info_func {lazy.irt}

R Documentation

Calculation of Various Information Functions

Description

This function calculates the information functions associated with the set of given weights,
and two types of locally optimam weights.

Usage

info_func(
  param,
  theta = NULL,
  weight = NULL,
  npoints = 31,
  thmin = -4,
  thmax = 4,
  thdist = "NORMAL",
  thmean = 0,
  thstd = 1,
  legend = 1,
  maxinfo = 0,
  numderiv = 0,
  smallP = 1e-09,
  print = 1,
  plot = 0,
  debug = 0
)

Arguments

param	Item Parameter Data Frame.
theta	Discrete theta values
weight	Weight data frame.
npoints	# of discrete points for theta.
thmin	Minimum value of discrete thata value.
thmax	Maximum value of discrete thata value.
thdist	= "NORMAL" or "UNIFORM" When "NORMAL" thmean and thstd will be used to generate thd. When "UNIFORM" thmin and thmax will be used to generate thd.
thmean	mean of theta distribution
thstd	std of theta distribution
legend	= 0 to skip printing legend.
maxinfo	= The maximum value of item information function for plot.
numderiv	= 1 to use numerical derivatives.
smallP	Minimum value of probability in irf and dirf functions.
print	= 1 to print the summary = 2 to print test information functions. = 3 to print item information functions. = 4 to print the locally best item categorie weights.
plot	= 1 to plot test information functions = 2 to plot item information functions. = 3 to plot the locally best item and item categorie weights.
debug	= 1 to print intemediate result.

Details

Note that, given category and item weights, information function is defined as
( slope of TRF at theta )^2 / (variance of x at theta)
where TRF and x is calculated with the given set of weights.

In general. the best weights depends on the value of theta. Therefore, the name locally best weight.
The best item weight given category weights is called as locally best item weight, or LOW.
When the category weights themselves are optimized it is called as the locally best weight, or, LO.
This weight is equivalent to the basic function of Samejima(1969) .

The information function with LO is defined as
\sum_j \sum_k (P'_{kj}(\theta))^2 / P_{kj}(\theta)
where P_{kj}(\theta) is the item category response function,
and P'_{kj}(\theta) is its derivative.

The information function with LOW is defined as
\sum_j (P_j^{*'}(\theta))^2 / var(U_j^{*} | \theta)
where U_j^{*} = \sum_k v_{kj} U_{kj} is the weighted item score,
and P_j^{*'}(\theta) is the derivative of the expected value of U_j^{*} at theta.


The Basic Function is defined as P`(theta) / P(theta) .
The Best Item Category Weight = Basic Function - Basic Function of the 0-th category.

Item Info Function is defined as \sum_k (P'_{kj}(\theta))^2 / P_{kj}(\theta) .

Value

A list of

theta: theta points
thd: theta distribution (to calculate average info)
info : information function defind as (slope_TRF)^2 / (stdx_t)^2
fromP, toP: location of each item category in item info
TRF: test response function (tcc)
TRF_LO: test response function (tcc) with locally best weight
icrf: item category response functions
dicrf: derivative of icrf
info_LOW: information function with locally best item weight
info_LO: information function with locally best category weight
info_item_LOW: item information function with locally best item weight
info_item_LO: item information function with locally best category weight
stdx_t: SEM at theta (conditional standard deviation of X given theta)

w_LO: the locally best item weight
v_LO: the locally best item category weight scaled so that v[0,j]=0, j=1,2,...,nitems.
A: The Basic Function

a_stdx_t: square root of the average of stdx_t^2
a_info: average of info
a_info_LOW: average of info_LOW
a_info_LO: average of info_LO
a_info_item_LOW: average of info_item_LOW (for each item)
a_info_item_LO: average of info_item_LO (for each item)
a_w_LO: average of w_LO (for each item)
a_v_LO: average of v_LO (for each item category)

References

Birnbaum, A.(1968) Some Latent Traint Models. In F. M. Lord and M. R. Novick, Statistical Theories of Mental Test Scores. Reading, Mass.: Addison-Wesley.

Samejima, F. (1969). Estimation of a latent ability using a response pattern of graded scores. Psychometrika Monographs, 34 (Suppl. 4).

Examples

resInfo <- info_func( paramS2, plot=2, print=1 )
resInfo <- info_func( paramS2, weight=weightS21, plot=1, print=0 )
resInfo <- info_func( paramS2, weight=weightS22, plot=1, print=0 )

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