### IQ & Skew, or Why Not to Log-Transform RTs

In his excellent book "Clocking the Mind," Arthur Jensen examines the striking correlations between intelligence and simple measures of reaction time. The word "intelligence" often invokes concepts like fluid reasoning ability, depth of spatial processing, and other sometimes ill-defined (but always high-level) constructs. In each chapter, however, Jensen demonstrates how simple reaction time measures - how long it takes you to press a lighted button, for example - are more strongly correlated with measures of general intelligence (g) than anyone might expect.

As it turns out, however, the variability of reaction time in such simple tasks is as much or even more predictive of IQ than the mean reaction time. Why should this be the case?

Variability in reaction times may index the integrity of dopaminergic thalamocortical connections. For example, in Kane & Engle's color-naming Stroop task, incongruent trials are characterized both by a positive shift in the distribution relative to neutral trials (thought to represent the time it takes to resolve interference from word-reading processes), but also by a positive skew in the distribution. This positive skew is even more pronounced among those with low working memory spans (which likely have decreased general intelligence as well).

Interestingly, the positive skew and positive shift of reaction times are uncorrelated. This suggests they index two distinct cognitive processes. Based on work from computational models of the Stroop task, the positive shift probably indexes to the efficiency with which the "ink color" representation can overpower the "word name" representation. However, positive skew has not been reproduced in such simple models, except for this one, which includes a term for the stochastic dopaminergic activity of subcortical areas that may be responsible for keeping information maintained in working memory. In this case, positive skew (termed "goal neglect" by Kane & Engle) could relate to the integrity of this dopaminergic circuit; in low-spans, this thalamocortical projection may have less predictable patterns of firing, leading to the spontaneous "neglect" of the current goal.

Positive skew in RT distributions could therefore reflect a very low-level aspect of neural architecture which may show large individual variation, just like IQ.

However, many datasets may fail to detect this correlation, or even the positive skew to begin with. Why? The positive skew of RT distributions is a well-known phenomenon, but it violates a primary assumption of statistical analysis: normality. If a dependent variables' distribution is non-normal, many statistics books will recommend a logarithmic transform, which essentially compresses the high end of the distribution. This returns the distribution to a nice gaussian curve.

This practice has unfortunate consequences beyond its primary effect (to erase the positive skew in a distribution, which as I've indicated above may be very important). Log-transforms will also mean that your data no longer conforms to an equal interval scale. This leads to problems in the calculation of variance - in other words, your measure of variance no longer corresponds to the variance of the actual data.

While the coefficients derived from ANOVAs of log-transformed data can always be interpreted in terms of equal-interval scales (by an inverse log transform), the p values are not so easily fixed, because they are based on a fundamentally unreal measure of variance.

The moral of the story: don't log-transform your RT data without performing a second analysis on the skew of the un-transformed RT data. You may just come up with something very interesting.

As it turns out, however, the variability of reaction time in such simple tasks is as much or even more predictive of IQ than the mean reaction time. Why should this be the case?

Variability in reaction times may index the integrity of dopaminergic thalamocortical connections. For example, in Kane & Engle's color-naming Stroop task, incongruent trials are characterized both by a positive shift in the distribution relative to neutral trials (thought to represent the time it takes to resolve interference from word-reading processes), but also by a positive skew in the distribution. This positive skew is even more pronounced among those with low working memory spans (which likely have decreased general intelligence as well).

Interestingly, the positive skew and positive shift of reaction times are uncorrelated. This suggests they index two distinct cognitive processes. Based on work from computational models of the Stroop task, the positive shift probably indexes to the efficiency with which the "ink color" representation can overpower the "word name" representation. However, positive skew has not been reproduced in such simple models, except for this one, which includes a term for the stochastic dopaminergic activity of subcortical areas that may be responsible for keeping information maintained in working memory. In this case, positive skew (termed "goal neglect" by Kane & Engle) could relate to the integrity of this dopaminergic circuit; in low-spans, this thalamocortical projection may have less predictable patterns of firing, leading to the spontaneous "neglect" of the current goal.

Positive skew in RT distributions could therefore reflect a very low-level aspect of neural architecture which may show large individual variation, just like IQ.

However, many datasets may fail to detect this correlation, or even the positive skew to begin with. Why? The positive skew of RT distributions is a well-known phenomenon, but it violates a primary assumption of statistical analysis: normality. If a dependent variables' distribution is non-normal, many statistics books will recommend a logarithmic transform, which essentially compresses the high end of the distribution. This returns the distribution to a nice gaussian curve.

This practice has unfortunate consequences beyond its primary effect (to erase the positive skew in a distribution, which as I've indicated above may be very important). Log-transforms will also mean that your data no longer conforms to an equal interval scale. This leads to problems in the calculation of variance - in other words, your measure of variance no longer corresponds to the variance of the actual data.

While the coefficients derived from ANOVAs of log-transformed data can always be interpreted in terms of equal-interval scales (by an inverse log transform), the p values are not so easily fixed, because they are based on a fundamentally unreal measure of variance.

The moral of the story: don't log-transform your RT data without performing a second analysis on the skew of the un-transformed RT data. You may just come up with something very interesting.

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