Statistical Methods in Medical Research 4ed

patternless way. We are putting forward a model, or conceptual framework, for
the random variation, and propose to make whatever statements we can about
the relevant features of this model, just as we wish to make statements about the
relevant features of a population in a strict sampling situation. Sometimes, of
course, the supposition that the data behave like a simple random sample is
blatantly unrealistic. There may, for instance, be a systematic tendency for the
earliest observations to be greater in magnitude than those made later. Such
trends, and other systematic features, can be allowed for by increasing the
complexity of the model. When such modifications have been made, there will
still remain some degree of apparently random variation, the underlying probability
distribution of which is a legitimate object of study.
The estimation of the population mean by the sample mean is an example of
the type of inference known as point estimation. It is of limited value unless
supplemented by other devices. A single value quoted as an estimate of a
population parameter is of little use unless it is accompanied by some indication
of its precision. In the following parts of this section we shall describe
various ways of enhancing the value of point estimates. However, it will be
useful here to summarize some important attributes that may be required for
an estimator:
1 A statistic is an unbiased estimator of a parameter if, in repeated sampling, its
expectation (i.e. mean value) equals the parameter. This is useful, but not
essential: it may for instance be more convenient to use an estimator whose
median, rather than mean, is the parameter value.
2 An estimator is consistent if it gives the value of the parameter when applied
to the whole population, i.e. in very large samples. This is a more important
criterion than 1. It would be very undesirable if, in large samples, where the
estimator is expected to be very precise, it pointed misleadingly to the wrong
answer.
3 The estimator should preferably have as little sampling error as possible. A
consistent estimator which has minimum sampling error is called efficient.
4 A statistic is sufficient if it captures all the information that the sample can
provide about a particular parameter. This is an important criterion, but its
implications are somewhat outside the scope of this book.
Likelihood
4.1 Statistical inference: tests and estimation 85
In discussing Bayes' theorem in §3.3, we defined the likelihood of a hypothesis as
the probability of observing the given data if the hypothesis were true. In other
words, the likelihood function for a parameter expresses the probability (or
probability density) of the data for different values of the parameter. Consider
a simple example. Suppose we make one observation on a random variable, x,
which follows a normal distribution with mean m and variance 1, where m is