MCMCordfactanal(x, factors, lambda.constraints=list(),
data=list(), burnin = 1000, mcmc = 10000,
thin=5, tune=NA, verbose = FALSE, seed = 0,
lambda.start = NA, l0=0, L0=0,
store.lambda=TRUE, store.scores=FALSE,
drop.constantvars=TRUE, ... )varname=list(d,c) which
will constrain the dth loading for the variable named lambda.start is set to a scalar the starting value for
all unconstrained loadings will be set to that scalar. If
lambda.start is a matrix of the same dimensionsLambda.Lambda.Let $i=1,\ldots,N$ index observations and $j=1,\ldots,K$ index response variables within an observation. The typical observed variable $x_{ij}$ is ordinal with a total of $C_j$ categories. The distribution of $X$ is governed by a $N \times K$ matrix of latent variables $X^*$ and a series of cutpoints $\gamma$. $X^*$ is assumed to be generated according to: $$x^*_i = \Lambda \phi_i + \epsilon_i$$ $$\epsilon_i \sim \mathcal{N}(0,I)$$
where $x^*_i$ is the $k$-vector of latent variables specific to observation $i$, $\Lambda$ is the $k \times d$ matrix of factor loadings, and $\phi_i$ is the $d$-vector of latent factor scores. It is assumed that the first element of $\phi_i$ is equal to 1 for all $i$.
The probability that the $j$th variable in observation $i$ takes the value $c$ is:
$$\pi_{ijc} = \Phi(\gamma_{jc} - \Lambda'_j\phi_i) - \Phi(\gamma_{j(c-1)} - \Lambda'_j\phi_i)$$ The implementation used here assumes independent conjugate priors for each element of $\Lambda$ and each $\phi_i$. More specifically we assume:
$$\Lambda_{ij} \sim \mathcal{N}(l_{0_{ij}}, L_{0_{ij}}^{-1}), i=1,\ldots,k, j=1,\ldots,d$$
$$\phi_i \sim \mathcal{N}(0, I), i=1,\dots,n$$
The standard two-parameter item response theory model with probit
link is a special case of the model sketched above.
MCMCordfactanal simulates from the posterior density using
standard Metropolis-Hastings within Gibbs sampling. The algorithm
employed is based on work by Cowles (1996). Note that
the first element of $\phi_i$ is a 1. As a result, the
first column of $\Lambda$ can be interpretated as item
difficulty parameters. Further, the first
element $\gamma_1$ is normalized to zero, and thus not
returned in the mcmc object.
The simulation proper is done in compiled C++ code to maximize
efficiency. Please consult the coda documentation for a comprehensive
list of functions that can be used to analyze the posterior density
sample.
M. K. Cowles. 1996. ``Accelerating Monte Carlo Markov Chain Convergence for
Cumulative-link Generalized Linear Models." Statistics and Computing.
6: 101-110.
Valen E. Johnson and James H. Albert. 1999. ``Ordinal Data Modeling."
Springer: New York.
Andrew D. Martin, Kevin M. Quinn, and Daniel Pemstein. 2003.
Scythe Statistical Library 0.4.
plot.mcmc, summary.mcmc,
factanal, MCMCfactanal,
MCMCirt1d, MCMCirtKddata(painters)
new.painters <- painters[,1:4]
cuts <- apply(new.painters, 2, quantile, c(.25, .50, .75))
for (i in 1:4){
new.painters[new.painters[,i]<cuts[1,i],i] <- 100
new.painters[new.painters[,i]<cuts[2,i],i] <- 200
new.painters[new.painters[,i]<cuts[3,i],i] <- 300
new.painters[new.painters[,i]<100,i] <- 400
}
posterior <- MCMCordfactanal(~Composition+Drawing+Colour+Expression,
data=new.painters, factors=1,
lambda.constraints=list(Drawing=list(2,"+")),
burnin=5000, mcmc=500000, thin=200, verbose=TRUE,
L0=0.5, store.lambda=TRUE,
store.scores=TRUE, tune=.6)
plot(posterior)
summary(posterior)Run the code above in your browser using DataLab