Encyclopedia of Environmental Science and Engineering, Volume I and II

1128 STATISTICAL METHODS FOR ENVIRONMENTAL SCIENCE

In situations where a two-way categorization of the data

exists, the expected values may be estimated from the mar-

ginals. For example, the formula for chi-square for the four-

fold contingency table shown below is

Classification II

Classification I A B

CD

x^2

2

2

NADBC





N

ABCD

⎛

⎝⎜

⎞

⎠⎟

⋅⋅⋅

.

(17)

Observe that instead of having independent expected values,

we are now estimating these parameters from the marginal

distributions of the data. The result is a loss in the degrees

of freedom for the estimate. A chi-square with four indepen-

dently obtained expected values would have four degrees of

freedom; the fourfold table above has only one. The con-

cept of degrees of freedom is a very general one in statistical

analysis. It is related to the number of observations which can

vary independently of each other. When expected values for

chi-square are computed from the marginals, not all of the

O  E differences in a row or column are independent, for their

discrepancies must sum to zero. Calculation of means from

sample data imposes a similar restriction; since the deviations

from the mean must sum to zero, not all of the observations in

the sample can be regarded as freely varying. It is important to

have the correct number of degrees of freedom for an estimate

in order to determine the proper level of significance; many

statistical tables require this information explicitly, and it is

implicit in any comparison. Calculation of the proper degrees

of freedom for a comparison can become complicated in spe-

cific cases, especially that of analysis of variance. The basic

principle to remember, however, is that any linear independent

constraints placed on the data will reduce the degrees of free-

dom. Tables for value of the x 2 distribution for various degrees

of freedom are readily available. For a further discussion of

the use of chi-square, see Snedecor.

Difference between Two Samples

Another common situation arises when two samples are

taken, and the experimenter wishes to know whether or not

they are samples from populations with the same parameter

values. If the populations can be presumed to be normal,

then the significance of the differences of the two means can

be tested by

t

s

N

s

N

mmˆˆ 12

1

2

1

2

2

2

−

(18)

where m^ 1 and m^ 2 are the sample means, s^21 and s^21 are the

sample variances, N 1 and N 2 are the sample sizes. and the

population variances are assumed to be equal. This is the

t -test, for two samples. The t -test can also be used to test the

significance of the difference between one sample mean and

a theoretical value. Tables for the significance of the t -test

may be found in most statistical texts.

The theory underlying the t -test is that the measures of

dispersion estimated from the observations within a sample

provide estimates of the expected variability. If the means are

close together, relative to that variability, then it is unlikely

that the populations differ in their true values. However, if

the means vary widely, then it is unlikely that the samples

come from distributions with the same underlying distribu-

tions. This situation is diagrammed in Figure 6. The t -test

permits an exact statement of how unlikely the null hypoth-

esis (assumption of no difference) is. If it is sufficiently

unlikely, it can be rejected. It is common to assume the null

hypothesis unless it can be rejected in at least 95% of the

cases, though more stringent criteria (99% or more) may be

adopted if more certainty is needed.

The more stringent the criterion, of course, the more likely

it is that the null hypothesis will be accepted when, in fact, it

is false. The probability of falsely rejecting the null hypoth-

esis is known as a type I error. Accepting the null hypothesis

when it should be rejected is known as a type II error. For a

given type I error, the probability of correctly rejecting the

null hypothesis for a given true difference is known as the

power of the test for detecting the difference. The function of

these probabilities for various true differences in the param-

eter under test is known as the power function of the test.

Statistical tests differ in their power and power functions are

useful in the comparison of different tests.

Note that type I and type II errors are necessarily related;

for an experiment of a given level of precision, decreasing

the probability of a type I error raises the probability of a

type II error, and vice versa. Thus, increasing the stringency

of one’s criterion does not decrease the overall probability

of an erroneous conclusion; it merely changes the type of

error which is most likely to be made. To decrease the over-

all error, the experiment must be made more precise, either

by increasing the number of observations, or by reducing the

error in the individual observations.

Many other tests of mean difference exist besides

the t-test. The appropriate choice of a test will depend on

the assumptions made about the distribution underlying the

observations. In theory, the t-test applies only for variables

which are continuous, range from ± infinity in value, and

X (σ UNITS)

f(X)

m 1 m 2 m 3

FIGURE 6

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