# Modeling reaction time and probability

This may be a naive question, but I could not find any related info so far. I am modeling a time-to-event problem where I try to capture the onset of a reaction given sensor data (from participants, sampled at 100 Hz). I know that the reaction always happened and it is just a matter of when it happened. I tried to treat this problem as a survival regression problem where the reaction onset = the “death”. However, I did not find a way to predict when the reaction would happen, only the probability of it happening (1 - survival probability). I used time-dependent Cox proportional hazards, by the way, so far only in a frequentist setting but I would like to go Bayesian eventually.

My question is: Could I, instead of modeling the probability of the event happening, model the reaction time directly with a Bayesian regression model and then infer the probability of the reaction happening from the posterior predictive distribution? I can certainly model the reaction time, but I am not certain if the uncertainty from the posterior predictive distribution gives the probability of the event happening at a certain time. I was imagining that I could, given a certain (new) time, predict the CDF of the posterior predictive distribution of the reaction time (given the inputs at that new time), and then extract the probability of the event from that CDF (i.e., the quantile at the given new time). The advantage I can see from this model is that I can do both, predicting the reaction time and probability (given a certain time). Does that sound valid/reasonable? Or is the survival approach more suitable for this kind of problem?

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In the cognitive science world, we rarely use a survival model for reaction time data, but I think there’s an equivalence between a survival model approach, which focuses on the CDF and the more typical distribution “fitting” approach which focuses on the PDF. My calculus sucks, but as far as I understand the CDF from the survival model could be differentiated to obtain a PDF and a PDF from a more standard model can be integrated to obtain a CDF. However, I suspect that the the model structure leading up to the CDF/PDF might make it so that a nice named distribution form in one might not have a similar nice/named distribution form in the other.

Yup, that’s a perfectly reasonable approach to me.

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Thank you very much for your fast response.

If I am able to use the same covariates in both types of models, would it be reasonable/possible to compare both on a common metric? I know that survival models (or others like logistic regression) that model the probability of the event are typically evaluated with AUC. Could there be a way to compute a similar AUC for the “distribution fitting” approach and compare both? Or is this more like comparing “apples and pears”?

Just came across a paper that seems to be discussing the relation between PDF-based and CDF-based approaches (though I suspect their preference for CDF/Hazard-based approaches stems from using fragile/nebulously-parametric MLE procedures):
https://journals.sagepub.com/doi/10.1177/2041669520978673

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