The identities hidden in the matching laws, and their uses.
The 'bias' and 'sensitivity' numbers in matching equations may be statistical artifacts—check simpler models first.
01Research in Context
What this study did
Thorne (2010) looked at the math inside the matching law. The paper asks: are the 'bias' and 'sensitivity' numbers real traits or just math ghosts?
No new data were collected. The author re-examined the algebra of the generalized matching equation and showed that hidden identities can fake those parameters.
What they found
The study proves that if the model is misspecified, the equation will still spit out bias and sensitivity values even when none exist.
In plain words: you may be chasing traits that are only arithmetic leftovers.
How this fits with other research
Rosenberg (1986) and Allen (1981) defended the power-form matching equation as mathematically solid. Thorne (2010) does not overturn the equation; it warns that the extra parameters we tack on can be artifacts.
Whitehouse et al. (2014) extends the warning by giving a real-world reason for deviations: food reinforcers trigger background activities that compete with the target response. Together, the two papers show both algebraic and behavioral reasons for 'bias' to appear.
Oliver et al. (2002) applied matching to problem behavior and found tight fits. Thorne (2010) implies those fits should be rechecked without assuming the bias term is meaningful.
Why it matters
Before you label a child as 'bias prone' or 'under-sensitive,' rerun your model without the bias term and see if the data still demand it. If the numbers collapse, you have saved time and avoided false traits. Always test simpler forms first.
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02At a glance
03Original abstract
Various theoretical equations have been proposed to predict response rate as a function of the rate of reinforcement. If both the rate and probability of reinforcement are considered, a simple identity, defining equation, or "law" holds. This identity places algebraic constraints on the allowable forms of our mathematical models and can help identify the referents for certain empirical or theoretical coefficients. This identity can be applied to both single and compound schedules of reinforcement, absolute and relative measures, and to local, global and overall rates and probabilities. The rate matching equations of Hernstein and Catania appear to have been approximations to, and to have been evolving toward, one form of this algebraic identity. Estimates of the bias and sensitivity terms in the generalized ratio and logarithmic matching models are here held to be averaging artifacts arising from fitting procedures applied to models that violate or conceal the underlying identities.
Journal of the experimental analysis of behavior, 2010 · doi:10.1901/jeab.2010.93-247