Bivariate data

ERDC TN-DOER-C15

July 2000

Parameters of interest in a CDF are likely to include percent sand, percent clay, and

contaminant concentrations. The spatial distribution of each of these parameters can be

examined individually, but by looking at the relationships between these parameters, it is

likely that much can be determined about the material distribution within the CDF using

physical parameters and more limited, targeted, chemical analysis. Bivariate data analysis

methods provide the means to do this.

Bivariate data Bivariate data analysis methods permit the comparison of two parameter

distributions to determine whether a functional relationship exists between them (Isaaks and

Srivastava 1989). Likely to be of interest in determining the distribution of recoverable

materials in a CDF is the relationship of percent sand and percent clay to contaminant levels.

Summary statistics and tests for normality should be calculated for each distribution individu-

ally. A relative location map can be employed, as for the univariate data, giving the values

of each parameter as a function of spatial distribution. A scatter plot of the two parameters,

one plotted on the ordinate and the other on the abscissa, may illustrate any functional

dependence that exists. The linearity of the relationship of the variables can be evaluated

using the correlation coefficient ρ, defined in Appendix I. The correlation coefficient varies

between -1 and +1; +1 indicates a straight line with a positive slope (positive correlation), -1

indicates a straight line with a negative slope (negative correlation), and values near zero

indicate little or no correlation between the variables (Isaaks and Srivastava 1989). For

example, one would expect particle size and contaminant concentration to be negatively

correlated and percent clay and contaminant concentration to be positively correlated,

contaminant level decreasing with increasing particle size. If the correlation coefficient is

unduly influenced by a few extreme values, the rank correlation coefficient may be a more

useful statistic. This is further described in Isaaks and Srivastava (1989).

Censored data In environmental sampling, a high percentage of samples may have no

measurable contaminants (nondetects). Concentrations of these analytes, known as censored

values, are normally reported as less than the method detection level (<MDL). The actual

concentration of the contaminant lies somewhere in the range from zero to the MDL. There

are several approaches to handling censored values. One approach is to ignore these values,

which results in an overestimate of the mean and underestimate of the standard deviation

(McBean and Rovers 1998). This alternative is acceptable only when the number of

nondetects is very small. Alternatively, the censored values can be assumed to be equal to

the detection limit, but this also introduces bias into the summary statistics. This alternative

is preferred when the values are not highly variable and are near the MDL. A third alternative

is to assume the censored values to be equal to MDL/2; this is the preferred alternative when

the contaminant is present in highly variable concentrations. There are a number of statistical

methods, parametric and nonparametric, for dealing with censored data; these are further

described in McBean and Rovers (1998).

Spatial analysis Several variations of data groupings are possible based on the relative

location map previously described. It may be visually instructive to identify the lowest and

highest values on the map, or to replace individual data points with symbols based on

assignment to certain ranges. An indicator map uses only two symbols, designating those

data points falling above and below a specified threshold (Isaaks and Srivastava 1989). The