conversion: Functions to Convert from and to CTT and IRT Models
In irtProb: Utilities and Probability Distributions Related to Multidimensional Person Item Response Models

Description Usage Arguments Details Value Author(s) References See Also Examples

ctt2irt and irt2ctt are converter functions to change the parametrization of item parameters from and to classical test theory (difficulty and discrimination parameters) and item response theory (difficulty and discrimination parameters). Consequently, the conversion is only valid between ctt and 2 parameters logistic or normal models.

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ctt2irt(rpbis = 0.7071068, difficulty = 0.5)

irt2ctt(a = 1, b = 0, c = 0, d = 1, model = "LOGISTIC")

`rpbis`	numeric; vector of discrimination parameters: point biserial correlation between the item response and the total score.
`difficulty`	vector of difficulty parameters: proportion of corrected responses.
`a`	numeric; vector of discrimination parameters.
`b`	numeric; vector of difficulty parameters.
`c`	numeric; vector of pseudo-guessing parameters (not used for the moment).
`d`	numeric; vector of inattention parameters (not used for the moment).
`model`	character; if NORMAL the constant D (1.702) is used. Default to LOGISTIC with constant D=1.

Eventually the 3 and 4 parameters logistic and normal models will be taken in account according to Urry approximation (1974).

`For ctt2irt`	...................................
`note`	character; warnings about the use of the `c` and `d` item parameters.
`normal.parameters`	numeric; vector returning difficulty `b` and discrimination `a` parameters from the normal model.
`irt.parameters`	numeric; vector returning difficulty `b` and discrimination `a` parameters from the logistic model.
`For irt2ctt`	...................................
`parameters`	numeric; vector returning difficulty `p` and discrimination `rpbis` parameters from the normal model.

Gilles Raiche, Universite du Quebec a Montreal (UQAM),

Departement d'education et pedagogie

Raiche.Gilles@uqam.ca, http://www.er.uqam.ca/nobel/r17165/

Bartholomiew, D. J. (1987). Latent variable models and factor analysis. London, U. K.: Charles Griffin and Company.

Lord, F. M. (1980). Applications of item response theory to practical testing problems. Mahwah, New Jersey: LEA.

Lord, F. M. and Novick, M. R. (1968). Statistical theories of mental test scores, 2nd edition. Reading, Massacusett: Addison-Wesley.

Urry, V. W. (1974). Approximations to item parameters of mental tests models and their uses. Educational and psychological measurement, 34, 253-269.

gr4pl, ggr4pl, ctt2irt, irt2ctt

## ....................................................................
# Values of p and rbis according to de a, b, c and d values
#  MODEL means that item parameters are from a NORMAL or LOGISTIC model
#  type
 irt2ctt()
 nItems <- 5
 b      <-  seq(-3, 3, length=nItems)
 a      <-  rep(1, nItems)
 c      <-  rep(0, nItems)
 d      <-  rep(1, nItems)
 
# Difference between classical item parameters and IRT ones
 irt2ctt(b=b,a=a,c=c,d=d,model="LOGISTIC")
 irt2ctt(b=b,a=a,c=c,d=d,model="NORMAL")

# Default values of a and b according p and rpbis
 ctt2irt()
 
# Verification of the recovery of original ctt item parameters
 nItems <- 5
 p      <- seq(0.10, 0.90, length=nItems)
 rpbis  <- seq(0.50, 0.95, length=nItems)
 irt    <- ctt2irt(dif=p, rpbis=rpbis)
 clas   <- irt2ctt(b=irt$irt[6:10], a=irt$irt[1:5], model="LOGISTIC")
 data.frame(NORMAL=irt$normal, IRT=irt$irt, CTT=clas,ORIGINAL=c(rpbis,p))
 clas   <- irt2ctt(b=irt$normal[6:10], a=irt$normal[1:5], model="NORMAL")
 data.frame(NORMAL=irt$normal, IRT=irt$irt, CTT=clas,ORIGINAL=c(rpbis,p))
## ....................................................................

Loading required package: lattice
Loading required package: moments
     rpbis difficulty 
 0.8621946  0.5000000 
     rpbis1      rpbis2      rpbis3      rpbis4      rpbis5 difficulty1 
0.862194552 0.862194552 0.862194552 0.862194552 0.862194552 0.995153368 
difficulty2 difficulty3 difficulty4 difficulty5 
0.902044927 0.500000000 0.097955073 0.004846632 
     rpbis1      rpbis2      rpbis3      rpbis4      rpbis5 difficulty1 
 0.70710678  0.70710678  0.70710678  0.70710678  0.70710678  0.98305257 
difficulty2 difficulty3 difficulty4 difficulty5 
 0.85557782  0.50000000  0.14442218  0.01694743 
$note
[1] For the moment, c and d parameters don't seem possible to be recovered from p and rpbis. These models cannot be compared for the moment.

$normal.parameters
a b 
1 0 

$irt.parameters
        a         b 
0.5875441 0.0000000 

       NORMAL        IRT    CTT ORIGINAL
a1  0.5773503  0.3392187 0.5000   0.5000
a2  0.7748549  0.4552614 0.6125   0.6125
a3  1.0526333  0.6184685 0.7250   0.7250
a4  1.5326552  0.9005025 0.8375   0.8375
a5  3.0424349  1.7875646 0.9500   0.9500
b1  2.5631031  2.5631031 0.1000   0.1000
b2  0.8561641  0.8561641 0.3000   0.3000
b3  0.0000000  0.0000000 0.5000   0.5000
b4 -0.6261499 -0.6261499 0.7000   0.7000
b5 -1.3490016 -1.3490016 0.9000   0.9000
       NORMAL        IRT    CTT ORIGINAL
a1  0.5773503  0.3392187 0.5000   0.5000
a2  0.7748549  0.4552614 0.6125   0.6125
a3  1.0526333  0.6184685 0.7250   0.7250
a4  1.5326552  0.9005025 0.8375   0.8375
a5  3.0424349  1.7875646 0.9500   0.9500
b1  2.5631031  2.5631031 0.1000   0.1000
b2  0.8561641  0.8561641 0.3000   0.3000
b3  0.0000000  0.0000000 0.5000   0.5000
b4 -0.6261499 -0.6261499 0.7000   0.7000
b5 -1.3490016 -1.3490016 0.9000   0.9000