The Errors of Lattice Sampling

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Abstract
Lattice samples are samples selected from a multiple classification of units with one unit in each subclass in such a way that they include equal numbers of units from each main class or from each combination of two, three or more main classes in different classifications. The present paper provides a discussion of the errors of such samples. Methods for determining the sampling variances from the complete data, when these are available, for purposes such as a comparison of different sampling methods, are described for several methods of selection. The estimation of the sampling variances from the sample data is also considered. In general estimates of error can be obtained (a) from samples consisting of special patterns of units or (b) from random samples of several mutually exclusive sets each of which itself constitutes a lattice sample. If suitable elements of randomization are introduced method (a) is satisfactory for selecting samples of 2p units from a p × p lattice, samples of 2p2 units from a p × p × p lattice and, in general, 2pn–1 units from a pn lattice. In other cases, however, the estimated means and errors may be inaccurate. Method (b) is most useful when several lattices have to be sampled. The case of the square lattice is considered in detail and rigorous proofs of the various formulae are given. In the section on cubic lattices a method is given whereby the required expressions can be written down immediately in terms of components of variance. This method is extended to the general case of sampling a pn lattice. Examples of rectangular lattices and multistage sampling are also briefly considered.

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