# Difference vs. Ratio of evidence in sequential sampling models

In sequential sampling models - for instance Ratcliff and Smith, (2006) - participants' responses in a binary choice experiment are modelled by a particle, which moves up or down towards the boundaries for selecting each response over time according to the evidence in favour of each, or, analogously, their expected utility (Busemeyer & Townsend, 1993), in a way that looks something like this: My question is if anyone knows, and preferably can provide a reference for, whether in such a model responses are best predicted by

• the difference in the evidence for/expected utility of each response (i.e. $P(Response\ A) = Evidence\ for\ A - Evidence\ for\ B$) or
• the ratio of evidence/expected utility (i.e. $P(Response\ A) = \frac{Evidence\ for\ A}{Evidence\ for\ B}$)

My intuition is that it's the ratio between the two responses, rather than the absolute difference, which should best predict responses, but I can't find a reference for this, and I'm sure this question has been answered somewhere before.

Has anyone any ideas here?

## 1 Answer

Unless I'm misunderstanding what you mean, ratio of evidence is actually a terrible method because that would end up with an immediate decision as soon as any evidence is encountered for either side, giving a ratio of infinity for that side (something/0 is greater than any finite decision threshold).

Try Gold and Shadlen 2007 for a review http://synapse.princeton.edu/~sam/gold_shadlen07_annu_rev_neurosci.pdf

• Partial credit (+1): the reference is a good one (thanks), but as I understand it, it suggests the ratio to be best predictor in principle: "The decision requires the construction of a DV [decision variable] from e [the evidence]. For binary decisions, the DV is typically related to the ratio of the likelihoods of $h_1$ and $h_2$ given e: $l_{12}(e) ≡ P(e | h_1)/P(e | h_2)$". – Eoin Nov 16 '15 at 14:57
• I think what you say about the case of $Something/0$ requires that participants begin each trial with a prior value $0$ for each option, which isn't true - unbiased participants should start with a prior of $.5$ on each option, and adjust this belief in response to new evidence. – Eoin Nov 16 '15 at 15:01
• nice. yes, ratio is only bad if you don't have a prior. – honi Nov 16 '15 at 18:36
• A benefit of the absolute difference method is that it becomes more likely that a decision will be made as time goes on (random walks tend to increase in range over time). This can be added to the ratio method by adding in an "urgency" value that decreases the decision threshold over time – honi Nov 16 '15 at 18:40