Available methods for joint modelling of longitudinal and survival data typically have only one failure type for the time to event outcome. We extend the methodology to allow for competing risks data. We fit a cause-specific hazards sub-model to allow for competing risks, with a separate latent association between longitudinal measurements and each cause of failure.The method is applied to data from the SANAD trial of anti-epileptic drugs (AEDs), as a means of investigating the effect of drug titration on the relative effects of lamotrigine (LTG) and carbamazepine (CBZ) on treatment failure. Concern had been expressed that differential titration rates may have been to the disadvantage of CBZ. The beneficial effect of LTG on unacceptable adverse events leading to drug withdrawal did not lessen and indeed increased slightly when a calibrated dose was accounted for in the joint model. Adjustment for the titration rate of LTG relative to CBZ resulted in an unchanged effect of the former on drug withdrawals due to inadequate seizure control. LTG remains the AED of choice from this analysis.