compute item information under the nominal response model

Usage

compute_ii_nrm(theta, bank, D = 1)

Arguments

theta: ability
bank: matrix of item parameters
D: metric constant kept for the same interface as the other files and not used

Value

Ii item information for each item
dIi first derivative of the item information for each item
d2Ii second derivative of the item information for each item
d3Ii third derivative of the item information for each item

Note

this function should return the same result as catR::Ii(model="NRM")
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 nominal response model this sum is the variance at theta of the
slope alpha_jk of the category chosen
the information is therefore large where categories with very different
slopes are both probable and it depends on the gaps between the slopes rather
than on their order
test information is the sum of the item informations
dIi and d2Ii are the first and second derivatives of the item information
with respect to theta obtained by differentiating each category term
dP_jk^2/P_jk and summing over categories
dIi=sum_k 2*dP*d2P/P-dP^3/P^2
d2Ii=sum_k (2*d2P^2+2*dP*d3P)/P-5*dP^2*d2P/P^2+2*dP^4/P^3
d3Ii=sum_k (6*d2P*d3P+2*dP*d4P)/P-(12*dP*d2P^2+7*dP^2*d3P)/P^2+
18*dP^3*d2P/P^3-6*dP^5/P^4
catR stops at d2Ii so d3Ii has no catR counterpart

Examples

bank<-matrix(c(0.614,0.782,0.297,1.036,0.521,
               1.187,0.906,1.523,0.418,0.694,
               1.395,1.487,1.103,0.688,1.214,
               1.812,1.093,2.176,1.407,1.032,
               2.106,2.391,1.724,2.012,1.873,
               1.264,0.517,1.936,0.793,0.186,
               2.918,3.087,2.215,2.604,3.382,
               0.097,-0.814,1.012,-0.296,-1.618),
             nrow=5,
             dimnames=list(c(1,2,3,4,5),
                           c("alpha1","c1","alpha2","c2","alpha3","c3","alpha4","c4")))
li_catr<-catR::Ii(th=0,bank,model="NRM") # this will work with catR package installed
li<-compute_ii_nrm(theta=0,bank)
li_catr
#> $Ii
#> [1] 0.5509014 0.6793970 0.4151954 0.5674988 0.5029803
#> 
#> $dIi
#> [1]  0.03268822  0.11239248 -0.06326216  0.22564642  0.28274374
#> 
#> $d2Ii
#> [1] -0.1263039 -0.2779506 -0.1592400 -0.2688705  0.3498580
#> 
li
#> $Ii
#> [1] 0.5509014 0.6793970 0.4151954 0.5674988 0.5029803
#> 
#> $dIi
#> [1]  0.03268822  0.11239248 -0.06326216  0.22564642  0.28274374
#> 
#> $d2Ii
#> [1] -0.1263039 -0.2779506 -0.1592400 -0.2688705  0.3498580
#> 
#> $d3Ii
#> [1] -0.01432536 -0.35685103  0.16250230 -0.80383781  0.49688954
#> 
data.frame(catr=li_catr[[1]],li=li[[1]],equal=li_catr[[1]]==li[[1]])
#>        catr        li equal
#> 1 0.5509014 0.5509014  TRUE
#> 2 0.6793970 0.6793970  TRUE
#> 3 0.4151954 0.4151954  TRUE
#> 4 0.5674988 0.5674988  TRUE
#> 5 0.5029803 0.5029803  TRUE
data.frame(catr=li_catr[[2]],li=li[[2]],equal=li_catr[[2]]==li[[2]])
#>          catr          li equal
#> 1  0.03268822  0.03268822  TRUE
#> 2  0.11239248  0.11239248  TRUE
#> 3 -0.06326216 -0.06326216  TRUE
#> 4  0.22564642  0.22564642  TRUE
#> 5  0.28274374  0.28274374  TRUE
data.frame(catr=li_catr[[3]],li=li[[3]],equal=li_catr[[3]]==li[[3]])
#>         catr         li equal
#> 1 -0.1263039 -0.1263039  TRUE
#> 2 -0.2779506 -0.2779506  TRUE
#> 3 -0.1592400 -0.1592400  TRUE
#> 4 -0.2688705 -0.2688705  TRUE
#> 5  0.3498580  0.3498580  TRUE