How to compare linear and categorical models of perceptual comparisons?
This sounds a lot like tests of categorical perception done in psycholinguistics. Categorical perception states that people perceive phonemes categorically, even though the acoustic signal varies continuously.
For example, the phonemes $/b/$ and $/p/$ differ on only one acoustic feature: voice onset time. Using computer synthesized speech, one can experimentally manipulate the voice onset time along a continuous dimension. In a prototypical experiment, subjects are asked to identify each sound as a $/b/$ or $/p/$. If perception were continuous, you might expect that the proportion of $/b/$ responses varies linearly with the voice onset time-- but it doesn't. As you can see in the red line, there is a sharp drop around 20ms in which people perceive everything to the left as $/b/$ and everything to the right as $/p/$.

Another way to see this is to plot discriminability (blue line). Here, participants are presented two stimuli from adjacent VOT bins and asked to judge which one matches a 'target' stimulus. Generally, people perform near chance at discriminating between the two phonemes unless the two stimuli cross the VOT boundary. Again, if perception were linear, you might expect a straight blue line rather than a peaked one.
Most researchers will use t-tests or ANOVA to test for differences of between and within category discrimination. For instance, one might conduct a series of t-tests on discriminability to show that only when stimuli cross the VOT boundary can subjects reliably discriminate them.
For an example of these tests in practice, see Morford et al. (2008). Note that some researchers have found conflicting results when using different dependent measures. For example, Massaro &amp; Cohen (1983) asked participants to make continuous judgments of stimuli and found no evidence of categorical perception.
Another alternative might be to find the best fitting line and the best fitting sigmoid curve, and compute the Bayesian Information Criterion for each. The one with the lower BIC is the preferred model. For regression analysis, BIC can be calculated as

$BIC = n\log(\frac{RSS}{n}) + p\log(n)$

where n is the number of data points in the model, p is the number of free parameters, and RSS is the residual sum of squares.
The drawback of using BIC is that it only tells you something about comparative model fit. If both linear and sigmoidal models were a bad fit for the data, BIC would not tell you that.

Massaro, D. W., &amp; Cohen, M. M. (1983). Categorical or continuous speech perception: A new test. Speech Communication, 2(1), 15-35. PDF
Morford, J. P., Grieve-Smith, A. B., MacFarlane, J., Staley, J., &amp; Waters, G. (2008). Effects of language experience on the perception of American Sign Language. Cognition, 109(1), 41-53. PDF