# Is location and scale of IRT parameter estimates arbitrary?

There is a thing I’ve never understood in IRT completely.

Packages in R such as mirt or ltm are usually consistent in their estimates and seem to be unaffected with starting values. However, I am convinced there is the infinity of the best solutions.

Consider a minimalistic example:

#matrix of responses (5 persons, 4 items)
X <- matrix(c(1,1,1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0,0), nrow=5, ncol=4)

#some random starting values
beta <- rnorm(dim(X)[2]) #items difficulty
theta <- rnorm(dim(X)[1]) #level of measured trait in persons

#compute loglikelihood for given beta and theta
get_log_likelihood <- function(X, theta, beta)
{
p <- outer(theta,beta, FUN = function(x,y) 1/(1+exp(y-x)) )
return(sum(log( (1-p)*X + p*(1-X) )))
}

#naive joint maximum likelihood
loglikelihood <- 0
difference <- 1
while(difference > 1e-05)
{
D1 <- nlm(function(par){-get_log_likelihood(X, theta, par)},p=beta)
beta <- D1$estimate D2 <- nlm(function(par){-get_log_likelihood(X, par, beta)},p=theta) theta <- D2$estimate
difference <- abs(loglikelihood+D2$minimum) loglikelihood <- -D2$minimum
}

#results
loglikelihood
beta
theta


The loglikelihood of estimate is -10.40133. However, the location of beta and theta estimates can drift together along the whole real axis. If we employ discriminant parameter, also the scale of estimates starts flowing.

The question is which solution is the prefeed one. Does it have any features that make it the best one or is the unambiguity of results in R only an artefact produced by estimation method?

Your approach is not identified, which is why there's an infinite number of solutions for the estimates to land on. This is why software fix the mean and variance of the latent trait distribution beforehand. Note also that mirt and ltm use marginal maximum likelihood, so the theta distribution is integrated over rather than treated as parameters. This is the common way to deal with random effects, which you'll find in basically all latent variable modelling frameworks (e.g., structural equation modelling).