  We propose a new mapping tool for supervised and unsupervised analysis of
multivariate binary data with multiple items, questions, or response variables.
The mapping assumes an underlying proximity response function, where
participants can have multiple reasons to disagree or say ``no'' to a question.
The probability to endorse, or to agree with an item depends on an item
specific parameter and the distance in a joint space between a point
representing the item and a point representing the participant. The item
specific parameter defines a circle in the joint space around the location of
the item such that for participants positioned within the circle the
probability is larger than 0.5. For map estimation, we develop and test an
MM-algorithm in which the negative likelihood function is majorized with a
weighted least squares function. The weighted least squares function can be
minimized with standard algorithms for multidimensional unfolding, except that
negative working dissimilarities may occur in the iterative process. To
illustrate the new mapping, two empirical data sets are analyzed. The mappings
are interpreted in detail and the unsupervised map is compared to a
visualization based on correspondence analysis. In a Monte Carlo study, we test
the performance of the algorithm in terms of recovery of population parameters
and conclude that this recovery is adequate. A second Monte Carlo study
investigates the predictive performance of the new mapping compared to a
similar mapping with a monotone response function.
