Large numbers of explanatory variables, a semi-descriptive analysis

Abstract

3. The formal probabilistic behavior of the procedure in this paper under idealized conditions will be discussed in a separate paper. This paper adopts a different, less formal, and more exploratory approach in which judgment is needed at various stages. In this the conclusion is typically that a number of different simple models fit essentially equally well and that any choice between them requires additional information, for example new or different data or subject-matter knowledge. That is, informal choices are needed at various points in the analysis. Although the choices could be reformulated into a wholly automatic procedure this has not been done here.The combinatorial arrangements used in the method are essentially partially balanced incomplete block designs (4), in particular so-called cubic and square lattices, first developed in the context of plant breeding trials involving a very large number of varieties from which a small number are to be chosen for detailed study and agricultural use.

Large numbers of explanatory variables: a probabilistic assessment

“…The special features of the procedure with binary responses will not be explored in the present paper. Note, however, that agreement of the outcome with the lasso solution should not be expected with binary responses for the reasons outlined by Cox & Battey [ 3 ].…”

“…It is assumed that v is roughly k d for k ≤ 15 and d a small positive integer. In the method as outlined by Cox & Battey [ 3 ], the indices of these variables are arranged in a k × k × k cube for d = 3 or k × k × k × k hypercube for d = 4, etc. Sets of variables are then selected for test k at a time by traversing the cube by rows, columns, etc.…”

“…The final phase of the analysis is to find small subsets of variables among the augmented set of retained variables and any square or interaction terms suggested at the informal phase. See Cox & Battey [ 3 ] for a more detailed description of the exploratory and subset selection phases.…”

“…It is possible, however, that many different choices of explanatory variables are essentially equally effective and have quite different subject–matter implications. Cox & Battey [ 3 ] outline a different approach aiming to specify those small sets of potential signal variables that give essentially the same fit; choice between different sets requires either more data or specific subject–matter information. Their procedure also allowed informal checks standard in much statistical work, such as those for nonlinearity, interaction or anomalous individuals.…”

Big data: Some statistical issues

“…Another particular situation has a very large number of variables of the same type, such as genomic variables, possibly on a relatively small number of individuals, in which if variables are considered simultaneously, interpretation relies on an assumed sparsity of effects Tibshirani (1996) , Cox and Battey (2017) .…”