Multidimensional Models of Perception and Cognition

By F. Gregory Ashby | Go to book overview

various aspect of the data that they explain. This table lists a series of data effects and for each model indicates which parameters must be examined to measure the effect. Although no single model is optimally suited to represent all characteristics of the data, among them a wide variety of effects can be examined.

For many of the parameters, separate values are estimated for each stimulus condition. The parameters describing the distributions in the bivariate Gaussian model are a good example. For example, the pattern of differences among the four high-frequency means μhl express the effects of both signals and of their interaction. Any of these sets of parameters can be reparameterized to isolate the influence of each signal and of their interaction. This transformation is essentially the same as that used to express the means in a 2 × 2 Analysis of Variance as main effects and interactions. Using a notation adapted from the log-linear models, μhl is

μhl = μ + μH(h) + μL(l) + μHL(hl). (11)

Summation-to-zero constraints in the manner of Equation 10 are required to identify the parameters. With this reparameterization, the four terms on the right describe the overall center of the means, the main-effect shift in mean due to the high-frequency signal, the main-effect shift due to the low-frequency signal, and the interaction of these effects. This form of parameterization is conventional for the log-linear models used in the tests of the conditional-independence hypotheses, but to simplify our notation we have not parameterized the bivariate Gaussian model and the multiplicative-association model in this way, preferring to emphasize the pattern of values, as in Figure 9.1. However, where this type of reparameterization is applicable the revised parameter is noted in brackets in Table 9.7. Only for the parameter μhl have we listed the main effect and interaction terms separately.

The techniques we have described are essentially data-driven. Except for the broad signal detection model and the context of General Recognition Theory, we do not appeal to any facts from visual theory in our analysis. This atheoretical character increases the applicability to other senses and other forms of perceptual interaction of both the concurrent detection design and its analysis. Of course, we believe that the effects we identify have specific implications for visual theory. An exciting class of models, which we have only begun to explore, links that theory to the statistical models.


ACKNOWLEDGMENT

This work was supported in part by a UCLA Faculty Research Grant to TDW and by US Public Health Service Grant No. EY00360 from the National Institutes of Health to James P. Thomas (renewal, Thomas & LAO). We thank Greg Ashby, Eric Holman, and an anonymous reviewer for comments and suggestions.

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