where 1 and 2 correspond to the new treatment and the reference treatment, respectively.
Thus, when the therapy being assessed is more expensive than the reference treatment, and is also less effective, the new treatment is dominated. When the treatment is more effective than the reference treatment, and also less expensive, the new treatment is dominant over the reference treatment. In the remaining cases, the ICER must be compared with the willingness to pay (k) to assess whether the new treatment is cost-effective (ICER < = k) or not cost-effective (ICER > k) [
1,
2]. The choice of the threshold of efficiency remains an unresolved issue in many health systems. Theoretically, the threshold of efficiency should be related to the value that society accords to a health outcome in, for example, life years gained, depending on the available resources [
1,
2].
One limitation of the ICER is that, as it is a ratio, the expected value of the ICER represents the difference in the expected cost divided by the difference in the expected effectiveness. Therefore, the main difficulty in making inferences from cost-effectiveness studies is that the random variable obtained is not necessarily normal or symmetric [
14]. However, even when cost and effectiveness data are distributed normally, there is no guarantee that the ratio between them will also behave normally and, ultimately, it is not possible to calculate the confidence intervals (CI). Thus, there is a need to use alternative methods, such as bootstrapping or incremental net benefits (INB) for this purpose [
15].
Where s is the number of the study and 1 and 2 the new treatment and reference treatment, respectively:
Identification of marginal distributions and copula
Copulas are bivariate distributions that provide dependent structures for two statistical variables, with any type of univariate distribution [
17‐
19]. In its construction, a C(u,v) copula is a multivariate distribution function defined from U and V random variables with uniform distribution in the interval [0,1], which verifies the following properties:
2.
C(u, 1) = u and C(1, v) = v
3.
For u1, u2, v1, v2 in [0, 1] so that u1 ≤ u2 and v1 ≤ v2, C(u2, v2) − C(u2, v1) − C(u1, v2) + C(u1, v1) ≥ 0.
Since the relationship between random variables is not based on the distribution, but rather constructed from mathematical structures between random variables, dependence can be evaluated using non-parametric correlation statistics (Spearman’s ρs and Kendall’s τ), which are independent on marginal distributions.
Copula distributions are little used in health economics, and are mainly applied to specify the distribution of regression models with more than one dependent variable [
20‐
23]. We used copulas to describe the dependence structure of cost-effect random variables, allowing random generation of a patient cohort under such a distribution in order to make different simulations. By applying copulas to EEHT, the joint distribution of each treatment can be simulated, since C
j(c
ij,e
ij) is a copula for treatment j, where costs (c) and effects (e) are known for each patient i. In addition, Sklar’s theorem [
24] shows that, given a copula (the joint distribution), the copula can be reconstructed through the marginal theoretical or empirical inverse distribution function, according to the specific case. However, recent studies show that there is no justification for the claim that a specific copula may be the most appropriate for the combination of costs and effects, not even when costs are broken down into direct and indirect costs, or if effects are represented by the following measurements: therapeutic success, life years, or quality-adjusted life years (QALY) [
25].
To create a hypothetical population to test the validity of the COMER methodology, we used data from a Spanish prospective observational study of patients with allergic rhinitis (n = 498) with direct costs (c) and mean utilities from the SF-12v1 (e) [
26]. The study served to obtain a copula structure which, by adding some theoretical marginal distributions, allowed the generation of a hypothetical population on which to apply the COMER methodology. The empirical copula was compared to the parametric estimate of the potential copula (Additional file
1) to evaluate the goodness of fit of the data with the theoretical copulas [
27]. The copula bound to evaluate the goodness of fit was developed using the inversion of Kendall’s τ. The p-value of the test was calculated by simulation of size-100 bootstrap, due to the lack of an analytical construction.
The following copulas were evaluated [
19,
25,
28]:
Independent copula: this can be generated automatically since, given U and V random variables, the joint distribution function is C
0(u, v) = u * v. Thus, the association between this copula and some data indicates the stochastic independence of U and V; equivalently, the absence of structure.
Gaussian copula: this is defined as G
∅(u, v) = N
ρ
(Φ− 1(u), Φ− 1(v)) where N
ρ
(x, y) is the normal distribution function of parameters x: mean and y: the standard deviation, Φ− 1(x) is the marginal distribution function N(0,1) and ρ is Pearson’s correlation coefficient. The Gaussian copula is still a multivariate normal distribution.
T copula: this is derived from the Student’s t multivariate distribution and is defined as where |ρ| < 1, θ > 0. This copula shows a similar structure to the Gaussian copula, but presents tail dependence, points (0,1) and (1,0). One of its qualities is that it includes the Gaussian copula when θ → ∞.
Gumbel copula: this allows a positive dependence structure to be modelled, region (1,1), and is defined as where θ ∈ [1, ∞]. When θ = 1 it is equivalent to the independent copula and when θ → ∞ it behaves as a comonotonic copula (min(u,v)). Thus, the behaviour of the Gumbel copula is an interpolation between the independent copula and the copula of perfect positive dependence.
Clayton copula: this is defined as where θ > 0. Like the Gumbel copula, this copula is an interpolation but lies between the independent copula and the perfect negative dependence (point (0,0)).
Frank Copula: this has the quality of showing symmetric dependencies and not showing dependence at points (0,0) and (1,1). It is defined as where θ ∈ ℝ \{0}.
Plackett copula: this is defined as where θ ≥ 0. This structure shows both positive and negative maximum dependence in function of parameter θ.
Creation of cohorts
Once the joint distribution was known, some theoretical marginal distributions were associated. To simulate costs, a lognormal distribution was associated, and to simulate life quality, a gamma distribution was associated (disutilities), with respect to a baseline quality of life (0.9 utility) [
14] (Table
2). For the two alternatives, the same copula and the same randomization (bivariate uniforms [0,1]) were used. This ensured covariance and correlation between the values generated for the costs and effects. Different random samples per cohort were created with sample sizes between 15 and 500 individuals for each alternative. Individuals in the simulated cohort were randomly assigned to the studies. To ensure that there was a minimum variability per study, the random assignation was conditioned to ensure that, for each study, there were at least 3 individuals.
Table 2
Simulated scenarios
Scenario 1: Cost-effective
|
|
CONTROL
|
ACTIVE
|
Cost | Lognormal (6.214608, 0.75) | Lognormal (6.907755, 0.75) |
Effect | 0.9-Gamma (6.25, 0.024) | 0.9-Gamma (2, 0.05) |
ICER | 13,247.85 = (1,324.78 - 662.39)/(0.8-0.75) |
INB* | 837.61 |
Scenario 2: Non-cost-effective
|
|
CONTROL
|
ACTIVE
|
Cost | Lognormal (6.214608, 0.75) | Lognormal(6.214608, 0.75) |
Effect | 0.9-Gamma (6.25, 0.024) | 0.9-Gamma (4.5, 0.03) |
ICER | 44,159.49 = (1,324.78 - 662.39)/(0.765-0.75) |
INB* | −212.39 |
Scenario 3:
Dominant
|
|
CONTROL
|
ACTIVE
|
Cost | Lognormal (6.907755, 0.75) | Lognormal (6.214608, 0.75) |
Effect | 0.9-Gamma (6.25, 0.024) | 0.9-Gamma (2, 0.05) |
ICER | −13,247.85 = (662.39 – 1,324.78)/(0.8-0.75) |
INB* | 2,162.39 |
The COMER methodology was applied to the simulated data for each alternative. For each alternative the mean costs, mean effectiveness, differential variance in costs, differential variance in effects and covariance between the differences in costs and effects, and the INB were estimated, setting an efficiency threshold (k = 30,000 monetary units per QALY gained). For example, of a sample of 15 individuals, the seven first could be assigned to study 1, the following five to study 2, and the remaining three to study 3, and the COMER methodology applied (Additional file
2).
For each sample size generated, 500 replications were made, entailing 25,000 meta-analyses for each scenario, thus allowing the number of the times the methodology agreed with reality to be validated. A 2,000-monetary-unit/QALY-tolerance was assumed to calculate the ICER, and a 500-monetary-unit-tolerance was assumed for the TINB. Additionally, we estimated the minimum sample size required to obtain an adjusted estimate with a probability >70% and when simulations converged to the original Kendall’s τ.
The following are the scenarios evaluated by modifying the marginal distribution parameters (Table
2):
1.
The second alternative is cost-effective.
2.
The second alternative is more costly and more effective, although the ICER is above the willingness to pay threshold.
3.
The second alternative is dominant.
All the possibilities are covered by these scenarios since the remaining potential outcomes derived from an EEHT which have not explicitly been evaluated are obtained assuming that the current alternative is the second alternative instead of the first.
The analysis was made using the R 3.0.1 statistical package (Additional file
3). In addition, the COMER method is included as an Excel file in the supplementary materials (Additional file
4).