The variance of the length of stay and the optimal DRG outlier payments

Transcription

1 The variance of the length of stay and the optial DRG outlier payents Stefan Felder, Otto-von-Guericke University Magdeburg February 1, 7 Abstract: Prospective payent schees in health care often include supply-side insurance for cost outliers. In the early US Medicare and the current Geran DRG systes, the outlier schee fixes a length of stay (LOS) threshold, constraining the loss risk for the provider. This threshold is to increase with the standard deviation of the LOS distribution. The present paper addresses the adequacy of the DRG threshold rule for risk-averse hospitals with preferences depending on expected profits and its variance. It first shows that the optial threshold solves the hospital s tradeoff between higher profit risk and lower preiu loading payents. It then deonstrates for norally distributed LOS that the optial threshold generally decreases with an increase of the standard deviation. The intuition for this result is that a higher variance increases the profit risk, which in turn leads hospitals to insure a larger part of the LOS distribution. JEL Index: G, I11 Keywords: Optial outlier DRG payents, supply-side insurance in health care, stop loss insurance Institute of Social Medicine and Health Econoics (ISMHE), Faculty of Medicine, Leipziger Str. 44, 391 Magdeburg, Gerany. Fon , Fax, , Eail stefan.felder@ishe.de. I a grateful to Claudia Heinecke for technical assistance.

2 1. Introduction In the id-eighties US Medicare introduced the prospective payent syste, under which hospital reiburseents for patient discharges were based on their classification into diagnosis related groups (DRGs). Prospective payents replaced the old cost-based reiburseent syste which usually depended on a hospital s characteristics and its patients length of stay (LOS). The change fro a retro- to a prospective payent syste transferred the loss risk fro the insurers to the providers and gave the latter an incentive to econoize treatent costs. Still, Medicare saved part of the forer syste by introducing outlier payents. For patients staying longer than a stated threshold, hospitals could charge the costs of treatent based on the actual LOS, while for patients discharged within the threshold, pure prospective payent applied. The new schee resebled an insurance contract with a deductible, i.e. the hospital insures only the part of the LOS distribution beyond the threshold. Like any other industrialized countries, Gerany followed US Medicare and introduced its own DRG payent syste for hospitals about twenty years later. Thereby it decided to use the original Medicare outlier ethod which couples the threshold to the standard deviation of LOS according to 1 : = exp µ +, (1) where µ and is the ean and standard deviation of the log of LOS, respectively. Table 1 presents characteristics for the top 3 Geran DRGs in 5, calculated fro a saple of roughly 7, hospital cases in the state of Saxony-Anhalt. The average LOS is 7.9 days and its average standard deviation 4.1 days. The next colun shows the actual threshold for the individual DRGs as published in 5. On average the actual threshold is days, which is close to two standard deviations above the ean. When I calculate the threshold based on the logs of LOS as required by the DRG outlier ethodology, the values in the fifth colun arise, the average being days. The last two coluns in Table 1 present the effect of an increase of the standard deviation on the hospital s arginal risk (details follow in Section 3), based on the actual and the calculated thresholds, respectively. It shows that for alost all 1 The specific rule is = in ( exp ( µ + ), exp ( µ ) + b) where b is a policy paraeter to be set such that the LOS threshold covers between 94 and 95 percent of all cases. For log-norally distributed LOS, the paraeter b will deterine the threshold, as µ + covers close to 98 percent of total cases. But since the cuulative density function of the noral distribution is decreasing in for > µ (see section 3), the threshold also increases with if b is binding.

3 3 DRGs, the aount of arginal risk born by the hospital increases when the standard deviation of LOS rises. Table 1: The top 3 Geran DRGs (5) DRG Mean (µ) LOS LOS threshold The change of risk Std. dev. () Actual ( ) calculated according to (1) () ( ) k k B7B F6B C8Z G67C F67B I68B O6C F6C E71B G6B B69B E77C G49Z n.d 3 n.d.7 E77B I44Z G54Z B8Z F66B D3Z E65B F49C n.d 3 n.d -.7 D63Z G4Z I48Z I69Z B76D LZ F6D G48Z G67B Mean Max Min Source: own calculation based on DRG cases of AOK patients in Saxony-Anhalt, 5. A higher risk would indicate that hospitals want to extend the insurance coverage, i.e. to decrease the LOS threshold. In fact this is the ain result of this paper, contradicting the actual Geran DRG outlier ethodology. The health econoics literature dealt with insurance aspects of DRG outlier payents early on when Medicare introduced its prospective reiburseent syste. Ellis and McGuire (1988) applied Arrow s principle that full insurance after a deductible is the optial structure

4 4 of insurance (see Arrow, 1963) to hospital reiburseent. They show that outlier payents should be based on the hospital s average loss per case rather than on individual case-level losses, since the hospital can itself pool the loss risk of individual cases. Keeler et al. (1988), cobining optial the deductible with coinsurance on the arginal costs of expensive cases to reduce oral hazard, cae to the sae conclusion. Outlier payents serve as an insurance for hospitals against excessive losses and they itigate probles of access and underprovision of care for the patients in need of costly treatent. Keeler et al. also studied the optial policy for paying ore than one DRG when the outlier payents have to be ade case by case. If the hospital s utility is quadratic, they show that the optial schee is deductibles that are the sae for all DRGs if there are no coinsurance restrictions and a stop equal average loss policy to each DRG per outlier under a constant coinsurance rate. Since with concave utility the arginal value of oney is higher when losses are greater, equalizing the expected loss on each DRG by adjusting the deductible correspondingly is the optial outlier payent policy. This paper s focus is on the relationship between the LOS standard deviation and the optial threshold. Section presents the optial risk sharing between the hospital and the insurer. The existence of a positive threshold arises since I assue loading on the insurance net preiu and risk aversion on the part of the hospitals. I derive the optial threshold for a odel that represents the hospital s utility function using the ( µ, ) criterion and coparative-static results regarding the degree of risk aversion, the loading factor and the costs of stay per die. Section 3 deals with the coparative statics referring to the LOS standard deviation, assuing norally distributed LOS truncated fro below at zero. It shows that the optial threshold usually decreases with an increase of the standard deviation. Section 4 discusses and concludes.. Optial risk sharing between the hospital and the insurer To begin with, I set the costs per die of a hospital stay equal to one. The insurer is assued to pay the hospital depending on a patient s LOS t according to the following rule: E, if t < t, if t, ()

5 5 where is the outlier threshold ( > ), E tf () t and = dt f t is the density function of LOS, with f () tdt= 1 and. With F t = f t dt as the cuulative density function, t f () t () () F( ) is the share of cases and E F is the average LOS within the LOS threshold. The reiburseent schee () includes an insurance contract covering LOS beyond the outlier threshold. Assuing that the insurer loads the insurance preiu by the factor l ( < l <1), with zero-profits, the insurer s per patient preiu aounts to ( 1 ) p E l E = + +, (3) where E = tf () t dt and E 1 F is the average LOS of the insured part of the LOS distribution. The hospital s utility function is assued to be r =, (4) EU µ π π with r as the degree of absolute risk aversion ( r > ), µ π as the expected profits per patient and π as the variance of profits per patient. If profits are norally distributed, (4) is a general representation of the preference of risk-averse agents (see Meyer, 1987). The expected profits equal the difference between the expected reiburseents and the preiu payents (3): µ π = E + E = le. p (5) The variance of profits is zero in the insured part of the LOS distribution. Given (), the variance of the expected profits per patient, thus, aounts to: π () (). (6) = E t f t dt = t E f t dt Inserting (5) and (6) into the expected utility function (4), yields: r EU = le ( t E ) f () t dt. (7)

6 6 Using the Leibniz rule and noting that E E f arginal increase of the threshold: = = holds, I derive for a EU r = lf ( E ) f + ( t E )( f ) f () t dt r = lf f ( E ) tf () t dt E f () t dt ( E ) r = f l E ( 1 F ), An increase of the threshold, on the one hand, reduces the preiu by f (8) l, which in turn increases expected utility. On the other hand, it increases the profit risk by f ( E ) E ( 1 F ), which lowers expected utility. Let ( E) ( k E 1 F ) (9) easure the change of the profit risk when the threshold arginally increases and rewrite (8) as: r EU = f l k. (1) This equation illustrates the two key factors governing the tradeoff for an increased threshold: the loading factor l, deterining the benefits, and the additional profit risk k, capturing the costs. The optial threshold balances the two opposite effects, giving rise to: l k( ) =. (11) r Assuing that a solution exists, (11) requires that condition for axial expected utility is k > since lr>,. The second-order EU <. For =, l ( r ) k( ) thus ( ) ( EU = f l r k ). It follows that k( ) expected utility axiu, given f ( ),,l,r >. = and > in the

7 7 Proposition 1: The optial threshold i) decreases with an increase of the degree of risk aversion r and ii) increases with an increase of the loading factor l. Proof: i) For infinitesial sall changes of and r it holds around the axiu: k d = l r r dr, or d l r = < dr k, since k( ) >. ii) As ( lr) 1 l dl = dl >, it follows that r d 1 r = >. dl k With higher loading, the optial LOS threshold increases. In other words, the hospital opts for a lower insurance coverage when the preiu becoes ore expensive. Furtherore, the threshold decreases when the degree of risk aversion increases. A risk-neutral hospital ( r = ), by coparison, would not choose any insurance at all (i.e. = ). I have assued that the cost per die of a stay is one. If the costs per die are β, the effect of an increase of the threshold on expected utility becoes β ( β ) EU = f l r k (see (9) and (1)). The optial threshold, thus, requires: l k( ) =. (1) βr Proposition : The optial threshold decreases with an increase of the cost per die β. If the LOS distribution is the sae across all DRGs, proposition iplies that β is the sae for all DRGs. The current rule sets the upper threshold depending on the ean and the standard deviation of LOS (see (1)), but not on the costs of treatent. According to proposition, the optial LOS threshold should c. p. decrease with the cost of treatent, reflecting the corresponding increase of the hospital s profit risk.

8 8 3. LOS standard deviation and the optial threshold In order to approach the Geran outlier rule, which sets the LOS threshold two standard deviations above the ean, I paraeterize the distribution and assue a norally distributed LOS according to: t = µ + ε, with E ε =, Var ( ε ) = 1. (13) The density function of the noral distribution is () f t 1 = e π 1 = e π (( t µ ) ) ε. (14) Since LOS is non-negative, I consider the noral distribution truncated below at point. The density function of the truncated distribution writes:, - t f () t = f () t, t, 1 F (15) = dt () where F f t and f t is defined as in (14). Using (13)-(15), I find: ( F ) ( a) ( a) 1 ε E = tf () t dt = ( µ + ε ) e dε π 1 a 1 ( µ ) ε ε = µ e dε e (16) π 1 a ( F ) a 1 F µ F F f f = For the non-truncated distribution, the truncation is at. Since F f follows that E E µ F f = =.. = =, it With the truncated distribution, the optial LOS threshold (see (9) and (11)) slightly changes to:

9 ( ) 9 E l k( ) E ( 1 F( ) + F ) =. (17) r Proposition 3: The optial threshold decreases with an increase of the LOS standard deviation, provided the latter increases the arginal profit risk. Proof: At the optial threshold, k( ) = holds. Since k ( ) d d k >, it follows that d k sign = sign d. In order to calculate k( ), I need to derive (16) I find: E, as well as F. Fro F F f f F + µ f f E E = 1 F.(18) Moreover, it holds that: () F t () = f t t µ and (19) () f t = ( µ ) 1 () ( t ) f t. () Inserting these equations into (18) leads to: ( µ ) ( µ ) E f + + f + E = 1 F. (1) Furtherore, fro (17) I derive the effect of a change in the standard deviation on the hospital s additional risk at the LOS threshold: k F F E = E ( E F + F ). () This equation reveals two effects of an increase of on the hospital s additional risk. First, a higher spread of the distribution changes the ass of LOS in the interval where the hospital bears the risk. (19) yields:

10 F F µ µ = f 1 f. (3) Given > µ, the first effect of the derivative () is negative. This is because an increase of the standard deviation tends to lower the risk, as a higher share of LOS will be covered by insurance. The second effect, however, likely cuts in the opposite direction. First, I observe E F + F > as E,F( ),F 1, given > µ. Hence, the sign of the second effect has the inverse sign of E. For the non-truncated distribution, I find E < fro (1). Hence, the second effect is indeed positive in this case. The hospital s arginal risk increases as the threshold oves further away fro the ean when the standard deviation increases. For the truncated distribution, the second effect cannot be signed. The total effect of an increase of the LOS standard deviation on the hospital s arginal risk is unclear since the difference of the two entioned effects cannot be signed even when the distribution is non-truncated. Hence, I have to depend on siulations based on the factual distribution of LOS to evaluate the total effect. Figure 1 shows the value of the derivative () for truncated and non-truncated LOS as a function of the threshold. µ and are three exaples taken fro the Geran top 3 DRG list (see Table 1). For the non-truncated distributions, the derivative is positive, declining to zero for large thresholds. For both the truncated and the non-truncated distributions, the derivative is positive over the whole range of thresholds. When the threshold is close to the ean, an increase of the threshold raises the arginal profit risk. The axiu is attained within one to three days, depending on the LOS distribution characteristics. For larger thresholds the effect, driven by the density function, fast converges to zero. Alternatively, one can evaluate the DRG threshold rule at the given thresholds and study whether the arginal risk increases with an increase of the standard deviation. The result of this is presented in two final coluns of Table 1, showing the values of k / for the current and the calculated Geran thresholds. Notice that except for two DRGs the sign of the derivative is positive. Thus, in alost all cases the hospital arginal risk increases, so that a risk-averse hospital would like to extend insurance coverage, i.e. to lower the LOS threshold. If I interpret the observed ean as the ean of the truncated distribution ( E ), the ean of the non-truncated (µ) can be derived using (16): µ E f ( F) = 1. If I eploy µ to calculate the derivative (), the results regarding the sign of () reain unchanged.

11 11 Figure 1: The effect of arginal profit risk fro an increase of for different µ and ; truncated (tr) and non-truncated (non-tr) noral distribution 1.5 Effect on arginal risk threshold µ= 3; =; tr µ= 6; =4 - tr µ=1; =6 - tr µ= 3; =; non-tr µ= 6; =4 - non-tr µ=1; =6 - non-tr 5. Discussion Since a hospital can influence a patient s LOS, the assuption of an exogenous LOS is unrealistic. An endogenous LOS raises probles of oral hazard. Beyond the threshold, reducing the LOS lowers preiu loading payents by of the LOS, de de l, while within the threshold a reduction, translates one to one in higher profits. In other words, insurance coverage dilutes the incentive to reduce the LOS. The optial policy then needs to approach the tradeoff between risk spreading and appropriate incentives (Zeckauser, 1979), which can be solved by cobining a threshold with a coinsurance on the arginal costs of stay beyond the threshold. DRG systes reflect this, as outlier days are usually reibursed with a 4% rebate on the average cost per die. An endogenization of the LOS will not necessarily change the coparative statics for the optial threshold regarding the variance of the LOS. Consider first the case where the productivity of efforts to reduce the LOS is independent of the initial LOS. Abstracting fro the discontinuity at the threshold, efforts will then only shift the LOS distribution to the left and have no effect on the LOS variance. Consequently, although the optial threshold will increase,

12 1 the qualitative relationship between the standard deviation and the optial threshold will not change. A higher LOS variance c. p. will coe with a decrease of the optial threshold. The productivity of efforts to reduce LOS ay, however, increase with the initial LOS, so that efforts will have an effect on the LOS variance. Still, it appears that the following hierarchy of instruents would apply in this case. Moral hazard can be restrained using a non-linear coinsurance rate that increases with the observed LOS to give a stronger incentive to reduce the LOS when it is less expensive to do so. A higher LOS variance will have no effect on the efforts to reduce LOS. The optial threshold will reflect the efforts to reduce LOS but the coparative statics regarding the variance of LOS will not change its qualitative nature. To prove this conjecture would be difficult, given the discontinuity of the LOS distribution, arising when efforts to reduce LOS are introduced. Ellis and McGuire (1988) criticized the existing outlier payents based on individual cases and proposed an insurance schee based on the average case. Risk pooling within the hospital will reduce the variance of the profit per patient and, thus, decrease insurance deand. This qualification, however, does not affect the optial threshold rule. A hospital which shoulders a higher risk due to a large case-ix index c. p. will deand a lower threshold copared to a hospital with a lower case-ix load. The new generation of outlier payent systes in the USA is no longer based on the LOS, but on the patients costs of stay 3. This reflects the epirical observation that after controlling for DRG, the costs of stay are only weakly related to LOS aong very long stay cases (see Keeler et al., 1988). Interestingly, the cost outlier schees do not define the thresholds as a function of the variance. Rather, a cost-to-charge factor deterines the threshold. In this case a ean-preserving increase in the standard deviation will not affect the threshold. This rule is better than the forer one that sets the threshold two standard deviations above the ean. Australian outlier payents do not depend on paraetric distribution, arguing that the LOS is not norally distributed. 4 The threshold, called the high tri point, is often or 3 ties the average length of stay. Like the US Medicare cost outlier, this schee appears to doinate the original threshold rule, as it does not further aggravate the hospitals profit risk, by increasing the threshold when the LOS standard deviation increases. 3 See for instance, Departent of Health and Huan Services, Centers for Medicare and Medicaid Services, 4 CFR Part 41, Federal Register, Vol. 68, No. 43, March 5, 3, p For New South Wales, see for Victoria, see and for South Australia, see

13 13 6. References Arrow, K. (1963), Uncertainty and the welfare econoics of edical care, Aerican Econoic Review 53, Ellis, R. P. and Th. G. McGuire (1988), Insurance principles and the design of prospective payent systes, Journal of Health Econoics 7, Keeler, E. B., F. M. Carter and S. Trude (1988), Insurance aspects of DRG outlier payents, Journal of Health Econoics 7, Meyer, J. (1987), Two-oent decision odels and expected utility, Aerican Econoic Review 77, Zeckhauser, R. (197), Medical insurance: a case study of the tradeoff between risk spreading and appropriate incentives, Journal of Econoic Theory,, 1-6.