Statistical Methods in Medical Research 4ed

264 Experimental design
type of design called a set of balanced lattice squares may be useful here. For a
brief description, see Cox (1958, §11.3(iii)); for details, see Cochran and Cox
(1957, Chapter 12).
In a balanced incomplete block design all treatments are handled in a symmetric
way. All contrasts between pairs of treatments are, for example, estimated
with equal precision. Some other incomplete block designs retain some, but not
all, of the symmetry of the balanced designs. They may be adopted because of a
deliberate wish to estimate some contrasts more precisely than others. Or it may
be that physical restrictions on the size of the experiment do not permit any of
the balanced designs to be used. Lattice designs (not to be confused with lattice
squares), in particular, are useful when a large number of treatments are to be
compared and where the smallest balanced design is likely to be too large for
practical use.
Sometimes it may be necessary to use incomplete block designs which have
no degree of symmetry. For some worked examples, see Pearce (1965, 1983).
Fractional replication and confounding
If the rows and columns of a Latin square represent different treatment factors
and the Latin letters represent a third treatment factor, we have an incomplete
factorial design. As we have seen in discussing the analysis of the Latin square,
one consequence is that the main effects of the factors can be studied only if the
interactions are assumed to be absent. There are many other incomplete or
fractional factorial designs in which only a fraction of all the possible combinations
of factor levels are used, with the consequence that not all the main effects
or interactions can be separately investigated.
Such designs may be very useful for experiments with a large number of
factors in which the number of observations required for a complete factorial
experiment is greater than can conveniently be used, or where the main effects
can be estimated sufficiently precisely with less than the complete number of
observations. If by the use of a fractional factorial design we have to sacrifice the
ability to estimate some of the main effects or interactions, it will usually be
convenient if we can arrange to lose information about the higher-order interactions
rather than the main effects or lower-order interactions, because the
former are unlikely to be large without the latter also appearing large, whereas
the converse is not true. A further point to remember is that SSq for high-order
interactions are often pooled in the analysis of variance to give an estimate of
residual variance. The sacrifice of information about some of these will reduce
the residual DF, and if this is done too drastically there will be an inadequately
precise estimate of error unless an estimate is available from other data.
Fractional factorial designs have been much used in industrial and agricultural
work where the simultaneous effects of large numbers of factors have to be