Statistical Methods in Medical Research 4ed

Are these results consistent with the hypothesis that, in a large enough series of counts, the
mean for preparation B will be 10 times that for preparation A? If the counts on B are
divided by 10 and denoted by x2, the counts on A being denoted by x1, an equivalent
question is whether the means of x1 and x2 differ significantly.
Preparation
A B
Counts
x1
x2
0 1 0
0 1 3
1 1 3
1 1 4
1 1 9
1 2 0
2 2 1
2 2 6
3 2 9
n1 ˆ 9 n2 ˆ 9
x1 ˆ 1 2222 x2 ˆ 1 8333
s2 1 ˆ 0 9444 s2 2 ˆ 0 4100.
The estimates of variance are perhaps not sufficiently different here to cause great
disquiet, but it is known from experience with this type of data that estimates of variance
of pock counts, standardized for dilution as we did for x2, tend to decrease as the original
counts increase. The excess of s2 1 over s2 2 is therefore probably not due to sampling error.
We have
1 2222 1 8333
d ˆ r ˆ
0 9444 0 4100
‡
9 9
0 6111
ˆ 1 58:
0 3879
Note that, when, as here, n1 ˆ n2, d turns out to have exactly the same numerical value as
t, because the expression inside the square root can be written either as
or as
s2 1
n ‡ s2 2
n
1
2 …s21 ‡ s2 1 1
2 † ‡
n n :
Using Satterthwaite's approximation, the test statistic of 1.58 can be referred to the t
distribution with DF given by
n ˆ
0 15042 0 10492 =8 ‡ 0 04552 ˆ 13 8:
=8
4.3 Comparison of two means 111
Using 14 DF the significance level is 0 14. The 95% confidence limits for the difference
between the two means are