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Abstract Conjoint analysis is a popular experimental design used to measure multidimensional preferences. Many researchers focus on estimating the average marginal effects of each factor while averaging over the other factors. Although this allows for straightforward design-based estimation, the results critically depend on the ways in which factors interact with one another. An alternative model-based approach can compute various quantities of interest, but requires correct model specifications, a challenging task for conjoint analysis with many factors. We propose a new hypothesis testing approach based on the conditional randomization test (CRT) to answer the most fundamental question of conjoint analysis: Does a factor of interest matterin any waygiven the other factors? Although it only provides a formal test of these binary questions, the CRT is solely based on the randomization of factors, and hence requires no modeling assumption. This means that the CRT can provide a powerful and assumption-free statistical test by enabling the use of any test statistic, including those based on complex machine learning algorithms. We also show how to test commonly used regularity assumptions. Finally, we apply the proposed methodology to conjoint analysis of immigration preferences. An open-source software package is available for implementing the proposed methodology. The proposed methodology is implemented via an open-source software R packageCRTConjoint, available through the Comprehensive R Archive Networkhttps://cran.r-project.org/web/packages/CRTConjoint/index.html.