Statistical Methods in Medical Research 4ed

90 Analysing means and proportions
3 To test the direction of a difference from a critical value. Suppose we are
interested in whether a certain parameter, u, has a value greater or less than
some value u0. We could test the null hypothesis that u is precisely u0. It may
be quite clear that this will not be true. Nevertheless we give ourselves the
opportunity to assert in which direction the difference lies. If the null hypothesis
is significantly contradicted, we shall have good evidence either that
u > u0 or that u < u0.
Finally, it must be remembered that the investigator's final judgement on any
question should not depend solely on the results of a significance test. He or she
must take into account the initial plausibility of various hypotheses and the
evidence provided by other relevant studies. The balancing of different types of
evidence will often be a subjective matter not easily formulated in clearly defined
procedures. Formal methods based on Bayes' theorem are described in Chapters
6 and 16.
Confidence intervals
We have noted that a point estimate is of limited value without some indication
of its precision. This is provided by the confidence interval which has a specified
probability (the confidence coefficient or coverage probability) of containing the
parameter value. The most commonly used coverage probability is 0 95. The
interval is then called the 95% confidence interval, and the ends of this interval
the 95% confidence limits; less frequently 90% or 99% limits may be used.
Two slightly different ways of interpreting a confidence interval may be
useful:
1 The values of the parameter inside the 95% confidence interval are precisely
those which would not be contradicted by a two-sided significance test at
the 5% level. Values outside the interval, on the other hand, would all be
contradicted by such a test.
2 We have said that the confidence interval contains the parameter with probability
0 95. This is not quite the same thing as saying that the parameter has
a probability of 0 95 of being within the interval, because the parameter is not
a random variable. In any particular case, the parameter either is or is not in
the interval. What we are doing is to imagine a series of repeated random
samples from a population with a fixed parameter value. In the long run, 95%
of the confidence intervals will include the parameter value and the confidence
statement will in these cases be true. If, in any particular problem, we
calculate a confidence interval, we may happen to be unlucky in that this may
be one of the 5% of cases in which the interval does not contain the
parameter; but we are applying a procedure that will work 95% of the time.
The first approach is akin to the system of interval estimation used by R.A.
Fisher, leading to fiducial limits; in most cases these coincide with confidence