MD Pierre Deharo
email: deharopierre@gmail.com

MD Alizée Porto

MD Thierry Bourguignon

MSc, C Saint Etienne Julien Herbert

Christophe Saint

MD Carl Semaan

MD Thibaud Genet

MD Nicolas Jaussaud

MD Pierre Morera

MD Alexis Theron

Etienne Md

MD Vlad Gariboldi

MD Frederic Collart

MD, V Gariboldi MD Thomas Cuisset

MD. Laurent Fauchier

Myocardial Revascularization Strategies in ST Elevation Myocardial Infarction Without Urgent Revascularization: Insight From a Nationwide Study

Keywords: STEMI, CABG, PCI

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

It is recommended to offer urgent revascularization in patients presenting with ST elevation myocardial infarction (STEMI), even in patients presenting late after symptoms onset (12 to 48 hours) [START_REF] Ibanez | ESC Guidelines for the management of acute myocardial infarction in patients presenting with ST-segment elevation: The Task Force for the management of acute myocardial infarction in patients presenting with ST-segment elevation of the European Society of Cardiology (ESC)[END_REF]. However, routine percutaneous coronary intervention (PCI) in an infarct related artery is not recommended after 48h [START_REF] Ibanez | ESC Guidelines for the management of acute myocardial infarction in patients presenting with ST-segment elevation: The Task Force for the management of acute myocardial infarction in patients presenting with ST-segment elevation of the European Society of Cardiology (ESC)[END_REF]. Therefore, revascularization strategies in patients presenting with a STEMI who did not undergo revascularization within the first 48 hours remain a challenge. PCI and coronary artery bypass graft (CABG) represent the two revascularization options that could be offered to those patients.

In non-STEMI myocardial infarction, the currently available evidence indirectly suggests that the criteria applied to patients with chronic coronary syndrome, to guide the choice of revascularization modality, should be applied to stabilized patients [START_REF] Collet | ESC Guidelines for the management of acute coronary syndromes in patients presenting without persistent ST-segment elevation[END_REF]. However, no data or study are available regarding the optimal revascularization strategy in STEMI patients.

Patients who undergo CABG after myocardial infarction have higher risk of short term mortality compared to patients with elective or delayed CABG [START_REF] Voisine | Influence of time elapsed between myocardial infarction and coronary artery bypass grafting surgery on operative mortality[END_REF][START_REF] Weiss | Optimal timing of coronary artery bypass after acute myocardial infarction: a review of California discharge data[END_REF]. Deik et al revealed that CABG performed >7 days after STEMI had similar post-operative outcomes to elective CABG [START_REF] Deeik | Appropriate timing of elective coronary artery bypass graft surgery following acute myocardial infarction[END_REF]. However, the timing of surgical revascularization remains unclear after a STEMI and should be determined individually according to guidelines.

The French Programme de Médicalisation des Systèmes d'Information (PMSI), a mandatory administrative database, offers a unique opportunity to assess exhaustive and comprehensive data on all consecutive patients hospitalized in France [START_REF] Fauchier | Outcomes in patients with acute myocardial infarction and new atrial fibrillation: a nationwide analysis[END_REF][START_REF] Deharo | Transcatheter Valve-in-Valve Aortic Valve Replacement as an Alternative to Surgical Re-Replacement[END_REF]. Therefore, based on this large, nationwide, administrative French database, we aimed to compare the outcomes of CABG versus PCI in patient with STEMI who did not undergo revascularization within the first 48hours of hospitalization.

METHODS

Data collection

Methods have been described previously [START_REF] Fauchier | Outcomes in patients with acute myocardial infarction and new atrial fibrillation: a nationwide analysis[END_REF]. All patients admitted with STEMI in France from January 2010 to June 2019 were screened using the national administrative PMSI database, through The French Data Protection Authority, covering hospital care from the entire French population, and as previously described [START_REF] Deharo | Transcatheter Valve-in-Valve Aortic Valve Replacement as an Alternative to Surgical Re-Replacement[END_REF][START_REF] Deharo | Impact of Sapien 3 Balloon-Expandable Versus Evolut R Self-Expandable Transcatheter Aortic Valve Implantation in Patients With Aortic Stenosis: Data From a Nationwide Analysis[END_REF]. The study was conducted retrospectively, with no impact on patient care. Data collection and management were approved by the Commission Nationale de l'Informatique et des Libertés, which ensures that all information remains confidential and anonymous, in compliance with the Declaration of Helsinki (authorization number 1897139).

Study population

From January 1, 2010, to June 31, 2019, a total of 520,258 patients with an age of 18 years or more were hospitalized with a STEMI (ICD-10 codes: I210, I211, I212, I213) as the principal (i.e., the health problem that justified admission to hospital), the related (i.e., potential chronic disease or health state during the hospital stay), or a significantly associated (i.e., comorbidity or associated complication) diagnosis.

For the purpose of this analysis, we enrolled patients who underwent either a PCI or a CABG between 48 hours to 90 days after the index presentation for a STEMI. We did exclude all patients with revascularization during the first 48 hours of hospitalization, patients who had concomitant valvular cardiac surgery and patients with a history of previous CABG or PCI.

Information regarding demographics, medical history, procedures and events during hospitalization or follow-up was collected.

Outcomes

Patients were followed until June 2019 for the occurrence of outcomes, starting from date of STEMI. Clinical outcomes for the analysis were: all-cause death, cardiovascular death, hospitalization for heart failure (HF), all-cause stroke and major bleeding. Outcomes were identified using their respective ICD-10 or procedure codes. The endpoints were evaluated with follow-up starting from date of either PCI or CABG until date of each specific outcome or date of last news in the absence of the outcome. Information on outcomes during follow-up was obtained by analyzing the PMSI codes for each patient. Mode of death (cardiovascular or non-cardiovascular) was identified based on the main diagnosis during hospitalization resulting in death. Hospitalization was considered to be related to heart failure when heart failure was recorded as the main diagnosis. We defined major bleeding using the modified International Society on Thrombosis and Haemostasis (BARC) definitions [START_REF] Schulman | Subcommittee on Control of Anticoagulation of the Scientific and Standardization Committee of the International Society on Thrombosis and Haemostasis. Definition of major bleeding in clinical investigations of antihemostatic medicinal products in non-surgical patients[END_REF]. Major bleeding, based on International Society on Thrombosis and Haemostasis definition and modified to take into account the availability of information in the database, was defined as a hospitalisation with one of the three following diagnoses: haemorrhagic stroke, other critical organ or site bleeding (intraspinal, intraocular, retroperitoneal, intraarticular or pericardial, or intramuscular), other bleeding with a transfusion during hospital stay, or resulting in death.

Causes of death and haemoglobin result are not available to take into account not hospitalized fatal bleeding and bleeding causing a fall in haemoglobin level of 20g/l. To increase the validation of our analysis, we also evaluated incidence rates of cancer and urinary infection as negative control endpoints.

Statistical analyses

Qualitative variables are described as frequency and percentages and quantitative variable as means ± standard deviation (SD). Comparisons were made using chi-square tests for categorical variables and the Student t test or non-parametric Kruskal-Wallis test, as appropriate, for continuous variables.

Owing to the non-randomized nature of the study, and considering for significant differences in baseline characteristics, propensity-score matching was used to control for potential confounders of the treatment outcome relationship. Propensity scores were calculated using logistic regression with revascularization strategy as the dependent variable. The propensity score included all variables of baseline characteristics (listed in table 1). For each patient with a PCI, a propensity score-matched patient with CABG was selected (1:1) using the one-toone nearest neighbor method (with a caliper of 0.001 of the SD of the propensity score on the logit scale) and no replacement. Incidence rates for outcomes in each group were estimated, and cumulative incidences curves were obtained using Kaplan Meier method. A Cox regression model on all baseline parameters described in Table 1 was used to obtain adjusted hazard ratios (HR). A multivariate analysis was performed to estimate adjusted hazard ratios with their 95% confidence intervals. A p value <.05 was considered statistically significant.

All analyses were performed using Enterprise Guide 7.1 (SAS Institute Inc., SAS Campus Drive, Cary, NC, USA) and STATA version 12.0 (Stata Corp, College Station, TX, USA).

RESULTS

Baseline characteristics

Between 1 January 2010 and 1 June 2019, 71,655 STEMI patients with no revascularization during the first 48 hours, and either PCI or CABG between 48hours to 90 days were identified in the database (Figure 1). In the unmatched population of 59,340 STEMI treated with PCI and 12,025 STEMI treated with CABG, patients treated with CABG were younger, more often male and presented less often with heart failure or cardiogenic shock (Table 1).

After propensity score matching, there were 12,012 patients in each group. Baseline characteristics in these populations were well matched (Table 2, supplementary Figure 1). Mean (SD) time between STEMI and PCI or CABG in the matched population was 16.7 (19.2) days (median 8, interquartile range [IQR] 4-21).

Clinical outcomes

Mean (SD) follow-up was 2.6 (2.7) years, (median 1.8, IQR 0.1-4.6). In the whole population, all-cause death was recorded in 11,477 patients and cardiovascular death was recorded in 5142 patients. Predictors of all-cause death are reported in table 3 and predictors of all-cause death are in table 4. Treatment with CABG (vs. PCI) was independently associated with a lower risk of all-cause death (HR 0.73, 95% CI 0.69-0.78) and of cardiovascular death (HR 0.88, 95% CI 0.82-0.96).

In the matched population, all-cause death was recorded in 2202 (7.1%) patients who underwent PCI versus 1596 (5.1%) who underwent CABG (HR 0.70, 95% CI 0.66-0.75; Table 5 and Figure 2). The incidence of cardiovascular death was also lower in the CABG group (2.6% versus 3.1%; HR 0.83, 95% CI 0.76-0.91; Table 5 and Figure 2). Incidences of rehospitalization for worsening heart failure was reported lower after CABG (1569 patients, 5.5%) versus PCI (2026 patients, 7.5%) (HR 0.76, 95% CI 0.71-0.81; Table 5 and Figure 3) while all-cause stroke were not statistically different between the two groups (2.1% versus 2.3%; (HR 0.89, 95% CI 0.80-1.00; Table 3 and Figure 3).

Major bleeding was reported in 1309 (4.6%) patients in the PCI arm versus 1658 (6.1%) patients in the CABG arm (HR 1.31, 95%1.22-1.41; Table 3, Figure 4).

DISCUSSION

Our nationwide analysis showed that:

-In real life setting, 13% of patients admitted with STEMI in France between 2010 to 2019 underwent either PCI or CABG outside the criteria of urgent myocardial revascularization (<48h) -In patients with STEMI who did not undergo a PCI within the first 48 hours, surgical revascularization was associated with better long-term outcomes Patients with STEMI should be revascularized as soon as possible within a 48 hours window from symptoms onset. Indeed, timing of intervention is crucial in STEMI, where myocardial salvage in the infarct related artery territory is usually possible within 48hours. However, it is not uncommon that patients do present very late and urgent revascularization could not be achieved in this time frame. In our national database around 13% of STEMI did undergo either PCI or CABG between 48hours to 90 days following admission.

Outcomes following CABG in STEMI patients are controversial. A recent analysis comparing outcomes of CABG in STEMI and non-STEMI treated within the first 48hours of hospitalization showed twice higher in hospital mortality after CABG for STEMI with similar long-term outcomes [START_REF] Malmberg | Outcomes After ST-Segment Versus Non-ST-Segment Elevation Myocardial Infarction Revascularized by Coronary Artery Bypass Grafting[END_REF]. Another study showed improved long-term survival after CABG in STEMI versus non-STEMI when performed during the first 48 hours [START_REF] Grothusen | Outcome of Stable Patients With Acute Myocardial Infarction and Coronary Artery Bypass Surgery Within 48 Hours: A Single-Center, Retrospective Experience[END_REF].

A large US nationwide analysis confirmed that observed mortality rates after CABG during the index hospitalization of a STEMI patient appeared low [START_REF] Pi | Utilization, Characteristics, and In-Hospital Outcomes of Coronary Artery Bypass Grafting in Patients With ST-Segment-Elevation Myocardial Infarction: Results From the National Cardiovascular Data Registry Acute Coronary Treatment and Intervention Outcomes Network Registry-Get With The Guidelines[END_REF], while another recent US dataset showed poorer outcomes for STEMI treated with CABG during the first 24 hours after admission [START_REF] Lemaire | Delay in coronary artery bypass grafting for STEMI patients improves hospital morbidity and mortality[END_REF]. Pooled together those data suggest that CABG might be safely performed in selected STEMI patients in a timely fashion, which should be individually evaluated in Heart Team.

When compared to PCI, a large monocentric registry (14) compared outcomes of PCI vs CABG in STEMI and showed that despite higher risk and unfavourable in-hospital course, long-term outcomes of patients treated with early CABG were similar to patients treated with PCI. Our data are in favour of lower rates of cardiovascular death and rehospitalisation for heart failure in patients with STEMI treated with CABG between 48hours to 90 days when compared to PCI. Major bleeding appear to be higher after CABG during the whole follow up. While short-term bleeding are related to the surgery, this difference in bleeding tends to become less significant over time.

In this nationwide study, timing of intervention demonstrates that when the 48 hours windows is outdated, revascularization, even percutaneous, is usually performed after few days (mean delay of 10 days for PCI). In addition to surgery timing [START_REF] Lemaire | Delay in coronary artery bypass grafting for STEMI patients improves hospital morbidity and mortality[END_REF], several other factors emerged in the literature as predictors for adverse outcomes in STEMI patients undergoing CABG such as age, female gender, preoperative cardiogenic shock or heart failure, CABG after failed PCI or fibrinolysis, and the degree of preoperative myocardial injury as quantified by cardiac troponin [START_REF] Malmberg | Outcomes After ST-Segment Versus Non-ST-Segment Elevation Myocardial Infarction Revascularized by Coronary Artery Bypass Grafting[END_REF][START_REF] Grothusen | Outcome of Stable Patients With Acute Myocardial Infarction and Coronary Artery Bypass Surgery Within 48 Hours: A Single-Center, Retrospective Experience[END_REF][START_REF] Pi | Utilization, Characteristics, and In-Hospital Outcomes of Coronary Artery Bypass Grafting in Patients With ST-Segment-Elevation Myocardial Infarction: Results From the National Cardiovascular Data Registry Acute Coronary Treatment and Intervention Outcomes Network Registry-Get With The Guidelines[END_REF][START_REF] Lemaire | Delay in coronary artery bypass grafting for STEMI patients improves hospital morbidity and mortality[END_REF][START_REF] Merdler | Long-Term Outcomes in ST Myocardial Infarction Patients Undergoing Coronary Artery Bypass Graft Versus Primary Percutaneous Coronary Intervention[END_REF]. In our cohort, we observed that in France patients who underwent CABG between 48 hours to 90 days were sicker than those who had PCI. Even if slightly younger, frailty index was higher in CABG patients, while heart failure and cardiogenic shock at presentation more frequently observed.

We assume that improved technology and equipment, in combination with optimized medical treatment prior to the procedure, contributed to the improved outcomes in the patients who underwent CABG. In addition, improvement in medical infrastructure and treatment of shortterm post-CABG complications also contributed to improved long-term outcomes.

Multi vessel disease is a frequent finding in STEMI patients [START_REF] Park | Extent, location, and clinical significance of non-infarct-related coronary artery disease among patients with ST-elevation myocardial infarction[END_REF]. In chronic coronary syndrome or stabilized non-STEMI with multivessel disease, CABG has shown to be an efficient revascularization strategy with better long-term outcomes, mostly regarding the rate of re intervention, when compared to PCI [START_REF] Mohr | Coronary artery bypass graft surgery versus percutaneous coronary intervention in patients with three-vessel disease and left main coronary disease: 5-year follow-up of the randomised, clinical SYNTAX trial[END_REF]. Moreover, complete revascularization after STEMI is associated with better prognosis [START_REF] Mehta | COMPLETE Trial Steering Committee and Investigators. Complete Revascularization with Multivessel PCI for Myocardial Infarction[END_REF] and more often achieved with CABG.

Following the COMPLETE study results, a multivessel percutaneous revascularization in STEMI appeared to be preferred to medical therapy for non-culprit coronary lesions [START_REF] Mehta | COMPLETE Trial Steering Committee and Investigators. Complete Revascularization with Multivessel PCI for Myocardial Infarction[END_REF]. Indeed, complete revascularization was associated with lower risk of myocardial infarction and ischemia driven revascularization. Therefore, the optimal management of STEMI who have not been revascularized in emergency poses a challenge to the cardiologist. On one arm, PCI represents a less invasive approach to allow semi-urgent revascularization in those patients. On the other arm, CABG may achieve more complete revascularization and may be associated with improved long-term outcomes. However, no direct comparison of PCI versus CABG in this setting is existing so far.

Limitations

We recognize several limitations to our analysis. A main limitation is inherent to the retrospective, observational nature of the study and its potential biases. Specifically, the study was based on administrative data, with limitations inherent to such methodology (6 -8). The PMSI database makes use of diagnoses coded using ICD-10; such diagnoses are the physician's responsibility and were not comprehensively checked, leading to a potential information bias. Nevertheless, the large scale of the database and the regular audit of coded diagnoses (linked with reimbursement) are expected to mitigate possible bias and errors.

The large population of patients admitted for STEMI is likely to represent a heterogeneous group of patients, each with various kinds of illnesses and severities, which may have affected prognosis. Moreover, relevant clinical or procedural information, as well as the usage of pharmacotherapies were not evaluated. Furthermore, the complete evaluation of the clinical circumstances (such as failed PCI or salvage CABG) that may have influenced treatment decisions on CABG was not available. However, the time frame (48 hours after hospital admission) should reduce this bias.

Analyses were restricted to the variables coded in the database, excluding precise results of coronary angiograms, biological parameters, echocardiographic measurements or use of medications. Comparisons between groups might not be fully appropriate, as the presence of other missing confounding factors cannot be ruled out. Coronary anatomy and SYNTAX score, which are pivotal element in the decision between PCI and CABG in stable and stabilized non-STEMI patients, were not available for this analysis. Despite having no access to a precise evaluation of left ventricular ejection fraction, we do report heart failure and cardiogenic shock at baseline. Moreover, CABG details (off pump versus on pump, arterial versus venous graft, complete revascularization) were not available for this analysis.

Conclusions

In patients with STEMI who did not undergo urgent revascularization (i.e. within 48hours after presentation) surgical revascularisation was associated with better outcomes and should be considered as an alternative to percutaneous coronary intervention after individualized Heart Team evaluation. Values are n (incidence rate, %/year). CI=confidence interval; HF=heart failure; HR=hazard ratio; STEMI = ST-segment elevation myocardial infarction.

Figures legends

Figure 1 :Figure 2 :

 12 Figure 1: Flow chart of the study Figure 2: Cumulative incidence of all-cause death (top panel) and of cardiovascular death (lower panel) in matched patients with STEMI treated with PCI or CABG in day 3-90 after STEMI.

Figure 3 :

 3 Figure 3: Cumulative incidence of rehospitalization for heart failure (top panel) and of ischemic stroke (lower panel) in matched patients with STEMI treated with PCI or CABG in day 3-90 after STEMI.

Figure 4 :

 4 Figure 4: Cumulative incidence of major bleeding in matched patients with STEMI treated with PCI or CABG in day 3-90 after STEMI.

Table 1 .

 1 Characteristics of patients with STEMI treated with PCI or CABG in day 3-90 after STEMI.

					Standardized
		PCI	CABG	p value	difference,	Total
					CABG vs PCI
		(n=59,340)	(n=12,025)		(%)	(n=71,365)
	Age, years	68.5±13.7	66.2±10.8	<.001	-18.2	68.1±13.3
	Sex (male)	41292 (69.6) 9678 (80.5)	<.001	25.4	50970 (71.4)
	Charlson comorbidity index	3.9±2.9	4.2±2.9	<.001	9.7	3.9±2.9
	Frailty index	5.3±3.1	5.6±2.8	<.001	12.0	5.3±3.1
	Obesity	9469 (16.0) 2296 (19.1)	<.001	8.3	11765 (16.5)
	Dyslipidemia	19486 (32.8) 4986 (41.5)	<.001	17.9	24472 (34.3)
	Diabetes mellitus	15023 (25.3) 3775 (31.4)	<.001	13.5	18798 (26.3)
	Hypertension	32121 (54.1) 6965 (57.9)	<.001	7.6	39086 (54.8)
	Smoker	12737 (21.5) 2815 (23.4)	<.001	4.7	15552 (21.8)
	Alcohol related diagnoses	2957 (5.0)	664 (5.5)	.01	2.4	3621 (5.1)
	Illicit drug use	274 (0.5)	48 (0.4)	.35	-1.0	322 (0.5)
	Dilated cardiomyopathy	2679 (4.5)	648 (5.4)	<.001	4.0	3327 (4.7)
	Heart failure with congestion	3752 (6.3)	710 (5.9)	.08	-1.7	4462 (6.3)
	Aortic stenosis	1737 (2.9)	390 (3.2)	.06	1.8	2127 (3.0)
	Aortic regurgitation	1063 (1.8)	262 (2.2)	.01	2.8	1325 (1.9)
	Mitral regurgitation	3437 (5.8)	979 (8.1)	<.001	9.2	4416 (6.2)
	Tricuspid regurgitation	268 (0.5)	75 (0.6)	.01	2.4	343 (0.5)
	Ischemic stroke	1805 (3.0)	330 (2.7)	.08	-1.8	2135 (3.0)
	Atrial fibrillation	9023 (15.2) 2081 (17.3)	<.001	5.7	11104 (15.6)
	Previous pacemaker or ICD	1193 (2.0)	142 (1.2)	<.001	-6.6	1335 (1.9)
	Vascular disease	5598 (9.4)	1324 (11.0)	<.001	5.2	6922 (9.7)
	Abnormal renal function	3185 (5.4)	551 (4.6)	<.001	-3.6	3736 (5.2)
	Liver disease	1635 (2.8)	391 (3.3)	.01	2.9	2026 (2.8)
	Lung disease	6924 (11.7) 1338 (11.1)	.09	-1.7	8262 (11.6)
	COPD	4172 (7.0)	804 (6.7)	.18	-1.4	4976 (7.0)
	Sleep apnea syndrome	2390 (4.0)	525 (4.4)	.09	1.7	2915 (4.1)
	Anaemia	5386 (9.1)	1314 (10.9)	<.001	6.2	6700 (9.4)
	Previous cancer	5966 (10.1)	855 (7.1)	<.001	-10.5	6821 (9.6)
	Thyroid diseases	3222 (5.4)	486 (4.0)	<.001	-6.5	3708 (5.2)
	Cognitive impairment	2284 (3.8)	212 (1.8)	<.001	-12.7	2496 (3.5)
	Depression	3639 (6.1)	589 (4.9)	<.001	-5.4	4228 (5.9)
	Poor nutrition	2455 (4.1)	322 (2.7)	<.001	-8.1	2777 (3.9)
	Anterior MI	25306 (42.6) 4664 (38.8)	<.001	-7.9	29970 (42.0)
	Inferior MI	18797 (31.7) 3478 (28.9)	<.001	-6.0	22275 (31.2)
	MI with other location	15237 (25.7) 3883 (32.3)	<.001	14.6	19120 (26.8)
	HF at the acute phase	16367 (27.6) 4196 (34.9)	<.001	15.8	20563 (28.8)
	Pulm. edema / shock at the acute phase	2851 (4.8)	850 (7.1)	<.001	9.6	3701 (5.2)
	Time between STEMI and PCI/CABG (days)	10.4±15.0	17.0±17.9	<.001	40.1	11.5±15.7
	Death during follow-up	9881 (16.7) 1596 (13.3)	<.001	-	11477 (16.1)
	Cardiovascular death	4326 (7.3)	816 (6.8)	.05	-	5142 (7.2)

Values are mean ± standard deviation, or n (%). AF = atrial fibrillation; CABG, coronary artery bypass graft; COPD, chronic obstructive pulmonary disease; ICD, implantable cardiac defibrillator; STEMI = ST-segment elevation myocardial infarction; PCI, percutaneous coronary intervention; SD, standard deviation.

Table 2 .

 2 Characteristics of matched patients with STEMI treated with PCI or CABG in day 3-90 after STEMI.

					Standardized
		PCI	CABG	p value	difference,	Total
					CABG vs PCI
		(n=12,012)	(n=12,012)		(%)	(n=24,024)
	Age, years	66.1±13.7	66.2±10.8	.25	1.5	66.2±12.3
	Sex (male)	9721 (80.9)	9665 (80.5)	.36	-1.1	19386 (80.7)
	Charlson comorbidity index	4.3±3.0	4.2±2.9	.6	-2.4	4.2±3.0
	Frailty index	5.7±3.0	5.6±2.8	.23	-1.5	5.6±2.9
	Obesity	2296 (19.1)	2291 (19.1)	.93	-0.1	4587 (19.1)
	Dyslipidemia	5015 (41.7)	4973 (41.4)	.58	-0.7	9988 (41.6)
	Diabetes mellitus	3773 (31.4)	3768 (31.4)	.94	-0.1	7541 (31.4)
	Hypertension	6920 (57.6)	6953 (57.9)	.67	0.6	13873 (57.8)
	Smoker	2821 (23.5)	2810 (23.4)	.87	-0.2	5631 (23.4)
	Alcohol related diagnoses	686 (5.7)	662 (5.5)	.5	-0.9	1348 (5.6)
	Illicit drug use	44 (0.4)	48 (0.4)	.68	0.5	92 (0.4)
	Dilated cardiomyopathy	681 (5.7)	647 (5.4)	.34	-1.3	1328 (5.5)
	Heart failure with congestion	719 (6.0)	710 (5.9)	.81	-0.3	1429 (5.9)
	Aortic stenosis	404 (3.4)	390 (3.2)	.61	-0.7	794 (3.3)
	Aortic regurgitation	276 (2.3)	261 (2.2)	.51	-0.9	537 (2.2)
	Mitral regurgitation	960 (8.0)	974 (8.1)	.74	0.5	1934 (8.1)
	Tricuspid regurgitation	80 (0.7)	75 (0.6)	.69	-0.6	155 (0.6)
	Ischemic stroke	359 (3.0)	330 (2.7)	.26	-1.4	689 (2.9)
	Atrial fibrillation	2078 (17.3)	2076 (17.3)	.97	0.0	4154 (17.3)
	Previous pacemaker or ICD	128 (1.1)	142 (1.2)	.39	0.9	270 (1.1)
	Vascular disease	1337 (11.1)	1320 (11.0)	.73	-0.5	2657 (11.1)
	Abnormal renal function	545 (4.5)	551 (4.6)	.85	0.2	1096 (4.6)
	Liver disease	386 (3.2)	390 (3.2)	.88	0.2	776 (3.2)
	Lung disease	1367 (11.4)	1337 (11.1)	.54	-0.8	2704 (11.3)
	COPD	836 (7.0)	803 (6.7)	.4	-1.1	1639 (6.8)
	Sleep apnea syndrome	512 (4.3)	524 (4.4)	.7	0.5	1036 (4.3)
	Anaemia	1328 (11.1)	1310 (10.9)	.71	-0.5	2638 (11.0)
	Previous cancer	838 (7.0)	855 (7.1)	.67	0.5	1693 (7.0)
	Thyroid diseases	468 (3.9)	486 (4.0)	.55	0.7	954 (4.0)
	Cognitive impairment	212 (1.8)	212 (1.8)	.99	0.0	424 (1.8)
	Depression	594 (4.9)	589 (4.9)	.88	-0.2	1183 (4.9)
	Poor nutrition	340 (2.8)	322 (2.7)	.48	-0.8	662 (2.8)
	Anterior MI	4740 (39.5)	4658 (38.8)	.28	-1.4	9398 (39.1)
	Inferior MI	3402 (28.3)	3478 (29.0)	.28	1.4	6880 (28.6)
	MI with other location	3870 (32.2)	3876 (32.3)	.93	0.1	7746 (32.2)
	HF at the acute phase	4231 (35.2)	4185 (34.8)	.53	-0.8	8416 (35.0)
	Pulm. edema / shock at the acute phase	859 (7.2)	849 (7.1)	.80	-0.4	1708 (7.1)
	Time between STEMI and PCI/CABG (days)	16.5±20.6	17.0±17.8	.05	3.0	16.7±19.2
	Death during follow-up	2202 (18.3)	1596 (13.3)	<.001	-	3798 (15.8)
	Cardiovascular death	951 (7.9)	816 (6.8)	0.01	-	1767 (7.4)
	Values are mean ± standard deviation, or n (%).
	AF = atrial fibrillation; CABG, coronary artery bypass graft; COPD, chronic obstructive pulmonary disease;
	ICD, implantable cardiac defibrillator; STEMI = ST-segment elevation myocardial infarction; PCI,
	percutaneous coronary intervention; SD, standard deviation.

Table 3 .

 3 Predictors of all-cause death during the whole follow-up in unmatched patients with STEMI treated with PCI or CABG in day 3-90 after STEMI.

		Univariate analysis		Multivariable analysis
		HR, 95%CI	p	HR, 95%CI	p
	Age, years	1.053 (1.052-1.055)	<.001	1.042 (1.039-1.044)	<.001
	Sex (male)	0.774 (0.744-0.804)	<.001	1.070 (1.026-1.116)	.01
	Charlson comorbidity index	1.195 (1.189-1.200)	<.001	1.082 (1.073-1.090)	<.001
	Frailty index	1.050 (1.049-1.052)	<.001	1.012 (1.010-1.014)	<.001
	Obesity	0.989 (0.941-1.040)	.67	0.914 (0.866-0.964)	.01
	Dyslipidemia	0.974 (0.938-1.012)	.18	0.858 (0.823-0.894)	<.001
	Diabetes mellitus	1.601 (1.541-1.663)	<.001	1.090 (1.044-1.140)	<.001
	Hypertension	1.717 (1.651-1.786)	<.001	0.962 (0.920-1.005)	.08
	Smoker	0.683 (0.649-0.718)	<.001	0.979 (0.925-1.036)	.47
	Alcohol related diagnoses	1.482 (1.377-1.594)	<.001	1.352 (1.246-1.468)	<.001
	Illicit drug use	0.809 (0.565-1.158)	.25	1.218 (0.849-1.746)	.29
	Dilated cardiomyopathy	1.872 (1.747-2.005)	<.001	1.037 (0.965-1.113)	.32
	Heart failure with congestion	3.293 (3.130-3.465)	<.001	1.713 (1.608-1.825)	<.001
	Ischemic stroke	2.007 (1.844-2.185)	<.001	1.101 (1.009-1.200)	.03
	Atrial fibrillation	2.273 (2.182-2.368)	<.001	1.176 (1.126-1.229)	<.001
	Previous pacemaker or ICD	2.233 (2.027-2.460)	<.001	1.002 (0.907-1.108)	.96
	Vascular disease	2.356 (2.248-2.470)	<.001	1.233 (1.170-1.300)	<.001
	Abnormal renal function	2.731 (2.577-2.895)	<.001	1.187 (1.114-1.265)	<.001
	Liver disease	2.651 (2.450-2.869)	<.001	1.360 (1.248-1.483)	<.001
	Lung disease	2.128 (2.033-2.226)	<.001	1.209 (1.125-1.299)	<.001
	COPD	2.161 (2.047-2.281)	<.001	1.085 (0.996-1.181)	.06
	Sleep apnea syndrome	1.202 (1.102-1.311)	<.001	0.874 (0.798-0.957)	0.003
	Anaemia	2.584 (2.464-2.710)	<.001	1.221 (1.159-1.287)	<.001
	Previous cancer	2.496 (2.379-2.617)	<.001	1.396 (1.325-1.470)	<.001
	Thyroid diseases	1.457 (1.356-1.566)	<.001	0.944 (0.876-1.017)	.13
	Cognitive impairment	2.250 (2.088-2.425)	<.001	1.091 (1.008-1.181)	.03
	Depression	1.472 (1.376-1.576)	<.001	0.967 (0.899-1.039)	.36
	Poor nutrition	2.945 (2.747-3.157)	<.001	1.226 (1.139-1.319)	<.001
	Anterior MI	1.104 (1.064-1.145)	<.001	0.944 (0.904-0.985)	0.01
	Inferior MI	0.705 (0.676-0.735)	<.001	0.868 (0.826-0.912)	<.001
	HF at the acute phase	1.821 (1.754-1.890)	<.001	1.402 (1.341-1.465)	<.001
	Pulm. edema / shock at the acute
	phase	4.434 (4.212-4.668)	<.001	2.946 (2.784-3.118)	<.001
	CABG (vs PCI)	0.727 (0.690-0.767)	<.001	0.734 (0.695-0.775)	<.001
	AF = atrial fibrillation; CABG, coronary artery bypass graft; COPD, chronic obstructive pulmonary disease;
	ICD, implantable cardiac defibrillator; STEMI = ST-segment elevation myocardial infarction; PCI,
	percutaneous coronary intervention.

Table 4 .

 4 Predictors of cardiovascular death during the whole follow-up in unmatched patients with STEMI treated with PCI or CABG in day 3-90 after STEMI. CABG, coronary artery bypass graft; COPD, chronic obstructive pulmonary disease; ICD, implantable cardiac defibrillator; STEMI = ST-segment elevation myocardial infarction; PCI, percutaneous coronary intervention.

		Univariate analysis		Multivariable analysis
		HR, 95%CI	p	HR, 95%CI	p
	Age, years	1.057 (1.054-1.060)	<.001	1.050 (1.046-1.053)	<.001
	Sex (male)	0.665 (0.628-0.704)	<.001	0.959 (0.901-1.020)	.18
	Charlson comorbidity index	1.127 (1.119-1.136)	<.001	0.984 (0.971-0.997)	.02
	Frailty index	1.035 (1.032-1.038)	<.001	0.996 (0.992-1.000)	.03
	Obesity	1.019 (0.947-1.096)	.62	0.944 (0.872-1.022)	.15
	Dyslipidemia	1.019 (0.963-1.079)	.52	0.875 (0.822-0.930)	<.001
	Diabetes mellitus	1.715 (1.621-1.814)	<.001	1.408 (1.319-1.503)	<.001
	Hypertension	1.794 (1.691-1.904)	<.001	0.996 (0.932-1.064)	.90
	Smoker	0.606 (0.560-0.655)	<.001	0.942 (0.863-1.028)	0.18
	Alcohol related diagnoses	1.233 (1.097-1.385)	<.001	1.330 (1.170-1.512)	<.001
	Illicit drug use	0.603 (0.334-1.090)	.09	1.169 (0.644-2.119)	.61
	Dilated cardiomyopathy	1.988 (1.802-2.194)	<.001	1.049 (0.948-1.162)	.35
	Heart failure with congestion	3.529 (3.282-3.793)	<.001	2.304 (2.098-2.530)	<.001
	Ischemic stroke	2.148 (1.907-2.420)	<.001	1.377 (1.218-1.557)	<.001
	Atrial fibrillation	2.487 (2.343-2.639)	<.001	1.220 (1.145-1.301)	<.001
	Previous pacemaker or ICD	2.415 (2.107-2.766)	<.001	1.068 (0.928-1.230)	.36
	Vascular disease	2.525 (2.359-2.703)	<.001	1.545 (1.431-1.668)	<.001
	Abnormal renal function	2.745 (2.524-2.986)	<.001	1.352 (1.232-1.483)	<.001
	Liver disease	2.513 (2.237-2.823)	<.001	1.625 (1.433-1.842)	<.001
	Lung disease	1.908 (1.781-2.045)	<.001	1.289 (1.161-1.432)	<.001
	COPD	1.803 (1.655-1.963)	<.001	0.997 (0.877-1.134)	.97
	Sleep apnea syndrome	1.094 (0.957-1.250)	.19	0.834 (0.725-0.958)	.01
	Anaemia	2.264 (2.106-2.433)	<.001	1.117 (1.032-1.209)	.01
	Previous cancer	1.552 (1.430-1.684)	<.001	1.098 (1.005-1.199)	.04
	Thyroid diseases	1.535 (1.383-1.703)	<.001	0.970 (0.871-1.080)	.58
	Cognitive impairment	2.102 (1.881-2.349)	<.001	1.176 (1.045-1.324)	.01
	Depression	1.411 (1.274-1.562)	<.001	1.007 (0.904-1.122)	.89
	Poor nutrition	2.381 (2.140-2.649)	<.001	1.092 (0.977-1.221)	.12
	Anterior MI	1.176 (1.113-1.242)	<.001	0.978 (0.917-1.043)	.49
	Inferior MI	0.668 (0.626-0.712)	<.001	0.860 (0.797-0.927)	<.001
	HF at the acute phase	2.147 (2.032-2.268)	<.001	1.767 (1.652-1.891)	<.001
	Pulm. edema / shock at the acute
	phase	5.881 (5.492-6.297)	<.001	3.824 (3.544-4.127)	<.001
	CABG (vs PCI)	0.856 (0.794-0.922)	<.001	0.884 (0.818-0.956)	.01
	AF = atrial fibrillation;

Table 5 .

 5 Clinical outcomes during the whole follow-up (mean [SD] 2.6 [2.7], median [IQR] 1.8 [0.1-4.6] years) in matched patients with STEMI treated with PCI or CABG in day 3-90 after STEMI.

	PCI	CABG	HR (95% CI) for	p (uncorrected)
	(n=12012)	(n=12012)	CABG vs PCI

Acknowledgments

None.

LF: consultant or speaker activities for Bayer, BMS/Pfizer, Boehringer Ingelheim, Medtronic, and Novartis. CSE reports honoraria from Abbott and Biotronik. TB reports consultant activities for Edwards Lifesciences. TC

Funding

None.

Abbreviations


Theodoros Perdikis

Giovanni Celano

Stelios Psarakis

Philippe Castagliola

An Exponentially Weighted Moving Average Control Chart Based on Signed Ranks for Finite Horizon Processes

Keywords: Quality control, Statistical Process Monitoring, Markov processes, Distribution-free, Wilcoxon signed rank statistic

Distribution-free control charts are an efficient quality monitoring tool to inspect lots of parts manufactured within a finite production horizon. In this work, the performance of the Exponentially Weighted Moving Average chart based on the Wilcoxon signed rank statistic is investigated for on-line monitoring of finite production runs. The chart's on-target performance is evaluated through a specific non-homogeneous Markov chain model under different process scenarios. A numerical analysis is conducted for determining its optimal design and a performance comparison with other available schemes is presented for different symmetric distributions of observations. Finally, an illustrative example is presented to show a practical implementation of the investigated chart.

Introduction

Statistical process monitoring (SPM), through the use of control charts, is widely implemented in industries as an efficient tool for the on-line monitoring of a product's characteristic of interest. The design of conventional control charts such as the Shewhart-type (see [START_REF] Shewhart | Statistical Method from the Viewpoint of quality control[END_REF]), the EWMA (Exponentially Weighted Moving Average, see [START_REF] Roberts | Properties of Control Chart Zone Tests[END_REF]) or the CUSUM (Cumulative Sum, see [START_REF] Page | Continuous Inspection Schemes[END_REF]) control charts is generally tackled by assuming on-line monitoring of long run processes. In particular, the chart's statistical design relies on the assumption that the process either never stops or it does after a very large number, say hundreds or thou-CONTACT Stelios Psarakis. Email:psarakis@aueb.gr sands, of inspections scheduled during the production horizon. However, in practice, in several manufacturing industries, process flexibility allows consecutive productions of lots of different part codes to be produced within a very short production horizon. For example, production of customized parts with the recent advanced manufacturing technologies, e.g. additive manufacturing based on 3D printing machines, is limited to delivering a finite number of parts within a short time horizon. Quality practitioners are required to monitor production by collecting a small number of samples and the main objective of monitoring is checking if the process location and/or dispersion is centered on a target value or not. At the end of each production run, the process resources are restored and reconfigured by performing proper set-up activities to begin manufacturing of the next scheduled lot of part codes. Frequently, in this manufacturing scenario only a few tens of inspections can be scheduled by quality practitioners for each production lot. As a result, a stream of SPM research has been developed on the design of control charts under the so called Finite Horizon Process (FHP) framework, see [START_REF] Chakraborti | Finite Horizon Process Monitoring[END_REF].

When a control chart should be selected to run in a FHP quality practitioners need to consider the following challenges:

• Under the FHP framework it is assumed that a finite lot of N parts is produced during a production horizon of finite length equal to H time units. To monitor the quality characteristic of interest, practitioners can schedule only I inspections during the production horizon H. Then, the sampling frequency between two consecutive inspections is defined as h 0 = H I time units. At each inspection, a sample of size n from a sub-population of N h = N I parts is collected to run a control chart. The value of I could be restricted to a few tens due to constraints on the lot size and/or the rate of inspection. • It should be noted that, due to the fact that the process setup frequently switches to a new part code, the underlying distribution of the quality characteristic to be monitored is not known: very often, the number of scheduled inspections is too small to allow this distribution to be identified. As a result, the use of distribution-free schemes is necessary (Chakrabortiand Graham (2019)). • Usually, in productions where a large number of inspections is scheduled, the chart's statistical design is determined by its average run length properties, ARL = E(RL), where the run length (RL) is the random variable defined as " the count of points plotted on the control chart". However, these metrics are valid under the assumption that an infinite number of inspections is available.

On the other hand, under the FHP framework, only a small number of inspections can be scheduled by the end of the production horizon. As a result, the distribution of the RL is truncated by the end of the production run. Therefore, different performance metrics have to be considered to design the control chart and evaluate its statistical performance.

As a consequence, with all being said, it is worth of interest to investigate the performance of conventional schemes (parametric or not), originally designed for long runs, under the FHP framework and to provide practitioners with guidelines about their design and statistical performance. Many researchers have investigated the performance of control charts monitoring FHP. Under the assumption of normal observations, there are many contributions for monitoring shifts in the process mean (see, [START_REF] Nenes | Evaluation of CUSUM Charts for Finite-Horizon Processes[END_REF], [START_REF] Castagliola | The Variable Sample Size t Control Chart for Monitoring Short Production Runs[END_REF], [START_REF] Nenes | The Variable Sampling Interval Control Chart for Finite-Horizon Processes[END_REF], [START_REF] Nenes | Economic and Statistical Design of Vp Control Charts for Finite-Horizon Processes[END_REF]). Similarly, many nonparametric control charts have been widely investigated in literature under the FHP framework.

For instance, there are many publications focusing on Shewhart control charts monitoring process location, which are based on distribution-free statistics such as the Sign control chart (Celano et al. (2016), [START_REF] Celano | An EWMA Sign Control Chart with Varying Control Limits for Finite Horizon Processes[END_REF]), the Wilcoxon signed rank control chart (Celano et al. (2016)) or the Mann-Whitney control chart [START_REF] Celano | A distribution-Free Shewhart-type Mann-Whitney Control Chart for Monitoring Finite Horizon Productions[END_REF]). Additionally, Bayesian control charts are another type of control schemes that can efficiently monitor shifts in the process location or scale in a FHP framework. In particular, at each inspection these charts allow the design parameters to be varied adaptively, based on prior knowledge and observed outcomes, see ?, [START_REF] Wang | Multistate Bayesian Control Chart Over a Finite Horizon[END_REF], [START_REF] Nikolaidis | New Indices for the Evaluation of the Statistical Properties of Bayesian X Control Charts for Short Runs[END_REF].

In this work, we aim to investigate the performance of the EWMA control chart based on the Wilcoxon signed rank statistic introduced by [START_REF] Graham | A Nonparametric Exponentially Weighted Moving Average Signed-Rank Chart for Monitoring Location[END_REF] under the FHP framework. With this control chart, the location deviation of the process from a nominal target value specified by the quality practitioner can be monitored for observations having a symmetric distribution.

The remainder of this work is organised as follows. In Section 2, the operation and design of the EWMA chart based on the Wilcoxon signed rank statistic is presented. Furthermore, its statistical design under the FHP framework is introduced and appropriate metrics are discussed regarding the determination of its performance. In Section 3, a numerical analysis is conducted on the selection of its design parameters and their effect on the investigated control chart's performance. In Section 4, the performance of the proposed chart is compared with existing nonparametric schemes under different process scenarios. Finally, in Section 5 an illustrative example is presented to show a practical implementation of the proposed scheme, while in Section 6 some concluding remarks and suggestions for future works are discussed.

Implementation of the EWMA chart based on Signed Ranks for FHP

The nonparametric EMWA chart based on the Wilcoxon signed rank statistic has been originally introduced by [START_REF] Graham | A Nonparametric Exponentially Weighted Moving Average Signed-Rank Chart for Monitoring Location[END_REF] for long run processes. In particular, its statistical performance has been investigated for several symmetric distributions under the assumption that the process operates indefinitely. Therefore, its statistical design and implementation were entirely based on the fact that the process theoretically never stops, meaning that practitioners have at their disposal thousands of potential inspections. Under the FHP framework, this assumption does no longer hold and the effect of varying control limits for the EWMA control chart must be investigated by calculating proper performance metrics defined for a FHP. In this work, we aim to revisit the EWMA control chart based on Wilcoxon signed rank statistic by providing practitioners with detailed guidelines about its design, performance and implementation under the FHP framework.

Implementation in a FHP

Suppose that, at each scheduled inspection i = 1, 2, . . . , I during the production run, a subgroup of observations {X i,1 , X i,2 , . . . , X i,n } having size n is collected from a subpopulation having size N h . We assume that the observations X i,j , for i = 1, . . . I, j = 1, . . . n are i.i.d. random variables with unknown continuous symmetric distribution. Without loss of generality, we also assume that N h is an even number, as in Celano et al. (2016), and θ t denotes the process median, which is the location parameter of interest at time t = h 0 , 2 × h 0 , . . . , H. In general, during the production run the process median is equal to

θ t =    θ 0 , t = h 0 , 2 × h 0 , . . . , (i -1) × h 0 θ 1 , t = i × h 0 , (i + 1) × h 0 , . . . , H
, where θ 1 = θ 0 +δσ, δ is the shift magnitude and i-1 is the inspection index immediately before the shift. Similarly to what was suggested by [START_REF] Zimmer | Guidelines for the Application of Adaptive Control Charting Schemes[END_REF] for long run processes, different operating conditions can be considered for the process finite production horizon:

-on-target condition: the process starts and remains on-target ( θ t = θ 0 for t = h 0 , 2 × h 0 , . . . , H) without any location shift during the production run. In this case, the control chart should not signal any alarm. Any signal triggering is a false alarm; -off-target condition: the process starts off-target ( θ t = θ 1 for t = h 0 , 2 × h 0 , . . .) due to an imperfect set-up. The location shift is conventionally assumed immediately after the start of the production run. In this case, all I scheduled inspections are available to the control chart to trigger a signal by the end of the production run.

-on-to-off-target condition: the process starts on-target ( θ t = θ 0 for t = h 0 , 2 × h 0 , . . . , (i -1) × h 0 ) with a location shift occurring during the production run before inspection i. In this case, only I -i + 1 scheduled inspections are available to the control chart to trigger a signal by the end of the production run.

At each inspection i, the Wilcoxon signed rank statistic (SR i ) from sample i of observations is computed as follows:

SR i = n j=1 sign(X i,j -θ 0 )L i,j , (1)
where sign(X i,j -θ 0 ) = -1, 0 or 1, if X i,j < θ 0 ,X i,j = θ 0 or X i,j > θ 0 , respectively. Additionally, L i,j ∈ {1, 2, . . . , n} denotes the rank of the absolute value of the differences |X i,j -θ 0 | , j = 1, 2, ..., n, for inspection i = 1, 2, . . . , I. It is worth stretching that, due to the assumption that each sample follows a continuous distribution, the condition X i,j = θ 0 , corresponding to an observation tied to the target median is very unlikely to occur. For the computation of the sign statistic, [START_REF] Castagliola | The Shewhart Sign Chart with Ties: Performance and Alternatives[END_REF] provide some practical solutions to the problem of tied observations that can be easily adapted to the WSR statistic.

Generally, the SR i statistic presented in (1) can be alternatively expressed as:

SR i = 2SR + i - n(n + 1) 2 , (2)
where SR + i is the sum of positive signed ranks and is often called as the null Wilcoxon distribution. Moreover, by definition, the SR i statistic is defined on

{-n(n+1) 2 , -n(n+1) 2 + 2, . . . , n(n+1) 2 -2, n(n+1) 2 }.
A simple and efficient way to compute the distribution of SR i is through the null Wilcoxon distribution, SR + i using the relation presented in (2), see Gibbons and Chakraborti (20121). Finally, at each inspection i, the charting statistic of the two-sided EWMA chart based on signed ranks (WSR EWMA chart) is defined by the following recursive formula:

Z i = λ × SR i + (1 -λ)Z i-1 , (3)
where Z 0 = E 0 (SR i ) and E 0 (SR i ) is the on-target mean of SR i . Considering a finite horizon process for which the number of inspections is small (say I < 50), the timevarying (TV) control limits of the proposed chart under the FHP framework are equal to:

UCL(i) = E 0 (SR i ) + K V 0 (SR i ) λ (2 -λ) (1 -(1 -λ) 2i ) LCL(i) = E 0 (SR i ) -K V 0 (SR i ) λ (2 -λ) (1 -(1 -λ) 2i )
When the process is on-target the mean and variance of SR i are equal to [START_REF] Graham | A Nonparametric Exponentially Weighted Moving Average Signed-Rank Chart for Monitoring Location[END_REF]):

E 0 (SR i ) = 0, V 0 (SR i ) = n(n + 1)(2n + 1) 6 .
Additionally, for a sufficiently large number of I inspections, the above limits converge to the asymptotic (A ∞ ) control limits defined as

UCL ∞ = E 0 (SR i ) + K V 0 (SR i ) λ (2 -λ) LCL ∞ = E 0 (SR i ) -K V 0 (SR i ) λ (2 -λ)
It is worth remembering that for a finite horizon process the control limits rarely converge to their asymptotic values due to the small values of I. For this reason, we discourage practitioners from implementing the EWMA control chart with (LCL ∞ , UCL ∞ ) in a FHP. Later, in the Numerical Analysis Section we also prove that, as expected, the TV control limits boost the control chart's detection capability vs. the A ∞ control limits.

Selection of design parameters

Generally, the design and performance of conventional EWMA charts for long-run processes are derived through their RL (Run Length) distribution. In particular, the most commonly used metric for the charts' performance is the ARL (Average Run Length), defined as "the expected number of plotted points until the first signal is triggered by the control chart" (out-of-control state) or as "the expected number of plotted points between two successive false alarms" (in-control state). However, under the FHP framework, this metric is inefficient due to the fact that its operation assumes that I → ∞. In this work, the chart's operation and statistical design will be derived though the following metrics [START_REF] Celano | An EWMA Sign Control Chart with Varying Control Limits for Finite Horizon Processes[END_REF]):

• Let us assume that the process is on-target. The chart's design parameters (λ, K) are selected by considering the FAP (False Alarm Probability) metric: it is defined as the probability of at least one false alarm occurring by the end of the production run, given a finite number of I scheduled inspections. The design parameters (λ, K) of the EWMA control chart are selected by fixing at a nominal value FAP 0 the expected probability of false alarm during the production run.

For the computation of the FAP value, a discrete Markov chain approach origi-nally introduced by [START_REF] Brook | An Approach to the Probability Distribution of CUSUM Run Length[END_REF] for long run processes is adapted to FHP. A detailed presentation regarding the computation of FAP is provided in the Appendix A. • The chart's off-target statistical performance is determined through the SP(i, g) (signal probability) defined as "the probability to get from the control chart a true signal by the g-th inspection after the event shifting the process location, assumed to occur before inspection i". In particular, assuming that the assignable cause occurs before the i-th inspection, the signal probabilities SP(i, g) are calculated as follows:

SP(i, g) = g k=1 P(Z k > UCL(k) or Z k < LCL(k)|θ = θ 1 )
where g ≤ (I -i)+1 denotes the g-th scheduled inspection after the occurrence of the assignable cause. For instance, assuming that a shift occurs before inspection i = 1 i.e. for an off-target process, then g = {1, 2, . . . , I}. Therefore, when g = 1, SP(1, 1) denotes the probability that the control chart triggers a true off-target signal by the first inspection after the occurrence of the assignable cause; when g = 3, SP(1, 3) denotes the probability that the control chart triggers a true off-target signal by the third inspection after the occurence of the assignable cause; when g = I, SP(1, I) is the probability that the control chart triggers a true off-target signal by the end of the production run. Thus, given the process location shift, a plot of SP(i, g), can be generated. This plot is known as the signal probability profile, as defined by [START_REF] Celano | A distribution-Free Shewhart-type Mann-Whitney Control Chart for Monitoring Finite Horizon Productions[END_REF]. Finally, assuming that the process shifts to the off-target state before inspection i, an efficient metric to quantify the chart's overall statistical performance during a production run is the SP(i) (Average Signal Probability), defined as:

SP(i) = 1 G i Gi g=1 SP(i, g)
For the computation of every SP(i, g) profile, we consider a Monte Carlo simulation because considering the process change-point at any time during the production run condition does not allow the vector of initial probabilities to be defined for the Markov chain. Furthermore, a steady-state condition cannot be assumed within a few inspections. Neither the zero state nor the FIR feature can be considered, because they would assume either an on-target or an off-target process. Therefore, we have decided to simulate N sim runs, for each process location change-point during the production run to overcome the problem and to perform the study. We considered N sim = 10 6 simulation runs to get reliable estimates of the off-target WSR EWMA chart's performance.

It should be clarified the fact that, even though the chart's off-target performance is determined through a simulation-based method, the obtained results are not affected by the selected number of simulation runs, (see Appendix B).

Numerical analysis

The proper determination of the parameters (λ, K) is essential to design an efficient EWMA chart. In particular, the couple of design parameters (λ, K) should be obtained under the condition that the corresponding FAP metric is equal to a nominal value FAP 0 . The correct estimation of FAP values can be affected by the number of subintervals S chosen for the Markov chain, see Appendix A. Thus, a preliminary study has been performed on the correct tuning of the Markov chain. In Figure 1, several plots present FAP values as a function of the number of Markov chain's subintervals S = {251, 301, . . . , 1001} for (I = 20, N h = 200) and n ∈ {5, 10, 15, 20} using the A ∞ control limits. Additionally, for each case, an error bandwidth FAP S=1001 ± 3% (dashed lines) is used where FAP S=1001 (solid lines) represents the FAP value computed for S = 1001, (i.e a substantially large number of subintervals). Finally, the couple (λ, K) = (0.2, 2.5) has been randomly selected only for illustration purposes.

From Figure 1 we may conclude that the FAP values are inside the error bandwidth and, as n increases, they become steady around the value FAP S=1001 . Looking at the plots in Figure 1, setting S ≈ 501 can be considered as a reasonable choice regarding the number of subintervals.

On-target and off-target performance

Similarly to any distribution-free control chart, our proposed scheme can be designed without any assumption about the underlying distribution of observations. In particular, for a given process scenario (n, N h , I), practitioners want to select a couple of design parameters (λ, K) that meets the constraint on false alarm probability FAP = FAP 0 , (on-target performance study). In particular, it should be mentioned that, for a distribution-free control chart, due to the discrete nature of the statistic to be monitored (such as the Sign or the Wilcoxon statistics), it is not always possible to find design parameters to get exactly FAP = FAP 0 , see [START_REF] Graham | A Nonparametric Exponentially Weighted Moving Average Signed-Rank Chart for Monitoring Location[END_REF]. As a result, for a given process scenario (n, N h , I), couples (λ, K) can be selected if:

|FAP -FAP 0 | ≤ τ
where τ is a fixed constant. Without loss of generality, we set τ = 0.001. In Table 1 several couples of the design parameters (λ, K) are reported for the WSR EWMA control chart under different process scenarios (n, N h , I). In particular, for a given value of λ ∈ {0.01, 0.025, 0.05, 0.1, 0.2} and FAP 0 ∈ {0.01, 0.1}, the corresponding values of K have been computed for a population size N h ∈ {50, 150} between two consecutive inspections, sample size n ∈ {5, 10, 15} and the scheduled number of inspections is I ∈ {10, 20}. In general, in conventional EWMA schemes, as λ increases the corresponding value of K also increases. From Table 1, it may be concluded that this trend is also valid for the proposed chart implemented in the FHP scenario. In particular, for the process scenario (n, N h , I, FAP 0 )=(5, 50, 10, 0.01) we may see that for λ ∈ {0.05, 0.1, 0.15, 0.2, 0.25} we get K ∈ {2.67, 2.761, 2.832, 2.873, 2.893}.

The off-target performance of the WSR EWMA chart is examined under the following symmetric distributions of observations:

• normal distribution, N(0, 1)

• Student t distribution, t(8) • logistic distribution, L 0, √ 3 π • contaminated normal distribution CN (0.05, 2)
where the location and scale parameters for each distribution have been chosen in order to have zero mean and unit variance.

Before we proceed to any conclusions or performance evaluations for the WSR EMWA control chart in the FHP scenario, for the sake of completeness it is important to investigate the effect of the two types of control limits (i.e. the TV and A ∞ control limits), described at the beginning of this Section, on the chart's best off-target performance. Under the design of a conventional WSR EMWA chart for long runs, due to the availability of a very large number of inspections the chart's control limits become steady and so the A ∞ limits can be used. On the other hand, under FHP, only a few inspections are available by the end of the production. As a consequence, the "steady" control limits cannot be achieved and the use of the TV limits is crucial. In Figure 2, assuming the N(0, 1) distribution, for δ ∈ {0.25, 0.5, 0.75, 1} the corresponding probability profiles SP(1, g) of the WSR EWMA chart are presented using the A ∞ , and TV control limits, considering the process scenario (I, n, N h ) = (10, 10, 100). Moreover, setting λ = 0.05, we get K = 2.69 (using A ∞ limits) and K = 2.17 (using the TV limits). Finally, the design parameters (λ, K) have been chosen in order to get that FAP ≈ 0.01. From Figure 2, it can be clearly seen that using the TV control limits the signal probabilities are significantly higher than for the A ∞ limits. As an immediate consequence, under the FHP framework the use of TV limits significantly increases the chart's detection power for any shift magnitude in the process median. This is an expected finding since the number of scheduled inspections is small and the narrower TV limits at the beginning of the process allow a faster detection speed than the wider A ∞ limits.

In Table 2, for each distribution listed above, the corresponding off-target performance for the WSR EWMA chart with TV limits is presented, considering the process scenario (n, N h , I, FAP 0 )=(10, 150, 10, 0.1) for λ ∈ {0.05, 0.1, 0.2} in terms of the SP(1) metric for an off-target process. For each case, as expected, we may observe that as δ increases SP(1) → 1. Values of SP(1) close to one mean that the WSR EWMA control chart immediately signals the process location shift from the target since the first scheduled inspection. For instance, when the underlying distribution is N(0, 1) for λ = 0.05 and δ ∈ {0.25, 0.5, 0.75, 1.25, 1.5, 2}, the corresponding values for the average signal probabilities are SP(1) ∈ {0.5348, 0.8548, 0.9475, 0.9947, 0.9989, 1}. To show the effect of λ on the chart's off-target performance, Figure 3 plots the SP(1) values presented in Table 2. Within each plot, each dotted line represents the offtarget performance of the WSR EWMA chart for λ ∈ {0.05, 0.1, 0.2}. It can be seen that, for small to moderate shifts, the corresponding SP(1) values are maximized by setting λ = 0.05. On the other hand, as δ increases, the probabilities tend to be the same regardless the value of λ. As a result, choosing λ = 0.05 can be considered as a reasonable choice.

Similarly, it is important to investigate the effect of the sample size n on the off-target performance of the WSR EWMA chart under the FHP framework. In general, as the sample size increases the chart's shift detection ability is significantly improved. However, in real-life applications, quality practitioners may not have at their disposal a large set of observations at each inspection. In Figure 4, the SP(1) values using TV control limits are reported, under different process scenarios for small to moderate sample sizes n ∈ {1, 5, 10, 15}. From Figure 4 it can be seen that, as the sample size becomes larger, the SP(1) values increase. In particular, for small shifts (δ = 0.5), when n = 1 we obtain SP(1) < 0.1 (FAP = 0.01) and SP(1) = 0.14 (FAP 0 = 0.1). On the other hand, when n = 15 we obtain SP(1) ≈ 0.8 (FAP = 0.01) and SP(1) ≈ 0.91 (FAP 0 = 0.1). For moderate to large shifts (δ > 1) and n > 10 the corresponding SP(1) values tend to be the same. As consequence, it can be concluded that the use of moderate sample sizes (n ≈ 10) yields to an efficient overall performance for detecting small to large shifts. It is worth noting that a location shift can occur at any time during the production run, (on-to-off-target process condition). In the numerical analysis presented so far, only for illustration purposes it has been assumed that an assignable cause occurs immediately for each investigated distribution of observations, the percentage difference between the Sign and the WSR control charts is presented and computed as:

∆ = SP(1) SR -SP(1) SN SP(1) SN × 100% (4)
where SP(1) SR and SP(1) SN are the corresponding average probabilities for the WSR and Sign EWMA charts, respectively. Additionally, for each scheme, setting λ = 0.05, the corresponding value of K has been computed for FAP ∈ {0, 01, 0.1}. It is clear that values of ∆ > 0 show the superiority of the WSR EWMA chart because they correspond to an higher average signal probability. From Table 4 we may conclude that for FAP = 0.1 the WSR outperforms the Sign EWMA chart except for large shifts with n = 5. On the other hand, for small to moderate shifts the WSR has a better performance, regardless of the sample size or the underlying distribution.

Similarly, for FAP = 0.01, the WSR EWMA chart has a better performance for small to moderate shifts (δ < 1.25). For n > 15 (large sample size) and δ > 1.25 (large shifts) the Sign EWMA chart performs slightly better under the t( 8) and L 0,

√ 3 π
distributions. As a consequence, for small to moderate sample sizes and shift mangitudes the WSR EWMA chart outperforms the Sign EWMA control chart. It is worth stretching that these findings are consistent with the perfomance comparisons for long run processes performed by [START_REF] Graham | A Nonparametric Exponentially Weighted Moving Average Signed-Rank Chart for Monitoring Location[END_REF] between these two schemes in terms of the ARL metric.

Table 5 shows values of ∆ between the WSR EWMA chart and the WSR Shewhart control chart. Note that, in order to perform fair comparisons, the design of the EWMA chart is selected as to get the same FAP value as the Shewhart chart. In fact, it is well acknowledged that a main disadvantage of Shewhart distribution-free schemes is that frequently these charts cannot be designed to get a FAP exactly equal to the fixed nominal value FAP 0 (i.e. close to 0.01 or 0.1). Moreover, in Table 5, the coefficient C denotes the width of the control limits for the Shewhart control chart. From the results in Table 5 we may conclude that, for small shifts, the WSR EWMA chart has the best performance. On the other hand, for large shifts, the WSR Shewhart chart performs slightly better. In general, for large shifts and using a small value of λ in the EWMA chart, Shewhart-type schemes perform slightly better. Of course, by setting λ ≈ 0.2 the EWMA chart will be superior for detecting large shifts. Nevertheless, in most cases, the differences are close to zero because the signal probabilities rapidly tend to one for both charts. As a consequence, both schemes can be considered for detecting large shifts in the process.

It should be noted that, for the performance comparisons performed above, it has been assumed that the shift occurs at the start of the process (i = 1, off-target condition). However, as it was highlighted in Section 3, a shift in a process can occur at any time during the production run. As a consequence, before we make any conclusions regarding the superiority of the proposed scheme, in Table 6 the performance of the WSR EWMA chart is compared to the Sign EWMA chart under different on-to-offtarget process scenarios (δ, I, i, n). In particular, for (N h = 150, I = 10, n = 15), λ = 0.05 and FAP 0 = 0.1 the differences ∆ defined in (4) between the Sign and the WSR EWMA charts are reported for i ∈ {3, 5, 7, 9}, for different shift magnitudes. It can be clearly seen that the results presented in Table 6 are consistent with those in Table 4.

In particular, Table 6 shows that, regardless of the time when the shift occurs and even for cases when an assignable cause occurs close to the end of the production run, i.e. before inspections (i ∈ {7, 9} in our case), the WSR EWMA chart performs the better for small to moderate shifts. As a result, for all the investigated process scenarios, we may conclude that the WSR EWMA chart has an overall better performance against its competitors for detecting small to moderate shifts in the process location.

δ i = 3 i = 5 i = 7 i = 9 i = 3 i = 5 i = 7 i =

Illustrative example

In this Section, we discuss a practical implementation of the WSR EWMA chart under the FHP framework on a real-life dataset related to a filling process of carbonated beverages into cans or polyethylene terephthalate (PET)/glass bottles, where frequent machine changeovers are scheduled to change the soft drink to be filled and the packaging size (see [START_REF] Celano | A distribution-Free Shewhart-type Mann-Whitney Control Chart for Monitoring Finite Horizon Productions[END_REF]). In these processes, the measurement rate of some important quality characteristics is very slow: among them, monitoring the carbon dioxide content of the soft drink filling a bottle/can is worth of attention by practitioners. Usually, only a few inspections can be scheduled between two consecutive changeovers. For the considered production horizon, I = 10 inspections have been scheduled. In particular, at each inspection i ∈ {1, 2, . . . , 10} a sample of n = 10 observations of the carbon dioxide content within the soft drink is collected, according to the company's inspection policy. Then, the deviation X from the target value of the carbon dioxide content is calculated. The nominal value of the median deviation is θ 0 = 0.095. It is worth stretching that the distribution of the carbon dioxide content is unknown. Based on past process knowledge, practitioners know that it is symmetric around its median value. The original dataset along with the values of the charting statistic Z i at each inspection are presented in Table 7. Moreover, setting λ = 0.05 and FAP 0 = 0.1, we get K = 1.88. In Figure 5, the WSR EWMA control chart with time-varying limits (right panel) and asymptotic limits (left panel) are plotted for the I = 10 scheduled inspections during the production run. Implementing a control chart with time-varying limits allows process location shifts to be quickly detected. In fact, for the control chart with asymptotic limits, an off-target signal is triggered only at inspection #9. On the other hand, using time-varying limits, the chart triggers a signal at inspection #6. Furthermore, at inspections #4 and #5 the plotted points next to the lower control limit already reveal a process shift leading to a lower amount of carbon dioxide within the carbonated drink to be packaged. A root cause search origi-shift sizes of the process location.

As a suggested future research directions in the field of distribution-free control charts for monitoring finite horizon production processes, many topics can be suggested. First of all, further research is required on joint monitoring schemes for location and scale. Particular attention should be paid to situations when a target value cannot be defined a priori and a reference sample should be considered at the beginning of the production run. An extension to multivariate observations must be carefully investigated as well, because current sensor technologies allow many process parameters to be contemporarily recorded. To boost the control chart's sensitivity, improved adaptive schemes can be taken into account. Finally, it is worth remembering that, currently, there aren't any software packages implementing control charts for the FHP scenario. Efforts in this direction are welcome to bridge the gap between academic research and industrial practice. Open source software like the Shiny package for R can be a good programming environment to develop distribution-free SPM web applications to be shared by quality practitioners within manufacturing companies.

Appendix A

The statistical on-target performance for the WSR EWMA control chart is evaluated through a modified version of the method of [START_REF] Brook | An Approach to the Probability Distribution of CUSUM Run Length[END_REF] as discussed by [START_REF] Celano | An EWMA Sign Control Chart with Varying Control Limits for Finite Horizon Processes[END_REF] for the Sign EWMA chart. In particular, the wider control interval [LCL(I), UCL(I)] is divided into 2m + 1 subintervals of width 2ρ = UCL(I)-LCL(I) 2m+1

. The Markov chain has S = 2m+3 states. Note that for the I-th control interval all these states are transient from s ∈ {2, . . . , 2m + 2}. Additionally, let H a , a ∈ {2, . . . , 2m+2}, represents the midpoint of the a-th subinterval. The regions below LCL(I) (denoted as state s = 1) and above LCL(I) (denoted as state s = 2m + 3) correspond to the absorbing states at inspection I. On the other hand, due to the fact that for i < I the control interval is tightened due to time-varying limits, some originally transient states 2 ≤ s ≤ 2m + 2 fall outside the interval and they become absorbing states. Similarly to [START_REF] Celano | An EWMA Sign Control Chart with Varying Control Limits for Finite Horizon Processes[END_REF], a state is considered as absorbing if at the i-th inspection H s -ρ < LCL(i) or H s + ρ > UCL(i), with s = {1, . . . , S}. Finally, the transient probabilities matrix takes the following form: After some simple calculations the above probabilities can be derived as:

P(i) =                  
p a,b =F SR + i 1 2 H b + ρ -(1 -λ)H a λ + n(n + 1) 2 n -F SR + i 1 2 H b -ρ -(1 -λ)H a λ + n(n + 1) 2 n
where F SR + i (. . .) is the c.d.f. of SR + i . Furthermore, let

W(i) = i t=1 P(t)
denote the the on-target matrix containing the i-step transition probabilities for i =∈ {1, 2, . . . , I}. Finally, the FAP value, defined as the probability to have a false alarm by the end of the production run, can be derived as:

Figure 1 .

 1 Figure 1. FAP values vs. the number of subintervals S for (I = 20, N h = 200), (λ, K) = (0.2, 2.5) and n ∈ {5, 10, 15, 20}

Figure 2 .

 2 Figure 2. SP(1, g) values (off-target process) using A∞ and TV control limits vs. g for n = 10, when (I, N h ) = (10, 100) and δ ∈ {0.25, 0.5, 0.75, 1} under N(0, 1)

Figure 3 .Figure 4 .

 34 Figure 3. Effect of λ: SP(1) values using TV control limits considering the process scenario (n, N h , I, FAP 0 )=(10, 150, 10, 0.1) for λ ∈ {0.05, 0.1, 0.2} under the t(8) (left) the L 0, √ 3 π (right) distributions

  The transient probabilities p a,b are computed as:• for j(i) + 1 > a or S -j(i) < a: p a,b = 1 (a = b) or p a,b = 0 (otherwise) • j(i) + 1 ≤ a ≤ S -j(i): p a,b = P(H b -ρ ≤ λSR i + (1 -λ)Z i-1 ≤ H b + ρ|Z t-1 = H a ).

Table 1 .

 1 Values of K when I ∈ (10, 20), FAP 0 ∈ {0.01, 0.1} under different process scenarios (n, N h , I) FAP 0 = 0.01

				I ∈ (10, 20)
					λ
	N h n	0.05	0.1	0.15	0.2	0.25
	50	5	(2.67,2.85) (2.761,3.019) (2.832,3.098) (2.873,3.127) (2.893,3.126)
		10 (2.69,2.86)	(2.791,3.06)	(2.88,3.148) (2.941,3.209) (2.98,3.217)
		15 (2.69,2.87)	(2.8,3.07)	(2.891,3.181) (2.961,3.241) (3.01,3.263)
				FAP 0 = 0.1
				I ∈ (10, 20)
					λ
	N h n	0.05	0.1	0.15	0.2	0.25
	150 5	(1.88,2.07) (2.011,2.298) (2.11,2.419) (2.199,2.489) (2.25,2.518)
		10 (1.88,2.08)	(2.02,2.302) (2.128,2.443) (2.219,2.523) (2.278,2.57)
		15 (1.88,2.079)	(2.02,2.31)	(2.129,2.451) (2.227,2.532) (2.287,2.581)

Table 2 .

 2 Off-target performance in terms of SP(1) for the WSR EWMA chart with TV limits for (n, N h , I, FAP 0 )=(10, 150, 10, 0.1) and λ ∈ {0.05, 0.1, 0.2}. Off-target process scenario, i = 1.

									N(0, 1)	t(8)
					δ	λ = 0.05 λ = 0.1 λ = 0.2	λ = 0.05 λ = 0.1 λ = 0.2
					0.25		0.5348	0.4835	0.4118	0.5702	0.5202	0.4502
					0.5		0.8548	0.8270	0.7874	0.8725	0.8471	0.8117
					0.75		0.9475	0.9313	0.9097	0.9538	0.9390	0.9195
					1.25		0.9947	0.9902	0.9816	0.9941	0.9894	0.9820
					1.5		0.9989	0.9974	0.9937	0.9981	0.9960	0.9922
					2		1.0000	0.9999	0.9997	0.9998	0.9994	0.9987
									L 0,	√ π 3	CN (0.05, 2)
					δ	λ = 0.05 λ = 0.1 λ = 0.2	λ = 0.05 λ = 0.1 λ = 0.2
					0.25		0.5724	0.5222	0.4524	0.5508	0.4999	0.4282
					0.5		0.8732	0.8478	0.8129	0.8645	0.8377	0.8005
					0.75		0.9538	0.9389	0.9197	0.9524	0.9369	0.9162
					1.25		0.9940	0.9891	0.9816	0.9959	0.9919	0.9843
					1.5		0.9981	0.9959	0.9920	0.9992	0.9981	0.9950
					2		0.9998	0.9995	0.9988	1.0000	1.0000	0.9998
		0.4
	SP(1,g)	0.2 0.3
		0.1
		0.0
		1	2	3	4	5	6	7	8	9	10
						g

Table 4 .

 4 Performance comparisons in terms of SP(1) between the Sign and WSR EWMA charts for (N h = 150, I = 20) with TV limits. Off-target process scenario, (i = 1).

					FAP0 = 0.01
			N(0, 1)			t(8)
	δ	n = 5	n = 10	n = 15	n = 20	n = 5	n = 10	n = 15	n = 20
	0.25 17.66% 23.43% 15.94% 10.46%	10.75% 14.18% 8.73%	4.94
	0.5 12.56%	6.37% 3.32%	1.93%	6.7%	3.28%	1.45%	0.65%
	0.75 4.18%	2.14%	0.86%	0.38%	2.04% 0.93%	0.1%	-0.17%
	1.25 0.66%	-1.1%	0.02%	0.09%	0.2%	-1.61%	-0.24%	-0.03%
	1.5	0.23%	-2.39%	0.06%	0.03%	0.03%	-2.63%	-0.13%	0%
	2	0.01%	-4.01%	0.01%	0%	-0.01%	-3.92%	-0.04%	0
			L 0,	√ 3 π			CN (0.05, 2)
	δ	n = 5	n = 10	n = 15	n = 20	n = 5	n = 10	n = 15	n = 20
	0.25 9.87% 12.41% 7.46%	3.8%	17.3% 23.18% 15.55%	9.87
	0.5	5.76%	2.82%	1.22%	0.44%	11.47% 5.76%	3.05%	1.77%
	0.75 1.78%	0.77%	0%	-0.23%	3.82%	1.93%	0.77%	0.33%
	1.25 0.19%	-1.6%	-0.25%	-0.04%	0.56%	-1.33%	0.03%	0.08%
	1.5	0.03% -2.59%	-0.13%	0%	0.18%	-2.6%	0.07% 0.02 %
	7	-0.01%	-3.89%	-0.02%	0 %	0.01%	-4.14%	0.01%	0%
					FAP0 = 0.1
			N(0, 1)			t(8)
	δ	n = 5	n = 10	n = 15	n = 20	n = 5	n = 10	n = 15	n = 20
	0.25 10.1% 13.09% 8.17%	6%	6.98%	8.59%	4.79%	3.5%
	0.5	5.49%	4.32%	2.57%	1.91%	2.92%	2.87%	1.63%	1.17%
	0.75 1.36%	2.26%	1.29%	0.87%	0.12%	1.59%	0.81%	0.52%
	1.25 -2.04%	1.02%	0.27%	0.07%	-2.55%	0.71%	0.16%	0.03%
	1.5	-3.13%	0.59%	0.07%	0.01%	-3.42%	0.4%	0.04%	0%
	2	-4.41%	0.1%	0%	0%	-4.37%	0.09%	0%	0
			L 0,	√ 3 π			CN (0.05, 2)
	δ	n = 5	n = 10	n = 15	n = 20	n = 5	n = 10	n = 15	n = 20
	0.25 6.23%	7.76%	4.36%	3%	10.3% 12.71% 7.93%	5.76%
	0.5	2.39%	2.61%	1.47%	1.08%	5.04%	4.11%	2.46%	1.83%
	0.75 -0.05%	1.51%	0.73%	0.47%	1.13%	2.16%	1.21%	0.81%
	1.25 -2.56%	0.7%	0.15%	0.03%	-2.24%	0.94%	0.22%	0.05%
	1.5	-3.37%	0.42%	0.05%	0%	-3.31%	0.51%	0.05%	0 %
	2	-4.34%	0.11%	0%	0%	-4.5%	0.07%	0%	0%

Table 5 .

 5 Performance comparisons between the WSR EWMA chart with TV limits and the Shewhart WSR chart. Off-target process scenario, (i = 1).

				N h = 150, I = 10, FAP 0 ≈ 0.01
			(n, C) = {(10, 49), (15, 88), (20, 134)}
			N(0, 1)			t(8)
	δ	n = 10	n = 15	n = 20	n = 10	n = 15	n = 20
	0.25 65.17% 70.14% 73.82%	66.22% 69.49% 69.84%
	0.5 66.64% 27.05% 13.87%	52.84% 20.49% 10.13%
	0.75 18.94% 2.65%	0.26%	12.91% 1.54%	-0.04%
	1.25 -0.98%	-0.65%	-0.1%	-0.98%	-0.67%	-0.12%
	1.5	-1.54%	-0.18%	-0.01%	-1.27%	-0.31%	-0.02%
	2	-0.55%	0%	0%	-0.58%	-0.05%	0%
			L 0,	√ π 3			CN (0.05, 2)
	δ	n = 10	n = 15	n = 20	n = 10	n = 15	n = 20
	0.25 65.17% 70.14% 73.82%	66.22% 69.49% 69.84%
	0.5 66.64% 27.05% 13.87%	52.84% 20.49% 10.13%
	0.75 18.94% 2.65%	0.26%	12.91% 1.54%	-0.04%
	1.25 -0.98%	-0.65%	-0.1%	-0.98%	-0.67%	-0.12%
	1.5	-1.54%	-0.18%	-0.01%	-1.27%	-0.31%	-0.02%
	2	-0.55%	0%	0%	-0.58%	-0.05%	0%
				N h = 150, I = 10, FAP 0 ≈ 0.05
			(n, C) = {(10, 38), (15, 69), (20, 105)}
			N(0, 1)			t(8)
	δ	n = 10	n = 15	n = 20	n = 10	n = 15	n = 20
	0.25	0.84% 5.72%	6.4%	3.49%	7.38%	6.78%
	0.5	6.86%	2.3%	0.44%	5.44%	1.54%	0.15%
	0.75 0.53%	-0.4%	-0.48%	0.24%	-0.42%	-0.41%
	1.25 -0.57%	-0.07%	-0.01%	-0.51%	-0.09%	-0.01%
	1.5	-0.23%	-0.01%	0%	-0.28%	-0.02%	0%
	2	-0.01%	0% L 0, √ π 3	0%	-0.05%	0% CN (0.05, 2)	0%
	δ	n = 10	n = 15	n = 20	n = 10	n = 15	n = 20
	0.25 3.04%	6.35%	6.31%	1.33%	6.78%	6.69%
	0.5	5.13%	1.51%	0.06%	6.12%	1.86%	0.21%
	0.75	0.2%	-0.46%	-0.43%	0.21%	-0.5%	-0.44%
	1.25 -0.53%	-0.1%	-0.01%	-0.5%	-0.06%	-0.01%
	1.5	-0.28%	-0.02%	0%	-0.18%	0%	0%
	2	-0.05%	0%	0%	-0.01%	0%	0%

Table 6 .

 6 Performance comparisons in terms of SP(i) for i ∈ {3, 5, 7, 9} between the Sign and WSR EWMA charts for (N h = 150, I = 10) with TV limits. On-to-off-target process scenario.

					FAP0 = 0.1
			N(0, 1)				t(8)
	δ	i = 3	i = 5	i = 7	i = 9	i = 3	i = 5	i = 7	i = 9
	0.25 14.41% 14.56%	9.94% 0.45%	24.73% 29.48% 25.68% 15.91%
	0.5	6.12%	10.09% 15.13% 11.39%	9.28%	16.42% 26.24% 26.47%
	0.75 2.45%	3.59%	6.96%	7.87%	3.51%	5.73%	11.63% 19.09%
	1	1.01%	0.77%	1.82%	1.24%	1.48%	1.69%	3.57%	7.08%
	1.5	-0.59%	-1.85% L 0,	-3.29% √ 3 π	-9.69%	-0.41%	-1.75% CN (0.05, 2) -2.81%	-8.63%

  = max[s : H s -ρ < LCL(i)].If we consider as an example the case where i = I, then the transient probabilities matrix is equal to:

	1		0	. . .	0	0	
	 p (S-j(i)+1),1 p (S-j(i)+1),2 . . . p (S-j(i)+1),S-1 p (S-j(i)+1),S 0 1 . . . 0 0 . . . . . . . . . . . . . . . p (j(i)+1),1 p (j(i)+1),2 . . . p (j(i)+1),S-1 p (j(i)+1),S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 . . . 1 0	                 
	0		0	. . .	0	1
	where j(i) P(i) =	      	1 p 2,1 . . .	0 p 2,2 . . .	. . . . . . . . .	0 p 2,S-1 . . .	0 p 2,S . . .
		

Table 8 .

 8 Simulated values for SP(1, g) and SP(1) under the N(0, 1) distribution, for (λ, K = 0.05, 1.88) and n in{10, 15}. N sim = {10 5 , 5 × 10 5 , 10 6 , 1.5 × 10 6 }

					n = 10
			δ = 0.5			δ = 75
	g	10 5	5 × 10 5	10 6	1.5 × 10 6	10 5	5 × 10 5	10 6	1.5 × 10 6
	1	0.3299 0.3304 0.3307	0.3304	0.6027 0.6015 0.6010	0.6015
	2	0.6134 0.6129 0.6131	0.6126	0.9014 0.8999 0.8992	0.8994
	3	0.7977 0.7957 0.7954	0.7960	0.9803 0.9801 0.9797	0.9801
	4	0.8964 0.8976 0.8971	0.8974	0.9964 0.9966 0.9964	0.9965
	5	0.9501 0.9501 0.9501	0.9503	0.9995 0.9994 0.9994	0.9994
	6	0.9767 0.9766 0.9764	0.9765	0.9999 0.9999 0.9999	0.9999
	7	0.9895 0.9890 0.9892	0.9893	1.0000 1.0000 1.0000	1.0000
	8	0.9952 0.9950 0.9952	0.9952	1.0000 1.0000 1.0000	1.0000
	9	0.9980 0.9978 0.9979	0.9979	1.0000 1.0000 1.0000	1.0000
	10	0.9991 0.9990 0.9991	0.9990	1.0000 1.0000 1.0000	1.0000
	SP(1) 0.8546 0.8544 0.8544	0.8545	0.9480 0.9478 0.9476	0.9477
					n = 15
			δ = 0.5			δ = 75
	g	10 5	5 × 10 5	10 6	1.5 × 10 6	10 5	5 × 10 5	10 6	1.5 × 10 6
	1	0.4422 0.4445 0.4440	0.4442	0.7717 0.7706 0.7704	0.7707
	2	0.7780 0.7801 0.7801	0.7803	0.9800 0.9786 0.9789	0.9788
	3	0.9212 0.9222 0.9223	0.9225	0.9986 0.9985 0.9984	0.9986
	4	0.9752 0.9747 0.9747	0.9746	0.9999 0.9999 0.9999	0.9999
	5	0.9925 0.9923 0.9923	0.9923	1.0000 1.0000 1.0000	1.0000
	6	0.9977 0.9976 0.9977	0.9977	1.0000 1.0000 1.0000	1.0000
	7	0.9993 0.9994 0.9993	0.9993	1.0000 1.0000 1.0000	1.0000
	8	0.9998 0.9998 0.9998	0.9998	1.0000 1.0000 1.0000	1.0000
	9	0.9999 1.0000 0.9999	0.9999	1.0000 1.0000 1.0000	1.0000
	10	1.0000 1.0000 1.0000	1.0000	1.0000 1.0000 1.0000	1.0000
	SP(1) 0.9106 0.9111 0.9110	0.9111	0.9750 0.9748 0.9748	0.9748

Table 3. SP(i, g) values assuming that an assignable cause occurs before inspection i under N (0, 1) for (n = 15, N h = 200, I = 10, FAP 0 = 0.1) with TV limits. On-to-off-target scenario. δ = 0.5 δ = 0.75 after the production begins (i.e. for i = 1, off-target condition). In order to investigate the control chart's signal probability profile for the on-to-off-target process condition, in Table 3 several scenarios are considered under N(0, 1) for (n = 15, N h = 200, I = 10, FAP 0 = 0.1, i ∈ {1, 3, 5, 7, 9}) using (λ, K) = (0.05, 1.88). In particular, for δ = 0.5, when i = 9, (i.e. we assume that the shift has occurred before inspection 9), the corresponding probabilities of detecting a shift by inspection g = {1, 2} after the occurrence of the shift, are SP(i, g) = {0.067, 0.24}. As expected, these probabilities are quite low, since the control chart should detect a small shift by only two available inspections. On the other hand, for δ = 1.5 and i = 9 the corresponding probabilities are SP(i, g) = {0.2235, 0.8023}. For the other investigated cases i ∈ {3, 5, 7} it is evident that the signal probabilities converge to 1 regardless of the shift magnitude.

Performance comparisons

In this Section we aim to investigate the performance of the WSR EWMA chart versus other nonparametric control charts under the FHP framework: in particular, we consider the Sign EWMA control chart with TV limits [START_REF] Celano | An EWMA Sign Control Chart with Varying Control Limits for Finite Horizon Processes[END_REF]) and the Shewhart Wilcoxon Signed Rank control chart (Celano et al. (2016)).

In Table 4, performances comparisons between the Sign and the WSR EWMA control charts are presented under different process scenarios (n, N h , I, FAP 0 ). In particular, UCL(i) LCL(i) 1 0.11 0.14 0.09 0.11 0.09 0.08 0.12 0.08 0.11 0.12 35.00 1.75 1.84 -1.84 2 0.10 0.10 0.11 0.12 0.07 -0.04 -0.04 0.04 -0.05 -0.05 -33.00 -1.65 2.54 -2.54 3 0.07 0.06 0.14 0.06 0.05 0.11 0.05 0.06 0.06 0.08 -31.00 -3.12 3.04 -3.04 4 0.06 0.14 0.03 0.03 0.03 0.10 0.03 0.10 0.02 0.08 -39.00 -1.95 3.43 -3.43 5 0.10 0.10 0.14 0.08 0.08 0.12 0.14 0.05 0.05 -0.01 -5.00 -2.10 3.74 -3.74 6 0.10 0.10 0.04 0.06 0.06 0.12 0.11 0.11 0.05 0.05 -25.00 -3.25 4.00 -4.00 7 0.06 0.06 0.15 0.14 0.14 0.14 0.11 0.11 0.15 0.15 41.00 2.05 4.23 -4.23 8 0.15 0.15 0.15 0.12 0.12 0.14 0.14 0.05 0.05 0.13 37.00 3.80 4.42 -4.42 9 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.13 0.15 0.15 55.00 2.75 4.59 -4.59 10 0.14 0.15 0.14 0.14 0.14 0.14 0.12 0.12 0.13 0.07 53.00 5.26 4.73 -4.73

nating from the control chart's signal has revealed a problem at the pressure regulator.

After eliminating the problem, a second signal is triggered at inspection #9 due to a pressure overcompensation. 7 with asymptotic limits (left panel) and time-varying limits (right panel)

Conclusions

In this work the performance of an EWMA chart based on the Wilcoxon signed rank statistic has been investigated for monitoring finite horizon processes. Using appropriate metrics of statistical performance and carrying out an extensive numerical analysis, the chart's design parameters have been selected and its off-target performance has been investigated. Additionally, performance comparisons have been discussed among the WSR EWMA, the Sign EWMA and WSR Shewhart charts. The obtained results show that the WSR EWMA chart can be considered as an efficient tool for monitoring small to moderate shifts in the process location regardless of the process scenario being considered. In fact, for on-line process monitoring of a FHP, the WSR EWMA control chart overperforms its competitors for small and medium FAP = q W(I)r

where q = (q 1,1 , . . . , q 1,S ) is the (2m + 1, 1) vector of initial probabilities which is set to q = (0, . . . , 1, . . . , 0) . Moreover, r denotes a vector with elements r = (1, . . . , 0, . . . , 1) .

Appendix B

In Table 8, we present the off-target performance of the WSR EWMA chart assuming the N(0, 1) distribution. The off-target performance is calculated by means of simulation. Here, we investigate the effect of the number of simulation runs N sim on the accuracy of estimation of the chart's detection capability. We have considered N sim = {10 5 , 5 × 10 5 , 10 6 , 1.5 × 10 6 }. For illustration purposes we randomly use (λ, K) = (0.05, 1.88) and n ∈ {10, 15}. Each column in Table 8 represents a different value of N sim for Monte Carlo simulation. The entries in each table's row show that, neither the SP(1, g) nor the SP(1) values are significantly affected by N sim , with some minor difference in the fourth decimal place for some cases. As a consequence, the simulation method is a robust approach for determining the chart's off-target performance. For our investigation we have chosen N sim = 10 6 .