Statistical Methods in Medical Research 4ed

110 Analysing means and proportions
However, this method is no more exact for finite values of n1 and n2 than
would be the use of the normal approximation to the t distribution in the case of
equal variances. The appropriate analogue of the t distribution is both more
complexthan the t distribution and more contentious. One solution, due to
B.L. Welch, is to use a distribution for d (tabulated, for example, in Pearson and
Hartley, 1966, Table 11). The critical value for any particular probability level
depends on s2 1 =s2 2 , n1 and n2. Another solution, similarly dependent on s2 1 =s2 2 , n1
and n2, is that of W.V. Behrens, tabulated as Table VI in Fisher and Yates
(1963). The distinction between these two approaches is due to different
approaches to the logic of statistical inference. Underlying Welch's test is an
interpretation of probability levels, either in significance tests or confidence
intervals, as long-term frequencies in repeated samples from the same
populations. The Behrens test was advocated by R.A. Fisher as an example of
the use of fiducial inference, and it arises also from the Bayesian approach (§6.2).
A feature of Welch's approach is that the critical value for d may be less than
the critical value for a t distribution with n1 ‡ n2 2 DF, and this is unsatisfactory.
A simpler approximate solution which does not have this disadvantage is to
test d against the t distribution with degrees of freedom, n, dependent on s2 1 =s2 2 , n1
and n2 according to the following formula (Satterthwaite, 1946):
…s2 1 =n1 ‡ s2 2
2 =n2†
n
ˆ …s2 2
1 =n1†
=n2† 2
n1 1 ‡ …s22 : …4:11†
n2 1
Although this test uses the t distribution, it should not be confused with the more
usual two-sample t test based on equal variances. This approximate test is
included in some statistical software packages.
Example 4.5
A suspension of virus particles is prepared at two dilutions. If the experimental techniques
are perfect, preparation B should have 10 times as high a concentration of virus particles
as preparation A. Equal volumes from each suspension are inoculated on to the chorioallantoic
membrane of chick embryos. After an appropriate incubation period the
membranes are removed and the number of pocks on each membrane is counted. The
numbers are as follows:
Preparation
Counts
A B
0 10
0 13
1 13
1 14
1 19
1 20
2 21
2 26
3 29