Randomized response (RR) is an interview technique that ensures confidentiality when questions are sensitive. In RR the answer to a sensitive question depends to a certain extent on a probability mechanism. As a result the observed data are partially misclassified, and the true status of the respondent is obscured. RR data are commonly analysed in a univariate way, with models that relate the observed responses to the prevalence of the sensitive characteristic, and with the more recent logistic regression models that relate the sensitive characteristic to a set of covariates. In an RR design with multiple sensitive questions, interest is usually not confined to the univariate prevalence and regression parameter estimates. Additional multivariate information may be obtained from an RR sum score variable, assessing the sum of sensitive characteristics that are associated with the respondent. However, the construction of an RR sum score variable is by no means straightforward, which might explain why sum scores have not yet been used within the context of RR. We present two models for RR sum score variables: the RR sum score model that relates the observed sum scores to the true sum scores and the RR proportional odds model that relates the true sum scores to covariates. The models are applied to RR data from a Dutch survey on non-compliance with social security regulations.