positive
                                     3.8%

Iron
Negative
                                     -124%
Christensen MU3
TDS
Positive
                                      20%

Iron
Negative
                                     -60%
Crow Butte MU1
Iron
Positive
                                      20%

Uranium
Positive
                                      13%

TDS
Positive
                                     2.1%
Highland A
Chloride
Negative
                                     -69%
Highland B
Uranium
Positive
                                      91%

Radium
Positive
                                      51%

TDS
Negative
                                     -13%

Chloride
Negative
                                     -57%
Irigaray MU1-9
Iron
Positive
                                     139%

Selenium
Negative
                                     -108%

Verify that Contaminants and Hazardous Constituent Concentrations are Below Required Restoration Levels

This section describes traditional statistical methods for verifying that contaminants and hazardous constituent concentrations are below required restoration levels. These methods involve a comparison of well data collected in the baseline and after restoration is complete. Several types of statistical hypothesis tests are recommended for conducting this comparison.

Although a geostatistical analysis is not required, these methods provide a way to convert the baseline and post-restoration sample data into 2- and 3-D graphical representations of the characteristics of the ore zone. For those operators with an understanding of geostatistical software and analytical procedures, these procedures may provide better insight into the differences between baseline characteristics and post-restoration conditions. Areas showing the greatest differences between baseline and post-restoration conditions may be identified with these procedures.

Geostatistical methods and other models are useful in demonstrating that the potentiometric surface has returned to baseline conditions. This demonstration requires more than a point-by-point analysis. For example, at least a 2-D assessment is required to assess if the direction of the hydraulic gradient has changed. This could be detected in multiple ways, including geostatistical models or ModFlow analysis. Point-by-point comparisons, which do not consider the location of the wells, are not capable of detecting a change of this type.

In this document, the hypothesis testing framework described in Section 7.3 is used to verify that contaminants and hazardous constituent concentrations are below required restoration levels. A hypothesis test is used to compare the post-restoration conditions to the baseline. The comparison may be based on a statistical parameter (e.g., a mean or median) of a probability distribution selected to best represent the population, or it may be a distribution-free comparison of the two populations. With small sample sizes, it is difficult to demonstrate conclusively that a particular distribution represents both populations adequately. Tests that do not assume a known family of probability distributions (e.g., normal or lognormal) to represent the populations are called distribution-free or nonparametric tests. A nonparametric statistical test may be more useful for comparing two populations than one which assumes a specific distribution, because the nonparametric tests are less sensitive to deviations from the assumed distribution.

The threshold value for the statistical test may be zero, in which case, the comparison is used to determine whether the post-restoration well values are less than baseline levels, or the threshold value may be a positive number representing the maximum allowable difference between the two populations. This threshold Δ is defined as a "substantial difference." It is anticipated that Δ will be different for each analyte.

When the baseline and post-restoration samples are not collected from the same wells, the test involves a comparison of two independent populations. Several statistical approaches are presented for comparing an individual well to a baseline consisting of pooled data from many wells. Two parametric statistical methods -- the two-sample Student t-test and PLs -- are designed to test for a significant difference in mean concentrations. One nonparametric test is also presented -- the WRS test.

Parametric Method for Determining Compliance of Individual Wells

Two-sample t-test

The two-sample t-test is a parametric test for a significant difference in the means of two data sets, when it can be assumed that the population variances are approximately equal and the data are approximately normally distributed or the sample sizes are large (at least 30 in both data sets). If this is not the case, then the nonparametric WRS test procedure described below is an alternative.

Limitations and Robustness: The two-sample t-test with equal variances performs well with moderate violations of the assumption of normality, but not with large inequalities of variances. The Welch-Satterthwaite version of the two-sample t-test is an alternative parametric method for use if unequal variances are encountered (see Satterthwaite 1946 and Welch 1947). The t-test is not robust against outliers because sample means and standard deviations are sensitive to outliers. The data should be screened for outliers using the method of Section 7.5.

--------------------------------------------------------------------------------
      Procedure for the Two-Sample t-Test (Equal Variances)
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
      Step 1. Assume there are n compliance samples, X1, ... Xn and N baseline samples Y1, ... YN. Calculate the sample means, MX and MY, and the sample standard deviations, SX and SY of the two populations. Also compute the pooled estimate of the standard deviation using the equation:
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------

--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
      Step 2. Compute the test statistic  where Δ may be 0.
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
      Step 3. Find the critical value of the t-distribution with degrees of freedom n+N-2 and cumulative probability (1−α) for a confidence level of 100(1−α)%, tn+N−2(1−α). The values of Student's t-distribution are shown in Table 7-31.
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
      Step 4. If t0 > tn+N−2(1−α) then reject the null hypothesis that the true difference between population means is less than Δ.

Prediction Limit for a Future Mean

Prediction limits (PLs) are designed to provide an upper bound for the mean of a future sample with a specified probability equal to (1 - α), known as the confidence level of the PL. It represents the chance (over repeated applications of the limit to many similar data sets) that the PL will contain the mean of a future (post-operations) sample from the monitoring wells. The PL for a future mean is similar to the parametric two-sample t-test and the nonparametric WRS test, because the mean of the compliance samples is compared to a limit calculated using the baseline mean. Prediction limits (PLs) assume that the future distribution and the baseline distributions have similar shapes and differ only by a shift in concentration to a higher level. Similar assumptions apply to the other two tests. If the baseline and future distributions have significantly different shapes, then the Welch-Satterthwaite form of the (parametric) two-sample t-test and the Brunner-Munzel generalization of the (nonparametric) WRS test may exhibit better performance.

A set of N baseline samples, pooled over all baseline wells and, if necessary, screened of outliers using the tolerance method described in Section 7.5, is used to construct the PL. The mean of n compliance samples from one or more wells is then compared to the PL to determine compliance. The PL has the same mathematical form as a tolerance limit, using the mean M and the standard deviation S of N baseline samples:
                                  if Δ=0 or
                                   if Δ>0

Here κN,n(1 - α) is a multiplier that depends on the number of baseline samples N, the number of compliance samples n, and the desired level of confidence (1 - α).
The PL on a future mean assumes that the baseline data used to construct the limit are either normally distributed or can be normalized by a transformation. If a transformation is used (e.g., the natural logarithm) and the PL is built on the transformed values, the PL should not be back-transformed before comparing it to the compliance point data. Rather, the compliance samples should first be transformed, and the future mean computed from the transformed compliance measurements. Then the mean of the transformed values (e.g., log-mean) should be compared to the PL.

--------------------------------------------------------------------------------
      Procedure for Calculating a Prediction Limit
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--------------------------------------------------------------------------------
      Step 1. Calculate the sample mean M and the standard deviation S from the set of N baseline samples.
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
      Step 2. If the background data are approximately normal, calculate the PL using the equation
--------------------------------------------------------------------------------
       where Δ may be 0.
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      The Student's t value used in the equations has degrees of freedom (N - 1) and the cumulative probability (1 - α) for a confidence level of 100(1 - α)%. The values of Student's distribution for selected values of N - 1 are shown in Table 7-31.
--------------------------------------------------------------------------------

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      Step 3. Using the PL computed in Step 2, compare the mean of the compliance samples against the PL. If the future mean is below the PL, then the null hypothesis of no significant difference is accepted and compliance is indicated. If the future mean exceeds the PL, there is statistically significant evidence of an increase in concentration over baseline levels.

Table 7-31.	Critical Values of the Student's t-Distribution
                                    Degrees
                                      of
                                 Freedom (N-1)
                         Confidence Level =100(1-α)%

                                                                            90%
                                                                            95%
                                                                          97.5%
                                                                            99%
                                                                          99.5%
                                                                          99.9%

                 Probability of Exceeding Critical Value (α)

                                                                            0.1
                                                                           0.05
                                                                          0.025
                                                                           0.01
                                                                          0.005
                                                                          0.001
                                       1
                                                                          3.078
                                                                          6.314
                                                                         12.706
                                                                         31.821
                                                                         63.657
                                                                        318.313
                                       2
                                                                          1.886
                                                                          2.920
                                                                          4.303
                                                                          6.965
                                                                          9.925
                                                                         22.327
                                       3
                                                                          1.638
                                                                          2.353
                                                                          3.182
                                                                          4.541
                                                                          5.841
                                                                         10.215
                                       4
                                                                          1.533
                                                                          2.132
                                                                          2.776
                                                                          3.747
                                                                          4.604
                                                                          7.173
                                       5
                                                                          1.476
                                                                          2.015
                                                                          2.571
                                                                          3.365
                                                                          4.032
                                                                          5.893
                                       6
                                                                          1.440
                                                                          1.943
                                                                          2.447
                                                                          3.143
                                                                          3.707
                                                                          5.208
                                       7
                                                                          1.415
                                                                          1.895
                                                                          2.365
                                                                          2.998
                                                                          3.499
                                                                          4.782
                                       8
                                                                          1.397
                                                                          1.860
                                                                          2.306
                                                                          2.896
                                                                          3.355
                                                                          4.499
                                       9
                                                                          1.383
                                                                          1.833
                                                                          2.262
                                                                          2.821
                                                                          3.250
                                                                          4.296
                                      10
                                                                          1.372
                                                                          1.812
                                                                          2.228
                                                                          2.764
                                                                          3.169
                                                                          4.143
                                      11
                                                                          1.363
                                                                          1.796
                                                                          2.201
                                                                          2.718
                                                                          3.106
                                                                          4.024
                                      12
                                                                          1.356
                                                                          1.782
                                                                          2.179
                                                                          2.681
                                                                          3.055
                                                                          3.929
                                      13
                                                                          1.350
                                                                          1.771
                                                                          2.160
                                                                          2.650
                                                                          3.012
                                                                          3.852
                                      14
                                                                          1.345
                                                                          1.761
                                                                          2.145
                                                                          2.624
                                                                          2.977
                                                                          3.787
                                      15
                                                                          1.341
                                                                          1.753
                                                                          2.131
                                                                          2.602
                                                                          2.947
                                                                          3.733
                                      16
                                                                          1.337
                                                                          1.746
                                                                          2.120
                                                                          2.583
                                                                          2.921
                                                                          3.686
                                      17
                                                                          1.333
                                                                          1.740
                                                                          2.110
                                                                          2.567
                                                                          2.898
                                                                          3.646
                                      18
                                                                          1.330
                                                                          1.734
                                                                          2.101
                                                                          2.552
                                                                          2.878
                                                                          3.610
                                      19
                                                                          1.328
                                                                          1.729
                                                                          2.093
                                                                          2.539
                                                                          2.861
                                                                          3.579
                                      20
                                                                          1.325
                                                                          1.725
                                                                          2.086
                                                                          2.528
                                                                          2.845
                                                                          3.552
                                      21
                                                                          1.323
                                                                          1.721
                                                                          2.080
                                                                          2.518
                                                                          2.831
                                                                          3.527
                                      22
                                                                          1.321
                                                                          1.717
                                                                          2.074
                                                                          2.508
                                                                          2.819
                                                                          3.505
                                      23
                                                                          1.319
                                                                          1.714
                                                                          2.069
                                                                          2.500
                                                                          2.807
                                                                          3.485
                                      24
                                                                          1.318
                                                                          1.711
                                                                          2.064
                                                                          2.492
                                                                          2.797
                                                                          3.467
                                      25
                                                                          1.316
                                                                          1.708
                                                                          2.060
                                                                          2.485
                                                                          2.787
                                                                          3.450
                                      26
                                                                          1.315
                                                                          1.706
                                                                          2.056
                                                                          2.479
                                                                          2.779
                                                                          3.435
                                      27
                                                                          1.314
                                                                          1.703
                                                                          2.052
                                                                          2.473
                                                                          2.771
                                                                          3.421
                                      28
                                                                          1.313
                                                                          1.701
                                                                          2.048
                                                                          2.467
                                                                          2.763
                                                                          3.408
                                      29
                                                                          1.311
                                                                          1.699
                                                                          2.045
                                                                          2.462
                                                                          2.756
                                                                          3.396
                                      30
                                                                          1.310
                                                                          1.697
                                                                          2.042
                                                                          2.457
                                                                          2.750
                                                                          3.385
                                      31
                                                                          1.309
                                                                          1.696
                                                                          2.040
                                                                          2.453
                                                                          2.744
                                                                          3.375
                                      32
                                                                          1.309
                                                                          1.694
                                                                          2.037
                                                                          2.449
                                                                          2.738
                                                                          3.365
                                      33
                                                                          1.308
                                                                          1.692
                                                                          2.035
                                                                          2.445
                                                                          2.733
                                                                          3.356
                                      34
                                                                          1.307
                                                                          1.691
                                                                          2.032
                                                                          2.441
                                                                          2.728
                                                                          3.348
                                      35
                                                                          1.306
                                                                          1.690
                                                                          2.030
                                                                          2.438
                                                                          2.724
                                                                          3.340
                                      36
                                                                          1.306
                                                                          1.688
                                                                          2.028
                                                                          2.434
                                                                          2.719
                                                                          3.333
                                      37
                                                                          1.305
                                                                          1.687
                                                                          2.026
                                                                          2.431
                                                                          2.715