Political scientists have long been interested in citizens' support level for such actors as ethnic minorities, militant groups, and authoritarian regimes. Attempts to use direct questioning in surveys, however, have largely yielded unreliable measures of these attitudes as they are contaminated by social desirability bias and high nonresponse rates. In this paper, we develop a statistical methodology to analyze endorsement experiments, which recently have been proposed as a possible solution to this measurement problem. The commonly used statistical methods are problematic because they cannot properly combine responses across multiple policy questions, the design feature of a typical endorsement experiment. We overcome this limitation by using item response theory to estimate support levels on the same scale as the ideal points of respondents. We also show how to extend our model to incorporate a hierarchical structure of data in order to uncover spatial variation of support while recouping the loss of statistical efficiency due to indirect questioning. We illustrate the proposed methodology by applying it to measure political support for Islamist militant groups in Pakistan. Simulation studies suggest that the proposed Bayesian model yields estimates with reasonable levels of bias and statistical power. Finally, we offer several practical suggestions for improving the design and analysis of endorsement experiments.