This work considers Phase I cancer dual-agent dose-escalation clinical trials in which one of the compounds is an immunotherapy. The distinguishing feature of trials considered is that the dose of one agent, referred to as a standard of care, is fixed and another agent is dose-escalated. Conventionally, the goal of a Phase I trial is to find the maximum tolerated combination (MTC). However, in trials involving an immunotherapy, it is also essential to test whether a difference in toxicities associated with the MTC and the standard of care alone is present. This information can give useful insights about the interaction of the compounds and can provide a quantification of the additional toxicity burden and therapeutic index. We show that both, testing for difference between toxicity risks and selecting MTC can be achieved using a Bayesian model-based dose-escalation design with two modifications. Firstly, the standard of care administrated alone is included in the trial as a control arm and each patient is randomized between the control arm and one of the combinations selected by a model-based design. Secondly, a flexible model is used to allow for toxicities at the MTC and the control arm to be modeled directly. We compare the performance of two-parameter and four-parameter logistic models with and without randomization to a current standard of such trials: a one-parameter model. It is found that at the cost of a small reduction in the proportion of correct selections in some scenarios, randomization provides a significant improvement in the ability to test for a difference in the toxicity risks. It also allows a better fitting of the combination-toxicity curve that leads to more reliable recommendations of the combination(s) to be studied in subsequent phases.