Description Usage Arguments Details Value Note Author(s) References Examples

`information`

computes the Fisher (test, item, and category) and observed (test only) information for an item response model. Fisher information can be computed at any specified value of `zeta`

but observed information is computed only at the MLE.

1 | ```
information(fmodel, y, zeta, observed = FALSE, ...)
``` |

`fmodel` |
Function with first argument |

`y` |
Vector of length m for a single response pattern, or matrix of size s by m of a set of s item response patterns. In the latter case the posterior is computed by conditioning on the event that the response pattern is one of the s response patterns. Elements of |

`zeta` |
The value of the latent trait at which to compute Fisher information. Observed information is always computed at the MLE regardless of |

`observed` |
Logical to determine if the observed information is computed. The default is FALSE, but only the observed information can be computed if |

`...` |
Additional arguments to be passed to |

The Fisher information is defined here as the negative of the expected value of the second-order derivative of the log-likelihood function for *ζ_i*. This is the test information function. The item and category Fisher information functions are defined by decomposing this quantity by item and category, respectively (see Baker & Kim, 2004). The observed information is the second-order derivative of the log-likelihood evaluated evaluated at the MLE of *ζ_i* which is computed using `postmode`

. The observed information function is only computed here for the test. The Fisher information cannot be computed by `information`

if `y`

has more than one row (i.e., more than one response pattern).

`test ` |
Test information at |

`item ` |
Item information at |

`category ` |
Category informtation at |

For generality `information`

computes Fisher and observed information using numerical (partial) differentiation even when closed-form solutions exist. Thus even though it does not depend on `y`

that argument must still be provided for computational purposes. General and some model-specific closed-form formulas for test/item/category Fisher information are given by Baker and Kim (2004).

Timothy R. Johnson

Baker, F. B. & Kim, S. H. (2004). *Item response theory: Parameter estimation techniques* (2nd ed.). New York, NY: Marcel-Dekker.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
alph <- c(1.27,1.34,1.14,1,0.67) # discrimination parameters
beta <- c(1.19,0.59,0.15,-0.59,-2) # difficulty parameters
gamm <- c(0.1,0.15,0.15,0.2,0.01) # lower asymptote parameters
# Fisher information of a three-parameter logistic binary model
information(fmodel3pl, y = c(0,1,1,1,1), apar = alph, bpar = beta, cpar = gamm)
# plot of Fisher information functions for each item
zeta <- seq(-5, 5, length = 100)
info <- matrix(NA, 100, 5)
for (j in 1:100) {
info[j,] <- information(fmodel3pl, c(0,1,1,1,1), zeta = zeta[j],
apar = alph, bpar = beta, cpar = gamm)$item
}
matplot(zeta, info, type = "l", ylab = "Information", bty = "n", xlab = expression(zeta))
legend(-3, 0.3, paste("Item", 1:5), lty = 1:5, col = 1:5)
# observed information given a sum score of 4
information(fmodel3pl, patterns(5, 2, 4), apar = alph, bpar = beta, cpar = gamm,
observed = TRUE)
``` |

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