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Survival analysis and competing risk models of hospital length of stay and discharge destination: the effect of distributional assumptions

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Abstract

The literature on length of stay and hospital discharge is often used to inform policy regarding hospital payment and quality. This literature has evolved from the use of ordinary least squares estimation of linear and log-linear models to the use of survival and competing risk models that control for unobserved patient and hospital heterogeneity. However, the authors tend to adopt different distributional assumptions and often motivate the choice of specific functional forms for the baseline hazard based on the visual inspection of the hazard rate plots. We contribute to this literature by showing that parameter estimates for patient and hospital characteristics in length of stay models are particularly sensitive to underlying assumptions regarding the hazard function. Moreover, we demonstrate that the inability to distinguish between competing risks of discharge destination may lead to distortions in the effect of important explanatory variables such as intensive care utilization.

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Notes

  1. While it is the case that reimbursement based on DRGs affects length of stay depending on the mode in which it is used (per patient payment versus hospital budget setting), this issue is beyond the scope of this particular paper. See Dismuke and Guimaraes (2002) for a discussion of the DRG system in Portugal.

  2. For example, Canoodt and Knickman (1984) found that teaching status of hospitals does not significantly influence length of stay, while Burns and Wholey (1991) found that teaching status significantly increases, but Shi (1996) concluded that teaching status significantly reduces length of stay.

  3. \({\phi \left({{\bf x},{{\varvec {\beta}}}}\right)}\) is a non-negative function of the covariates. As typically done in the literature we let this function be exponential and will henceforth replace it by \({\exp \left({{{\bf x}^{\prime}}{\varvec {\beta}}}\right)}\) .

  4. The parameter α assumes only positive values. If α > 1 then the hazard function increases monotonically, if α < 1 then it decreases monotonically and if α = 1 the model collapses to the exponential case.

  5. A duration is right censored if all that is known about that duration is that it lasted longer than a certain time.

  6. So far, we avoided using the subscript i for individual observations in order to facilitate the exposition. We only use it in presenting the likelihood function.

  7. Again, we only use the individual subscript in presenting the likelihood function, but leave it out for the remainder of the text.

  8. For a more detailed discussion on competing risk models, see, for instance, Cox (1959), David and Moeschberger (1978), and Prentice et al. (1978).

  9. The Instituto de Gestão Informatica e Financeira, the entity responsible for the management of the information technology and financial resources of the Portuguese Ministry of Health, provided all public hospital discharges classified into DRG 14, Cerebrovascular Disorders Except Transient Ischemic Attack, for the January 1992–December 1994 time period. DRG 14-Cerebrovascular Disorders includes the following ICD-9-CM diagnosis codes: 430 Hemorrhage, subarachnoid; 431 Hemorrhage, intercerebral; 432 Hemorrhage, intracranial, other and unspecified; 433.01 Occlusion and stenosis, basilar artery with cerebral infarction; 433.11 Occlusion and stenosis, cartoid artery with cerebral infarction; 433.21 Occlusion and stenosis, vertebral artery, with cerebral infarction; 433.31 Occlusion and stenosis, multiple and bilateral arteries, with cerebral infarction; 433.81 Occlusion and stenosis, other specified precerebral artery, with cerebral infarction; 433.91 Occlusion and stenosis, unspecified precerebral artery, with cerebral infarction; 434.01 Thrombosis, cerebral, with cerebral infarction.

  10. Although in practice, time is always measured in discrete units, “when these units are very small, it is usually acceptable to treat time as if it were measured on a continuous scale” Allison (1984), p. 14.

  11. The Charlson Index contains 19 categories of comorbidity, which are primarily defined using ICD-9-CM diagnoses codes (a few procedure codes are also employed). Each category has an associated weight, taken from the original Charlson paper (1987), which is based on the adjusted risk of 1-year mortality. The overall comorbidity score reflects the cumulative increased likelihood of 1-year mortality; the higher the score, the more severe the burden of comorbidity.

  12. According to practice guidelines, Computerized Tomography of the head is critical for the emergent evaluation of patients with acute stroke. It is important for excluding or documenting inter-cranial hemorrhaging as the stroke mechanism and to identify other features that directly or indirectly impact the diagnostic evaluation and management of stroke (Culebras et al. 1997).

  13. Portuguese hospitals are classified as either central, district or level one depending on the availability of specialties. Central hospitals generally provide all specialties, district hospitals a moderate number and level one, a basic level of specialties. The level of technological capacity also varies from greatest in central to least in the level one hospitals.

  14. The Casemix variable is a measure of the severity of all patients treated in a given hospital. To compute this variable, each discharge in a hospital is weighted with the relative weight of its DRG, and the weighted sum of all discharges is divided by the number of discharges in that hospital.

  15. Most models were estimated using the STREG command in Stata. The exception were the Cox models that were estimated with the STCOX procedure and models with Heckman and Singer heterogeneity that required programming the likelihood function using ML commands. A list with the Stata code used in this paper is available at http:// www2.eeg.uminho.pt/economia/cangelica/downloads/stata_commands.pdf.

  16. The parameter on duration dependence is not significantly different from 1 at the 1% level.

  17. A likelihood ratio test of the hypothesis of equality for the seven ancillary parameters of the piecewise constant model, with a chi-squared test statistic of 2,013.4, is rejected at the 1% significance level.

  18. There are 78 hospitals in our data, so that models with hospital fixed effects include 77 additional dummy variables.

  19. The Akaike information criterion (AIC) for the models in Table 4 is 101,397.8, 101,553.2, 101,228.8 and 100,838.8 versus 97,119.4, 97,113.4, 98,176.8 and 97,554.2 for Table 5. Similarly, the Bayesian information criterion (BIC) is 101,516, 101,679.8, 101,389.2 and 101,016 versus 97,837, 97,847.8, 98,945 and 98,339.3.

  20. We also estimated models that treated the hospital effects as random, namely the Weibull, piecewise-constant hazard and Cox model all with Gamma shared frailty. All models pointed to the existence of unobserved hospital heterogeneity but provided estimates that were similar in sign and significance to models that accounted exclusively for unobserved patient heterogeneity.

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Acknowledgements

The first two authors gratefully acknowledge financial support by the Portuguese Foundation for Science and Technology (FCT; refs. SFRH/BD/5054/2001; POCT/ECO/34666/2000). This revised version has benefited considerably from comments from the editor and two anonymous referees.

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Correspondence to Clara E. Dismuke.

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Sá, C., Dismuke, C.E. & Guimarães, P. Survival analysis and competing risk models of hospital length of stay and discharge destination: the effect of distributional assumptions. Health Serv Outcomes Res Method 7, 109–124 (2007). https://doi.org/10.1007/s10742-007-0020-9

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