CALCULATING MARKOV TRANSITION PROBABILITIES WHEN TREATMENT EFFECTS ARE REPORTED AS RELATIVE RISKS WITH A DIFFERENT CYCLE TIME

Purpose: It is frequently necessary to populate Markov models that have a cycle time T₁, using Relative Risks (RRs) or Odds Ratios (ORs) estimated from data collected over a different cycle time T₂. We show how this should be done for multi-state models, and estimate the error introduced by failing to do so correctly.

Method: RRs or ORs estimated from RCTs are typically defined over the length of the trial. However, probabilities are non-linear functions of time even if the hazard rate is constant. The ratio of event probabilities is therefore specific to the duration of the trial, which will usually be different to the cycle length used in a cost-effectiveness analysis. Therefore, estimates of RR and ORs reported in RCTs are not suitable for adjusting baseline transition probabilities defined over a different period of time. In the absence of competing risks we present a method in which RR estimates defined for the correct time period may be obtained using adjusted formulae in the form of ratios of power transformed event probabilities derived from the approach of Craig and Sendi(2002). An alternative, equivalent, approach is to estimate the hazard ratio (HR) by modelling the log-rates. This approach is essential when summarising results from several trials reporting RRs or ORs with different durations. In the presence of competing risks, or with interval-censored observations in more complex models, it is necessary to adjust the baseline rates using the HR. The functional relationship between the rates and probabilities is expressed through Kolmogorovs Forward Equations (Welton et al 2005). In all cases, uncertainty in the estimates may be correctly propagated into a decision model by using Markov-chain Monte-Carlo simulation in WinBUGs.

Result: The error introduced by using trial length specific estimates of RR was found to be substantial, and in most cases was many times that of the error introduced by not including a half-cycle correction. The magnitude of the error depends on the absolute and relative sizes of the event rates in each trial arm, the difference between the study duration and cycle length, the time horizon of the study, and the size of the baseline transition probability.

Conclusion: Correct adjustment of baseline transition parameters for the effect of treatment is crucial to the implementation of cost-effectiveness analysis.