General discussion

A priori considerations (Section 1.1) argue for the idea that complex processes and devices may be composed of relatively independent parts, and psychological research has revealed abundant evidence for modular mental processes, composed of separately modifiable and functionally distinct sub-processes. Some of this evidence is included among the examples here discussed. One reasonable starting point for understanding a complex process is therefore to determine whether it is composed of such modules, and if so, try to identify them. An initial step in doing so is to divide the process into just two parts (Section 1.3), perhaps followed later by dividing those parts in turn. Ultimately this should lead to analysis of the functions of each such module, as well as an understanding of how they work together. Also, we would hope to find that different tasks are implemented by different subsets of the same set of basic modules, an idea associated with both the task-comparison method (Appendix A.1) and functional brain localization (Section 11).

In this paper I have explored the idea of separate modifiability as a criterion for modularity, by considering the shared and distinguishing features of a set of diverse examples. The examples fall into three categories, depending on whether the measures of the hypothesized modular processes are believed to be pure or composite and, among the pure measures, whether they depend on different data (direct pure) or different aspects of the same data (derived pure). In each case, evidence for separate modifiability consists of finding experimental manipulations (factors) that influence the hypothesized modules selectively. Such evidence consists of pure measures that are selectively influenced by the factors, or of a composite measure for which the effects of the factors combine appropriately. Selective influence is also, in itself, evidence for the modules being functionally distinct.

That the discovery and identification of modules is a central issue in psychology is reflected by the broad range of examples; they include different substantive areas, species, responses (e.g., stimulus-elicited vs operant), measures, combination rules, experimental designs, and relations between modules; they lead to the decomposition of neural as well as functional processes. By considering such examples together, within a unified structure that highlights their inferential logic, we come to understand them better, and applications in one area benefit from lessons learned in another.

Inferential logic. In the course of this exploration I have suggested rational reconstructions of the inferential logic associated with the identification of modules, using measures hypothesized to be pure (Table 2) and composite (Tables 3, 4), with variant reconstructions in Appendix A.2.3. They are "rational reconstructions" because research is often not explicitly guided by such formal considerations, and experimental results may predate the formulation of such hypotheses. Attempts to make the inferential logic explicit should help in thinking about what a set of findings may mean, and may suggest preferences for some kinds of experiment over others.

Several lessons follow from the reconstructions:
(1) Along with the researcher's good fortune in finding suitable experimental manipulations (factors), what is tested in every example is a joint hypothesis: two or more distinct individual hypotheses, such that all gain support when evidence for modularity is revealed, while only one need be faulty for us to fail in an attempt to acquire such evidence. More generally, failure of the prediction of a joint hypothesis may be less informative than success.
(2) While we are interested in processes, all we have is measures of them (Section 2.1); whereas change in a measure implies change in the process, the converse need not be true.
(3) Nonetheless, to establish separate modifiability, process invariance is critically important; it is indicated by invariance of the value of a pure measure across levels of a factor, or of the effect (or p.effect) on a composite measure of one factor across levels of another. Implications of such invariance are sufficiently strong that it should not be asserted simply because the evidence against it is unconvincing. The consequences of this for statistical tests and reliability assessment are mentioned in Section 1.4, Appendix A.11.2, and elsewhere (and illustrated by use of the S.E.s of relevant contrasts); like much else in the present paper, these consequences need further consideration.
(4) While modularity can be demonstrated (assuming sufficiently precise data), non-modularity cannot: as illustrated by Ex. 3, it is always possible that "we didn't look enough" for appropriate experimental manipulations. Of course, the search for such factors is not blind, but depends on knowledge related to the functions that might be carried out by hypothesized modules; as attempts fail we should become increasingly convinced that these modules reflect an erroneous partition of the process under study, or that the process is simply too integrated to be modular.
(5) A poor choice of measure or measures can cause a search for modules to fail (Section 13.3, Appendix A.7.2).

Experimental design. Another consequence of the importance of invariance is the desirability of increasing the chance of revealing any systematic violation, by using factorial experiments even with pure measures (illustrated in Section 7 and Appendix A.6, and discussed in Appendix A.9.1 and elsewhere), and by using factors with more than two levels (illustrated by Exs. 3, 6, 7, and 9, and discussed in Appendix A.9.2 and elsewhere). Though not required with pure measures, factorial experiments permit testing the generality of findings of invariance. And factors with multiple levels permit separating systematic from non-systematic deviations, the former more damaging to claims of invariance. In Exs. 3 and 9, where multiple-level factors were employed, I illustrated the use of one-dimensional indices of invariance violation, based on numerically scaled factor levels, to create focused tests for one interesting and plausible kind of systematic deviation. We need to consider further the meaning of "systematic deviation", as well as the merit of the meanings proposed in Sections 7.2 and 15.1.

Brain measures. The use of brain measures to identify modular neural processes may be informative about the structure of functional processes as well, but brain measures introduce complexities as well as opportunities. Some of the issues are discussed in Section 1.6 and Appendix A.1.2, and in the context of the examples (Sections 6, 10, 11 and 14; Appendices A.10 and A.11). It is advantageous to study effects of the same factors within the same task on both brain and behavioral measures (Sections 6, 10, 11; Appendix A.6).

Process decomposition vs task comparison. Because the separate-modifiability approach to process decomposition has formal similarities to the popular task-comparison method and the associated pattern of double task-dissociation, I consider the latter method in Section 11 and Appendix A.1. The two methods have different purposes and strengths; I suggest that task comparison is not especially helpful in identifying the parts of a complex process.

Limitations. For simplicity, this paper is limited in at least two ways: First, in illustrating the inferential logic and treatment of data I discuss the examples largely as if they are isolated cases, rather than considering them in relation to existing knowledge and related studies; conclusions normally depend on more context. Second, the examples are almost entirely restricted to dividing a complex process into only two parts. With a caveat (footnote 3), however, this is a reasonable starting point in decomposing a complex process, and because modules may themselves be modular, further partitioning can follow.

Open questions. Among others, these include the following:
Are there useful further elaborations of the inferential logic?
What is the best statistical method for testing a prediction of invariance or additivity?
In which patterns of deviation should we be especially interested?
How can we strengthen the inferences from brain measures to the structure of functional processes?
What is the relation between the modularity of functional processes and of the brain processes that implement them?
What are the relative merits of using pure vs composite measures?
For composite measures are there other combination rules that might be of interest?
Can we be more specific about what to do when a partitioning attempt fails?
How interesting are cases of partial modularity (Section 2.3) or approximate modularity (Section 7.2)?
Is separate modifiability too strong or too weak to be a useful criterion for partitioning a process?
What are the relative merits of alternative criteria for modularity, and alternative approaches to module identification?
Does the present approach lead to modules that have other desirable properties?
How does it compare to module discovery in other sciences?