compute category response probabilities and derivatives under the nominal response model

returns the category response probabilities for a given ability value and a
given matrix of item parameters under Bock nominal response model (NRM)
returns the first second and third derivatives of the category response
probabilities

Usage

compute_pi_nrm(theta, bank, D = 1)

Arguments

theta: ability
bank: item parameters one row per item
the columns come in pairs alpha_jk and c_jk for the categories 1 to m
so the columns are alpha1 c1 alpha2 c2 up to alpha_m c_m
alpha_jk is the slope and c_jk the intercept of category k
category 0 is the reference category with slope 0 and intercept 0 so it
has no columns
the categories do not need to be ordered which is what makes the model
nominal
an item with fewer categories than the widest item is padded with NA pairs
a mirt model fitted with itemtype="nominal" converts with
pars<-mirt::coef(model,simplify=TRUE)$items
m<-length(grep("^ak",colnames(pars)))-1
alpha<-pars[,"a1"]*pars[,paste("ak",1:m,sep="")]
cj<-pars[,paste("d",1:m,sep="")]
bank<-cbind(alpha,cj)[,order(rep(1:m,2))]
colnames(bank)<-paste(c("alpha","c"),rep(1:m,each=2),sep="")
mirt renumbers the observed categories of each item from 0 so the
responses must be coded the same way
D: metric constant kept for the same interface as the other files
the nominal response model has no metric constant because any constant
multiplying the exponent is absorbed in the slopes and the intercepts so D
is not used and catR ignores it too

Value

Pi category response probabilities one row per item one column per category
dPi first derivatives of the category response probabilities
d2Pi second derivatives of the category response probabilities
d3Pi third derivatives of the category response probabilities
d4Pi fourth derivatives of the category response probabilities which catR
does not return and which the third derivative of the information needs

Note

this function should return the same result as catR::Pi(model="NRM")
the number of response categories is ncol(bank)/2+1 so 5 response categories
coded 0 1 2 3 4 need a bank with 8 columns
the probability of category k is
P(X=k|theta)=exp(alpha_jk*theta+c_jk)/sum over all categories
with alpha_j0*theta+c_j0 equal to zero
log(P(X=k)/P(X=0))=alpha_jk*theta+c_jk is a straight line so the most
probable category at theta is the category with the highest line
at very high theta that is the category with the largest slope and at very low
theta the category with the smallest slope
when two categories share that slope their lines are parallel and the one
with the larger intercept wins
as theta grows the categories take turns being the most probable in the order
of their slopes and not of their codes
a category is skipped when the next category overtakes it before it overtakes
the previous one and this can happen with ordered slopes too
with D=1 the partial credit model is the nominal model with alpha_jk=k and
c_jk=-(delta_j1+...+delta_jk) and the generalized partial credit model
multiplies both by the item discrimination
for a bank used with D=1.702 both are multiplied by D as well because the
nominal model ignores D
in the code dj holds alpha_jk*theta+c_jk and gamma holds exp(dj) for each
category while v holds the slope alpha_jk
the exponent of category k grows by alpha_jk when theta grows by one so the
derivative of gamma is gamma*v and is exact
each further derivative multiplies by v again so gamma*v^2 and gamma*v^3 give
the second and third derivatives of Pi through the quotient rule

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")))
pi_cat<-catR::Pi(th=0,bank,model="NRM") # this will work with catR package installed
pi<-compute_pi_nrm(theta=0,bank)
pi_cat
#> $Pi
#>             cat0      cat1      cat2      cat3       cat4
#> Item1 0.06648350 0.2178820 0.4070572 0.2353218 0.07325553
#> Item2 0.11658155 0.2884700 0.3477873 0.1955060 0.05165521
#> Item3 0.04152992 0.1904547 0.3659198 0.2878427 0.11425288
#> Item4 0.10464180 0.1589426 0.4273242 0.2312601 0.07783125
#> Item5 0.13867515 0.2775869 0.3892159 0.1670234 0.02749857
#> 
#> $dPi
#>              cat0        cat1         cat2      cat3       cat4
#> Item1 -0.09380632 -0.17364578 -0.006501355 0.1635553 0.11039816
#> Item2 -0.15967681 -0.16952159  0.040810176 0.1996785 0.08870968
#> Item3 -0.05022984 -0.17378714 -0.038965175 0.1480992 0.11488292
#> Item4 -0.11789275 -0.01440521 -0.187437888 0.2047504 0.11498544
#> Item5 -0.14185996 -0.13933920  0.074353479 0.1419756 0.06487007
#> 
#> $d2Pi
#>             cat0        cat1       cat2        cat3       cat4
#> Item1 0.09573221  0.01835925 -0.2241445 -0.01596350 0.12601659
#> Item2 0.13949741 -0.09636499 -0.2314969  0.07111397 0.11725049
#> Item3 0.04350924  0.07950235 -0.1477790 -0.04331177 0.06807917
#> Item4 0.07343759 -0.08889417 -0.1602898  0.05003971 0.12570672
#> Item5 0.07536704 -0.06967723 -0.1815639  0.03667460 0.13919947
#> 
#> $d3Pi
#>               cat0        cat1         cat2       cat3        cat4
#> Item1 -0.033892584  0.16956944 -0.002562816 -0.1989930  0.06587899
#> Item2  0.012801319  0.25455269 -0.121705687 -0.2206637  0.07501539
#> Item3 -0.008286124  0.08381506  0.071241527 -0.1270552 -0.01971524
#> Item4  0.027458848 -0.01145833  0.186625566 -0.2402707  0.03764459
#> Item5  0.026398083  0.09665937 -0.219529905 -0.1588720  0.25534450
#> 
pi
#> $Pi
#>             cat0      cat1      cat2      cat3       cat4
#> Item1 0.06648350 0.2178820 0.4070572 0.2353218 0.07325553
#> Item2 0.11658155 0.2884700 0.3477873 0.1955060 0.05165521
#> Item3 0.04152992 0.1904547 0.3659198 0.2878427 0.11425288
#> Item4 0.10464180 0.1589426 0.4273242 0.2312601 0.07783125
#> Item5 0.13867515 0.2775869 0.3892159 0.1670234 0.02749857
#> 
#> $dPi
#>              cat0        cat1         cat2      cat3       cat4
#> Item1 -0.09380632 -0.17364578 -0.006501355 0.1635553 0.11039816
#> Item2 -0.15967681 -0.16952159  0.040810176 0.1996785 0.08870968
#> Item3 -0.05022984 -0.17378714 -0.038965175 0.1480992 0.11488292
#> Item4 -0.11789275 -0.01440521 -0.187437888 0.2047504 0.11498544
#> Item5 -0.14185996 -0.13933920  0.074353479 0.1419756 0.06487007
#> 
#> $d2Pi
#>             cat0        cat1       cat2        cat3       cat4
#> Item1 0.09573221  0.01835925 -0.2241445 -0.01596350 0.12601659
#> Item2 0.13949741 -0.09636499 -0.2314969  0.07111397 0.11725049
#> Item3 0.04350924  0.07950235 -0.1477790 -0.04331177 0.06807917
#> Item4 0.07343759 -0.08889417 -0.1602898  0.05003971 0.12570672
#> Item5 0.07536704 -0.06967723 -0.1815639  0.03667460 0.13919947
#> 
#> $d3Pi
#>               cat0        cat1         cat2       cat3        cat4
#> Item1 -0.033892584  0.16956944 -0.002562816 -0.1989930  0.06587899
#> Item2  0.012801319  0.25455269 -0.121705687 -0.2206637  0.07501539
#> Item3 -0.008286124  0.08381506  0.071241527 -0.1270552 -0.01971524
#> Item4  0.027458848 -0.01145833  0.186625566 -0.2402707  0.03764459
#> Item5  0.026398083  0.09665937 -0.219529905 -0.1588720  0.25534450
#> 
#> $d4Pi
#>              cat0        cat1       cat2        cat3        cat4
#> Item1 -0.09279934 -0.12093656 0.42253604 -0.09823979 -0.11056034
#> Item2 -0.21561247  0.18415967 0.54046104 -0.38330315 -0.12570509
#> Item3 -0.04709228 -0.17816151 0.22735926  0.06252010 -0.06462558
#> Item4 -0.04802192  0.20486694 0.43281201 -0.37434507 -0.21531195
#> Item5 -0.06891532  0.07769496 0.03279227 -0.36924907  0.32767716
#> 
isTRUE(all.equal(pi_cat[[1]], pi[[1]]))
#> [1] TRUE
isTRUE(all.equal(pi_cat[[2]], pi[[2]]))
#> [1] TRUE
isTRUE(all.equal(pi_cat[[3]], pi[[3]]))
#> [1] TRUE
isTRUE(all.equal(pi_cat[[4]], pi[[4]]))
#> [1] TRUE