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      Joint distribution for number of crossings and longest run in independent Bernoulli observations. The R package crossrun

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      Author(s): 1 , 2 , * , 3
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      Publication date (Electronic):
      Journal: PLoS ONE
      Publisher: Public Library of Science

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          Abstract

          The R package crossrun computes the joint distribution of the number of crossings and the longest run in a sequence of independent Bernoulli observations. The main intended application is statistical process control where the joint distribution may be used for systematic investigation, and possibly refinement, of existing rules for distinguishing between signal and noise. While the crossrun vignette is written to assist in practical use, this article gives a hands-on explanation of why the procedures works. The article also includes a discussion of limitations of the present version of crossrun together with an outline of ongoing work to meet these limitations. There is more to come, and it is necessary to grasp the basic ideas behind the procedure implemented both to understand these planned extensions, and how presently implemented rules in statistical process control, based on the number of crossings and the longest run, may be refined.

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          Run Charts Revisited: A Simulation Study of Run Chart Rules for Detection of Non-Random Variation in Health Care Processes

          Jacob Anhøj, Anne Olesen (2014)
          Background A run chart is a line graph of a measure plotted over time with the median as a horizontal line. The main purpose of the run chart is to identify process improvement or degradation, which may be detected by statistical tests for non-random patterns in the data sequence. Methods We studied the sensitivity to shifts and linear drifts in simulated processes using the shift, crossings and trend rules for detecting non-random variation in run charts. Results The shift and crossings rules are effective in detecting shifts and drifts in process centre over time while keeping the false signal rate constant around 5% and independent of the number of data points in the chart. The trend rule is virtually useless for detection of linear drift over time, the purpose it was intended for.
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            Diagnostic Value of Run Chart Analysis: Using Likelihood Ratios to Compare Run Chart Rules on Simulated Data Series

            Jacob Anhøj (2015)
            Run charts are widely used in healthcare improvement, but there is little consensus on how to interpret them. The primary aim of this study was to evaluate and compare the diagnostic properties of different sets of run chart rules. A run chart is a line graph of a quality measure over time. The main purpose of the run chart is to detect process improvement or process degradation, which will turn up as non-random patterns in the distribution of data points around the median. Non-random variation may be identified by simple statistical tests including the presence of unusually long runs of data points on one side of the median or if the graph crosses the median unusually few times. However, there is no general agreement on what defines “unusually long” or “unusually few”. Other tests of questionable value are frequently used as well. Three sets of run chart rules (Anhoej, Perla, and Carey rules) have been published in peer reviewed healthcare journals, but these sets differ significantly in their sensitivity and specificity to non-random variation. In this study I investigate the diagnostic values expressed by likelihood ratios of three sets of run chart rules for detection of shifts in process performance using random data series. The study concludes that the Anhoej rules have good diagnostic properties and are superior to the Perla and the Carey rules.
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              Sense and sensibility: on the diagnostic value of control chart rules for detection of shifts in time series data

              Jacob Anhøj, Tore Wentzel-Larsen (2018)
              Background The aim of this study was to quantify and compare the diagnostic value of The Western Electric (WE) statistical process control (SPC) chart rules and the Anhoej rules for detection of non-random variation in time series data in order to make recommendations for their application in practice. Methods SPC charts are point-and-line graphs showing a measure over time and employing statistical tests for identification of non-random variation. In this study we used simulated time series data with and without non-random variation introduced as shifts in process centre over time. The primary outcome was likelihood ratios of combined tests. Likelihood ratios are useful measures of a test’s ability to discriminate between the true presence or absence of a specific condition. Results With short data series (10 data points), the WE rules 1–4 combined and the Anhoej rules alone or combined with WE rule 1 perform well for identifying or excluding persistent shifts in the order of 2 SD. For longer data series, the Anhoej rules alone or in combination with the WE rule 1 seem to perform slightly better than the WE rules combined. However, the choice of which and how many rules to apply in a given situation should be made deliberately depending on the specific purpose of the SPC analysis and the number of available data points. Conclusions Based on these results and our own practical experience, we suggest a stepwise approach to SPC analysis: Start with a run chart using the Anhoej rules and with the median as process centre. If, and only if, the process shows random variation at the desired level, apply the 3-sigma rule in addition to the Anhoej rules using the mean as process centre. Electronic supplementary material The online version of this article (10.1186/s12874-018-0564-0) contains supplementary material, which is available to authorized users.
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                Author and article information

                Contributors
                Tore Wentzel-Larsen:
                ORCID: http://orcid.org/0000-0002-0318-4162
                Role: ConceptualizationRole: Formal analysisRole: MethodologyRole: Writing – original draft
                Jacob Anhøj:
                ORCID: http://orcid.org/0000-0002-7701-1774
                Role: ConceptualizationRole: Formal analysisRole: MethodologyRole: Writing – original draft
                Ferdinando Chiacchio: Role: Editor
                Journal
                Journal ID (nlm-ta): PLoS One
                Journal ID (iso-abbrev): PLoS ONE
                Journal ID (publisher-id): plos
                Journal ID (pmc): plosone
                Title: PLoS ONE
                Publisher: Public Library of Science (San Francisco, CA USA )
                ISSN (Electronic): 1932-6203
                Publication date (Electronic): 1 October 2019
                Publication date Collection: 2019
                Volume: 14
                Issue: 10
                Electronic Location Identifier: e0223233
                Affiliations
                [1 ] Centre for Child and Adolescent Mental Health, Eastern and Southern Norway, Oslo, Norway
                [2 ] Norwegian Centre of Violence and Traumatic Stress Studies, Oslo, Norway
                [3 ] Rigshospitalet, University of Copenhagen, Copenhagen, Denmark
                University of Catania, ITALY
                Author notes

                Competing Interests: The authors have declared that no competing interests exist.

                Author information
                http://orcid.org/0000-0002-0318-4162
                http://orcid.org/0000-0002-7701-1774
                Article
                Publisher ID: PONE-D-19-14277
                DOI: 10.1371/journal.pone.0223233
                PMC ID: 6772032
                PubMed ID: 31574115
                SO-VID: 974fc066-bd59-4d43-a887-31b912bc10dc
                Copyright © © 2019 Wentzel-Larsen, Anhøj
                License:

                This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

                History
                Date received : 20 May 2019
                Date accepted : 17 September 2019
                Page count
                Figures: 2, Tables: 3, Pages: 11
                Funding
                The authors received no specific funding for this work.
                Categories
                Subject: Research Article
                Subject: Physical Sciences
                Subject: Mathematics
                Subject: Probability Theory
                Subject: Probability Distribution
                Subject: Physical Sciences
                Subject: Mathematics
                Subject: Probability Theory
                Subject: Statistical Distributions
                Subject: Engineering and Technology
                Subject: Signal Processing
                Subject: Statistical Signal Processing
                Subject: Engineering and Technology
                Subject: Signal Processing
                Subject: Noise Reduction
                Subject: Physical Sciences
                Subject: Mathematics
                Subject: Statistics
                Subject: Statistical Data
                Subject: Computer and Information Sciences
                Subject: Data Visualization
                Subject: Infographics
                Subject: Charts
                Subject: Physical Sciences
                Subject: Mathematics
                Subject: Algebra
                Subject: Polynomials
                Subject: Binomials
                Subject: Biology and Life Sciences
                Subject: Organisms
                Subject: Eukaryota
                Subject: Animals
                Subject: Vertebrates
                Subject: Amniotes
                Subject: Mammals
                Subject: Wildebeest
                Custom metadata
                Data Availability The article describes software, the R package crossrun. No data except what is generated by the attached script crossrunart.R is used.

                ScienceOpen disciplines: Uncategorized
                Data availability:
                ScienceOpen disciplines: Uncategorized

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