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In our data world, a host of not necessarily trusted controllers gather data on individual subjects. To preserve her privacy and, more generally, her informational self-determination, the individual has to be empowered by giving her agency on her own data. Maximum agency is afforded by local anonymization, that allows each individual to anonymize her own data before handing them to the data controller. Randomized response (RR) is a local anonymization approach able to yield multi-dimensional full sets of anonymized microdata that are valid for exploratory analysis and machine learning. This is so because an unbiased estimate of the distribution of the true data of individuals can be obtained from their pooled randomized data. Furthermore, RR offers rigorous privacy guarantees. The main weakness of RR is the curse of dimensionality when applied to several attributes: as the number of attributes grows, the accuracy of the estimated true data distribution quickly degrades. We propose several complementary approaches to mitigate the dimensionality problem. First, we present two basic protocols, separate RR on each attribute and joint RR for all attributes, and discuss their limitations. Then we introduce an algorithm to form clusters of attributes so that attributes in different clusters can be viewed as independent and joint RR can be performed within each cluster. After that, we introduce an adjustment algorithm for the randomized data set that repairs some of the accuracy loss due to assuming independence between attributes when using RR separately on each attribute or due to assuming independence between clusters in cluster-wise RR. We also present empirical work to illustrate the proposed methods.