Remap the MCMC iterates in an ideal object via an affine
transformation, imposing identifying restrictions ex post (aka
post-processing).
postProcess(object, constraints="normalize", debug = FALSE)an object of class ideal
list of length d+1, each component providing
a set of d restrictions, where d is the dimension of the
fitted ideal model; or the character string
normalize (default). If a list, the name of each component should
uniquely match a legislator/subject's name. See Details.
logical flag for verbose output, used for debugging
An object of class ideal, with components suitably
transformed and recomputed (i.e., x is transformed and
xbar recomputed, and if the
ideal object was fit with store.item=TRUE,
beta is transformed and betabar is recomputed).
Item-response models are unidentified without restrictions on the
underlying parameters. Consider the d=1 dimensional case. The
model is
$$P(y_{ij} = 1) = F(x_i \beta_j - \alpha_j)$$
Any
linear transformation of the latent traits, say,
$$x^* = mx + c$$
can be exactly offset by applying the appropriate linear
transformations to the item/bill parameters, meaning that there is no
unique set of values for the model parameters that will maximize the
likelihood function. In higher dimensions,
the latent traits can also be transformed via any arbitrary rotation,
dilation and translation, with offsetting transformations applied to
the item/bill parameters.
One strategy in MCMC is to ignore the lack of identification at run
time, but apply identifying restrictions ex post,
“post-processing” the MCMC output, iteration-by-iteration. In
a d-dimensional IRT model, a sufficient condition for global
identification is to fix d+1 latent traits, provided the
constrained latent traits span the d dimensional latent space.
This function implements this strategy. The user supplies a set of
constrained ideal points in the constraints list. The function
then processes the MCMC output in the ideal
object, finding the transformation that maps the current
iteration's sampled values for x (latent traits/ideal points)
into the sub-space of identified parameters defined by the fixed
points in constraints; i.e., what is the affine transformation
that maps the unconstrained ideal points into the constraints? Aside
from miniscule numerical inaccuracies resulting from matrix inversion
etc, this transformation is exact: after post-processing, the
d+1 constrained points do not vary over the MCMC iterations.
The remaining n-d-1 ideal points are subject to (posterior)
uncertainty; the “random tour” of the joint parameter space of
these parameters produced by the MCMC algorithm has been mapped into a
subspace in which the parameters are globally identified.
If the ideal object was produced with store.item
set to TRUE, then the item parameters are also post-processed,
applying the inverse transformation. Specifically, recall that the
IRT model is $$P(y_{ij} = 1) = F(x_i'\beta_j)$$
where in this formulation \(x_i\) is a vector of
length d+1, including a -1 to put a constant term into
the model (i.e., the intercept or difficulty parameter
is part of \(\beta_j\)). Let \(A\) denote the
non-singular, d+1-by-d+1 matrix that maps the \(x\)
into the space of identified parameters. Recall that this
transformation is computed iteration by iteration. Then each
\(x_i\) is transformed to \(x^*_i = Ax_i\) and
\(\beta_j\) is transformed to \(\beta_j^* = A^{-1}
\beta_j\), \(i = 1, \ldots, n; j = 1,
\ldots, m\).
Local identification can be obtained for a one-dimensional model by
simply imposing a normalizing restriction on the ideal points: this
normalization (mean zero, standard deviation one) is the default
behavior, but (a) is only sufficient for local identification when
the rollcall object was fit with d=1; (b) is not
sufficient for even local identification when d>1, with
further restrictions required so as to rule out other forms of
invariance (e.g., translation, or "dimension-switching", a
phenomenon akin to label-switching in mixture modeling).
The default is to impose dimension-by-dimension normalization with
respect to the means of the marginal posterior densities of the
ideal points, such that the these means (the usual Bayes estimates
of the ideal points) have mean zero and standard deviation one
across legislators. An offsetting transformation is applied to the
items parameters as well, if they are saved in the ideal object.
Specifically, in one-dimension, the two-parameter IRT model is $$P(y_{ij} = 1) = F(x_i \beta_j - \alpha_j).$$ If we normalize the \(x_i\) to \(x*_i = (x_i - c)/m\) then the offsetting transformations for the item/bill parameters are \(\beta_j^* = \beta_j m\) and \(\alpha_j^* = \alpha_j - c\beta_j\).
Hoff, Peter, Adrian E. Raftery and Mark S. Handcock. 2002. Latent Space Approaches to Social Network Analysis. Journal of the American Statistical Association 97:1090--1098.
Edwards, Yancy D. and Greg M. Allenby. 2003. Multivariate Analysis of Mulitple Response Data. Journal of Marketing Research 40:321--334.
Rivers, Douglas. 2003. “Identification of Multidimensional Item-Response Models.” Typescript. Department of Political Science, Stanford University.
# NOT RUN {
data(s109)
f = system.file("extdata",package="pscl","id1.rda")
load(f)
id1Local <- postProcess(id1) ## default is to normalize
summary(id1Local)
id1pp <- postProcess(id1,
constraints=list(BOXER=-1,INHOFE=1))
summary(id1pp)
## two-dimensional fit
f = system.file("extdata",package="pscl","id2.rda")
load(f)
id2pp <- postProcess(id2,
constraints=list(BOXER=c(-1,0),
INHOFE=c(1,0),
CHAFEE=c(0,.25)))
tracex(id2pp,d=1:2,
legis=c("BOXER","INHOFE","COLLINS","FEINGOLD","COLEMAN",
"CHAFEE","MCCAIN","KYL"))
# }
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