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Kansas Geological Survey, Open-File Rept. 93-1B
Statistical Methods for Delineating Water Quality--Page 2 of 5

Statistical Methods

In this section we summarize the various tests, specify the criteria for the tests, and explain the hypothesis to be tested. The application of the tests to the Dakota water-chemistry data are presented in the "Comparison of Methods" section. Nonparametric tests were used when the parametric test assumption of a normal distribution was violated.

Kolmogorov-Smirnov Test

The nonparametric Kolmogorov-Smirnov test for normality was used on the original data, a log transform of the data, and a transformation by conversion of the original analyses from milligrams/liter (mg/L) to millequivalents/liter (meq/L) to percentage of total anions and total cations. Neither the transformed nor the untransformed data showed a normal distribution. Based on this finding, nonparametric tests were used to evaluate trends in the data, and factor analysis and discriminant analysis were used to classify and clarify the data, not to determine statistical significance.

Kendall's tau and Spearman's rho

The nonparametric rank-order tests of Kendall's tau and Spearman's rho were used to measure the degree of correlation between chloride concentration and depth. Both methods test the degree of independence between random variables. The hypotheses for a one-tailed test of positive correlation between variables are:

H0: The Xi and Yi are mutually independent; that is, there is no correlation.
H1: There is a tendency for the larger values of X and Y to be paired together.

Discriminant Analysis

Discriminant analysis is a commonly used method for differentiating among constituent sources for waters and for determining hydrologic boundaries that might affect the water chemistry (Riley et al., 1990; Williams, 1982; Davis, 1986). The method is useful as a multivariate statistic for placing objects into predefined groups. In using the discriminant analysis test, one does not need to assume that the data are multivariate normal unless the significance of the classification is needed. For this study the test was used as a classification tool and the results are presented for comparison with other water-typing methods.

Factor Analysis

Factor analysis is frequently used to reduce multiple variables for a single location to several factors that help to explain the relationship among the variables. The objective of factor analysis is to represent the relationships between a number of variables in terms of a smaller number of underlying variables or factors (Harbaugh and Demirmen, 1964). The method resolves the chemistry data into two dominant factors. Although in our tests the factors were not obviously useful for explaining the data statistically, plots of the factor loadings helped to support the findings of both the discriminant analysis and the Piper diagram classifications.

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Kansas Geological Survey, Dakota Aquifer Program
Original report available from the Kansas Geological Survey.
Electronic version placed online Nov. 1998
Scientific comments to P. Allen Macfarlane
Web comments to webadmin@kgs.ku.edu