80
Size of Experiments for Testing Sera or Vaccines
direction has to be as large or larger than 1.645 times the standard error. The
exact probability corresponding to any ratio of the observed departure to its
standard error is readily available from text books on statistics. In a sample
of 12 with none catching infection the departure is clearly 3 ; the ratio of the
departure to its standard error is 3/1.5 or 2 and the probability will be
found to be 0.023 as against the true value of 0.032 deduced earlier. A
sample of 24 giving a departure of four from the expected infected number
will be similarly found to give the probability of 0.030 against the value of
0.042 obtained earlier from first principles.
The difference in the values of the probability found by the two methods
is obviously due to the use of the approximation in place of the exact method
of the binomial law of distribution. The difference, is, however, small. It can
be shown that when the sample size is not too small, say, not smaller than 10,
and also when the rate of infection is not too small, say, not less than 10 per
cent or not too large, say, not more than 90 per cent, the normal distribution
gives a sufficiently close approximation to the binomial law of distribution. In
the problem before us the number of animals required for experimentation
will depend upon the normal rate of infection and the margin of difference in
percentage infection due to treatment it is considered desirable to detect.
The larger the margin the smaller will be the number of animals required. In
practice we shall rarely be concerned with testing a reduction of more than
about 25 per cent and we can, therefore, safely use the normal method of ap
proximation in place of the binomial method for working out the probability
and hence for finding out the minimum number of animals required.
It has, however, been assumed above that the normal rate of infection
remains unchanged. The normal rate of infection is calculated from a large
body of data collected in the past and cannot necessarily be realized in a given
sample. For the rate of infection besides being subject to secular changes
may vary with place and climate and may even differ in the two sexes. The
normal rate of infection based on past experience may not for these reasons
be comparable with the rate of infection observed in the experimental group
of animals and it is, therefore, safer and usual for the experimenter to have a
control group side by side with an experimental one in order to ensure that the
comparison is unbiased and reliable. The problem now is to determine the
size of each group so that a conclusive verdict on the question whether the
experimental treatment will reduce the normal rate by a given margin
will be possible.
With two groups of size n in each, the experimenter will observe two rates of
infection q 1 in the control group and q 2 in the experimental one and will be faced
with the question whether the difference d between q 1 and q 2 is more than is
likely to arise merely by chance. If the treatment was without effect and if
the experiment was repeated a large number of times q 1 would sometimes be
less and sometimes more than q 2 ; but in the long run the experimenter will
observe no significant difference between the two. He would indeed expect
the difference q 1 —q 2 to be distributed around zero with its own standard error.
This standard error can be shown to be given by the expression √2q 1 (1—Q 1 )/n