obscore {lazy.irt} | R Documentation |

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.

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

`param` |
Item Parameter Data Frame |

`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` |
Type of theta distribution |

`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 |

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.

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

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).

# 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]