Calculation of Observed Score Distribution and Posterior Distribution of Theta with Various Information Functions

Description

This function calculates the distributioin of the weighted observed score, the information function associated with it, and the posterior distribution of theta given the observed score.
Also, the information functions associated with two types of locally optimam weights will be calculated.

Usage

obscore(param, weight = NULL, npoints = 31, thmin = -4, thmax = 4,
  thdist = 1, alpha = 0.1, compress = 0, print = 1, plot = 0,
  debug = 0)

Arguments

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 are calculated with the given set of weights.

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

The information function with LO is defined as
∑_j ∑_k (P'_{kj}(θ))^2 / P_{kj}(θ)
where P_{kj}(θ) is the item category response function,
and P'_{kj}(θ) is its derivative.

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

Value

list( theta_stat, obs_stat, Px_t, Pt_x, etcetc )

where

theta_stat as data frame
theta theta points
Pt prior probability distribution of theta
TRF test response function
slope_TRF slope of TRF
stdx_t standard deviation of X (observed score) given theta
info information function defind as (slope_TRF)^2 / (stdx_t)^2
info_LOW information function with the locally optimal item weight given categoriy weights
info_LO information function with the locally optimal category weight
qt_L upper quantile of X given theta
qt_U lower quantile of X given theta
poststd posterior std of theta given X
as a function of posterior mean

obs_stat as data frame
score domain of X (observed score)
Px marginal probability of X
post_mean posterior mean of theta given X
post_std posterior standard deviation of theta given X
ci_L upper limit of conficence interval for theta given X
ci_U lower limit of conficence interval for theta given X
ci_hwid half the width of CI


Px_t score x theta conditional prob of X given theta
Pt_x score x theta conditional prob of theta given X
Px score x 1 marginal dist of X

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.

Mayekawa, S., & Arai, S. (2008). Distribution of the Sum of Scored Multinomial Random Variables and Its Application to the Item Response Theory. In K. Shigemasu, A. Okada, T.Imaizumi, & T. Hoshino (Eds.) New Trends in Psychometrics. Tokyo: University Academic Press.

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

Examples

# Define the observed raw score X
# and calculate the score distribution, information function, etc.
res <- obscore( paramS1, plot=1 )

# Define X using the item weights w stored in weightS11
res <- obscore( paramS1, weight=weightS11, plot=1 )

# Define X using the item and category weights w and v stored in weightS12.
res <- obscore( paramS1, weight=weightS12, plot=1 )