Statistical methods and spectral density (SD) calculation

Building the time series

Series were built for La Fuenfría Hospital’s admissions, discharges (Dis) and hospitalized patients, called “inpatients”, and shortened to InP in this paper. The Hospital depends on the Public Health System Network [Servicio Madrileño de Salud (SERMAS)] of the Autonomous Madrid Region, which, like most in Spanish autonomous regions, requires that all hospital information in the area is stored into a central data base. In the mid-term stay hospital this information is kept using a program called SELENE^©. SELENE^© is jointly developed by UTE Siemens– INDRA [40]. SELENE^© keeps data in a central database for SERMAS. Data were retrieved from SELENE^© as monthly spreadsheets for each kind of information, and were integrated in a single file covering our entire period of study for each type of information.

To build the sequences for this study, we used data collected monthly as SELENE^© Hospital’s spreadsheet files called “Cuadros de Mando” which we translate Command panels. In them, information appears as daily data for La Fuenfría Hospital from 5/1/2014 up to 12/31/2014, then they run from January 1 to December 31 for years 2015 and 2016, and from 1/1/2017 up to 12/19/2017.

Data was routinely hospital’s statistics, was not recorded purportedly for this study, and we found transcription errors in the SELENE^©-exported data base, which were dealt with as follows:

• January 1^st data corrections. An obvious error detected in the time series was that more than 900 admissions or discharges appeared for 1/1/2015 and 1/1/2016, which is impossible for a hospital with 192 beds. The source of such errors appears to be a bug in the software used to gather data which did not evaluate the first datum of the year correctly. Those points were replaced by averaging December 31 of the previous year with January 2 of the current year, respectively. This error affected 2 out of 1329 days, or ≈ 0.16% of data.
• Missing data corrections. In Admissions (Adm) and Discharges (Dis) series we found a gap without information from 5/29/2014 to 6/6/2014 (both inclusive), in daily in–patients. This gap (9 out of 1329 days, or ≈ 0.67% of data) was closed by filling the data on daily changes in the sequences for the gap days with a same value calculated as (1) where dates follow the “mm/dd/yyyy” style, and the double headed arrow at the left of Eq (1) points at the limits of the gap to be filled. In Eq (1), as well as elsewhere in this paper, brackets are used to indicate sets.
• Daily differences between Adm, Dis and InP. Once daily data admissions (DAdm), daily discharges (DDis) and daily inpatients (DInP) sequences were built, new series were developed by subtracting from each daily value, the immediately preceding day’s similar datum. A set of sequences of one-day differences (2) with t = 0, 1, …, 1328 days, was built. In Eq (2) each t is a given day, and Δt = 1 is a one-day time difference, in which the first day was arbitrarily taken as 0. The procedure was repeated for admissions, discharges and stays, to obtain the corresponding daily differences sequences for the three kinds of parameters shown in Fig 2A and 2B.

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Fig 2. Sequences of daily admissions and daily discharges from La Fuenfría Hospital.

Panels are: A- Daily admissions (Adm) sequence; B- Same as panel A, but scale modified to only to show data ⩾ 0; C- Daily discharge (Dis) sequence; D- Same as panel C, but the scale is modified to only show data ⩾ 0. In all cases abscissa is days elapsed since May 1 2014, Ordinate is number of daily admissions or discharges determined as indicated in Building the time series in Methods. Negative peak represent transcription errors of the data made available for this study, perhaps positive outliers could have the same origin, however this is harder to demonstrate.

https://doi.org/10.1371/journal.pone.0262815.g002

Ethics statement, author’s conflicts of interests and privacy of the information used in the study

No human or animal subjects were used in this study. Thus no animals were sacrificed. Neither personal nor animal pain or discomfort were induced at any point of the study.

In compliance with the European Data Protection Law [41] no patient’s personal or clinical information is revealed in this paper. La Fuenfría Hospital Ethics Committee approved this study as indicated in a letter submitted to PLoS One.

Though it is the author’s interest always to improve the efficiency of LFH. The authors state that they have no conflicts of interest or competing interests in the subject of this study.

Excluding personal or clinical patient information, everything stored in the SELENE system, including the La Fuenfría Hospital “Cuadros de Mando” which we translated here as “Command panels”, are public information according to Spanish Law. All the original La Fuenfría Hospital “Cuadros de Mando” which we translate “Command panels” spreadsheets cis be provided with this revised manuscript. There is also a short Readme text file explaining some issues of the LFH data file. The Command panels, and a short Readme text file, used here are provided to PLoS One, and are freely available to anyone interested in them.

Data availability and privacy of the information used in the study

All relevant data are within the manuscript and its S1 Data files. The Command panels data used in this study are provided as S1 Data and are freely available. A short Readme text file is also included, which explains the LFH data file.

Excluding personal or clinical patient information, everything stored in the SELENE system, including the La Fuenfría Hospital “Cuadros de Mando” which we translated here as “Command panels”, are public information according to Spanish Law. All the original La Fuenfría Hospital “Cuadros de Mando” which we translate “Command panels” spreadsheets cis be provided with this revised manuscript. The translated Command panels are provided to PLoS One, and are freely available to anyone interested in them.

LFH Command Panels also include names and ID codes of Hospital personnel for administrative purposes. Since these are also protected by Spanish and EU laws on personal information privacy they were remove prom data posted in PLoS ONE.

Ethics statement and author’s conflicts of interests

No human or animal subjects were used in this study. Thus no animals were sacrificed. Neither personal nor animal pain or discomfort were induced at any point of the study.

In compliance with the European Data Protection Law [41] no patient’s personal or clinical information is revealed in this paper. La Fuenfría Hospital Ethics Committee approved this study as indicated in a letter submitted to PLoS ONE.

Though it is the author’s interest always to improve the efficiency of LFH. The authors state that they have no conflicts of interest or competing interests in the subject of this study.

Analysis of runs above and below the median

During the period studied, the number of hospital in–patients were 150 (149–151) (median, , and 95% CI, n = 1329 days) ranging (102 –- 187) patients, meaning that Hospital’s is 0.77 (0.53–0.77) (median, , and 95% CI). Let us consider a random variable y = f(t) such that f(t) is independent from f(t − Δt) or from f(t + Δt) (where Δt is a short time lapse, ideally Δt → 0). This is, a variable for which its present is independent from its past, and whose present does not determine its future. Such variable produces a sequence of data which are uniformly distributed around their median . We say that a run occurs when one or more sequence points fall above or below the median, this is easily grasped by looking at Fig 3. In Fig 3C there are 5 runs, and in Fig 3D there are 9 runs. Points that are equal to are not considered in the analysis. Fig 3 presents four sets of monthly averages from the Function Recovery Uunit (FRU). Fig 3A, results from the total of inpatients (Adm) sequence (n = 22, n_runs = 11, probability of random run distribution about the median P_rnd is ≈ 0.15); Fig 3B, Total discharged patients’ (Dis) sequence (n = 22, n_runs = 8, P_rnd ≈ 0.04); Fig 3C, is the sum of lengths of stay (LoS) for all in–patients (InP) in the Unit during each month (n = 23, n_runs = 5, P_rnd ≈ 4.5 ⋅ 10⁻⁴); Fig 3D, is a sequence of the average stay duration (Mean LoS) of each in–patient at the Unit (n = 23, n_runs = 10, P_rnd ≈ 0.13); Fig 3E, during de observation period the Function Recovery Unit gained relevance and the number of beds were increased, panel shows the changes in beds at the beginning in several months, as expected, the analysis of this sequence was found to be non random (n = 23, n_runs = 5, P_rnd ≈ 4.5 ⋅ 10⁻⁴), actually n_runs may be just 3 in this case, but the algorithm probably found tiny decimal differences between data Fig 3 an in one segment; if n_runs = 3, should n runs be 3 then P_rnd ≪ 4.5 ⋅ 10⁻⁴. P_rnds are probabilities (obtained with the Wald–Wolfovitz test) that the sequences are not randomly distributed about their medians. In all cases abscissa is the month number during the study period (January 2016 = 1, November 2017 = 23). Out of these P_rnd values the only one indicating a statistically significant non zero number of runs, occurs for InP sequence in Fig 3C [42, 43]. The sequence non-randomness the Fig 3C could have several reasons, one of them could be some kind of ‘periodicity’ (which would be odd, since its period would be longer that 2 years), but it could also stem from some kind of random walk. During the period studied, the Hospital’s in–patients were 150 (149–151) (median, , and 95% CI) ranging (102–187) patients. If we consider a random variable y = f(t) such that f(t) is independent from f(t − Δ t) or from f(t + Δt) (where Δt is a short time lapse, ideally Δt → 0). In other words, a variable for which its present is independent from its past, and whose present does not determines its future. Such variable produces a sequence of data which are uniformly distributed about their median . We say that run occurs when one or more sequence points fall above or below the median, this is easily grasped by looking at Fig 3. In Fig 3C there are 5 runs, and in Fig 3D are 9 runs. Points that are equal to are not taken into account in the analysis. Fig 3 presents four sets of monthly averages from the Function Recovery Uunit (FRU). Fig 3A, results from the total of inpatients (Adm) sequence (n = 22, n_runs = 11, probability of random run distribution about the median P_rnd is ≈ 0.15); Fig 3B, Total discharged patients’ (Dis) sequence (n = 22, n_runs = 8, P_rnd ≈ 0.04); Fig 3C, is the sum of lengths of stay (LoS) for all in–patients (InP) in the Unit during each month (n = 23, n_runs = 5, P_rnd ≈ 4.5 ⋅ 10⁻⁴); Fig 3D, is a sequence of the average stay duration (Mean LoS) of each in–patient at the Unit (n = 23, n_runs = 10, P_rnd ≈ 0.13); Fig 3E, along de observation period the Function Recovery Unit gained relevance and the number of beds were increased, panel shows the changes in beds at the benign in several months, as expected, the analysis of this sequence was found to be non random (n = 23, n_runs = 5, P_rnd ≈ 4.5 ⋅ 10⁻⁴), actually n_runs may be just 3 in this case, the algorithm probably found very minor decimal differences between data Fig 3 an in one segment; if n_runs = 3, should n runs be 3 then P_rnd ≪ 4.5 ⋅ 10⁻⁴. Ps are probabilities (obtained with the Wald–Wolfovitz test) that the sequences are not randomly distributed about their medians. Fluctuations in Fig 3E, resulted from administrative decisions, thus a low P_rnd, indicative of non randomness, was to be expected. In all cases abscissa is month number in the study period (January 2016 = 1, November 2017 = 23). Out of these P_rnd values the only one indicating a statistically significant non zero number of runs, occurs for InP sequence in Fig 3C [42, 43]. The non-randomness of sequence Fig 3C could have several reasons, one of them is some kind of ‘periodicity’ (a strange one, since its period would be longer that 2 years), but it could also stem from some kind of random walk determined by the uncertainties in n_j and LoS_i, j [11, 37, 44]. It is difficult to prove, but the apparent ‘seasonality’ was probably determined by the increase in Unit’s bed capacity depicted in panel Fig 3E. The figure illustrates that relatively simple sequences of random variables having unknown pdfs such as n_j and LoS_i,j, may interact in strange manners to produce apparently less random “simi-periodic” sequences such as Fig 3C. determined by the uncertainties in n_j and LoS_i, j [11, 37, 44]. It is difficult to prove, but the apparent ‘seasonality’ was probably determined by the increase in Unit’s bed capacity depicted in panel Fig 3E. The figure illustrates that relatively simple sequences of random variables having unknown pdfs such as n_j and LoS_i,j, may interact in strange manners to produce apparently less random “semi periodic” sequences such as Fig 3C.

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Fig 3. Four examples of runs around the median () from La Fuenfría’s Function Recovery Uunit.

(FRU). A: Total admissions (Adm) sequence (n = 22, n_runs = 11, P_rnd ≈ 0.15); B, Total discharges (Dis) sequence (n = 22, n_runs = 8, P_rnd ≈ 0.04); C, is the sum of lengths of stay (LoS) for all in–patients (InP) in the Unit during each month (n = 23, n_runs = 5, P_rnd ≈ 4.5 ⋅ 10⁻⁴); D, is a sequence of the average stay duration (Mean LoS) of each in–patient at the Unit (n = 23, n_runs = 10, P_rnd ≈ 0.13); E, throughout the observation period the Function Recovery Unit gained relevance and the number of beds were increased, the panel shows the changes in beds at the behind ib several months, as expenses, the analysis of this senescence was found to be non random (n = 23, n_run = 5, P_rnd ≈ 4.5 ⋅ 10⁻⁴, actually n runs may be just in this case, the algorithm probably found very minor decimal differences between data an in one segment; if n runs = 3, should n runs be 3 than P_rnd 4.5 ⋅ 10⁻⁴. P_rnd s are probabilities (obtained with the Wald–Wolfovitz test) that the sequences are randomly distributed about their medians. As indicated in the text of this communication P_rnd stands for probability of random distribution about the median. Fluctuations in E resulted from administrative decisions, thus a low P_rnd was to be expected. In all cases abscissa is time expressed as month number in the sequence, (January 2016 = 1, November 2017 = 23).

https://doi.org/10.1371/journal.pone.0262815.g003

LaN Fuenfría Hospital data statistical characteristics.

Fig 2 presents sequences built in the preceding Building the time series Subsection. Panels Fig 2A (Adm) and Fig 2C (Dis) are sequences with a few negative, artefactual (see Building the time series Subsection, outliers. Panels Fig 2B and 2D are the same data with the scale magnified to see the positive part and better appreciate data detail when y_t ⩾ 0.

Testing with the Jarque-Bera [30], the robustified Jarque-Bera-Gel [31] and the Shapiro-Wilk [32] tests, we found that none of the data sequences collected during the study period from La Fuenfría Hospital, were Gaussian (also called normal) variables, (Probability of Gaussianity, P ≪ 10⁻⁶). InP sequences contained a large number of ties (points with same value). In ti InP sequence every value is repeated

In the expression: InP_max, is InP’s maximum values; InP_min, is InP’s minimum values and n_total, is total number of points in the sequences. Given the narrow range there are only 187 − 102 = 85 possible different values among 1329 integer data in the series, they are not all equally likely. Please notice that the number of ties increases as InP_range → 0, that is the more uniform bed occupation gets, the larger the number of ties, in the InP sequence, will get. This is to be expected when integers are sampled in a narrow range, under these conditions any statistical test, parametric or not, must be taken with caution, since test power decreases wit ties. Tie corrections were used for the nonparametric statistical procedures employed here [26]. Since the Hospital has 192 beds, data indicate that the Hospital operates with some leeway, having an occupancy ranging from 53.1 to 97.3%, with a 78.1% median.

With the data in the sequence of InP (shown in Fig 2A) and Eq (2) a curve of DInP at the Hospital was built, this is shown in Fig 1B. The sequence in Fig 1D was obtained by subtracting to each Adm the same day Dis. Fig 1D presents a sequence obtained by subtracting Dis—Adm (DDiAd) to the Hospital.

A direct visual comparison between sequences in Fig 1B (DInP) and Fig 1D (DDiAd) shows, that in spite of some differences, they are similar, and very different from the series in Fig 1A (InP). Sequences in Fig 1B (DInP) and Fig 1D (DDiAd) compared with the Smirnov test had the same distribution, that is, their distributions are statistically indistinguishable (Probability of identity P ⪅ 1, n = 1329, two tails). Yet, when Smirnov test was used to compare sequences Fig 2A and 2C, admission (Adm) and discharges (Dis) sequences were found to be highly statistically significantly different (Probability of identity P ≪ 10⁻⁶, n = 1329, two tails).

Empirical Hospital occupancy Probability Distribution Functions (PDF) [28] calculated from these data is shown in Fig 4. Daily hospital occupancy data set was found to be not Gaussian when tested with the Jarque-Bera [30], and subsequently with the robustified Jarque-Bera-Gel [31] and the Shapiro Wilk [32] tests (P ≪ 10⁻⁶, in the three cases). The perception of data lacking Gaussianity becomes intuitive when very disperse data (dots) are observed around a Gauss pdf line in the same panel. The Gauss pdf line (shown also as pdf, in Fig 1C) with the same mean () and variance [s²(y)] as the set of 1329 data points forming the sequence in Fig 1A (, s[y] = 24.4).

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Fig 4. Empirical Probability Distribution Functions (PDF) [28] of daily admissions and stays at La Fuenfría Hospital from May 12014 to December 19 017.

Panel A: Blue ine with ■ PDF of LFH in-patients (InP in Fig 1A); Red line with ⧫, PDFs for daily differences in admissions (Adm) and discharges (Dis), called DDiAd in Fig 1D. Panel B: Red line with ⧫ PDF of LFH daily Admissions in LFH (called Adm in Fig 2A); Red line with ■ PDF for LFH daily Discharged Dis in Fig 2C). Panel C: Same as Paanel B, but with abscissa expanded to exclude outliers. In all cases abscissa is the value ve aare interested in, and ordinate is the estimated accumulated probability from −∞ up to the abscissa value.

https://doi.org/10.1371/journal.pone.0262815.g004

Daily fluctuations in stay duration and average stay duration in operative units at La Fuenfría Hospital

Fractal analysis of sequences from Hospital

Ruling out sources of errors in sequences.

To analyze data in Table 2 we must consider the transcription errors in the SELENE^® data base and how were they corrected for this study, as indicated in Methods and the problem of ties mentioned in the La Fuenfría Hospital data statistical characteristics Sub-Subsection, also the narrow data rage determines further underestimations of D_h.

A case random noises in seen in [37] where much longer sequences needed for D_s to converge towards D_h, these sequences contain only decimal digits , not real numbers . Empiric data on Tables 1 and 2, may differ from D_s of a calibration white noise from a longer sequences.

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Table 1. Fractal dimension D_s estimated for La Fuenfría Hospital sequences.

https://doi.org/10.1371/journal.pone.0262815.t001

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Table 2. Statistical comparison between fractal dimensions (D_s) estimated for the sequences appearing in Table 1.

https://doi.org/10.1371/journal.pone.0262815.t002

Monte Carlo simulated sequences.

White Gaussian noise in Fig 5A is a sequence of and Fig 5C of . Brownian noise in Fig 5B is the result of (7) where b_i is the i^th point in Fig 5B and g_i−1 is the (i − 1)^th point in Fig 5A. The required random normal variates of type N[0, 1] were generated as indicated in thin the Mathematical Appendix. In contrast with the Gaussian white noise (Fig 5A and 5C), Brownian noise sequences have slow and fast meanders and relatively long periods with trends to increase and/or to decrease which are just local tendencies in the segment, generated as an example (Fig 5B and 5E), the segment did not indicate any periodicity. Segments such as the one in Fig 5B for 7 ⋅ 10³ ⩽ x ⩽ 10⁴ look very much like the InP curve at La Fuenfría Hospital in Fig 1A which also produces a false sensation of periodicity with a slight tendency to decrease, yet they are just chaotic trends, in spite of the fact that the underlying random processes (Figs 1A and 5A and 5C) have no periodicities at all. A Rw expresses a system that is out of control, just like epidemics [5, 11, 44], storms [2, 61–64] or earthquakes [65], which if described with statistical parameters based on a (short) periods of time leads to wrong conclusions.

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Fig 5. Two sequences generated with Monte Carlo simulation using Eq (23) as described by Box and Muller [59, 60].

Panels are: Panel A, Gaussian white noise and; B: Gaussian Rw, a Rw generated using Eq (7) from data on pane A; Panel C, Gaussian white noise and ; D: Gaussian Rw, a Rw generated using Eq (7) from data on pane C Abscissa and ordinate are arbitrary, Please notice [D_s ± s(D_s)] estimated with Eqs (14) and (15) are: Panel A, 1.66122 ± 7.60191 ⋅ 10⁻⁴; Panel B, 1.32692 ± 7.6926 ⋅ 10⁻⁴; Panel C, 1.6692 ± 7.64176 ⋅ 10⁻⁴; Panel D, 1.31723 ± 7.72806 ⋅ 10⁻⁴. Rw stands for random walk, more details in the communication text.

https://doi.org/10.1371/journal.pone.0262815.g005

Power spectrum analyzes

A classic form to show function periodicities in time domain, is to transform this them into frequency domain function, which is a sum of sines and cosines, using a fast Fourier transform (FFT) [39, 48]. Transformed values for each frequency (f) are ‘complex numbers’ (), c_f comprised of two parts or components: one is sometimes referred to just the ‘real component a_f’ () and a, so called, ‘imaginary’ () component (). A complex number (c_f) is of the form (8) where parentheses indicate sets of numbers, and ∈ reads as “belongs to”. The ‘energy’, expressed as either a steady curve’s nonzero curve values, or sharp increases or variations in amplitude (such a as the peak in Fig 6B, something we could informally call its ‘weight’), contributed by each frequency component to total signal energy in a fluctuating complex process is given by (9) plotting ψ_f against f id called a spectral density plot, and it is areal number.

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Fig 6. False periodicity analysis in Rws. Use of spectral density ().

Panels are: Panel A- A perfectly periodic signal, the sine trigonometric function ; Panel B- of the sine function shown in panel A; Panel C- of Gaussian white noise shown at Fig 5A; Panel D- of Brown Rw shown in Fig 5B; Panel E- pf daily inpatients (InP) sequence which in appears in Fig 1A. Ordinate of panels B through E, is power dissipated at each frequency. The sine function depicted dissipates all its energy at a single frequency observed at a single neat peak in its ψ_j (panel B). sequence for the number of hospitalized patientsSDf_j corresponds to a Gaussian Rw without any underlying periodicity. All SD plots were filtered using a Hann window [39, 66]. The rapid oscillations observed at right of signals in panel B to D are an artefact due to sampling discrete points in the signal for the Fourier transformation. Rw stands for random walk, further details in the text.

https://doi.org/10.1371/journal.pone.0262815.g006

An intuitive manner to understand this, is to consider a perfect periodic signal such as a trigonometric sine function like where π = 3.1415926… and T is a constant (called the waveform’s period) with same units (meters, seconds, grams, volts, or else) as x. The example is show in Fig 6A. Fig 6B, presents Fig 6A’s power spectrum consisting of a single peak frequency. Please notice that to calculate ψ_f the actual signal is sampled in Δx intervals (equivalent to multiplying by something called a Dirac comb and the FFT also contains the frequency components of the Dirac comb producing a burst of rapid vibrations in the right half of each trace [39].

Fig 6C is a ψ_f spectrum of the Gaussian white noise shown in Fig 5A; ordinates of panels B through E, in arbitrary units. In Fig 6C it is seen that all frequencies have the same ‘weight’, i.e., all frequencies contributions are equal. Fig 6D is a Brown Rw ψ_f spectrum,; in double logarithmic coordinates, the spectral density is a straight line which decays with a slope of . Fig 6E is ψ_f of daily inpatients (InP) sequence shown in Fig 1A.

Although the simulated sequences in Fig 6 are 10000 points long and Hospital data are only 1329 points long, there is an evident similarity between the Brownian noise ψ_f (Fig 6D), and ψ_f from the sequence of Hospital inpatients (Inp) between May 1 2013 and December 19 2017 (Fig 6E). In both cases ψ_f are characteristic of a Brown Rw and none of these ψ_fs has a peak which could indicate a predominant periodic component which could suggest for any periodicity, despite of short lapses in the Brown Rws that could give this impression.

Fig 6C is Gaussian white noise calculated for the signal in Fig 5A. In Fig 6C all frequencies have the same ‘weights’, the contributions of al frequencies are equal, which leads to the ‘white’ name of this kind of noise. Fig 6D shows of the Brownian Rw in Fig 5B, one may observe that in double logarithmic coordinates the spectral density follows a straight line which decays with a slope of . Finally, also in Fig 6E, we show corresponding to daily hospitalized patients sequence shown in Fig 1A. Except for that simulated sequences in Fig 6 are 10⁴ points long and Hospital data are only 1329 points long, there is an evident similarity between the Brownian noise , and that from the sequence of Hospital discharges between May 1 2013 and December 19 2017. In both cases are characteristic of a Rw and none of these has no peak which could suggest periodic components which could account for any periodicity, despite short periods in the Rws that could give this impression.

Daily fluctuations in stay duration and average stay duration in operative units at La Fuenfría Hospital

The Hospital is divided into the following Clinical Operative Units (COU): RCPU, Relapsing Chronic Patients Unit, this unit operated until May 2016 when its beds were transferred to the CCU; Chronic Care Unit; PCU, Palliative Care Unit; FRU, Function Recovery Unit; TU, Tuberculosis Unit and NTU, Neurorehabilitation Treatment Unit. Fig 7A presents data on daily number of in–patients (InP) at each unit, and their average stay duration on Panel Fig 7B). Daily fluctuations of in–patients and average length of stays (LoS) for each COU calculated every day from January 1 2015 uo to December 12 201. Abscissas are days from January 1 2014, ordinates are in decimal logarithmic scale to better visualize some of sequence large outliers, which are particularly evident in Panel Fig 7B. Traces in Fig 7 seem to be a kind of Rw. All have some outlying peaks, particularly RCPU and UTB, however, less prominent peaks, occur in other COU too. It it is important to note that in this paper we sometimes refer to products or quotients between random variables, this is the case of averages including number of patients and stay duration in each COU in Fig 7B. We know that our data are not Gaussian, however even in the case of Gaussian values these averages, may be quotients of random variables following a a “pathological” statistical distribution such as the Cauchy pdf (See the Mathematical Appendix Subsection On Cauchy distributed variables).

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Fig 7. Daily fluctuations of in-patients and average duration of stays for each Clinical Operative Unit recalculated every day from January 1 2015 until December 12 2017.

A- Number of in–patients (InP) at each Operative Unit. B- Average stay duration at each Operating Unit [Υ_d defined as in Eq (10)]. Abscissas are days elapsed from January 1 2014, ordinates are in logarithmic scale to better visualize some of sequence features. Unit initials mean: RCPU, Relapsing Chronic Patients Unit, this unit was suppressed on May 2016 and its beds were transferred to CCU; CCU, Chronic Care Unit; PCU, Palliative Care Unit; FRU, Function Recovery Unit; TU, Tuberculosis Unit; NTU, Neurrehabilitation Treatment Unit.

https://doi.org/10.1371/journal.pone.0262815.g007

Nonparametric statistical analysis of Hospital’s operative units sequences shown in Fig 7

“If a sample is too small you can’t show that anything is statistically significant,

whereas if sample is too big, everything is statistically significant.”… “you are doomed to significance.”

Norman and Steiner [72, pg. 24].

As said in Section sequences in Fig 7B include data such InP average patients’ stay duration in each operative unit (Υ). This number calculated as: (10) where d represent a given period and j is the j^th patient in the service, and N_d is the number of patients in each service at day d. Thus, Υ_d are quotients unknown distribution random variables. As mentioned in the Mathematical Appendix, those quotients of random variables have uncertain distributions, which in the case of Gaussian variables is the Cauchy pdf Eq (24) discussed in the Mathematical Appendix. These quotients follow unknown pdfs when terms in the ratio obey unknown pdfs. For these distributions sample mean () n and sample variance [s²(x)] are meaningless. Statistical sampling theory is based on the fact that and s²(x) converge towards the population mean (μ), and population variance [σ²]. Yet, μ and σ² do not exist for Cauchy and other, so called, “pathological distributions” for which and s²(x) have nowhere towards to converge to. Cauchy sample experimental estimates of and s²(x) vary wildly, specially when sample sizes grow and large outlier values become very likely [73, 74]. Due to these only nonparametric descriptive analysis provides meaningful information on sample localization and dispersion.

Caveat emptor: data samples from La Fuenfría Hospital, except for those in Fig 3, are rather large (986 ⩽ N ⩽ 1434). Under these circumstances small, but statistically significant changes, which lack relevance or meaning in practice, may be detected, [70, 75, 76]. Table 3 contains results where sequences in Fig 3 are subjects o a nonparametric Theil regression analysis [77]. As seen in Table 3, many of the hospital units sequences in Fig 6 seem to produce a small, but highly statistically significant, correlation between the sequence under study and time. All the time series in this figures are Rws (not shown), as was confirmed by their spectral density, characteristic of Brownian noise, without any evidence of periodicity, and resembling those in Fig 7D and 7E. As stated, in connection with the sequence in Fig 5B and 5D, Rws may some times exhibit rather long, trends to increase or decrease, that are independent from any variation in factors determining the Rw. The pdf of did not change in those periods. Due to these meanderings of time sequences, descriptive statistical parameters, even if optimally chosen, like in Table 3 fail to provide meaningful information on Rws like Fig 6B and some of the preceding figures.

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Table 3. Nonparametric linear Theil [67–70] regression parameters of LFH clinical units’ time sequences expressed for each unit as in Fig 7.

https://doi.org/10.1371/journal.pone.0262815.t003

Study purpose an empirical finding

At least during the study period, the La Fuenfría Hospital operated, in median, at 76.8% of its total capacity and fluctuated in what could be interpreted as slow periodic meanders (Fig 1A). Yet, the daily admissions (Fig 1B) or discharges from LFH (Fig 1D) are random sequences which show no periodicities whatsoever. Daily differences (Fig 1D) between patients admitted (Fig 2B) and discharged (Fig 2D) from the Hospital were also random non periodic sequences. It was somewhat surprising to find that the operating Hospital’s curve (Fig 1A)) could be almost perfectly (Fig 1E) reproduced when daily differences between admitted and discharged patients (Fig 1D) were added sequentially as indicated in the Random fractals Section’s Eq (3).

Meaning of lack of control of a process

Data on Figs 1A and 6E reflect that Hospital bed occupancy is described by a random fractal function, the so called Brownian Rw. Just like Brownian Rw, bed occupancy fluctuates with important periods of low occupancy.

The main characteristic of Brownian Rw. is that its complexity (reflected in its D_s) does not change if the number of daily inpatients (DInP) increases or decreases, this is called self affinity of the process [5, 44], it does not change with scaling DInP. This is shown by the Monte Carlo simulated traces of Fig 5 where the Gaussian white noise in panel Fig 5A looks very similar to the white noise in panel Fig 5C, exp of course for the ordinate scale; Something similar happens with the associated Brownian noises in panels Fig 5B and 5D, which have similar complexities, but the fluctuations in Panel Fig 5B are approximately 10 times wider in than in Panel Fig 5D.

Still, from a production efficiency view point if the differences (in Fig 5 ) are very important since the fluctuation range of the Brownian Rw diminishes when g_i → 0 and production of goods or services increases as the system becomes more predictable (controlled), uniformly productive and efficient.

At La Fuenfría Hospital InP would fluctuate less if the lag between Adm and Dis → 0. Identifying a problem is the first step to be able to solve it. This means, making the Hospital more efficient requires analyzing the factors that keep Adm and Dis out of synchrony.

Characteristics of La Fuenfría Hospital [78]

Hospital La Fuenfría (https://bit.ly/32FVmyf) is one of three medium and long stay SERMAS hospitals in Madrid’s Autonomous Region (MAC) together with Virgen de la Poveda Hospital and Guadarrama Hospital, under the Sanitary Chancellery (Consejería de Sanidad), LFH is in Cercedilla Municipality at the north of Madrid Region, at 60 km from Madrid city. The MAC comprises an area of of 8021.8 km² and a population of 6.6 M.

The long stay duration are related to local norms and procedures to handle patients, it is likely that in Spain the public health system has more leeway to tolerate longer stays than perhaps other EU nations, especially when dealing with needy patients. Concurrently, LFH manages many chronic cases, and it most important unit is FRU, responsible for recovering patients with disabilities stemming from bone surgery (prothesis, fractures, etc.) and motor deficiencies following cerebrovascular accidents (CVA or strokes). Thus perhaps it is to be expected that stays at LFHa may be longer than at hospitals dealing under ambulatory treatment.

LFH is located in a pleasant rural area, Madrid’s Sierra de Guadarrama, convenient for its goals, but distant from Madrid (population 3.3 M) city and MAC, in general. Patients are referred from other SERMAS hospitals, thus, delays between LFH discharges and new patients arrivals, are not surprising. The analyst of data in this paper (CS) was completely unaware of this factors, but they popped out sharply from data analysis discussed here.

Classical statistical analysis of institutional efficiency

Descriptive statistics, using parametric (Gaussian) methods, is probably the the most common manner used tools to gauge institutional efficiency. LFH SELENE^©’ include command panels parameters that are ratios of random variables. These ratios have unknown statistical distributions which like, the Cauchy pdf, have no central moments, therefore mean or variances do not exist, thus textbook recipes like for sample mean and variance, are, calculable, yet, absolutely meaningless [79–82] (See the section on the Cauchy distribution in the Mathematical Appendix. For these kinds of random samples the use of nonparametric statistical estimators such as medians and 95% CI must always be preferred to and which, are also, very sensitive to outliers and, as said, are meaningless for Cauychyan variables. Furthermore, in the analysis of data extracted from LFH command panels, no Gaussaian variables were found, further demanding the use of nonparametric statistical methods.

Another example of the failure of statistics to productively help analyze aspects of LFH operative efficiency, may be derived from Fig 7. The figure presents unit operation data are Rws with large outliers. However, when these Rw were subjected to linear nonparametric regressions (Table 3) apparent correlations with time appeared, with correlation coefficients statistically distinct from zero, i.e., apparently highly unlikely to differ from zero per chance. Large samples are a good way to discover statistically significant, but irrelevant, trends, which, as shown here, may just be temporary tendencies of a random walk. Such correlations could be used to create institutional (probably wrong) operation policies. Classical statistics is notoriously limited to study fractal processes [4, 5, 7, 35, 36, 44]. Therefor, of our findings regarding statistical evaluation of LFH operational curves are neither new nor surprising. The novelty of our findings relates to the commonality of statistical analyses and the lack of other analyzes (fractal or else) to study institutional efficiency.

Several caveats should be considered:

1. This communication is centered on maximizing the use of hospital facilities as an indication of efficiency. Yet, it does not consider other criteria such as the patient’s welfare, degree of recovery, speed of recovery, which are of fundamental importance, but also depend on the efficient of hospital resources.
2. The main theoretical tool we use is fractal analysis, but to accurately estimate fractal dimension very long data series [11, 37, 83]. this limitation, as done here, may be overcome py the use of sequences with known fractal dimensions as applied here and in previous studies [11, 37].
3. Spectral analysis confirmed the lack of periodicity in the InP time series.
4. Yet, the most important finding reported here is that that the wide fluctuations observed in the inpatients (InP, Fig 1A) may be reconstructed with identical naked-eye and fractal properties (Ξ_i sequence in Fig 1E) by adding the daily differences in LFH patients admitted and discharged (DDiAd, Fig 1D) using Eq (5). These are empirical finding result from any mathematical theory or artifact.

Rw in La Fuenfría Hospital operation curves

The most surprising finding in this study, is that curves representing in-patients, InP, and ΔInP are random Markovian processes [51]. In a Markovian processes the past determines what happens now, and what occurs now decides the future. If there were no random components in the sequence, known the present would allow to know past and future of the system. Yet, since uncertainty separates one instant from the next, the whole system becomes unpredictable. It dt drifts over time in ways that a naive observer can realize, one is driven to believe that the system is predictable and determined.

If a present drift trend pleases us, we tend to believe that whatever we are doing is “correct”, in the opposite case we would tend to believe that we are doing something “wrong”, even if in reality we have no “merit” in the first case nor are we “guilty” in the second case. An examples of this is seen at times from 0 to 6000 in Fig 5B. But after that time the Rw in Fig 5B which produces a false appearance of relative “stability”, even when at all those times it is exactly the same system under the same Gaussian conditions as shown by the plot in Fig 5A.

The contribution of our work

We know (both authors are trained MDs) that hospitals and medicine are complex systems, yet our study shows that the waveform meandering between full and insufficient occupancy at LFH may be perfectly, explained considering a single factor: admission and discharges are systematically and randomly out of phase. These random differences, small as they may be, constitute the de determining factor at LFH: To the best of our knowledge nobody has previously singled this small but important hidden factor. We know (both authors are trained Mds) that hospitals and medicine are complex systems, yet our study shows that the waveform meandering between full and insufficient occupancy at LFH may be, exactly, explained considering a single factor: admission and discharges are systematically and randomly out of phase. This random difference, is the determining factor at LFH: To the best of our knowledge nobody has previously singled out this small but important hidden factor.

Detecting problem indicators

Descriptive statistics and common Command panels, are not enough to detect hidden administration problems with impact on hospital efficiency. Descriptive statistics limitations mentioned in in the prior Subsection Rw in La Fuenfría Hospital operation curves open a question on how to evaluate and optimize the operation and administration of La Fuenfría Hospital whose operation curves are Markovian random walks.

Random walk curve meanders reduce system efficiency. In a hospital performance case, low occupation equals inefficient periods, for the following reasons:

1. The population demanding service, does not receive such service..
2. The cost of operating the hospital operation costs (infrastructure, salaries, etc.) are wasted during periods of low occupancy

At la Fuenfría Hospital occupation was as low as 45% (Fig 1A) of its capacity. As discussed in the La Fuenfría Hospital data statistical characteristics Sub-section oscillations between full and partial occupation seem related to a daily lag between Admissions and Discharges. If this lag is eliminated, fluctuations in occupancy would disappear should this lag be eliminated fluctuations in occupancy would disappear.

During the study period, the mean number of LFH in-patients was almost identical to its median , and ranged (102, 187). Since the hospital had 192 patient’s beds, thus the bed occupancy during the study period was and a median of ranging (53.1, 97.4)%. Thus over the observation period, on average 22% ranging (3, 47)% La Fuendría Hospital beds were vacant, which means that a similar proportion of the resources assigned to LFH was lost, and therefore this does not generate a lack of demand.

Can the problem be solved?

Our study indicates that LFH vacancies could be eliminated or greatly reduced if the daily difference Adm-Dis is nullified (see Meaning of lack of control Sub-section) of a process, Figs 2 and 5). Efficient service characterized by a Rw in time which may be increased by reducing broad meanders of the walk (Fig 5). In the case of La Fuenfría Hospital this study suggests that the key factor for reducing efficiency is the daily difference between admissions and discharges.

Nonetheless, the solution probably transcends the realms of La Fuenfría Hospital. We have no information on other SERMAS hospitals, yet the description, La Fuenfría Hospital realms. We have no information on other SERMAS hospitals, yet the description of LFH provided in Introduction Section illustrates that LFH is interwoven in the SERMAS network, seems necessary to consider current communications and required modifications within the network. The easy aspect may be to improve communications between admission and discharge departments within LFH, although this may imply some changes to the internal computer network. Thus, if the admissions department were to be given notice prior to the actual discharges, this would enable the possibility to notify the SERMAS network that LFH will have space for a new patient and ensure that the hospital is ready to occupy the space immediately when it actually becomes available.

However, this immediate availability is also hindered by the large area that SERMAS covers, the fact that there is a distance of 60 km from the city of Madrid (the largest set of SERMAS patients), and even more from other areas in Madrid’s Autonomous Region. Factors such as means and speed of transportation may be critical, institutional transportation within the SERMAS network is probably faster and easier to program, than solely relying on the patient’s and its family’s resources and efficiency, for example. However, if the efficiency of SERMAS hospitals is affected to the same extent as La Fuenfría Hospital’s, it may be worthwhile to study the system and to implement solutions between admission and discharges in SERMAS hospitals to improve the system’s efficiency.

Mathematical appendix

Using the inequality to calculate significance when

Theorem 1 holds even with heavily skewed distributions and puts bounds on how much of the data is, or is not “in the middle”. Setting ϵ = 0.05 then , for a two tailed test. By virtue of Eq (26) no matter which unimodal distribution, no matter how skewed, there will be <2.5% chance that a datum will belong to a population with mean μ and variance σ² if it lays farther than ±2.981σ from μ. Please note that if the probability distribution function of data is Gaussian ±1.96σ suffices to reach the same confidence level. We introduced the ϵ variable in Eq (26) for the statistical argumentation that follows.

Let and s²(D_s,1) be the mean and variance estimated for the random variable D_s,1, and and s²(D_s,2) be the mean and variance estimated for the random variable D_s,2 (linearly independent from D_s,1), then (27) and the estimated variance of is (28) By virtue of the V-P inequality [Eq (26)] (29) which means Eq (26) that (30)

Therefore, there is ≤ ϵ probability that due to random sampling variation, and with a confidence level P ≤ ϵ. Thus: (31) for a test with only one tail, since in this case we are only interested with one alternative, that . Again, when the probability distribution function is Gaussian, the value of λ for the one tailed case, would be ≈ 1.65 instead of the ≈ 2.108 demanded by Eq (26).

Using the inequality to calculate significance when

Theorem 1 does not hold, but you can still get an answer using Eq (26) as shown in the following Corollary 1 of the V-P inequality. We know propose and prove a corollary of Theorem 1:

Corollary 1. : If then P is not known but is bound as (32) Proof. As indicated by Eq (26), then if (33) then ϵ_th is the value of ϵ at the threshold required for Theorem 1 to hold. When the predicted value of ϵ ⩾ ϵ_th, and since there is no limit to this inequality, it may happen that the estimated ϵ > 1, but there is no probability P > 1, by definition. Thus when it follows that ϵ does not express a probability. But since if Eq (33) gives the highest P value which may be estimated with the V-P inequality is , P is undetermined but bound in the interval: (34)