Purpose: The widely used formula ‚i =1-(1-pi)s/t for converting a probability pi over time interval t into a transition probability ‚i for a Markov model with cycle length s ignores competing risks. We demonstrate anomalies with this approach and derive formulas that take into account the dynamics of competing risks and compute the bias resulting from using the traditional approach in a liver disease Markov model.
Methods: The three-state model with competing risks consists of patients starting at decompensated-cirrhosis (DeCirr), and moving to either hepatocellular carcinoma (HCC) or "Death"ócompeting risk (Figure 1). Using the relationship between the transition probability and rate matrices, we derived formulas to convert t-interval probabilities into transition probabilities for the model with a fixed cycle length s. Setting s = 1/52.18 (1 week) and t=1 (1 year), and assuming specific values for inputs (Figure 1a), we estimated the cumulative incidence of HCC and "Death", total discounted (at an annual rate of 0.03) cost and quality-adjusted life years (QALYs) using the corrected formula and the traditional approach.
Results: The formulas for converting t-interval probabilities into cycle-length s transition probabilities are shown in Figure 1b. The "uncorrected" approach gave ‚1=0.0013, ‚2=0.0029, and ‚3=0.0106, whereas the "corrected" approach yielded „1=0.0019, „2 =0.0025, and „3 =0.0106. The n-step (n=52.18) transition probability matrix (TPM) of weekly cycle-length did not yield back the original annual TPM with the "uncorrected" approach. The estimated total cost using the "uncorrected" and "corrected" approach were $45,870 and $53,594, respectively (bias= Ė 17%), and QALYs were 2.69 and 2.68, respectively (bias=1%) (Table 1).
Conclusions: The traditional approach of converting interval probabilities into different cycle lengths ignores competing risk states and results in biased estimates of disease outcomes, cost and QALYs, which can lead to different conclusions. Our method accounts for competing risk and can be generalized to other Markov model structures.