Variations in admissions to NHS hospitals are the source of a great deal of consternation. Over the long-run, admissions and the volume of activity required of the NHS have increased, without equivalent increases in funding or productivity. Over the course of the year, there are repeated claims of crises as hospitals are ill-equipped for the increase in demand in the winter. While different patterns of admissions at weekends relative to weekdays may be the foundation of the ‘weekend effect’ as we recently demonstrated. And yet all these different sources of variation produce a singular time series of numbers of daily admissions. But, each of the different sources of variation are important for different planning and research aims. So let’s decompose the daily number of admissions into its various components.
Daily number of emergency admissions to NHS hospitals between April 2007 and March 2015 from Hospital Episode Statistics.
A similar analysis was first conducted on variations in the number of births by day of the year. A full description of the model can be found in Chapter 21 of the textbook Bayesian Data Analysis (indeed the model is shown on the front cover!). The model is a sum of Gaussian processes, each one modelling a different aspect of the data, such as the long-run trend or weekly periodic variation. We have previously used Gaussian processes in a geostatistical model on this blog. Gaussian processes are a flexible class of models for which any finite dimensional marginal distribution is Gaussian. Different covariance functions can be specified for different models, such as the aforementioned periodic or long-run trends. The model was run using the software GPstuff in Octave (basically an open-source version of Matlab) and we have modified code from the GPstuff website.
The four panels of the figure reveal to us things we may claim to already know. Emergency admissions have been increasing over time and were about 15% higher in 2015 than in 2007 (top panel). The second panel shows us the day of the week effects: there are about 20% fewer admissions on a Saturday or Sunday than on a weekday. The third panel shows a decrease in summer and increase in winter as we often see reported, although perhaps not quite as large as we might have expected. And finally the bottom panel shows the effects of different days of the year. We should note that the large dip at the end of March/beginning of April is an artifact of coding at the end of the financial year in HES and not an actual drop in admissions. But, we do see expected drops for public holidays such as Christmas and the August bank holiday.
While none of this is unexpected it does show that there’s a lot going on underneath the aggregate data. Perhaps the most alarming aspect of the data is the long run increase in emergency admissions when we compare it to the (lack of) change in funding or productivity. It suggests that hospitals will often be running at capacity so other variation, such as over winter, may lead to an excess capacity problem. We might also speculate on other possible ‘weekend effects’, such as admission on a bank holiday.
As a final thought, the method used to model the data is an excellent way of modelling data with an unknown structure without posing assumptions such as linearity that might be too strong. Hence their use in geostatistics. They are widely used in machine learning and artificial intelligence as well. We often encounter data with unknown and potentially complicated structures in health care and public health research so hopefully this will serve as a good advert for some new methods. See this book, or the one referenced in the methods section, for an in depth look.