The distribution of patients' lengths of stay in English hospitals is measured by using routinely
collected data from 11 years. It is found to be well approximated by a power law distribution
spanning over more than 3 decades. To explain this observation, a theoretical resource allocation
model is presented. It is based on iterative long-term scheduling of hospital beds, and its main
assumption is that future beds are allocated preferentially. This represents a situation where
di®erent parts of the health care system compete for resources, with bargaining powers proportional
to current resource levels

Hellervik, A

Rodgers, GJ

English

Brunel University Research Archive

A power law distribution in patients’ lengths of stay in hospitalA. Hellervik∗ and G. J. Rodgers†Department of Mathematical Sciences,Brunel University, Uxbridge, Middlesex, UB8 3PH, UK(Dated: July 28, 2006)AbstractThe distribution of patients’ lengths of stay in English hospitals is measured by using routinelycollected data from 11 years. It is found to be well approximated by a power law distributionspanning over more than 3 decades. To explain this observation, a theoretical resource allocationmodel is presented. It is based on iterative long-term scheduling of hospital beds, and its mainassumption is that future beds are allocated preferentially. This represents a situation wheredifferent parts of the health care system compete for resources, with bargaining powers proportionalto current resource levels.PACS numbers: 87.23.Ge, 87.19.Xx, 89.75.DaKeywords: health care, power law distributions, preferential attachment, resource competition, rate equation∗Electronic address: Alexander.Hellervik@brunel.ac.uk†Electronic address: G.J.Rodgers@brunel.ac.uk1I. INTRODUCTIONAs a simple indicator of hospital activity, length of stay (LOS) has been used for manypurposes on several levels of management. It can be used to monitor quality and appropri-ateness of resource allocations both internally in hospitals and at a national level [1]. LOSis clearly related to resource consumption and can thus be used for planning and efficientsorting of patients into groups [2].However, it is well known that the distribution of LOS is highly skewed, which makesmany statistical estimates less robust. In a small sample, it can be very difficult to trackchanges over time since a single outlier can significantly change the mean and variance.A review of different commonly used distributions for LOS is presented in Ref. [1].Lognormal, Weibull, and Gamma distributions are compared with empirical data from anumber of countries. The results showed that both Weibull and lognormal are useful, andwhich one to choose depends on the specific country. In contrast to the study presented inthis paper, the authors considered specific diagnostic groups separately, here we are onlyconcerned with the aggregation of all diagnoses.In Ref. [3] it is pointed out that resources allocated can be excessive for a small subsetof users and at the same time, others get few resources or even none at all. The skewnessin the distribution of LOS reinforces this picture.There have been some attempts to explain the underlying cause for the skewed distri-butions. One example is described in Ref. [4], where a model, employing the transitionsbetween different states of severity in the illnesses of patients, is suggested.The hypothesis brought forward in this paper is that LOS is mainly determined at theresource allocation stage. This would mean that the competition for resources betweendifferent groups of patients is the significant underlying cause of the observed distributions.The competition takes place on all levels, between hospitals, between departments andbetween different types of patients within the same department.We find that the distribution of LOS is well described by a power law, which is a commonobservation in both natural and artificial systems and no specific underlying generativemodel can be inferred from the observation [5–7]. However the association between LOS andresource allocation, makes it possible that a competitive multiplicative process is creatingthe fat tailed distribution. Such processes are well known and historically trace back to the2Simon model [8]. More generally, it has been shown that exponential growth combined withrandom survival times are sufficient to produce power law distributions [9]. This broad classof models includes the one proposed in this paper.This paper is organised as follows: In section II the empirical findings are presented, insection III the model is formulated, analysed and compared to the empirical data, and inthe final section the conclusions are discussed.II. EMPIRICAL FINDINGSThe Hospital Episode Statistics (HES) database contains routinely collected data onpatients in the English National Health Service (NHS). The NHS is managed by the De-partment of Health and is funded by taxpayers and provides care mostly free of charge tothe individual patients. For the analysis presented in this paper, an extract of all patientrecords from 1989-90 to 2002-03 is considered. (The NHS years do not follow calendar years,instead they span between April 1 and March 31.) Some patients admitted before 1989 arepresent in the data set, but no discharge data exists on patients before 1991-92. Length ofStay (LOS) is calculated by taking, for every episode, the difference between the dischargedate (which can also mean the date of transfer or death) and the admission date. The dataset contains 130920010 valid values for LOS. This is about 95% of the recorded episodes.Here we consider all patients in the data set with a valid entry for LOS, which means thatdata from all diagnostic groups are aggregated. In Fig. 1 a double logarithmic histogramfor LOS is shown. It is obvious that the distribution of LOS is very fat tailed and that itcan be described fairly well as a power law over several decades. This means that P [LOS =x] ∼ x−γ. One of the most reliable ways to find a value for the exponent γ is to use themaximum likelihood estimate [5],γ = 1 +n∑ni lnxixmin, (1)where xi are the empirically measured LOS, xmin is a lower cutoff for the power law, and nis the sample size. LOS-values less than xmin are not used in the estimation of the exponent.We choose xmin = 5 (giving n = 33956629) and obtain γ = 2.12.3100 101 102 103 104 10510−410−310−210−1100101102103104105106relative N Treatment time in daysFIG. 1: An exponentially binned histogram, including all measured LOS in the database, 1989-2003. The reference line has a slope of -2. A MLE estimate gives an exponent of -2.12.III. MODELOur model is formulated from the viewpoint of a scheduler, which means that only futurehospital episodes are taken into account. Its essential ingredient is that planning is takingplace iteratively at the same time as parts of the plan are being realised. We have b bedsand a planning window of length T days. This gives a total number of available time slotsS = bT , where every time slot corresponds to one day of hospital stay for one patient. Fromnow on we do not consider the actual beds, just the fact that we have a pool of S timeslots that can be used for planning future hospital episodes. These time slots are allocatedbetween J planned episodes. We assume an arbitrary, initial allocation, where sj denotesthe number of time slots allocated to episode j with j = 1, 2, . . . , J . The initial condition isconstrained by the total number of available slots,∑Jj=1 sj = S. As time passes in the realworld, the planning window is shifted forward which adds new time slots that have to beallocated among sj. Episodes may also be added and removed. This results in an iterativeplanning process, where at every iteration one of the following events can take place (see4Arriving patientsNew plannedepisodes are addedTime slotsare addedto plannedepisodesPlanning pool with S time slots and J episodesRealizedepisodesDischargedpatients FIG. 2: A schematic overview of the model structure. The important dynamics take place at theplanning stage. When a planned episode is realised due to an arrival of a suitable patient at ahospital it is removed from the planning pool. As a consequence of these removals, new episodesare added to the planning pool. Planned episodes are also constantly competing to be lengthenedby attracting added time slots. What takes place after realisation (arrival of an actual patient) isnot taken into account in the model. This means that it is assumed that patients on average stayas long as originally planned.also figures 2 and 3):1. With probability p, a new planned episode is created, which means that S ← S + 1and J ← J + 1. One time slot is allocated to this episode, sJ ← 1.2. Otherwise, an episode is chosen randomly with a preferential probability, Πj =sj∑Jk=1 sk.An additional time slot is allocated to this planned episode, sj ← sj+1 and S ← S+1.At the same iteration, with probability p, an actual patient arrives at the hospital, whichmeans that one of the planned episodes is realised for this patient. This episode is chosenrandomly with a uniform probability, Πj =1J. After realisation, the episode is removedfrom the planning window and cannot be used by subsequent arrivals. In the model, thisis equivalent to taking S ← S − sj, J ← J − 1 and doing a re-labelling nk ← nk+1 fork = j, j + 1, . . . , J .52.1.3. FIG. 3: An example of the flow in and out of the planning pool. 1. Random episodes are realisedand removed from the pool. 2. New time slots are attached to planned episodes. Longer episodeshave a greater chance of attracting these time slots (preferential attachment). 3. New episodesconsisting of one time slot are added.The reason for having the same probability for a patient arriving and for a new plannedepisode being created is to keep the number of planned episodes, J , constant on average. Swill not be constant in this model, but it will find an equilibrium level as shown in Eq. (3)below.When a time slot is allocated to an existing planned episode according to event 2 above,the target episode is chosen with a preferential probability, i.e. proportional to the length ofthe episode. This represents an assumption that bargaining power is proportional to currentallocation of resources.If we denote the number of episodes with k time slots as Nk, then we get a rate equationdNkdt=1− pS[(k − 1)Nk−1 − kNk]− pNkJ+ pδk,1. (2)6Note that the time t in this equation is abstract and represents iterations in the planningprocess. To find a stationary solution for Nk, we first observe thatdSdt=∑kkdNkdt= 1− pSJ(3)which means that we must have J = pS. We can now solve0 =1− pS[(k − 1)Nk−1 − kNk]− NkS+ pδk,1 (4)by recursion to findNk =k − 1k + 11−pNk−1, for k > 1 (5)and N1 = Sp2−p . This yields an exact solutionNk = Sp2− pΓ(11− p + 2)Γ(k)Γ(1 + 11−p + k) (6)and asymptotics give, for large k,Nk ∼ k−2−p1−p . (7)To find the actual LOS of patients in the model we must consider realised patients. Thesecorrespond to a uniform sampling of the planned episodes, with some random noise repre-senting deviations. Hence, if we disregard the noise, the distribution of LOS is proportionalto Nk in Eq. (7). This means that the model will give an average LOS ofSpS= 1p. If theLOS is purely power law distributed we also have an approximate average stay of< k >= (1− γ)∫ ∞1kk−γdk =γ − 1γ − 2 . (8)This gives also another way of predicting the exponent γ,γ =2 < k > −1< k > −1 =2− p1− p. (9)To check the validity of this we can estimate p, from the number of available beds andthe total number of patients. To get the number of available beds (excluding those reservedfor day only patients) we use NHS data [10] to obtain an estimate of the total number ofavailable bed nights, nb, during the relevant years (1991-92 to 2002-03) of the data set. Theestimate is nb = 2435860 × 365 = 2.13 × 1010. The total number of patients in the dataset used for the empirical results in Sec. II with LOS > 0 (staying at least one night) isn = 82047768. We get an estimation of the arrival rate, pest =nnb= 0.092. Fitting this intoEq. (9) gives γ = 2.10, in good agreement with the MLE estimate in the previous section.7IV. DISCUSSIONWhen all different diagnoses and hospitals are taken into account, and aggregated statis-tics are produced, the distribution of length of stay becomes very broad. It is well approx-imated by a power law spanning over more than 3 decades. It is well known that thereexist a large number of models generating power laws. Here, a model based on a preferentialscheduling process is proposed. The micro mechanisms have not been verified, but the modelis falsifiable. To test if preferential allocation is present in the real scheduling process, wewould need access to low level data on planned capacity in different hospital departments.It would also be necessary to study how local planned capacities change over time.If the model is a good description of the actual micro mechanisms, then there are a numberof interesting conclusions to be made. First, the resource allocation is iteratively based onprevious allocations. This could clearly result in a sub-optimal allocation of resources.Second, a preferential allocation tends to favour parts of the health care system that alreadypossess large amounts of resources. It cannot be ruled out that this could be caused mainlyby the better opportunities for large departments or influential patient groups to lobby forresources.Regardless of whether or not the proposed mechanisms are actually present in the realsystem, the fact that the distribution of LOS is very wide is important to note in itself. Itmakes statistical estimations of average and standard deviation of the LOS very sensitive tosample size and clearly undesirable as performance indicators. A MLE estimate for a powerlaw exponent would be a much better option for tracking performance, if a single numberindicator is needed.AcknowledgmentsThis work was funded by the Leverhulme Trust. We also wish to thank T.P. Young foruseful discussions.[1] A. Marazzi, F. Paccaud, C. Ruffieux, and C. Beguin, Medical Care 36, 915 (1998).[2] J. R. Lave and S. Leinhardt, American Journal of Public Health 66, 959 (1976).8[3] L. C. N. MacLean and A. N. Richman, Health Care Management Science 4, 337 (2001).[4] M. J. Faddy and S. I. McClean, Applied Stochastic Models in Business and Industry 15, 311(1999).[5] M. E. J. Newman, Contemporary Physics 46, 323 (2005).[6] D. Sornette, International Journal of Modern Physics C 13, 133 (2002).[7] M. Mitzenmacher, Internet Mathematics 1, 226 (2003).[8] H. A. Simon, Biometrika 42, 425 (1955).[9] W. J. Reed and B. D. Hughes, Physical Review E 66, 67103 (2002).[10] UK Department of Health, Average daily number of available beds, by sector, England, 1987-88to 2004-05, www.performance.doh.gov.uk/hospitalactivity/data_requests/download/beds_open_overnight/beds_ts.xls.9

A power law distribution in patients' lengths of stay in hospital

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