Encyclopedia of Environmental Science and Engineering, Volume I and II

1130 STATISTICAL METHODS FOR ENVIRONMENTAL SCIENCE

becomes cumbersome for problems of any complexity, and

a number of computer programs are available for analyzing

various designs. The Biomedical Statistical Programs (Ed. by

Dixon 1967) are frequently used for this purpose. A method

recently developed by Fowlkes (1969) permits a particularly

simple specification of the design problem and has the flex-

ibility to handle a wide variety of experimental designs.

SPECIAL ESTIMATION PROBLEMS

The estimation problems we have considered so far have

involved single experiments, or sets of data. In environmen-

tal work, the problem of arriving at an estimate by combin-

ing the results of a series of tests often arises. Consider, for

example, the problem of estimating the coliform bacteria

population size in a specimen of water from a series of dilu-

tion tests. Samples from the water specimen are diluted by

known amounts. At some point, the dilution becomes so

great that the lactose broth brilliant green bile test for the

presence of coliform bacteria becomes negative (Fair and

Geyer, 1954). From the amount of dilution necessary to

obtain a negative test, plus the assumption that one organism

is enough to yield a positive response, it is possible to esti-

mate the original population size in the water specimen.

In making such an estimate, it is unsatisfactory simply

to use the first negative test to estimate the population size.

Since the diluted samples may differ from one another, it is

possible to get a negative test followed by one or more posi-

tive tests. It is desirable, rather, to estimate the population

from the entire series of tests. This can be done by setting

up a combined hypothesis based on the joint probabilities of

all the obtained results, and using likelihood estimation pro-

cedures to arrive at the most likely value for the population

parameter, which is known as the Most Probable Number

(MPN) (Fair and Geyer, 1954). Tables have been prepared

for estimating the MPN for such tests on this principle, and

similar procedures can be used to arrive at the results of a set

of tests in other situations.

Sequential testing is a problem that sometimes arises in

environmental work. So far, we have assumed that a con-

stant amount of data is available. However, very often, the

experimenter is making a series of tests, and wishes to know

whether he has enough data to make a decision at a given

level of reliability, or whether he should consider taking

additional data. Such estimation problems are common in

quality control, for example, and may arise in connection

with monitoring the effluent from various industrial pro-

cesses. Statistical procedures have been developed to deal

with such questions. They are discussed in Wald.

CORRELATION AND RELATED TOPICS

So far we have discussed situations involving a single vari-

able. However, it is common to have more than one type

of measure available on the experimental units. The sim-

plest case arises where values for two variables have been

obtained, and the experimenter wishes to know how these

variables relate to one another.

Curve Fitting

One problem which frequently arises in environmental work

is the fitting of various functions to bivariate data. The sim-

plest situation involves fitting a linear function to the data

when all of the variability is assumed to be in the Y variable.

The most commonly used criterion for fitting such a function

is the minimization of the squared deviations from the line,

referred to as the least squares criterion. The application of

this criterion yields the following simultaneous equations:

YnA Xi

i

n

i

i

n


   



 11

∑∑

(22)

and

XYii A X B X

i

n

i

i

n

i

i

n


   




 11

2

1

∑∑∑.

(22)

These equations can be solved for A and B, the intercept and

slope of the best fit line. More complicated functions may

also be fitted, using the least squares criterion, and it may be

generalized to the case of more than two variables. Discussion

of these procedures may be found in Daniel and Wood.

Correlation and Regression

Another method of analysis often applied to such data is

that of correlation. Suppose that our two variables are both

normally distributed. In addition to investigating their indi-

vidual distributions, we may wish to consider their joint

occurrence. In this situation, we may choose to compute the

Pearson product moment correlation between the two vari-

ables, which is given by

r

xy

xy

ii

xy

cov( )

ss

(23)

where cov( x i y i ) the covariance of x and y, is defined as

()()

.

xy

n

ixiy

i

n 

mm

1

∑

(24)

It is the most common measure of correlation. The square

of r gives the proportion of the variance associated with one

of the variables which can be predicted from knowledge of

the other variables. This correlation coefficient is appropri-

ate whenever the assumption of a normal distribution can be

made for both variables.

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