Regression Equation Development

The SRWS used in the Inorganic Blind Sample Project are analyzed through a round-robin evaluation program described by Farrar and Long (1997). A statistical summary of the round-robin results is prepared for each set of samples. Until 1988, the SRWS project used means and standard deviations in the summary reports. After 1988, the SRWS project used median and F-pseudosigma in the summary reports as a measure of location and variability. The mean or median data reported in the SRWS summaries have been referred to as most probable values (MPV's). The F-pseudosigma (Fs) is defined by Hoaglin and others (1983) as:

Equation 1

where, the Fourth-spread is analogous to the interquartile range of the data. In addition, they show that the F-pseudosigma yields an unbiased estimate of standard deviation when the data distribution is Gaussian. However, it is recognized that some of the SRWS project data sets may not have Gaussian data distributions, the regression equations will be derived from a combination of parametric and nonparametric results.

The SRWS project MPV's are used to estimate the MPV's of the Inorganic Blind Sample Project mixes based on the proportion of the SRWS used. Likewise, if deionized water is used in the preparation of a blind sample mix, then a sample concentration of zero is applied to estimate the resultant MPV based on the proportion used. The resulting MPV's are used in the Inorganic Blind Sample Project to compare with analytical results.

For each constituent, a regression equation was used to estimate the F-pseudosigma (Fs), over a continuous range of concentrations. The equations were derived by regressing the F-pseudosigma or standard deviation against the MPV using ordinary least-squares. Helsel and Hirsch (1992) provide a general model for estimation of ordinary least-squares regression,

Equation 2

whereFigure 4 = the ith observation of the response variable, Fs,
MPVi = the ith observation of the explanatory variable, MPV,
ß0 = the intercept,
ß1 = the slope,
ei = the random error or residual for the ith observation, and
n = the sample size.

SRWS project summary data for semi-annual round-robin sample studies conducted during the last 7 years were used to derive regression equations for each constituent. The table below presents the concentration range of SRWS samples used, the derived Fs equation, and the correlation.

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WY Constituent SRWS Source Slope = b1 Y-Inter = bo r-Squared Range
2008 Alkalinity Maj/Trc 0.0173 1.56 0.805 0.1 - 180
2008 Aluminum Maj/Trc 0.0871 3.60 0.475 10.3 - 70.6
2008 Antimony Maj/Trc 0.0556 0.546 0.521 3 - 33
2008 Arsenic Maj/Trc 0.0561 0.418 0.800 0.2 - 33
2008 Barium Maj/Trc 0.0417 0.411 0.932 10 - 185
2008 Beryllium Maj/Trc 0.0490 0.272 0.500 8 - 26
2008 Boron Maj/Trc 0.0514 2.92 0.702 10 - 280
2008 Cadmium Maj/Trc 0.0596 0.116 0.897 2 - 27
2008 Calcium Maj/Trc 0.0429 0 n/a 0.1 - 75
2008 Chloride Maj/Trc 0.0272 0.4245 0.874 0.1 - 142
2008 Chromium Maj/Trc 0.048 0.493 0.832 7 - 49
2008 Cobalt Maj/Trc 0.0413 0.397 0.434 1 - 27
2008 Copper Maj/Trc 0.0434 0.731 0.682 1 - 38
2008 ROE (DSRD 180) Maj/Trc 0 12.64 n/a 50 - 750
2008 Fluoride Maj/Trc 0.0582 0.0178 0.913 0.02 - 2.2
2008 Iron Maj/Trc 0 6.46 n/a 5 - 100
2008 Lead Maj/Trc 0.0533 0.514 0.808 4 - 46
2008 Lithium Maj/Trc 0.0562 0.757 0.809 1 - 54
2008 Magnesium Maj/Trc 0.0399 0 n/a 0.1 - 26
2008 Manganese Maj/Trc 0.0408 0.611 0.902 1 - 98
2008 Molybdenum Maj/Trc 0.0423 0.918 0.470 1 - 78
2008 Nickel Maj/Trc 0.0483 0.781 0.567 1 - 33
2008 Potassium Maj/Trc 0.0547 0.035 0.907 0.1 - 10.2
2008 Selenium Maj/Trc 0.0828 0.5403 0.654 2 - 10.2
2008 Silica Maj/Trc 0.0605 0 n/a 1 - 24
2008 Silver Maj/Trc 0.0518 0.412 0.725 2 - 20
2008 Sodium Maj/Trc 0.0423 0 n/a 0.1 - 154
2008 Strontium Maj/Trc 0.043 0 n/a 10 - 680
2008 Sulfate Maj/Trc 0.0361 0.358 0.971 0.1 - 240
2008 Thallium Maj/Trc 0.0731 0.511 0.751 3 - 51
2008 Uranium Maj/Trc 0.097 -0.029 0.389 1.1 - 4
2008 Vanadium Maj/Trc 0 1.633 n/a 3 - 35
2008 Zinc Maj/Trc 0.0494 1.27 0.767 3 -73
2008 Mercury Mercury 0.08 0 n/a 0.015 - 0.100
2008 Chloride Precip 0 0.1873 n/a 0.2 - 3
2008 Fluoride Precip 0 0.0246 n/a 0.025 - 0.35
2008 Potassium Precip 0.0664 0.0153 0.760 0.10 - 1.0
2008 Sulfate Precip 0 0.2473 n/a 0.3 - 3.0
2008 Ammonia Nutrient 0.0512 0.0102 0.901 0.04 - 1.4
2008 NH3 + Org N Nutrient 0.0667 0.0596 0.539 0.05 - 2.5
2008 NO2 + NO3 Nutrient 0.0554 0 n/a 0.036 - 1.75
2008 NO2 + NO3 Nut (Low Level) 0.0583 0 n/a 0.036 - 1.0
2008 Orthophosphate Nutrient 0.0467 0 n/a 0.038 - 1.6
2008 Orthophosphate Nut (Low Level) 0 0.00678 n/a 0.038 - 0.2
2008 Phosphorus Nutrient 0.0367 0.00434 0.939 0.04 - 1.6
2008 Phosphorus Nut (Low Level) 0.0689 0 n/a 0.04 - 0.2
 

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