Quantifying the effect of intertrial dependence on perceptual decisions

Today we discussed a paper on sequential effects in psychophysics by Fründ et al. Although inter-trial dependencies are known to exist, psychophysical responses are typically modeled as independent Bernoulli observations. One way to account for the effect of previous trial outcomes is to use logistic regression with terms that represent previous stimuli or responses (Busse et al). Fründ and colleagues extend this approach by including a lapse rate which captures responses where the subject was not doing the task.

They propose a latent variable on each trial, l_t that is in one of three possible states: 0,1,2 (“guesses left”, “guesses right”, “does task”). The probability of a response is dependent on the stimulus on that trial, the outcome of previous trials and the state of the latent variable:


where x_t is the concatenation of stimulus and history terms for that trial, \omega is weights on the stimulus and history terms and g(x) is \frac{1}{1+e^{-x}}

The authors use Expectation-Maximization to fit the weights, \omega, and the probabilities of the latent states, p. They then compare model fits that include history terms to fits without them for four different psychophysical tasks.

The main results are: a small gain in likelihood per trial for the full model with history terms, thresholds are a few percent lower if history is included, and the effect of history is strongest for low signal conditions and disappears with high signal strength. Probably the coolest figure is 4c where they demonstrate how history weights reveal different task strategies for the different psychophysical experiments.


The different colors are the four different tasks, each dot is a different subject.  Red and Blue (both visual 2AFC tasks) seem safely in ‘switch’ or ‘win stay – lose switch’. All subjects maintain a ‘stay’ strategy for Yellow (an Auditory Yes-No task).

Does it matter? The authors point out that all though the bias in estimated parameters introduced by inter-trial dependence is systematic, it is small. They suggest that it is particularly important to account for history effects in the analysis of neurophysiology and it’s correlation with choice.


  • Fründ I, Wichmann FA, Macke JH. (2014). Quantifying the effect of intertrial dependence on perceptual decisions. Journal of Vision. 14(7). pii: 9. doi: 10.1167/14.7.9.
  • Busse, L., Ayaz, A., Dhruv, N. T., Katzner, S., Saleem, A. B., Scholvinck, M. L., et al. (2011). The Detection of Visual Contrast in the Behaving Mouse. Journal of Neuroscience, 31(31), 11351–11361. doi:10.1523/JNEUROSCI.6689-10.2011

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