Characterizing Heterogeneity Bias in Cohort-Based Models

Abstract

Purpose

Previous research using numerical methods suggested that use of a cohort-based model instead of an individual-based model can result in significant heterogeneity bias. However, the direction of the bias is not known a priori. We characterized mathematically the conditions that lead to upward or downward bias.

Method

We used a standard three-state disease progression model to evaluate the cost effectiveness of a hypothetical intervention. We solved the model analytically and derived expressions for life expectancy, discounted quality-adjusted life years (QALYs), discounted lifetime costs and incremental net monetary benefits (INMB). 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 while keeping all others constant. We evaluated the curvature of outcome functions and used Jensen’s inequality to determine the direction of the bias.

Results

Both life expectancy and QALYs were underestimated by the cohort-based approach. If there was heterogeneity only in disease progression, total costs were overestimated, whereas QALYs gained, incremental costs and INMB were under- or overestimated, depending on the progression rate. INMB was underestimated when only efficacy was heterogeneous. Both approaches yielded the same outcome when the heterogeneity was only in cost or utilities.

Conclusion

A cohort-based approach that does not adjust for heterogeneity underestimates life expectancy and may underestimate or overestimate other outcomes. Characterizing the bias is useful for comparative assessment of models and informing decision making.

Appendix

The model can be represented by a system of ordinary differential equations and solved analytically to determine the number of persons in each state and the overall expected value of each outcome. The transition matrix Q and the transition probability matrix B of a continuous-time Markov chain are related according to the Kolmogorov forward differential equations:

$$ \frac{{{\text{d}}B\left( t \right)}}{{{\text{d}}t}} = B\left( t \right)Q, B\left( 0 \right) = I. $$

The matrices B and Q in this case are given by:

$$ B\left( t \right) = \left[ {\begin{array}{*{20}c} {B_{11} \left( t \right)} &\quad {B_{12} \left( t \right)} &\quad {B_{13} \left( t \right)} \\ 0 &\quad {B_{22} \left( t \right)} &\quad {B_{23} \left( t \right)} \\ 0 &\quad 0 &\quad 1 \\ \end{array} } \right] $$

$$ Q = \left[ {\begin{array}{*{20}c} { - \left[ {\left( {1 - h} \right)p + m} \right]} &\quad {\left( {1 - h} \right)p} &\quad m \\ 0 &\quad { - \left( {m + d} \right)} &\quad {d + m} \\ 0 &\quad 0 &\quad 0 \\ \end{array} } \right], $$

where p denotes the disease progression rate per year, h denotes efficacy, d denotes the disease-specific death rate per year and m denotes the all-cause mortality rate per year. The two matrices, B and Q, are related to each other according to:

$$ B\left( t \right) = \exp^{Qt} , $$

where exp denotes the matrix exponential.

The state probability distribution of the Markov chain at time t, $x\left( t \right) = \left[ {B_{11} \left( t \right),\,B_{12} \left( t \right),\,B_{13} \left( t \right)} \right] = \left[ {W\left( t \right),\, S\left( t \right),\,D\left( t \right)} \right],$ satisfies the following equation:

$$ \frac{{{\text{d}}x\left( t \right)}}{{{\text{d}}t}} = x\left( t \right)Q, x\left( 0 \right) = x_{0} , $$

where x ₀ is the initial distribution [20].

Assuming the Markov chain starts in the Well health state, $x\left( 0 \right) = \left[ {1,0,0} \right],$ the number of persons in a given health state at time t evolves over time according to:

$$ \frac{{{\text{d}}W\left( t \right)}}{{{\text{d}}t}} = - \left[ {\left( {1 - h} \right)p + m} \right]W\left( t \right) $$

(A.1)

$$ \frac{{{\text{d}}S\left( t \right)}}{{{\text{d}}t}} = \left( {1 - h} \right)pW\left( t \right) - \left( {m + d} \right)S\left( t \right) $$

(A.2)

$$ \frac{{{\text{d}}D\left( t \right)}}{{{\text{d}}t}} = mW\left( t \right) + \left( {m + d} \right)S\left( t \right) $$

(A.3)

$$ W\left( 0 \right) = 1,S\left( 0 \right) = D\left( 0 \right) = 0 $$

where W is the number of persons in the Well state, S is the number of persons in the Disease state and D is the number of persons in the Dead state.

This is a block-recursive system, which can be solved as follows (see reference [21], Chapter 14, Section 14.1). Equation (A.1) can be rewritten as:

$$ \frac{{{\text{d}}W\left( t \right)/{\text{d}}t}}{W\left( t \right)} = \frac{{{\text{d}}\ln W\left( t \right)}}{{{\text{d}}t}} = - \left[ {\left( {1 - h} \right)p + m} \right] $$

Using standard integration methods, we obtain $\ln W\left( t \right) - \ln W\left( 0 \right) = - \left[ {\left( {1 - h} \right)p + m} \right]t,$ noting that $\ln W\left( 0 \right) = \ln 1 = 0,$ we have:

$$ W\left( t \right) = e^{{ - \left[ {\left( {1 - h} \right)p + m} \right]t}} $$

(A.4)

Substituting Eq. A.4 into Eq. A.2 yields:

$$ \frac{{{\text{d}}S\left( t \right)}}{{{\text{d}}t}} = \left( {1 - h} \right)pe^{{ - \left[ {\left( {1 - h} \right)p + m} \right]t}} - \left( {m + d} \right)S\left( t \right) $$

(A.5)

This is a nonhomogeneous equation with a variable coefficient whose general solution is given by reference [21], Chapter 14, Section 14.3.

$$ S\left( t \right) = e^{{ - \mathop \int \nolimits \left( {m + d} \right){\text{d}}t}} \left( {A + \mathop \int \nolimits \left( {1 - h} \right)pe^{{ - \left[ {\left( {1 - h} \right)p + m} \right]t}} e^{{\mathop \int \nolimits \left( {m + d} \right){\text{d}}t}} {\text{d}}t} \right) = e^{{ - \left( {m + d} \right)t}} \left( {A + \mathop \int \nolimits \left( {1 - h} \right)pe^{{ - \left[ {\left( {1 - h} \right)p - d} \right]t}} {\text{d}}t} \right) = e^{{ - \left( {m + d} \right)t}} \left( {A + \frac{{\left( {1 - h} \right)pe^{{ - \left[ {\left( {1 - h} \right)p - d} \right]t}} }}{{d - \left( {1 - h} \right)p}}} \right), $$

where the arbitrary constant A can be determined from the initial condition, S(0) = 0, as:

$$ A = - \frac{{\left( {1 - h} \right)p}}{{d - \left( {1 - h} \right)p}} $$

Substituting the value of A, we obtain:

$$ S\left( t \right) = \frac{{\left( {1 - h} \right)p\left\{ {e^{{ - \left( {m + d} \right)t}} - e^{{ - \left[ {m + \left( {1 - h} \right)p} \right]t}} } \right\}}}{{\left( {1 - h} \right)p - d}} $$

(A.6)

The number of persons in the Dead state can be recovered from Eqs. (A.4) and (A.6), using the equation:

$$ D\left( t \right) = 1 - W\left( t \right) - S\left( t \right) $$

(A.7)

Assuming a lifetime horizon (i.e. infinite time), discounted (at rate r per year) the QALYs are:

$$ {\text{QALY}}\left( h \right) = \mathop \int \limits_{0}^{\infty } e^{ - rt} \left[ {W\left( t \right) + \left( {1 - q} \right)S\left( t \right)} \right]{\text{d}}t = \frac{{p\left( {1 - h} \right)\left( {1 - q} \right) + d + m + r}}{{\left( {d + m + r} \right)\left[ {p\left( {1 - h} \right) + m + r} \right]}} $$

where q denotes the decrement in the quality of life of a sick person.

Undiscounted life expectancy is obtained from the above expression by setting r = q = 0. The discounted disease cost is:

$$ {\text{COST}}\left( h \right) = \mathop \int \limits_{0}^{\infty } e^{ - rt} cS\left( t \right){\text{d}}t = \frac{{cp\left( {1 - h} \right)}}{{\left( {d + m + r} \right)\left[ {p\left( {1 - h} \right) + m + r} \right]}}, $$

where c denotes the cost of disease per year.

The incremental discounted QALYs are:

$$ \Delta {\text{QALY}}\left( h \right) = {\text{QALY}}\left( h \right) - {\text{QALY}}\left( 0 \right) = \frac{{hp\left[ {d + q\left( {m + r} \right)} \right]}}{{\left( {d + m + r} \right)\left( {m + p + r} \right)\left[ {p\left( {1 - h} \right) + m + r} \right]}} $$

Similarly, the incremental discounted disease costs are:

$$ \Delta {\text{COST}}\left( h \right) = {\text{COST}}\left( h \right) - {\text{COST}}\left( 0 \right) = - \frac{{chp\left( {m + r} \right)}}{{\left( {d + m + r} \right)\left( {m + p + r} \right)\left[ {p\left( {1 - h} \right) + m + r} \right]}} $$

Denoting the maximum WTP for a QALY by λ (also referred to as the cost-effectiveness threshold), the intervention has an INMB of:

$$ {\text{INMB}} = \lambda \times \Delta {\text{QALY}}\left( h \right) - {\text{COST}}\left( h \right) - I = \frac{{ph\left\{ {\lambda \left[ {d + q\left( {m + r} \right)} \right] + c\left( {m + r} \right)} \right\}}}{{\left( {p + m + r} \right)\left( {d + m + r} \right)\left[ {p\left( {1 - h} \right) + m + r} \right]}} - I, $$

where I denotes the one-time cost of the intervention.