Purpose: Because of the non-linear relationship between some model inputs and outcomes, previous research using numerical methods suggested that the choice of modeling technique — cohort versus individual — can have significant effects on model results. However, the direction of the effect is not known a priori. Our purpose was to derive mathematically (analytically) the conditions under which a cohort model (which does not capture baseline heterogeneity) compared with an individual-based simulation approach will under-estimate or over-estimate health outcomes.
Method: We used a three-state Markov model to estimate the cost-effectiveness of a hypothetical intervention, with efficacy e and cost I, to prevent disease developing at rate b resulting in disease-specific death at rate d, cost at c per period, and quality of life loss q. All parameters, including all-cause mortality at rate m, were constant. We solved the continuous-time model analytically and derived expressions for life expectancy, discounted quality-adjusted life years (QALYs), discounted lifetime disease costs, incremental cost-effectiveness ratio (ICER), and net monetary benefits (NMB). An outcome was calculated using the mean of the input under the cohort-based approach and the whole input distribution for all persons under the individual-based approach. We investigated the impact of heterogeneity on outcomes by varying one parameter at a time (e.g., b) while keeping all others constant (e.g., e, m, d, q, and c). We evaluated the curvature of outcome functions and used Jensen's inequality (i.e. if f is convex in X, f(E[X] £ E[f(X)]) to determine whether a cohort model under- or over-estimated a heath outcome (e.g., if f was convex, then a cohort model under-estimated the outcome).
Results:
Both life expectancy and QALYs were underestimated by the cohort-based approach (Figure1). If there was only heterogeneity in disease hazard, discounted costs were overestimated whereas QALYs gained, incremental costs, and ICER were under or overestimated depending on the value of b. ICER was overestimated and NMB underestimated when there was heterogeneity in efficacy only. Both approaches yielded the same outcome when there was only heterogeneity in c or q.
Conclusion: Use of a cohort-based approach that does not adjust for heterogeneity underestimates life expectancy and may under- or over-estimate other outcomes. Characterizing the bias is useful for calibration and predictions.
