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Data collected on a set of objects (cities, for instance) over a set of attributes (time points, for instance) can be subjected to a variety of aggregation schemes. For example, if a hypothesized pattern over the attributes is to be confirmed (a temporal increase in homicide rate, for instance), the data could be first aggregated over cities and then compared to the hypothesized pattern. Alternatively, the correspondence for each city could be separately assessed and the individual city indices aggregated. The stage at which aggregation takes place affects the size of the final measure of confirmation as well as its significance, but, unfortunately, in opposite ways. Preliminary data aggregation typically leads to larger summary statistics and larger significance levels. The conflicting notions of size and significance are first formalized in detail when the basic data are single numerical values obtained for each object-attribute pair. Extensions are presented to multigroup concordance, hierarchical aggregation schemes, and to object data defined by pairwise proximities between the attributes.

A major concern of geographers has been the explanation and prediction of spatial choice behavior. Several discrete choice models have been developed which attempt to account for the use of different transport routes, modes, shopping centers, and so on. One of the fundamental issues associated with these models is the problem of model specification; furthermore, each type of model can potentially give disparate results for the same spatial choice situation. In this paper the models and their characteristics are summarized, and a generalized method for comparing the results yielded by alternative model specifications, or results obtained from different model formulations, are presented. The procedure is an application of recent developments in heuristic methods for comparing related matrices.

An inference strategy is presented to assess the goodness-of-fit for logit models. The conventional approach has emphasized likelihood ratio tests in which a distribution for the test statistic is assumed. A more recent development, the prediction success table, is based upon a ratio of predicted to observed choice patterns, yet it is primarily a descriptive index not placed within the context of an inference strategy/hypothesis-testing framework. An alternative method is put forward which uses residuals. Although problematic in a parametric context, residuals can be used in an approach which incorporates both a nonparametric randomization procedure and a sample reuse design. A test statistic is defined and compared to a reference distribution created under the assumption of random variation in the responses with variation in the explanatory variable(s). A numerical example is carried out to investigate the efficacy of using this approach.

A variety of tests for randomness are reviewed based on simple product—moment statistics defined between two matrices, {aij} and {bij}. Typically, the first matrix, {aij}, contains proximity data on the spatial placement of nobservations, {x1….xn} the second matrix, {bij}, is obtained from the relationships among the nobservations themselves. Depending on the definitions of {aij} and {bij}, a variety of well-known tests of randomness and/or trend (those attributed to Mann, Daniels, Moore and Wallis, Cox and Stuart, and Goodman and Grunfeld) as well as various approaches to serial correlation and spatial autocorrelation can be encompassed. Finally, these notions of randomness can be extended to an assessment of spatial association between two variables. A numerical example is given within this latter context.