Statistical Methods in Medical Research 4ed

680 Statistical methods in epidemiology
1 543 1 96 0 0901 ˆ 1 366 and 1 720:
The corresponding limits for c are the exponentials, 3 92 and 5 58. The point
estimate and the confidence limits are very similar indeed to those derived using Woolf 's
method. Details of the calculation are not given here but (19.24) gives SE (ln RMH)
ˆ 0 0977 to give 95% confidence limits of 3 86 and 5 67, similar to both sets of limits
derived earlier.
Another frequently occurring situation is when the factor, instead of having
just two levels, presence or absence, has three or more levels that have a definite
order. For example, instead of classifying people as non-smokers or smokers, the
smokers could be divided further according to the amount smoked. Such a
situation was discussed in §15.7. Mantel (1963) extended the Mantel±Haenszel
procedure to provide a test for trend and to combine a number of subsets of the
data (15.21). An example of the method is given in Example 15.8.
The methods discussed above are powerful in the commonly occurring
situation where it is required to analyse the association between a disease and
a single exposure factor, making allowance for one other factor, or at most a few,
and where the effect of the other factors may be adequately represented by
subdivision of the data into strata. The methods are inconvenient where several
exposure factors are of interest and for each factor allowance must be made for
the others, as was the case in Example 14.1. Also, where several factors are to be
adjusted for, stratification would involve a large number of strata; in the limit
each stratum would contain only one subject, so that comparisons within strata
would be impossible. In these cases more general methods must be used and
logistic regression (§14.2) is appropriate. This is because of the combination of
two features. First, as observed above, the odds ratio can be estimated from a
case±control study and, secondly, the logarithm of the odds ratio is a difference
of logits. For a case-control study all the coefficients of (14.6) may be estimated
except for the constant term b 0, which is distorted by the investigator's choice of
the size of the case and control groups. The estimate of a regression coefficient
measures the logarithm of the odds ratio for a change of unity in the corresponding
variable, and so the exponentials of the regression coefficients give estimated
odds ratios or approximate relative risks.
It was observed above that, for a case±control study where a matched control
is chosen for each case, the Mantel±Haenszel method may be applied, and leads
to McNemar's test. When the number of cases available is limited, the precision
of a study may be increased, to some extent, by choosing more than one matched
control per case. The Mantel±Haenszel method can be applied, treating each set
of a case and its controls as a subset. The analysis may be simplified by grouping
together sets providing identical information; for example, with three controls
per case those sets where the case and just two of the three controls are positive
for the exposure factor all provide identical information. This is an extension of