mptspec: Specify a Multinomial Processing Tree (MPT) Model

Description

Returns the specification of an MPT model object for fitting with mpt.

Usage

mptspec(…, .restr = NULL)
# S3 method for mptspec
update(object, .restr = NULL, …)

Arguments

…

(named) expressions or a character string specifying the model. See Details.

.restr

a named list of parameter restrictions. See Details.

object

an object of class mptspec.

Value

An object of class mptspec that serves as input to mpt which fits the model to data. It consists of the following components:

par2prob

a function that takes a vector of parameter values and computes the response probabilities.

par2deriv

a function that takes a vector of parameter values and computes first and second derivatives of the model equations.

prob

a list containing expressions of the model equations.

deriv

a list containing expressions of the first and second derivatives of the model equations.

par

a named vector of parameter values.

restr

a list containing expressions of parameter restrictions.

treeid

a vector that identifies each tree.

Details

… is used to symbolically specify the MPT model equations by suitable expressions, for example, they could look like this

r + (1 - r)*b, (1 - r)*(1 - b), b, 1 - b

where each expression represents the probability of a response in the corresponding category (link probabilities are multiplied, branch probabilities are added). Thus, there usually are as many expressions as response categories.

Joint (or product) multinomial models consist of more than a single processing tree. To identify the trees in such a model, expressions may have optional names. Canonically, these names are of the form x.y, where x is the tree identifier (treeid) and y specifies the response category within a tree.

Alternatively, … may be a character string identifying one out of a list of prevalent MPT models. Currently implemented are the following models (other models have to be specified by explicit expressions as described above):

1HT: the one-high-threshold model (Blackwell, 1963; Swets, 1961).

2HT: the two-high-threshold model (Snodgrass & Corwin, 1988; see also Broeder & Schuetz, 2009).

PairAsso: the paired-associate learning model (Riefer & Batchelder, 1988).

prospec: the event-based prospective memory model (Smith & Bayen, 2004).

rmodel: the r-model of recognition heuristic use (Hilbig, Erdfelder, & Pohl, 2010).

SourceMon: the source-monitoring model (Batchelder & Riefer, 1990).

SR, SR2: the storage-retrieval model for pair clustering (Batchelder & Riefer, 1986). SR2 is the model without singleton items.

WST: the inference-guessing model with relaxed assumptions (Klauer, Stahl, & Erdfelder, 2007) for the Wason selection task.

If one of these models is selected, … may include an optional .replicates argument that specifies the number of replicates of the model equations, for example, when the same model is applied repeatedly in several experimental conditions. Accordingly, parameter names are augmented by numbers to make them unique.

Parameter restrictions included in .restr may be of the form b = r or b = 0.5 etc. Depending on the fitting algorithm employed in mpt (BFGS, but not EM), mathematical functions are permissible, for example, b = sqrt(r).

The update method is used to add parameter restrictions to an existing mptspec object.

References

Batchelder, W.H., & Riefer, D.M. (1986). The statistical analysis of a model for storage and retrieval processes in human memory. British Journal of Mathematical and Statistical Psychology, 39, 129--149. 10.1111/j.2044-8317.1986.tb00852.x

Batchelder, W.H., & Riefer, D.M. (1990). Multinomial processing models of source monitoring. Psychological Review, 97, 548--564. 10.1037/0033-295x.97.4.548

Blackwell, H.R. (1963). Neural theories of simple visual discriminations. Journal of the Optical Society of America, 53, 129--160. 10.1364/JOSA.53.000129

Broeder, A., & Schuetz, J. (2009). Recognition ROCs are curvilinear--or are they? On premature arguments against the two-high-threshold model of recognition. Journal of Experimental Psychology: Learning, Memory, and Cognition, 35, 587--606. 10.1037/a0015279

Hilbig, B.E., Erdfelder, E., & Pohl, R.F. (2010). One-reason decision-making unveiled: A measurement model of the recognition heuristic. Journal of Experimental Psychology: Learning, Memory, and Cognition, 36, 123--134. 10.1037/a0017518

Klauer, K.C., Stahl, C., & Erdfelder, E. (2007). The abstract selection task: New data and an almost comprehensive model. Journal of Experimental Psychology: Learning, Memory, and Cognition, 33, 680--703. 10.1037/0278-7393.33.4.680

Riefer, D.M., & Batchelder, W.H. (1988). Multinomial modeling and the measurement of cognitive processes. Psychological Review, 95, 318--339. 10.1037/0033-295x.95.3.318

Smith, R.E., & Bayen, U.J. (2004). A multinomial model of event-based prospective memory. Journal of Experimental Psychology: Learning, Memory, and Cognition, 30, 756--777. 10.1037/0278-7393.30.4.756

Snodgrass, J.G., & Corwin, J. (1988). Pragmatics of measuring recognition memory: Applications to dementia and amnesia. Journal of Experimental Psychology: General, 117, 34--50. 10.1037/0096-3445.117.1.34

Swets, J. A. (1961). Is there a sensory threshold? Science, 134, 168--177. 10.1126/science.134.3473.168

Examples

Run this code

# NOT RUN {
## Specify storage-retrieval model for pairs
spec1 <- mptspec(
  c*r,
  (1 - c)*u^2,
  2*(1 - c)*u*(1 - u),
  c*(1 - r) + (1 - c)*(1 - u)^2
)

## Specify storage-retrieval model with parameter restrictions
spec2 <- mptspec(
  c*r,
  (1 - c)*u^2,
  2*(1 - c)*u*(1 - u),
  c*(1 - r) + (1 - c)*(1 - u)^2,
  .restr = list(c = r/2, u = 0.3)
)

## Optional names identifying trees in joint MPT model
spec3 <- mptspec(
  "1.1" = r + (1 - r)*b,
  "1.2" = (1 - r)*(1 - b),
  "2.1" = b,
  "2.2" = 1 - b
)

## Fit one-high-threshold model to data in Broeder and Schuetz (2009)
m <- mpt(spec <- mptspec("1HT"), c(55, 35, 45, 765))

## Working with the mptspec object
spec$par2prob(c(0.5, 0.1))     # response probabilities
spec$par2deriv(coef(m))$deriv  # Jacobian matrix at ML estimate

## See data(package = "mpt") for application examples.
# }

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