# Non-independence of trial likelihoods in a staircase procedure?

In psychometrics,
we often want to know, for instance,
a given participants’ perceptual threshold:
the intensity of a stimulus that they can detect 50% of the time.

It’s common to use staircase procedure,
where we adjust the stimulus according to the participant’s previous response,
making it more intense if they failed to detect it on the previous trial,
and less intense if they successfully detected it.
This produces a staircase like the one pictured below,
which converges on an intensity that can be detected 50% of the time. We might then fit a logistic (or probit) regression model
$$P(y_i = 1) = text{logit}^{-1}(alpha + beta x_i)$$,
where $$y_i=1$$ if the stimulus was detected on trial $$i$$
and $$x_i$$ is the stimulus intensity on that trial,
to refine and obtain confidence intervals for our estimate of the threshold.

However, the standard GLM routines assume that each trial $$i$$ is independent,
$$log(P(Y|X, alpha, beta)) = sum_i log(P(y_i|x_i, alpha, beta))$$.

Does this assumption hold, and so are the GLM standard errors valid,
when the stimulus $$x_i$$ depends on the stimulus and response on the previous trial $$(x_{i-1}, y_{i-1})$$?

Cross Validated Asked on November 21, 2021

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