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We could have made similar points with reference to other sensory sciences, like auditory science, or animal learning. Among the primary outlets for research in vision are journals like Vision Research,Journal of the Optical Society of America, A, and more recently, Journal of Vision, in which the small-N design has tended to be the dominant paradigm. Published studies in these journals often report data from as few as two or three participants (typically styled “observers”), often members of the laboratory, and consequently far from naive. Our reading is that vision science shows no evidence of being in the grip of the kind of replication crisis that is currently afflicting cognitive and social psychology.
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In these situations, the value of averaging is primarily the pragmatic one of data smoothing, rather than the inferential one of estimating population parameters. The results for the weighted least squares analysis and the maximum likelihood method are qualitatively similar, and we make no further distinction between them except to note that the maximum likelihood estimation appears to be more sensitive to small differences from the null interaction. The next thing to note is that the individual analysis picks up the difference in individuals sampled from the null interaction and the positive interaction quite clearly, with the two distinct regions reflecting the separation between individuals.
In practice, because of the sequential effects that are found in many cognitive tasks (Luce, 1986; Treisman & Williams, 1984), models that assume independent trials can only approximate the true probability structure of such tasks. Sequential effects complicate the task of defining an appropriate goodness-of-fit measure for model evaluation, but the problems are not insurmountable. One way of dealing with the effects of uncontrolled (i.e., unmodeled) sources of across-trial variability in probability models is to use an appropriate model of statistical overdispersion (McCullagh & Nelder, 1989; Smith, 1998). Another way is to model the sequential dependencies directly (Jones, Curran, Mozer & Wilder, 2013; Little et al., 2016). We take vision science as an example, as the sensory science nearest to the areas of cognitive and mathematical psychology in which we as authors work.
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Individual- versus group-level inference: a simulation using additive factors
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These factors include not only variability across participants but also across the stimuli which are tested, equipment, location, time of day, and so on. Baribault et al., (2018) developed a novel method of determining whether or not any of these factors contributes to changes in a population parameter estimate by randomizing all of the various aspects of the experiment that might contribute to the effect. The specific effect of any factor was then determined by marginalizing over the other factors providing a way to isolate causal variables.
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Here we provide an intuitive overview of our simulation; details can be found in the Appendix. For each simulated participant, for each item condition, we sampled 400 RTs from a log-normal distribution. Four hundred trials is a fairly typical sample size for a small-N design in which the researcher is attempting to characterize the RT distributions in each condition for each participant individually. We chose the number 400 on this basis rather than on the basis of any formal power analysis. We attach no strong theoretical interpretation to the log-normal and use it simply as a flexible family of unimodal positively skewed distributions that has sometimes been used to characterize distributions of RT empirically (Woodworth & Schlosberg, 1954).
They noted that the foundational studies of animal learning by Pavlov and Thorndike, some decades before Fisher’s book, were carried using small samples and that this practice continued in the work of B. F. Skinner (1938)—who provided us with the epigraph for this article—and his contemporaries, and that the small-N design remains the mainstay of animal learning studies to this day. They also noted that the historical shift from small-N to large-N designs was traced by Boring (1954). He analyzed the studies published in the Journal of Experimental Psychology in the years 1916, 1933, and 1951 and reported that in 1916 not a single study used a large-sample design, but by 1951 the number had risen to over 50%. Until the current replication crisis, this history might reasonably have been read as one of psychology gradually learning to use more appropriate and reliable methods—a process that was largely complete by the time of the cognitive revolution of the 1960s.
The parameters of each item distribution were linked via a linear equation in which the interaction effect was either present or absent. This implies that a sufficient number of trials must be collected at the individual level in order for such verification to be meaningful. Whether averaging or aggregation leads to artifacts that can distort inference depends wholly on the domain and the models in question and it is difficult to give a priori, domain-independent prescriptions.
It is much more difficult to develop effective methods of strong inference in sparse environments, in which inference depends on significance tests of point hypotheses about means in one or two conditions. In sparse environment, increasing the number participants allows for more precise estimation of the between-participants error terms used to test hypotheses and reduces the likelihood of type I errors. Conversely, the combination of data-rich environments and the availability of methods that allow strong inference will often lead researchers to prefer small-N designs. If significant differences among participants in the interaction were found under these circumstances, this in itself, irrespective of any other experimental finding, would probably lead researchers to investigate other models of the cognitive architecture. Other models of cognitive architecture, such as McClelland’s (1979) cascade model and Schweickert’s (1978, 1980) PERT networks, relaxed the restrictive assumptions made by the additive factors method and can predict a richer variety of interactions that might be more consistent with significant individual differences. Sternberg’s (1969) model assumes serial, stage-dependent processing; that is, processing at any stage only commences once its predecessor has finished.
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