compute item information under the partial credit model

Usage

compute_ii_pcm(theta, bank, D = 1)

Arguments

theta: ability
bank: matrix of item parameters
D: metric constant can be 1 or 1.702

Value

Ii item information for each item

Note

this function should return the same result as catR::Ii(model="PCM")
the polytomous item information is the sum over categories of the squared
first derivative divided by the probability
I_j(theta)=sum_k (dP_jk/dtheta)^2/P_jk
under the partial credit model this sum is the variance of the item score at
theta times D^2 and with the catR derivative the D^2 is left out
there is no discrimination so for a given D the height and the shape of the
curve depend only on the number of steps and on their spacing and the mean of
the steps only moves the curve
steps spread over a short range of theta or reversed give a taller curve
around their mean and steps spread over a long range give a lower and wider
curve that splits into one bump per step when the gaps are large
the variance of a score from 0 to m is at most m^2/4 which caps the height
test information is the sum of the item informations

Examples

bank<-matrix(c(-1.853,-2.214,-0.936,-1.508,-0.412,
               -0.627,-0.504,-1.215,-0.183,0.338,
               0.418,0.692,0.284,0.735,1.102,
               1.636,1.927,1.405,2.118,2.285),
             nrow=5,
             dimnames=list(c(1,2,3,4,5),
                           c("deltaj1","deltaj2","deltaj3","deltaj4")))
li_catr<-catR::Ii(th=0,bank,model="PCM")
li<-compute_ii_pcm(theta=0,bank)
li_catr
#> $Ii
#> [1] 0.8407174 0.7383865 0.8952369 0.8363394 0.9121881
#> 
#> $dIi
#> [1] -0.02462767  0.06707017 -0.26667915  0.07422137  0.35380136
#> 
#> $d2Ii
#> [1] -0.24497371 -0.14918727  0.01489645 -0.28438800 -0.39974150
#> 
li
#> $Ii
#> [1] 0.8407174 0.7383865 0.8952369 0.8363394 0.9121881
#> 
#> $dIi
#> [1] -0.02462767  0.06707017 -0.26667915  0.07422137  0.35380136
#> 
#> $d2Ii
#> [1] -0.24497371 -0.14918727  0.01489645 -0.28438800 -0.39974150
#> 
data.frame(catr=li_catr[[1]],li=li[[1]],equal=li_catr[[1]]==li[[1]])
#>        catr        li equal
#> 1 0.8407174 0.8407174  TRUE
#> 2 0.7383865 0.7383865  TRUE
#> 3 0.8952369 0.8952369  TRUE
#> 4 0.8363394 0.8363394  TRUE
#> 5 0.9121881 0.9121881  TRUE
data.frame(catr=li_catr[[2]],li=li[[2]],equal=li_catr[[2]]==li[[2]])
#>          catr          li equal
#> 1 -0.02462767 -0.02462767  TRUE
#> 2  0.06707017  0.06707017  TRUE
#> 3 -0.26667915 -0.26667915  TRUE
#> 4  0.07422137  0.07422137  TRUE
#> 5  0.35380136  0.35380136  TRUE
data.frame(catr=li_catr[[3]],li=li[[3]],equal=li_catr[[3]]==li[[3]])
#>          catr          li equal
#> 1 -0.24497371 -0.24497371  TRUE
#> 2 -0.14918727 -0.14918727  TRUE
#> 3  0.01489645  0.01489645  TRUE
#> 4 -0.28438800 -0.28438800  TRUE
#> 5 -0.39974150 -0.39974150  TRUE