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There are other forms of small-N designs and many variations within each category of design. Identifying which design is the best fit for a particular research question or context depends on many factors. They can be useful in the early developmental phase of research as well as in refining the application of research findings to individual patients. While we fully understand the arguments in favor of such models, to us, many of the published examples of their use have tended to obscure rather than to emphasize the quality of the fits at the individual level. However, our ultimate goal throughout this article is not to criticize these or any other particular methods, but to highlight that psychology is not a homogeneous discipline. The lesson is that a common feature of small-N methods, and the increased power and precision of inference they offer, is only realizable in data-rich environments.
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In this form of replication, participants are matched as closely as possible on subject characteristics. The aim is to establish, as clearly as possible, that a given intervention can have an effect on a certain kind of patient within a specific setting. If a series of direct replication small-N studies produces consistently positive results, then the replication process moves to the next level. The figure is adapted from Carey and Matyas’ study17 on direct and transfer effects of stimulus-specific training on joint proprioception in five patients with stroke.
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Methodology
The second level of replication involves the systematic replication of the treatment across various participants, settings, clinicians, or a combination of these. Systematic replication helps to establish the generality of the findings over a wider range of situations than does direct replication. The final strategy of replication identified by Barlow and Hersen34 is clinical replication. Clinical replication involves establishing the generality of related components of the intervention.
Appendix: Details of the additive factors simulation
However, every clinician does have a responsibility to document the services they provide and changes in patient performance related to the person’s individual rehabilitation goals. 5Downsampling (of a larger sample) is typically used in bootstrapping when comparing two samples of unequal size (Racine et al., 2014); here, we applied downsampling to increase the variability of the individual distributions. This, of course, is the well-known interaction effect in a standard 2 × 2 ANOVA.
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As a corollary to this interpretation, we might expect that the areas of psychology that were slowest in taking advantage of Fisher’s methodological innovations would be those in which the replication crisis is now deepest. An experimenter who runs a small number of participants probably does so in the expectation of finding a high degree of interparticipant agreement, as is often found in sensory science, animal learning studies, and some areas of cognitive neuroscience. However, in situations like the one in our simulation in which there is appreciable heterogeneity in the underlying population, the expected consistency is unlikely to eventuate, or not completely.
The effect of increasing N on the individual- and group-level analyses
We then used model selection (i.e., a G2 test) to determine whether the more general model fit significantly better than the constrained model. The key strengths of the AB design are its applicability to almost any clinical setting or problem, as well as its simplicity in evaluating whether changes occurred in the outcome following the transition from baseline to intervention. The basic design in this classification involves an AB structure, wherein “A” represents the baseline (non-treatment) phase and “B” refers to the intervention phase.
One method estimated the size of the interaction effect using weighted linear regression. Here we also estimated the power using the proportion of bootstrapped samples which had a significant effect. The other method was to fit a log-normal distribution to the data using maximum likelihood estimation. We fit a general model, in which the interaction term was allowed to be non-zero, and a constrained model, which forced the interaction term to zero.
We conducted additional simulations in which we increased the sample N (see Fig. 2). As N increases, the power of the group-level analysis increases as expected, but it is only at large levels of N that the group-level analysis is comparable to the individual-level analysis. We would further note that at large N, the group-level analysis continues to obscure qualitative individual differences in the level of the effect. Furthermore, the group-level analysis provides no indication of the effect size at the individual level, which, by contrast, the individual-level analysis captures as its primary focus. For each bootstrapped set of N simulated participants, we averaged the data for each participant in each of the factorial conditions and conducted a 2 × 2 ANOVA.
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