Assessing bimodality to detect the presence of a dual cognitive process *Behavior Research Methods*, Vol. 45, No. 1. (March 2013), pp. 83-97 by Jonathan B. Freeman, Rick Dale

@article{freeman-and-dale-2013, abstract = {Researchers have long sought to distinguish between single-process and dual-process cognitive phenomena, using responses such as reaction times and, more recently, hand movements. Analysis of a response distribution's modality has been crucial in detecting the presence of dual processes, because they tend to introduce bimodal features. Rarely, however, have bimodality measures been systematically evaluated. We carried out tests of readily available bimodality measures that any researcher may easily employ: the bimodality coefficient ({BC}), Hartigan's dip statistic ({HDS}), and the difference in Akaike's information criterion between one-component and two-component distribution models ({AIC}(diff)). We simulated distributions containing two response populations and examined the influences of (1) the distances between populations, (2) proportions of responses, (3) the amount of positive skew present, and (4) sample size. Distance always had a stronger effect than did proportion, and the effects of proportion greatly differed across the measures. Skew biased the measures by increasing bimodality detection, in some cases leading to anomalous interactive effects. {BC} and {HDS} were generally convergent, but a number of important discrepancies were found. {AIC}(diff) was extremely sensitive to bimodality and identified nearly all distributions as bimodal. However, all measures served to detect the presence of bimodality in comparison to unimodal simulations. We provide a validation with experimental data, discuss methodological and theoretical implications, and make recommendations regarding the choice of analysis.}, author = {Freeman, Jonathan B. and Dale, Rick}, issn = {1554-3528}, journal = {Behavior Research Methods}, keywords = {bimodality, math, statistics}, month = mar, number = {1}, pages = {83--97}, pmid = {22806703}, posted-at = {2014-05-21 09:54:20}, priority = {2}, title = {Assessing bimodality to detect the presence of a dual cognitive process}, url = {http://view.ncbi.nlm.nih.gov/pubmed/22806703}, volume = {45}, year = {2013} }

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Freeman and Dale discuss three measures for detecting bimodality in an observed probability distribution:

- The bimodality coefficient (BC),
- Hartigan's dip statistic (HDS), and
- Akaike's information criterion between one-component and two-component distribution models (AID).⇒

Measures for detecting bimodality can be used to detect whether psychometric measurements include cases in which behavior was caused by different cognitive processes (like intuitive and rational processing).⇒

According to Freeman and Dale, Hartigan's dip statistic is more robust against skew than either the bimodality coefficent and Akaike's information criterion.⇒

The bimodality coefficient can be unstable with small sample sizes (n<10).⇒

Bimodality measures for probability distributions are affected by

- distance between modes,
- proportion (relative gain) of modes, and
- proportion of skew.⇒

Bimodality measures for probability distributions are affected by

- distance between modes,
- proportion (relative gain) of modes, and
- proportion of skew.

Of the three, Freeman and Dale found distance between modes to have the greatest impact on the measures they chose.⇒

In Freeman and Dale's simulations, Hartigan's dip statistic was the most sensitive in detecting bimodality.⇒

In Freeman and Dale's simulations, Hartigan's dip statistic was strongly influenced by proportion between modes.⇒

In Freeman and Dale's simulations, the bimodality coefficient suffered from interactions between skew and proportion between modes.⇒

According to Freeman and Dale, the bimodality coefficient uses the heuristic that bimodal distributions often are asymmetric which would lead to high skew and low kurtosis.

It therefore makes sense that it may detect false positives for uni-modal distributions with high skew and low kurtosis.⇒

Freeman and Dale `are inclined to recommend' Hartigan's dip statistic to detect bimodality.⇒

Intuitively, Akaike's information criterion between one-component and two-component distribution models (AID) tests whether a one model or another describes the data better, with a penalty for model complexity.⇒

Freeman and Dale found Akaike's information criterion between one-component and two-component distribution models (AID) to be very sensitive to but highly biased towards bimodality.⇒

Pfister et al. recommend using Hartigan's dip statistic and the bimodality coefficient plus visual inspection to detect bimodality.⇒