obscore {lazy.irt} R Documentation

## 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

 `param` Item Parameter Data Frame `weight` Weight data frame or a vector containing item weights. or a scalar or NULL. `npoints` # of discrete points for theta `thmin` Minimum value of discrete thata value `thmax` Maximum value of discrete thata value `thdist` Type of theta distribution = 0 to use uniform, = 1 to use N(0,1) `alpha` small prob for quantile and confidence interval `compress` = 1 to remove zero-probability weighted total observed scores `print` = 1 to print result `plot` = 1 to plot result `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 are calculated with the given set of weights.
This is the information function associated with the test scores
calculated using the given item and category weights.

The information can be increased by adjusting the weights:
However, 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 and defined as:.
w_j(θ)= P^{*'}_j(θ) / var( U_j^{*} | θ)
where U_j^{*} = ∑_k v_{kj} U_{kj} and P^{*'}_j(θ)=∑_k v_{kj} P_{kj}(θ) .
When the category weights themselves are optimized it is called as the locally optimal weights, or, LO,
which are related to the basic function of Samejima(1969) defined as:.
v_{kj}(θ)=P'_{kj}(θ) / P_{kj}(θ) - P'_{0j}(θ) / P_{0j}(θ) .

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.
This is the maximum information function for all the theta range.
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.

The test reliability coefficient is calculated as `rel = (SigmaX^2-SigmaE^2)/SigmaX^2`,
where SigmaE^2 is the square root of the average of stdx_t^2,
SigmaX^2 is the variance of the observed score.

### Value

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

where

theta_stat as data frame containing the following:
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 containing the following:
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

SigmaX2, SigmaT2, SigmaE2: (observed, true, and average error variances)
aSEM=sqrt(SigmaE2): average Standard Error of Measurement,
rel: test reliability
SigmaT_theta2, SigmaE_theta2, aSEM_theta, rel_theta
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

pdfname: item parameter data frame name
wdfname: item weight data frame name
npoints: # of theta points
thmin, thmax: the range of theta
thdist: Type of theta distribution
nitems: # of items
minscore_t: minimum score
maxscore_t: maximum score
alpha: small probability value

### 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. (2018) A Method to Estimate the Ability from the Weighted Total Score wich Minimizes the Standard Error of Estimation. DNC Research Note. RN-18-01.

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 )

```

[Package lazy.irt version 0.1.3 Index]