Page 163 - Statistics for Environmental Engineers

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19

Assessing the Difference of Proportions

KEY WORDS bioassay, binomial distribution, binomial model, censored data, effluent testing, normal
distribution, normal approximation, percentages, proportions, ratio, toxicity, t-test.

Ratios and proportions arise in biological, epidemiological, and public health studies. We may want to
study the proportion of people infected at a given dose of virus, the proportion of rats showing tumors
after exposure to a carcinogen, the incidence rate of leukemia near a contaminated well, or the proportion
of fish affected in bioassay tests on effluents. Engineers would study such problems only with help from
specialists, but they still need to understand the issues and some of the relevant statistical methods.
A situation where engineers will use ratios and proportions is when samples have been censored by
a limit of detection. A data set on an up-gradient groundwater monitoring well has 90% of all observations
censored and a down-gradient well has only 75% censored. Does this difference indicate that contami-
nation has occurred in the groundwater flowing between the two wells?

Case Study

Biological assays are a means of determining the toxicity of an effluent. There are many ways such tests
might be organized: species of test organism, number of test organisms, how many dilutions of effluent
to test, specification of response, physical conditions, etc. Most of these are biological issues. Here we
consider some statistical issues in a simple bioassay.
Organisms will be put into (1) an aquarium containing effluent or (2) a control aquarium containing
clean water. Equal numbers of organisms are assigned randomly to the control and effluent groups. The
experimental response is a binary measure: presence or absence of some characteristic. In an acute
bioassay, the binary characteristic is survival or death of the organism. In a chronic bioassay, the organisms
are exposed to nonlethal conditions and the measured response might be loss of equilibrium, breathing
rate, loss of reproductive capacity, rate of weight gain, formation of neoplasms, etc.
In our example, 80 organisms (n 1 = n 2 = 80) were exposed to each treatment condition (control and
effluent) and toxicity was measured in terms of survival. The data shown in Table 19.1 were observed.
Are the survival proportions in the two groups so different that we can state with a high degree of
confidence that the two treatments truly differ in toxicity?

The Binomial Model

The data from a binomial process consist of two discrete outcomes (binary). A test organism is either
dead or alive after a given period of time. An effluent is either in compliance or it is not. In a given
year, a river floods or it does not flood. The binomial probability distribution gives the probability of
observing an event x times in a set of n trials (experiment). If the event is observed, the trial is said to
be successful. Success in this statistical sense does not mean that the outcome is desirable. A success
may be the death of an organism, failure of a machine, or violation of a regulation. It means success in

© 2002 By CRC Press LLC