Encyclopedia of Environmental Science and Engineering, Volume I and II

EPIDEMIOLOGY 383

among the workers exposed to the putative causative factor,

whether that is a substance or a process that is used in certain

parts of a factory. In its most elementary form, the results of

an investigation can be put in the form of a 2  2 table, as

in Figure 10.

The fi rst two cells of this table, horizontally, include the

numbers of those who were exposed (
) to the presumed

hazard, the fi rst cell containing the number of those who devel-

oped the disease in question (a), the second those who did

not (b). In the lower line are those who were not exposed (),

the fi rst cell again including all those for this group who

developed the disease (c) and the second those who did

not (d). Clearly if the ratio of the left to the right is the same

in both rows, there is no evidence of an effect. This is an

example of the 2  2 or fourfold table to which the X^2 test

can easily be applied, with one degree of freedom, to assess

whether any difference in the proportion, horizontally or

vertically, attains statistical signifi cance, at whatever level

may be chosen. The conventional levels of signifi cance are

0.05 (5%), 0.01 (1%), and 0.001 (0.1%), each referring to the

probabilities that the observed result could have occurred by

chance alone (the levels are sometimes quoted in the form

of percentages, multiplying their probabilities by 100). For

each cell of the table of Figure 10 an “expected” fi gure can

be calculated from the marginal and grand totals, by divid-

ing, for instance, each row total in the proportions of the

column totals: thus the expectation for the top left cell is the

product of the fi rst row’s total divided by the grand total.

The difference between the observed number in each cell

is squared and divided by the expectation for that cell, and

the sum of these four quantities constitute X^2. Tables pro-

vided in almost all books on statistics will enable the level

of signifi cance to be obtained for the value of X^2 and for one

degree of freedom. Two caveats should be noted: fi rst, that

no expectation should be less than 5—if it is, a larger size

of sample is required—and second, that when numbers are

small (yet satisfy the proceeding conditions), Yates’s cor-

rection should be made, which reduces the absolute size of

the difference between observed and expected by the quan-

tity 1/2. It will have been noted that this difference is the

same magnitude in each of the four cells, though it changes

sign, but that is irrelevant since it is squared. It is the abso-

lute magnitude of this common difference that should be

reduced by 1/2.

ODDS RATIO AND RELATIVE RISK

In the circumstances set out above and in Figure 10, the ratio

c /( c 
 d ) is the risk of disease in the unexposed group, which

we can call P 0 , and P 1 / P 0 is known as the “relative risk,” RR

or r. In many cases the disease in question will be rare, even

among the exposed, so that a and c will be small relative to

b and d. If Q 0 
 1  P 0 and Q 1 
 1  P 1 express the risk

of not contracting the disease, then they will both be close

to 1, since P 0 and P 1 are supposed to be small. The quantity

( P 1 / Q 1 )/( P 0 / Q 0 ) is known as the “odds ratio,” since it is the

ratio of the odds of occurrence of the disease in the exposed

to the unexposed groups. Since we are presuming the Q ’s to

be close to 1, the odds ratio can be put as P 1 / P 0 , which is the

same as the relative risk, r. We shall see later that this fact

permits the estimation of the ratio of the incidence of disease

in the exposed and unexposed groups from a case-control

type of investigation, though their absolute incidences are

not obtainable.

ETIOLOGICAL STUDIES

A situation that is formally very similar to what we have just

been considering arises if we suspect a certain factor may be

one that is involved in the etiology of the disease. We shall

again be comparing persons with and without the disease

and those affected in the form of a 2  2 table like Figure

10, where we replace “Factor” for “Exposure.” If the factor

is indeed an etiological one, it will be found more frequently

Smoking

Asbestos

Relative risk

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

• 
  


• 
  
  
  
  • 
    
  
  
  
  
  
  
  • –
  
  
  
  

+

+

+

FIGURE 9 Cancer of the lung in relation to asbestos and smoking.

Disease

Exposure

+ –

+ ab

c d

FIGURE 10

C005_011_r03.indd 383C005_011_r03.indd 383 11/18/2005 10:25:43 AM11/18/2005 10:25:43 AM