# When to use shifted log-normal distributions

Hi,
I’m modelling some reaction time data, and stumbled upon the shifted log-normal family argument. It’s described as well-suited for modelling reaction time, so I was intrigued.

The residual plot and qqplot of my model show equal variance and generally a normal distribution. However, my summary statistics (Est. error, credible intervals, and R-hat) are way better, when I use a shifted log-normal distribution, than a Gaussian distribution.

I’ve searched the web for general knowledge on the shifted log-normal distribution, its assumptions or when to apply this instead of Gaussian or just log-normal distributions, but without luck.

So my question is:
WHEN do you use the shifted log-normal distribution? What kind of data is it suitable for? And why/when do you choose it over the Gaussian or log-normal distributions?

Best regards,

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Hi @IdaC,
The big thing for the shifted log-normal is that you assume that no valid reaction time can be smaller than the shift. This could be modelling something like an experiment where subjects should press a button after a light flashed. You know from the experiment design, that no reaction time smaller than the time it takes for the light to flash can be valid.
The brmsfamily: Special Family Functions for 'brms' Models in brms: Bayesian Regression Models using 'Stan' page also proposes exgaussian for reaction times so you could just try that out as well.
One reason the shifted log-normal is a better fit than the normal could be the symmetric nature of the normal distribution. It is trying to match the (probably existing) right tail of your data and through symmetry is then putting too much emphasis on the left tail. A nice way to visualize this is through the `pp_check` function. Just beware that the smoothing in the plot can make hard bounds look like smooth tails.

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Maybe the following resources can be helpful?
Reaction time distributions: an interactive overview https://lindeloev.shinyapps.io/shiny-rt/

I would usually not use a normal distribution for reaction times, because a) RTs cannot be negative, b) RT distributions tend to be right skewed. Both lognormal and shifted lognormal distributions seem generally good choices.
Another neat feature of lognormal and shifted-lognormal distributions is that the spread automatically scales with the size of the location parameter: ‘On the linear relation between the mean and the standard deviation of a response time distribution’ APA PsycNet)https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwjywbrI6YDwAhV9hv0HHSsMCMYQFjACegQIBBAF&url=http%3A%2F%2Fejwagenmakers.com%2F2007%2FWagenmakersBrown2007.pdf&usg=AOvVaw1U0Odl5dNke2qaN71uGpJD

On why you might want to add a shift:
Are unshifted distributional models appropriate for response time? https://link.springer.com/article/10.1007%2Fs11336-005-1297-7 [paywalled, let me know if you have trouble finding access elsewere]
However, the shifted-lognormal model can be very sensitive to your lowest data points, which may lead to problems using loo (see e.g. this post: Error using Loo with moment matching)

But of course it always depends on your specific data and questions.

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Thank you for your tips and advice! That makes a lot of sense, and I’ll look into it :-)

Thank you for all of the sources/references! That was exactly what I was looking for, so I’ll definetly look into it, thank you very much :-)

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