Costa Rica

The results of this section have shown that Dynamic Transition Matrices provide the greatest improvement in distribution-fit for the wards which are most likely to cause outliers. Given that fixed capacities are known to exist in the real system, these wards are the ones which exhibit the largest degree of negative

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skewness. The primary contribution that DTMs make in improving the fit of the simulated distributions with the data is a reduction in variance; reducing the likelihood that the simulated realisations of midnight occupancy take very low or very high values. Indeed, the three most negative-skew wards are also those which show the biggest improvement in distribution-fit (via visual inspection) with the data gathered from the AGH.

As negative skewness reduces (approaches symmetry), so too will the frequency with which the hospital will be required to turn away patients from the ward, thereby decreasing the impact of using DTMs. Three wards were identified as being moderately skewed; Intensive Care, Ward 5A and Ward 6D. For Intensive Care, the trend identified in the outputs generated by the simulation meant that DTMs offered no improvement in fit, over the use of STMs. For Ward 5A, the peculiarity of the empirical distribution from the PA data meant that neither STMs nor DTMs performed particularly well, suggesting other factors, not included in the simulation, influence the distribution of midnight occupancy on this ward. For Ward 6A, the DTMs offer a clear improvement in the fit of the midnight occupancy distributions.

For the wards on which the distribution of midnight occupancy is positive-skew or symmetric, the performance of the two routing policies is so similar that the modeller is likely to be indifferent. These wards include the Emergency Department, Ward 4K and Northside. The aggregate Other ward is positively

skewed, although the increase in average occupancy over time means that DTMs offer no improvement in terms of fit with the PA data, relative to the use of STMs.

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In summary, if a modeller is only interested in predicting measures of central

tendency over the course of the planning horizon, then given the choice between STMs and DTMs, and based on the estimates of mean Δℎ-occupancy shown in Table 5.4, STMs should be the routing policy of choice. They outperform DTMs in terms of absolute error of the simulation mean relative to the observed mean and require no statistical modelling to derive. However, if more information is required about the distribution of midnight occupancy then DTMs should be used. By using a routing model which is informed by occupancy, DTMs provide a better representation of the patient diversion strategy employed by many hospitals during peaks in bed demand, thereby improving the overall fit with the PA data, for the wards most frequently causing such diversions. Although most wards see a small overestimation in the value of mean/median midnight occupancy under DTMs (due to comparing skewed and symmetric distributions, in most cases), this is offset by improved estimation of the variability of the occupancy distributions. This in-turn improves estimates of the likelihood that a ward is found above a given capacity threshold during the planning horizon, on the assumption that the busyness-dependent routing behaviour estimated from the PA data will continue in a similar way.

5.5 Discussion and Conclusions

To answer the first part of Research Question 2, which is; “Can the effect of hospital busyness on patient-to-ward placement decisions be detected in

patient administrative data, and can this be incorporated in a simulation model?”, a statistical model (Multinomial Logistic Regression) was chosen

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which allowed the extent of the relationship between occupancy and transition probability to be quantified, for each ward. The use of a generalized linear model (along with appropriate statistical analysis software) means the detection of a relationship between the explanatory and response variables becomes part of the model fitting process. In this application, a relationship between ward- level occupancy and transition probability is detected by using a stepwise search, with AIC as the model selection criteria.

Tables 5.1 and 5.2 show that for all but two of the ward and patient type combinations, it is possible to improve the predictive capability of the transition model by incorporating occupancy information collected at a time just prior to

the occurrence of a transition between wards. Therefore, it is possible to

statistically detect the effect of hospital busyness on patient placement, when busyness is gauged by the number of occupied beds on each ward just prior to ward transition, and the transitions themselves are framed in terms of their probability of occurring.

As with any generalized linear model, once the regression coefficients have been estimated, the modeller is left with a functional relationship between the explanatory and response variables which can be used for prediction. Such an equation can be used in any simulation package which is flexible enough to define transition rules in terms of a mathematical equation. Therefore, the effect of hospital busyness on patient-to-ward placement decisions can always be incorporated in a simulation model provided the effects can be approximated by a set of formulae (such as Dynamic Transition Matrices) and the chosen simulation software offers sufficient flexibility when defining the routing policy.

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The second part of Research Questions 2 asks, “If so, what effect does it have?”. In terms of simulation run-time, using DTMs increases the real time

taken to run 100 parallel 560-day simulations (re-initialising once per simulated week) by approximately 22%, adding around 2.5 minutes to a 12-minute simulation experiment in which STMs are used. However, the full 560-day run is solely for validation purposes, and in practice, the simulation would only be run for the length of the planning horizon. Therefore, the impact of the additional run time will be inconsequential for a simulation running on time-scales such as this.

In terms of the simulation outputs, the empirical results from this chapter show that for the wards whose simulated distributions provide a better fit to the data, the improvement is largely the result of a reduction in variability once DTMs are implemented, meaning the wards are less likely to be found at both very low and very high occupancies. Reduced probability in the right-tail of the distributions may be an expected result since one of the primary reasons for using DTMs is to redistribute arrivals and transfers at times when the ward is experiencing busyness. However, there is also a reduction in probability in the left-tails of the simulated distributions (compared to the model in which STMs are used), indicating that wards experience fewer days at low occupancy once the routing policy allows for outlier patients to occupy alternative beds, along with those which would have spent time on the ward regardless of busyness. In addition to the model’s improved ability to represent the impact of outlier patients for the largest wards, the statistical framework used to produce the DTMs (Multinomial Logistic Regression) allows patient diversion behaviour to

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be estimated directly from the hospital’s PA database, rather than being derived from separately collected information, or assumed, for modelling convenience. A data-driven approach also allows the model to be easily recalibrated in the presence of new data, or if the routing procedures are believed to have changed appreciably after the model development phase. A data-driven approach to approximating the routing behaviour also means that a DTM recalibration procedure could theoretically be included within a so-called “auto-validation” module which automatically updates the simulation parameters as necessary.

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Chapter 6