To empirically illustrate our method for simulating the CCI under different deductible modalities we follow a four-step procedure:
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1.
Estimate an expenditure model;
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2.
Approximate the probability that healthcare expenses end up in the deductible range;
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3.
Approximate the expected expenses given that they end up in the deductible range;
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4.
Simulate the CCI.
In this paper we are interested in the CCI under a specific deductible modality relative to others; absolute figures of the CCI are of little significance. Empirical results are intended as an illustration of the method developed. First, we derive the CCI under a first-euro deductible of €500, €1000, €2000, €3000, €4000, €5000 and €10,000 in order to examine the effects of the deductible amount. After that, we examine the CCI under a doughnut hole of €1000 with a uniform starting point at €500, €1000, €2000, €2257 (i.e., the mean of actual healthcare expenses in the selected sample of 499,990 individuals), €3000, €4000 and €5000 in order to compare the CCI between a first-euro deductible and a doughnut hole. Average CCIs under the two deductible modalities are simulated for the full sample, and separately, for a group of high-risk individuals and the complementary group of low-risk individuals. Morbidity information is used to determine to which risk-group an individual belongs: those individuals with (without) a DCG, PCG, HCG and/or MHC are considered as a high-risk individual (low-risk individual). In this sample 72% is considered as a low-risk individual and 28% as a high-risk individual.
It is important to mention that—next to the assumption on rational behavior—our concept is based on some other (implicit) assumptions. For example, we assume a linear relationship between the probability that healthcare expenses end up in the deductible range and the CCI. Furthermore, we focus on the CCI regarding total healthcare utilization that is subject to the deductible and neglect the composition of the care that is used. The implications of these and other assumptions, will be discussed in the last section of this paper.
Estimate an expenditure model
First, to predict expected healthcare expenses E(Y) for each individual, an expenditure model is estimated with actual expenses in 2011 as dependent variable and age-gender classes, DCGs, PCGs, HCGs and MHCs as explanatory variables. We opted for a Generalized Linear Model (GLM) with a gamma distribution and a log-link function, which is considered to be an appropriate statistical method for modelling healthcare expenses in many studies (see, for example, [3, 5, 7, 11, 17]). Basically, all risk characteristics are statistically significant at the conventional level (given the large sample size). On average the expected healthcare expenses were €2537 with a standard deviation of €7762, and the R
2 of the model is 0.39. In the subsequent tables we show that our estimation approach provides an acceptable fit between the actual and predicted parameters of the CCI.Footnote 6
Approximate the probabilities that healthcare expenses end up in the deductible range
After estimating an expenditure model, the probability P that healthcare expenses Y remain below deductible amount d, starting point s and endpoint s + d is approximated. We follow the procedure as described by van Kleef and colleagues [17], who have identified the relevant parameters given the use of a gamma distribution with a log-link. The probabilities that we are interested in can be derived by Eqs. (3) till (5).
$$ P\left( {Y < d} \right) = \varGamma \left( {c_{d} ,k} \right), $$
(3)
$$ P\left( {Y < s} \right) = \varGamma \left( {c_{s} ,k} \right), $$
(4)
$$ P\left( {Y < s + d} \right) = \varGamma \left( {c_{s + d} ,k} \right), $$
(5)
where Γ(.) is the cumulative density function of the gamma distribution, the scale parameter k is 0.4969, and:
$$ \lambda = k / E(Y), $$
(6)
$$ c_{d} = d*\lambda , $$
(7)
$$ c_{s} = s*\lambda , $$
(8)
$$ c_{s + d} = (s + d)*\lambda . $$
(9)
Given the assumptions made and given our dataset, we check whether the results based on Formulae (3) till (9) are in line with the actual figures in the sample; the proportion ρ and probability P that healthcare expenses Y remain the deductible amount d under a first-euro deductible are compared. Table 1 shows that ρ(Y < d) and P(Y < d) follow the same pattern, specifically in case of a relatively high deductible amount.
Table 1 Proportions ρ and probabilities P that healthcare expenses Y remain below various deductible amounts d for the full sample
Approximate the expected expenses given that they end up in the deductible range
Given expected expenses E(Y) and the parameters calculated in the previous step, expected expenses given that expenses end up in the interval [0, d], [0, s], respectively [0, s + d] can be calculated by Eqs. (10), (11) and (12) [17].
$$ E\left( {Y|Y < d} \right) = E\left( Y \right)*\varGamma \left( {c_{d} ,k + 1} \right) / \varGamma (c_{d} ,k), $$
(10)
$$ E\left( {Y|Y < s} \right) = E\left( Y \right)*\varGamma \left( {c_{s} ,k + 1} \right) / \varGamma (c_{s} ,k), $$
(11)
$$ E\left( {Y|Y < s + d} \right) = E\left( Y \right)*\varGamma \left( {c_{s + d} ,k + 1} \right) / \varGamma (c_{s + d} ,k). $$
(12)
Table 2 shows the actual expenses and expected expenses given that expenses remain below first-euro deductible amount d. Our approach somewhat underestimates these expenses for the relatively small first-euro deductibles and somewhat overestimates them for the higher ones, but these deviations do not seem important.
Table 2 Mean of actual expenses Y and expected expenses E(Y) given that expenses Y remain below various deductible amounts d for the full sample
Based on the results presented in Tables 1 and 2, there seems to be no reason to believe that the overestimations of the mean and the standard deviation of expected healthcare expenses compared to the actual healthcare expenses have unacceptable effects on the key parameters of interest in this paper.
Simulate the CCI
As discussed in section “A method to simulate incentives for cost containment”, the CCI is conceptualized as a product of the probability that individual healthcare expenses end up in the deductible range and the expected expenses given that they end up in the deductible range. Therefore, parameters obtained in step 2 and step 3 are combined in order to determine the CCI for each individual. The CCI under a first-euro deductible with deductible amount d is calculated by Eq. (1). The CCI under a doughnut hole with starting point s and deductible amount d is approximated by Eq. (2). The CCI is presented in Euros and can be interpreted as the marginal amount of healthcare expenses for which a consumer is fully price sensitive. Hypothetically speaking, the CCI will be zero for a consumer who knows for sure his spending will exceed the deductible amount. For a consumer who knows for sure his spending will not exceed the deductible amount, the CCI will equal his expected spending.
Implications
At least three implications arise from the conceptual framework as described in section “A method to simulate incentives for cost containment”. These hypotheses are to be addressed in “Results” where the simulation results are presented. First, the CCI under a deductible increases when the deductible amount increases. If, ceteris paribus, the deductible amount increases (i.e., point d and, accordingly, point s + d is shifted to the right), the deductible range is broadened. As a result, both the probability that expenses end up in the deductible range and the expected expenses in the deductible range once they ended up in the deductible interval are expected to increase. This will result in a stronger CCI.
Second, we expect that different deductible modalities lead to different CCIs. Shifting the deductible influences the CCI. The direction of the effect is an interesting empirical question. On the one hand, a shift of the deductible to higher expenditure levels reduces the probability to reach the deductible range, which negatively affects the CCI. On the other hand, such a shift increases the expected expenses given that they end up in this range, which positively affects the CCI.
Third, we hypothesize that the CCI under a first-euro deductible and a doughnut hole will differ across risk-groups. Figure 6 shows P(Y < x) of a relatively low-risk individual under a first-euro deductible (left panel) and under a doughnut hole with a starting point at the mean of actual healthcare expenses in the population (right panel). E(Y) for this healthy individual are relatively low, but there is always a certain level of uncertainty whether or not this individual needs care. This implies that, under a first-euro deductible, there is a low probability that healthcare expenses exceed the deductible amount. In contrast, under a doughnut hole with a starting point at the mean of healthcare expenses, it is not very likely that this low-risk individual ends up in the doughnut hole. P(Y < s) and P(Y < s + d) both approximate 1. As a result of the relatively high P(Y < d) under a first-euro deductible compared to P(s < Y < s + d) under a doughnut hole, the CCI for this low-risk individual is relatively strong in case of a first-euro deductible in comparison to a doughnut hole.
Now consider a relatively high-risk individual, such as a chronically ill patient. P(Y < x) is depicted in Fig. 7. E(Y) for this relatively unhealthy individual are above average. Accordingly, under a first-euro deductible, P(Y < d) is low (Fig. 7, left panel). In contrast, P(s < Y < s + d) is relatively high when the starting point of the doughnut hole is set at the mean of actual healthcare expenses (Fig. 7, right panel). Consequently, for this high-risk individual the CCI is relatively strong in case of a doughnut hole in comparison to a first-euro deductible.
The previous consideration implies that, at the population level, it is not obvious whether a first-euro deductible leads to a stronger or weaker CCI than a doughnut hole. On the one hand, a shift of the starting point of the deductible to a higher expenditure level than €0 may increase the CCI for the high-risk individuals (a relatively small group with relatively high savings potential). On the other hand, such a shift may decrease the CCI for the low-risk individuals (a relatively large group with relatively low savings potential). In our empirical illustration we aim to simulate the net outcome of these two effects.