I have collected psychophysiological data measuring the subjects' (two groups) ability to perceive vibration. A vibrating probe moves against the skin at smaller and smaller displacements, and the subject indicates when they feel the vibration. Unfortunately, at high frequencies, the probe can only move a short distance, and sometimes the largest distance that the probe can move is still not large enough for the subjects to perceive. Thus, I have accurate threshold values for some subjects, but for some who never felt the vibration, I simply have a value that I know their threshold is greater than. Is there any way for me to still include this data? And what is the best way to analyze it?
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I like to use heterogeneous Mixture Models to describe combined effects from fundamentally different sources.
You might look at something like a "Zero Inflated Poisson" model in the style of Diane Lambert. "Zero-Inflated Poisson Regression, With an Application to Defects in Manufacturing", Diane Lambert, Technometrics, Vol. 34, Iss. 1, 1992
I find this idea particularly delightful because it seems to contradict the notion that application of statistical design of experiments to medicine cannot fully cure disease. Behind the notion is the idea that the scientific method cannot complete its purpose in medicine comes from the idea that there is no disease data from a "perfectly" healthy individual and so that data cannot inform remedy of disease. Without measurement there is no room to improve.
Using something like a zero-inflated model allows one to extract useful information from data that is partially "error free". It is using insight into the process to take the information that could be thought of as "silent" and make it speak. To me this is the kind of thing you are trying to do.
Now I can't begin to assert which combinations of models to use. I suspect that you could use a zero-inflated Gaussian Mixture Model (GMM) for starters. The GMM is a bit of an empirical universal approximator for continuous PDFs - like the PDF cousin of the Fourier Series approximation, but with the support of the central limit theorem to improve the global applicability and allow typically many fewer components in order to make a "good" approximation.
Best of Luck.
More on zero-inflated models:
Clustering the results and defining a scale might be a solution.
Make a category variable like so (or differently):
You could use this variable to do the analysis, but whether the results are meaningful depends on how well you define the categories.