compute the likelihood of a polytomous response pattern

Usage

compute_likelihood_pcm(theta, bank, x, D = 1)

Arguments

theta: ability
bank: matrix of item parameters
x: vector of item responses scored 0 to m
D: metric constant can be 1 or 1.702

Note

this is the polytomous counterpart of prod(Pi^x*(1-Pi)^(1-x)) used for the 4PL
model
compute_pi_pcm returns a matrix so the response of item i selects the column x[i]+1
the +1 is index bookkeeping because categories start at 0 and R columns start
at 1
the probabilities of the observed responses are multiplied across items under
the local independence assumption
under the partial credit model the only part of the likelihood that depends on
both theta and the responses is exp(D*theta*sum(x))
the denominators depend on theta but not on the responses so two patterns with
the same total score on the same items have likelihoods that differ by a
constant factor

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")))
response<-c(4,3,2,1,0)
compute_likelihood_pcm(theta=0,bank=bank,x=response)
#> [1] 0.0004145528