Multinomial Processing Tree Models

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Multinomial Processing Tree Models
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Agenda

• Questions?
• MPT model overview.
• MPT overview
• Parameters and flexibility.
• MPT Evaluation
• Batchelder Riefer, 1980.
• Assignment 1 solution(s).
• Math for next week.
• Assignment 2 problem statement.

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Uses of MPT Models

• Data-analysis tool capable of disentangling and
  measuring separate contributions of different
  cognitive processes.
• Provides a means for separately measuring latent
  processes that are confounded in observable data.
• Framework for developing and testing quantitative
  theories.

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Purview of MPTs

• Categorical data each observation falls into one
  and only one of a finite set of categories.
• Example 1 Correct or incorrect.
• Example 2 Reaction time lt 100, 100 RT lt 200,
  200 RT.

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Multinomial Distribution

• Consider a die with sides 1, 1, 1, 2, 2, 3.
• If we roll the die 10 times, what is the
  probability we get five 1s, three 2s, and two
  3s?

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Multinomial Distribution

• In general,
• Inverse goal is to determine the pis given the
  nis.

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Statistical vs Explanatory

• Multinomial models (log-linear, logit) are
  statistical.
• The parameters are used to explore main effects
  and interactions.
• MPT models are explanatory.
• The parameters of the MPT model represent the
  underlying psychological processes.

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MPT Models
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MPT Models

• The probability of each process state change is
  represented by a parameter.
• The parameters range from 0 to 1.
• The parameters are independent.
• There are other restrictions

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MPT Models

• The response probabilities are given by
  polynomials, e.g., P(E1)cr.
• The parameter estimators (and other important
  properties) are easy to find.

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Estimators

• A parameter is a descriptive measure in a model
• E.g., a is a parameter the describes how quickly
  the line increases in yaxb.
• An estimator is a function on the data that gives
  a parameter estimate, usually denoted with a hat,
  â.
• E.g., given some data, get an estimate of how
  quickly the linear trend increases.
• Estimators are usually picked to minimize the
  discrepancy between the predictions and the data.

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Parameters and Flexibility

• As a rule of thumb, a model should have fewer
  parameters than degrees of freedom in the data.
• This model is saturated.

p1
C1
Cond Data Model
1 .2 .2
2 .4 .4
3 .9 .9
C2
p2
p3
C3
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Parameters and Flexibility

• In general, the more parameters, the more
  flexible the model.

yaxb
yb
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But
1 parameter
2 parameters
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Further

• It is not always easy to count parameters.

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Parameters and Flexibility
Possible Events

• A more restricted a model is
• usually simpler.
• usually easier to interpret.
• usually more falsifiable.

Observed Events
Model 1
Model 2
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Nested Models

• Models 2 and 3 are submodels (nested in) Model 1.
• Models 2 are 3 are NOT nested.
• The main benefit of nested models is that it
  makes it relatively easy to compare the
  goodness-of-fit of the two models.

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Parameters and Flexibility

• What is important is the flexibility of the model
  relative to
• the data.
• competing models.

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Identifiability

• The parameters in a model are identifiable if
  there is a unique set of parameters that give
  rise to the model predictions.
• Identifiability is especially desirable if the
  parameter values are to be interpreted.

vs
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Batchelder and Modeling

• The assumption is clearly an approximation,
  but one that greatly simplifies the analysis of
  the model and still allows the model to reflect
  the main processing stages of the task (p. 59).

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Batchelder and Modeling

• there is usually a large number of parameters
  used to account for a small number of categories,
  leaving few, if any, degrees of freedom for
  testing the models fitHowever, it is the
  measurement of the cognitive processes in the
  form of parameter estimates, and not the
  data-fitting capacity, that characterizes the
  usefulness of MPT models (p. 81-82).

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Batchelder and Modeling

• there will often be psychologically
  uniterpretable MPT models that nevertheless fit a
  given set of data well. Thus, the process of
  developing a valid model requires that one fit a
  number of data sets in the same paradigm and that
  the resulting parameter estimates be
  interpretable in terms of the underlying
  processing assumptions (p. 76).

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Batchelder and Modeling

• Each MPT model is at best an approximation to a
  complete process description of categorical data,
  and the task of the modeler is to select the most
  important processes and capture them in a valid
  way (p. 75).

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Batchelder and Modeling

• the process of developing a valid model
  requires that one fit a number of data sets in
  the same paradigm and that the resulting
  parameter estimates be interpretable in terms of
  the underlying processing assumptions (p. 76).

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Batchelder and Modeling

• an even more crucial test of a models validity
  is to show that the model performs well under
  basic experimental manipulations. If the models
  parameters behave in a psychologically
  interpretable fashion, then the model gains
  credence as a valid measurement tool (p. 82).

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Batchelder and Modeling

• an acceptable MPT model must not only be able
  to fit data but its parameters must be globally
  identifiable, must be psychologically
  interpretable, and must pass appropriate
  validation experiments (p. 78).

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Batchelder and Modeling

• there are many models for categorical data that
  are not in the MPT class. If one of these models
  accounts for data in a a particular paradigm,
  then, technically one can infer that the MPT
  class is falsified in that paradigm. Of course,
  it may be possible to design an MPT model that
  closely mimics or approximates the successful
  fits of the non-MPT model thus, it may be
  difficult to argue that the MPT framework is
  falsifiable in practice (p. 78).