Hospital characteristics, rather than surgical volume, predict length of stay following colorectal cancer surgery

Objective: Length of hospital stay (LOS) is considered a vital component for successful colorectal surgery treatment. Evidence of an association between hospital surgery volume and LOS has been mixed. Data modelling techniques may give inconsistent results that adversely impact conclusions. This study applied techniques to overcome possible modelling drawbacks.

L ength of hospital stay (LOS) following colorectal surgery is considered an important proxy for successful treatment. 1,2 LOS is influenced by neoplasm (e.g. site and stage) and patient characteristics including comorbidities, social and geographic factors and discharge destination. 3 While a short hospital stay may negatively affect quality of care and health outcomes for patients, 2 a protracted LOS may increase the risk of -or signal complications including, but not limited to -cardiopulmonary, gastrointestinal and muscular morbidities, thromboembolic disease and hospital acquired infection. 2,4,5 Hospital-specific factors also influence LOS, including pre-and post-operative processes, the incorporation of multi-disciplinary teams into patient management practices and the degree of implementation of emerging evidence-based best practice. 6,7 In addition to the impact on patients, an unnecessarily prolonged LOS is an inefficient use of finite health system resources that could be allocated to other areas of patient need. This problem is not insignificant. In 2016, the Victorian Auditor General identified $125 million per year in potential statewide savings from efficient LOS performance. 2 LOS is an important component in the delivery of efficient, high-quality patient care. 8 Since Adams et al. 9 in 1973 and Lufte et al. 10 in 1979 reported a link between provider volume and successful surgical outcomes, provider volume has been used as a surrogate measure of hospital quality. Evidence of an association between hospital or surgeon cancer surgery volume and better patient outcomes has been mixed; 3,7,8, however, previous analyses have had important limitations. A 2002 review that examined seven statistical modelling techniques used to assess association between patient factors and LOS in a cardiovascular setting found that choice of model influenced the conclusion of Hospital characteristics, rather than surgical volume, predict length of stay following colorectal cancer surgery the analyses. Model results were inconsistent due to LOS being a complex phenomenon and unmet assumptions regarding distributional fit. A small proportion of patients with very long hospital stays made it inherently difficult for simpler parametric methods to effectively model the data. 32 These models have been employed also in colorectal cancer (CRC) studies of association between LOS and provider volume and hence some doubt is cast on both negative and positive conclusions. 7,13,14,20,26,30,31 Some studies have used arbitrary thresholds for the categorisation of LOS and volumes (low, medium and high), [12][13][14][18][19][20][21][22]25,31,33 which may reduce statistical power. 34 If the skewness of LOS changes as provider volume changes, then arbitrary categorisations of LOS may be problematic. [34][35][36] Additionally, heterogeneity due to arbitrary categorisations has prevented synthesis of findings. 29,37 A weakness of some hospital patient outcome studies is not accounting for the non-independence of the data if it is important to do so. Analyses may be biased if there are correlations between outcomes within a hospital that are unaccounted forsometimes referred to as random effects. 38 Furthermore, surgeries are performed in the context of a hospital with a distinct infrastructure and management that affect their outcome. 19 Modelling this contextual effect may yield important information regarding its association with patient outcomes such as LOS. [39][40][41] The systematic differences in patients' outcomes across hospitals that persist after differences in patients' risk profiles have been accounted for reflect differences in hospitals' quality of care. 42 In this study, we examined the relationship between LOS following CRC surgery and provider volume by using a quantile regression model that makes no distributional assumption about LOS or error terms 43,44 and avoids arbitrarily categorising LOS or provider volume. 34 The model takes the individual patient as the unit of analysis but uses the data structure to analyse the association between the hospital context and LOS. [39][40][41]

Methods
The Victorian Admitted Episode Dataset (VAED) includes all separations (discharges and transfers) undertaken within all Victorian hospitals. All separations between 1 July  2005 and 30 June 2015 recorded in the   VAED, which included one of 30 ICD-10-AM Australian Classification of Health  Interventions procedure codes for colorectal  surgery as the primary reason for admission,  were identified (Supplementary Table 1).  There were 62,774 admissions for 57,446 patients. Analysis was restricted to admissions whose principal diagnosis was for CRC, ICD-10-AM codes C18.x to C21.x, which resulted in a final data set of 28,343 admissions for 27,633 patients.
Length of stay was defined as the number of days from admission to discharge for the episode of care including transfers to other hospitals and geriatric and rehabilitation centres.
As it was conceivable that LOS and provider volume were not necessarily linearly related, we used an additive quantile regression (AQR) model that does not require a predetermined functional fit but instead determines the best fit from the data. 34,[44][45][46] Provider volume was defined as the number of colorectal surgical procedures performed by a hospital within a fiscal year (1 July to 30 June), whether patients had a principal diagnosis of CRC or not; that is, annual volume (AV). The model we used was based on a formulation by Mundlak 47 and is in a class commonly referred to as a within and between effects model. 39,41,48 It required that we enter both AV and mean annual volume (MAV). Mean annual volume (MAV) was defined for each hospital as the mean of all AV over the number of years the hospital operated within the 10-year study period. Not all hospitals performed colorectal surgical procedures in every study year. 19 The within effect was modelled by AV. It estimated the effect on LOS within hospitals as AV varied and its interpretation is equivalent to any fixed effect estimator. 48 The between effect was modelled by MAV. Due to the model formulation used, it estimated the effect on LOS if a patient were to attend another hospital with a different MAV, that is, the hospital contextual effect. 39,40,42 This method draws comparisons across hospitals and estimates the effect of hospital choice on patient LOS or, in other words, hospitals' quality of care or efficiency regarding LOS. 42 The model was adjusted for various patient and hospital factors that may confound the association between provider volume and LOS. 3,8,38,49 These included patient factors such as: sex; age; Elixhauser Comorbidity Index; the American Society of Anesthesiologists' physical status classification system (ASA); cancer site (colon, rectum or anus); whether the cancer was metastatic or not based on ICD-10-AM codes; admission type; separation mode; 3,50 and the colorectal procedure performed. Hospital factors were: number of daily colorectal surgical admissions; type of hospital (private for profit, private non-profit or public); and whether the hospital was colocated or not. 51 Possible seasonal variation in LOS was modelled by month of surgery. The Elixhauser comorbidity index was calculated based on the diagnoses codes (up to 40) present in the surgical episode. All other patient and hospital factors are recorded in the VAED except for colocation and private for profit hospitals. These were determined by checking hospital websites or The Victorian Department of Health and Human Services internal records.
We did not know a priori whether all variables in the model were confounders and hence may have over-specified the model. 52 We tested confounding for the continuous variables by running separate additive quantile regressions (AQR) with them as sole covariate and LOS and AV separately as outcomes. This was done for the 1 st , 5 th , 25 th , 50 th , 75 th , 95 th and 99 th percentiles of LOS. The results indicated that the continuous covariates were all associated with both AV and LOS and hence could be accepted as potential confounders. We tested the categorical variables for confounding by running Kruskal-Wallis rank sum tests for both AV and LOS. The results indicated that, except for sex, all the categorical variables were associated with both AV and LOS and therefore could also be accepted as potential confounders. Sex was associated with LOS but not with AV; however, we included it with the full model in the boosting process to assess its influence. We did not include ICU use or ICU hours in the model as they are directly on the causal pathway to LOS (use of ICU necessarily means a lengthening in LOS) and hence an over-adjustment bias. 52 We assessed possible random effects 33 by including a random intercept for hospital into the boosting process, where hospital was represented by a dummy variable.
Model building was aided with boosting, a statistical technique in the class of machine learning methods. 34,43,[53][54][55][56] Boosting assessed which variables and functional fit better predicted LOS. These results were used to build a second AQR model that did not use boosting but used shrinkage and penalisation for model estimates. 44,45 We tested how well

Epidemiology
A retrospective analysis of 62,774 colorectal surgeries 2005-2015 the models represented the data by using them to predict the empirical cumulative distribution (CDF) of LOS and assessed its fit. We refer to this as the recovered distribution (see Supplementary File: Statistical Details).
As the model formulation we used estimated the effect on LOS if a patient were to attend another hospital, 39,40,42 we used the model to simulate the change in LOS if CRC patients were to have counterfactually attended a hospital that the model indicated to be more LOS efficient. We predicted each percentile 1 to 99; these were combined to obtain the predicted CDF of LOS. 56 This was termed the counterfactual CDF. The area under the counterfactual CDF was calculated and compared to the area under the recovered CDF. As the area under each CDF directly related to the total sum of LOS days, the difference in areas estimated the change in total sum of LOS due to this hypothetical experiment. A 95% confidence interval (CI) was computed for the estimated change.
We calculated a dollar value for the change in total LOS by assuming an average of $1,000 per bed day. This was based on the 2016 report by the Victorian Auditor General who estimated an average cost of $864 per bed day due to direct costs such as daily labour, primarily for medical and nursing staff, and indirect costs such as lighting, heating and cleaning. The cost estimate excluded theatre costs, pharmacy and pathology costs, depreciation, capital and maintenance costs, as these costs are unlikely to be saved by reduced LOS. 2 The Auditor General relied on data up to 2014 and so -allowing for CPI increases of 2% per annum to 2019 -$864 converted to approximately $1,000 but, when applied to total LOS, the estimate was rounded down to the nearest million.
Statistical significance was set at the 0.05 level. All statistical analysis was performed using R, version 3.3.3 57 using the following packages: quantreg for the AQR 58 ; mboost for the boosting 59 ; and flux for computing the area under the graph using the auc function. 60 (Further details are provided in the Supplementary File).

Model Fit
Boosting indicated all entered variables may be important for predicting LOS except for Elixhauser score and co-location status. Separation mode and month were the strongest and most consistent predictors across the percentiles. Boosting suggested that non-linear fits for MAV, AV, age, and month better predicted LOS than linear, but a linear fit for year and number of daily colorectal surgery admissions better predicted LOS. Boosting also indicated that allowing for hospital random effects was not useful for predicting LOS. These results were used to build the second AQR model. We compared how well the models represented the data by using them to predict the empirical cumulative distribution (CDF) of LOS and assessed their fits. We chose the second model as a better fit to the data (see Supplementary File: Statistical Details).

Mean Annual Volume Association with LOS
The model graphs ( Figure 1) indicate that hospitals' performances (contextual effect) regarding patient LOS varied greatly for all percentiles and that this variation was not systematically associated with MAV. The graphs display initial falls in LOS for MAV up to approximately 33, which are then followed by marked variation, with low points in LOS at MAV of 122.1 and 245.8 and a high point at 105.6 MAV. The former two MAV had the lowest LOS over all percentiles 40 and 20 times, respectively, while the latter had the highest 89 times. In a sensitivity analysis, the admissions to the three hospitals with these three MAV were removed from the analysis. This resulted in very much flattened graphs (not shown here). For all percentiles, the p-values from an F statistic for model fit were less than 1×10 -6 .

Annual Volume Association with LOS
The model graphs ( Figure 2) indicated that the association between AV and LOS varied with percentile of LOS. For percentiles 10 to 95, rising AV was generally associated with rising LOS. For percentiles 10-60 this association was generally attenuated or reversed for AV greater than about 100. This was similar for percentiles 65-95, except the attenuation occurred later at AV of approximately 200. The magnitude of these variations in association generally ranged between 0.5 and 4 days and increased with increasing LOS percentile. For all percentiles, the p-values from an F statistic for model fit were less than 1×10 -6 .    Figure 3 demonstrates that, for the second counterfactual experiment, the predicted savings mainly came from reduced LOS for patients who had LOS between percentiles 14 and 94.

Counterfactual prediction of change in LOS contingent on change in MAV
Please see the Supplementary File for an in-depth display of the results for the associations between all other patient and hospital factors in our model and LOS.

Discussion
Our model indicated that there was marked variation across hospitals' performances regarding LOS and that it was not contingent on MAV. This contextual effect in our model reflects hospitals' quality of care or efficiency regarding LOS. 42 It estimated change in patient LOS if patients were admitted, counterfactually, to another hospital. 39,42 If the hospital contextual effect was negatively associated with LOS, for example, the graphs in Figure 1 would have shown consistent falls in LOS with increasing MAV. Instead they showed increased or decreased LOS was largely independent of change in MAV. Therefore, the hospital contextual effect on LOS was associated with factors other than MAV.
The substantial size of the predicted change produced by the counterfactual experiment and its statistical significance indicated that the hospital contextual effect was important in explaining LOS. Both counterfactual predictions implied a statistically significant recommended analytical approaches that respect the multilevel structure of hospital data and allow for both between and within hospital variation. 38 Our first model was multilevel and allowed for between and within effects via the Mundlak formulation. The second model allowed for between and within effects via the Mundlak formulation but was not multilevel in the sense of allowing for correlation between outcomes within a hospital -referred to as random effects. However, our model building process indicated that, for our data, random effects were not important for predicting LOS and the second model (without random effects) that we used for the final analysis was superior. The weakness in not allowing for random effects is that, although coefficient

Figure 1B: Model estimates for between hospital differences (contextual effect) in the association between annual volumes and LOS for percentiles 75-95 in intervals of 5.
estimation is still consistent and close to the estimate when random effects is employed, coefficient standard errors may be underestimated. 61 That is, statistical significance is over optimistic. However, even if our model building process was inaccurate, the p-values for model fit for MAV and AV were so low -less than 1×10 -6 for all percentiles -it would be unlikely that standard errors would be increased to the extent that statistical significance at the 0.05 level would be lost. As our modelling respected the multilevel structure in the data by incorporating a Mundlak formulation and checking the importance of random effects, we compared our findings to other findings from multilevel models.
Other studies using multilevel models have made similar findings to our study. Zheng

Vicendese et al. Article
ideal (≤5 days postoperative) or prolonged (≥90 th percentile, 21 days postoperative) as outcomes following CRC surgery for 240,873 patients. 8 This study produced funnel plots based on their analyses that, similarly to our study, clearly displayed marked variation between providers in both outcomes but with no systematic association with provider volume.
We found no evidence that higher AV was associated with lower LOS. AV was the within hospital effect in our model. The model indicated that over the 10-year study period, increased volume in a single year was generally associated with higher LOS. This association may be explained by the demands on management and service staff resources that may increase incommensurately compared to increases in AV and which eventually effect operational efficiencies in the short term. 62 We also found the association between AV and LOS was non-linear. Increase in LOS was attenuated after AV of around 100 for percentiles 60 and lower of LOS and AV of around 200 for higher percentiles of LOS. Increasing AV beyond these threshold points generally showed little change in LOS. This implies that, in the face of growth in provider volume, lower-volume hospitals would be more subject to increased LOS, but higher-volume hospitals would show little change. This may be due to relative capacities to absorb the demands on management and service staff resources. These results concord with Faiz, who noted that there was little evidence that increasing volumes of colorectal surgery hospitals was a warranted strategy for improving quality of care. 63 In our model, the association between AV and LOS was independent of the hospital contextual effect, as these factors have been mutually adjusted for in our model. 39 Therefore, the effect of increased AV could be augmented or attenuated depending on the hospital's contextual effect. This adds further evidence to the conclusion by Burns et al. 19 mentioned above -that centralisation may be ineffective in improving results unless the higher volume is directed towards highquality providers. This recommendation has been echoed by others. 50,64 There have been conflicting results from many studies regarding the relationship between colorectal surgery hospital volumes  although the initial impression is that the literature overwhelmingly substantiates a benefit from high hospital volume, analysis of the quality of the data suggests that this conclusion is unsafe. 29 They suggest that studies had been biased by not including the extra LOS if a patient was transferred to another hospital to recover. We included this contribution to LOS in our study. It has also been suggested that there may be publication bias that may contribute to the impression of the importance of hospital volume. 50 Some of the uncertainty regarding the association between provider volume and LOS may be partially explained by reverse causality. Market forces may have directed high volume referrals to hospitals with known quality outcomes, or perceived as so, rather than the high volumes causing the high quality outcomes as such. 63 That is, hospital volume has an endogenous and exogenous quality. It can be both a predictor and outcome of hospital quality. Ash et al. pointed out this endogeneity and exemplified it from the other end of the volume spectrum. The current low volume of a hospital may be a result of poor quality and hence in the causal pathway between the hospital and the outcome being considered. 38 Therefore, because of the combined exogeneity and endogeneity of provider volume, careful consideration needs to be given to its inclusion as a way of characterising hospital performance.
As with all observational studies, a weakness of this study is the risk of bias due to inadequate adjustment for included confounders or non-adjustment due to non-included confounders, especially for confounders that may be differentially distributed across patients and hospitals. 38,42 These could be such factors as: hospital waiting list pressure 62 ; tumour stage; socioeconomic status (SES); patient health behaviour; multidisciplinary teams; a broader range of specialist and technology-based services at high-volume hospitals; better coordination of personnel or a higher level of collaboration between physicians and nurses; the presence of more or bettertrained nurses 65 ; use of specific clinical pathways; the method by which laboratory tests are reported to physicians; policies and operations of hospitals 20 ; and surgeon volume and specialisation. 29,66 We had no data for any of these factors. Omitted variable bias has been controlled to a certain extent due to the Mundlak model formulation; however, it is not a panacea. 38  volume and specialisation are important confounders for the association between hospital volume and CRC surgical outcomes, internationally and in Australia. 29,[66][67][68] There is evidence from a systematic review and meta-analysis that indicated surgeon volume and experience may be more important than hospital volume. 29,37 It may be possible that high volume can be caused by a large group of low-volume surgeons in one hospital, while a hospital volume may be low if a single -perhaps more experienced -surgeon performs all the procedures. This infers that interpreting the association between hospital volume and outcomes is difficult without controlling for surgeon volume. 29 Our model would have benefited from including surgeon as another level. 38 As the hospital contextual effect seemed largely due to factors other than provider volumes, it would be beneficial for patients and the health system if some hospitals were directly investigated to determine how they are functioning more efficiently regarding LOS and conversely why some hospitals are functioning less efficiently. 37,42,69,70 This would provide valuable information that may greatly benefit the patient and health system and be a check on our model. These could be such factors as systemised protocols and procedures, multi-disciplinary teams and ERAS (Enhanced Recovery After Surgery), for which data was not available. As LOS efficiency was not dependent on provider volume, this implies that it may be possible for all hospitals, regardless of volume, to operate more efficiently in regard to LOS and hence realise the important benefits to the patient and healthcare system alike. 71 However, it may be unlikely that hospitals with very low volumes could justify the allocation of resources to achieve operational efficiencies.
An inconsistent association between MAV and LOS, the general positive association between AV and LOS and possible endogeneity, suggest it is inadvisable to use provider volume as a marker of quality regarding LOS following CRC surgery in Victoria. The Victorian Auditor General also found provider volume was not a reliable indicator of LOS efficiency following surgery for many conditions other than CRC. 2 We found it was more pertinent to use the hospital contextual effect to analyse the performance of hospitals in regard to LOS.
Using the hospital contextual effect, our model presents itself as a useful method for assessing provider performance regarding LOS, adjusted for pertinent patient and hospital factors. 38 The model can produce graphs such as that in Figure 1 to guide analysis. More importantly, by using the model's counterfactual prediction capacity, an index of performance with a 95% CI can be obtained for any hospital. The prediction would immediately indicate if the hospital was performing better, worse or at about the same level as all other hospitals. If performing the counterfactual experiment -using the hospital being assessed as the basis of comparison -resulted in a 5% drop in total LOS, then that hospital would have an index of 0.95 with an associated confidence interval. An index lower than 1 indicates superior efficiency, higher than 1 indicates inferior and 1 shows no difference. This index is independent of any distributional or model fit assumptions or arbitrary categorisation. For the analysis of LOS, the Victorian Auditor General's report resorted to using trimmed data in a linear regression. This method may be subject to statistical objections if a mean is not representative of the whole data and because information in the tail(s) of the distribution, that may represent patients who may not fit the profile of a mean patient, is discarded 2,44,72,73 (see Supplementary File).
Our model could be used in national or international settings as it can allow for nesting in those levels, and so help assess hospital efficiency in regard to LOS in broader contexts. This would assist with synthesis of future international studies when, in the past, diverse categorisation methods had impeded synthesis. It can be extended to analyse variation between hospitals regarding other outcomes such as mortality and readmission following CRC surgery. However, our model requires very large data sets, especially if many patient factors were entered into the analysis.
Our model was further complemented by shrinkage and penalisation to obtain more accurate estimates and reduce statistical error 38

Conclusion and implications for public health
Our model showed marked variation between Victorian hospitals regarding LOS efficiency following CRC surgery between 2005 and 2015. This efficiency was not predicted by provider volume. Patients and the health system would benefit from an analysis of the function of identified efficient hospitals to clarify factors contributing to their relative efficiency regarding LOS and conversely to rectify factors that cause some hospitals to function less efficiently.
As volume was not a determining factor for LOS efficiency, all providers could benefit -assuming capacity to allocate required resources. This has the potential to improve convalescent times for CRC surgical patients and save the Victorian health system more than $3 million per year in present day terms.
Due to its endogeneity, we recommend not using raw provider volume as an indicator of LOS operational efficiency. Further study is required to determine if LOS efficiency may be associated with increased risk of postoperative complications.