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The Quadruple Process Model (Quad Model) is a multinomial processing tree designed to distinguish four operating principles (i.e., processes) underlying biased responses on the IAT

A Multinomial Process Tree (MPT)

MPTs are measurement models designed to distinguish multiple processes underlying responses on performance tasks, like the IAT. The model is made up of a series of equations, one equation for every unique response (what we call a response category) in  a performance task.

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With 16 responses categories in the IAT, there are 16 equations which make up the Quad model for measuring IAT data. Each equation reflects the probability of a participant selecting its respective response category. In a binary task like the IAT, in which each response is either correct or incorrect, the 16 equations can be thought of as 8 pairs of complementary probabilities. For example, the complement of the probability that participants correctly categorize a Black target on an incompatible trial is reflected by the equation for the probability that they incorrectly categorize that target on an incompatible trial.

P (correct | compatible, Black) = AC + [1-AC] x D + [1-AC] x [1-D] x [1-G]

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P (incorrect | compatible, Black) = AC x [1-D] x G

P (correct | incompatible, Black) = AC x D x OB + [1-AC] x D + [1-AC] x [1-D] x G

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P (incorrect | incompatible, Black) = AC x D x [1 - OB] + AC x [1-D] + AC x [1-D] x [1- G]

For example, on a Black-White Race IAT, the standard Quad model includes separate equations for the correct and incorrect categorization of Black targets on prejudice-congruent trials (left) and prejudice-incongruent trials (right). Congruency is based on the presumed racially prejudiced associations participants hold about Black (Black-bad) and White (White-good) people.

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As you can see, these equations map onto paths along the tree diagram. This makes sense, as the diagram is derived by the relationships of the parameters in the equations. For example, incorrectly categorizing a Black target with the bad key is accounted for in the model by the activation of the Black-Bad association, so long as the biasing association (it presumably influences responses toward incorrect responses) is not overcome in favor of a correct response which was detected (AC x D x [1-OB] + AC x [1-D]).

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The model also accounts for the chance that the incorrect response wasn't driven by prejudiced associations, but rather a simple response bias in the absence of all other processes for which the model account ([1-AC] x [1-D] x [1-G]). The complementary paths reflect the complementary response category (i.e., correctly categorizing the Black target on the incompatible trial). Equations for White targets, bad words, and good words, are derived in the same manner.

For more details regarding the instantiation and benefits of the quad model, please check out a chapter I wrote about implicit social cognition

If you'd like to figure out how to implement the quad model for your own data, I suggest you start by walking through the template I've put together. If you click the link above, you'll be taken to my OSF page, which has template code, sample data, and the standard quad model equations file. The template code is commented to walk you through the analysis for a 2-group between-participants estimation and analysis of quad model parameters.

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I have limited time to update this material, so please contact me with any questions that cannot be answered with the material I've put together already. I hope this has been helpful and that I'll see your own quad model work published someday soon :)

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