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.

翻译：我们提出一种新的映射工具，用于对包含多个项目、问题或响应变量的多元二值数据进行监督与无监督分析。该映射假设存在隐含的邻近响应函数，其中参与者可能因多种原因对问题表示不同意或回答“否”。赞同或同意某个项目的概率取决于项目特定参数，以及项目代表点与参与者代表点在联合空间中的距离。项目特定参数在联合空间中围绕项目位置定义了一个圆形区域，使得位于该圆形区域内的参与者，其赞同概率大于0.5。为进行映射估计，我们开发并测试了一种MM算法，其中负似然函数通过加权最小二乘函数进行优化上界逼近。该加权最小二乘函数可利用标准的多维展开算法进行最小化，但迭代过程中可能出现负的工作差异度。为展示新映射，我们分析了两个实证数据集。对映射结果进行了详细解释，并将无监督映射与基于对应分析的可视化结果进行比较。通过蒙特卡洛研究，我们测试了算法在群体参数恢复方面的性能，结果表明恢复效果良好。第二个蒙特卡洛研究考察了新映射相较于具有单调响应函数的相似映射的预测性能。