Statistics graphics for the laboratory Applications Internal quality

Statistics & graphics for the laboratory Applications Internal quality control Dietmar Stöckl Dietmar@stt-consulting. com Linda Thienpont Linda. thienpont@ugent. be In cooperation with AQML: D Stöckl, L Thienpont & • Kristian Linnet, MD, Ph. D Linnet@post 7. tele. dk • Per Hyltoft Petersen, MSc Per. hyltoft. petersen@ouh. fyns-amt. dk • Sverre Sandberg, MD, Ph. D Sverre. sandberg@isf. uib. no Statistics & graphics for the laboratory. Courses-Overview

Prof Dr Linda M Thienpont University of Gent Institute for Pharmaceutical Sciences Laboratory for Analytical Chemistry Harelbekestraat 72, B-9000 Gent, Belgium e-mail: linda. thienpont@ugent. be STT Consulting Dietmar Stöckl, Ph. D Abraham Hansstraat 11 B-9667 Horebeke, Belgium e-mail: dietmar@stt-consulting. com Tel + FAX: +32/5549 8671 Copyright: STT Consulting 2007 Statistics & graphics for the laboratory 2

Content Introduction IQC in the broader context Origin of internal quality control Quality management Technical competence Management of the analytical process Purpose of IQC and regulation Manufacturers IVD 98/79/EC Directive & CEN-documents Manufacturer’s recommendations Laboratories Belgian Guidelines (Koninklijk Besluit/Praktijkrichtlijn) Summary and conclusion Fundamentals of IQC Analytical paradigm and IQC paradigm IQC – practical aspects IQC materials • Nature • Target mean & CV – Statistical excursion: uncertainty of mean & SD • Frequency and location Presentation of results Software & Documentation Statistics & graphics for the laboratory 3

Content Statistical basis of IQC Introduction Statistics Basic calculations Gaussian distribution (standard and cumulated) Gaussian distribution and %-age of observations Statistical probabilities “Rare events” Summary outside probabilities Control rules From outside probabilities to control rules Control rules – An example Control rules – basic monitoring principles Selection of control rules: Fundamental problems Power function graphs Construction of the power function for 13 s and SE Construction of the power function for 12 s and RE Power of control rules and s-limits Power of control rules and n Comparison of the power of control rules Control rules: The problem of false rejection (Pfr) Circumventing the increase in Pfr with n: control rules with variable limits The ideal control rule Magnitude of errors detected by IQC Conclusions from statistical considerations Statistics & graphics for the laboratory 4

Content Metrological basis of IQC Introduction The error concept chosen here The total error concept Total error – calculations Instability of the analytical process Instability and analytical process specifications Instability – how much can be tolerated? Analytical process specifications - TEa IQC and TEa The error model for IQC Basic formula Critical errors Graphical presentation of critical errors Calculation of critical errors Special topic: The TEa problematic Control rules based on TEa Automatic selection of rules based on TEa: The Validator OPSpecs®-Charts Critical error graphs Selection of a control rule based on TEa with the Validator: an example Other selection tools: the TEa/CVa, tot ratio; the IQC decision tool Summary and Conclusion Statistics & graphics for the laboratory 5

Content IQC policy Introduction Software Samples Frequency (& location) of IQC measurements Performance (State-of-the-art) IQC rule selection • Patient release • Process control • Examples IQC rules for state-of-the-art performance • Screening with TEa/CVa, tot • STT IQC decision tool • EZ rules/Validator Special topics • Calculation of an actual TEa • Rule “n” and patient release • Dealing with a bias • IQC rules with wide limits (e. g. 6 s) and lot variations • “Fine-tuning” of IQC according to instability Remedial actions & • Pfr of the IQC rule and frequency of remedial actions • CVa, tot /CVa, w ratio • Inspecting IQC charts Summary EXCEL files Data IQC decision Statistics & graphics for the laboratory 6

Content Annex Checklists Glossary of terms References Statistics & graphics for the laboratory 7

Introduction Statistics & graphics for the laboratory 8

IQC in the broader context Origin of internal quality control The idea of internal quality control (IQC) was applied first in the industrial manufacturing process (Shewart). In clinical chemistry (= laboratory medicine), IQC was introduced by Levey & Jennings. Since the 70 ies, IQC is very much related with the name of Westgard. Before going into the details of IQC, some general aspects will be addressed, of: • Quality management • Technical competence • Management of the analytical process Quality management All activities of the overall management function that determine the quality policy, objectives and responsibilities, and implement them by means such as quality planning, quality control, quality assurance and quality improvement within the quality system. While this text focuses on IQC, it shall be stressed that IQC should not be viewed isolated. IQC is embedded in the overall effort of the laboratory for quality. This relates to its technical competence as well as its management competence. Management competence In the quality management system, IQC is part of the circle quality-planning, assurance, -control, and -improvement (note: a glossary of terms is found in the Annex). Thus, IQC is an integral part of the quality system. Technical competence relates to all aspects of the analytical process. Naturally, any test in the medical laboratory has to be proven medically useful. Technical competence starts with the knowledge of the principles for the establishment of medically relevant analytical performance specifications. On that basis, the adequate test is selected, installed, and run in daily routine. However, routine performance of a test needs adequate analytical quality management, IQC being one part of it. Thus, IQC is an integral part of the analytical process. References • Shewart WA. Economic control of manufactured products. Van Nostrand: 1931. • Levey S, Jennings ER. The use of control charts in the clinical laboratory. Am J Clin Pathol 1950; 20: 1059 -66. • Westgard JO, Groth T, Aronsson T, Falk H, de Verdier C-H. Performance characteristics of rules for internal quality control: probabilities for false rejection and error detection. Clin Chem 1977; 23: 1857 -67. Statistics & graphics for the laboratory 9

IQC in the broader context Technical competence (ctd. ) IQC, an integral part of the analytical process ("Westgard") An analytical process has two major parts: Measurement procedure … necessary to obtain a measurement on a patient's sample. Control procedure … necessary to assess the validity of a measurement result. In the words of Westgard, it is made absolutely clear that a measurement result that was obtained without IQC, "is no result". IQC is a "sine-qua-non" for reporting a result. Thus, a well established IQC system is an important part of the technical competence of the laboratory. Management of the analytical process “IQC should be imbedded in the overall quality philosophy of the laboratory” It is important that the laboratory does not elaborate "stand-alone" solutions for IQC is only one means of managing daily routine quality. For example, if it has chosen a robust test that easily fulfills the performance specifications, IQC may be quite easy. Also, work according to the motto: "prevention is better than curation". And, make use of the information available through external quality assessment (EQA). References • Westgard JO, Barry PL. Cost-effective quality control. AACC Press, 1995 Statistics & graphics for the laboratory 10

IQC in the broader context Management of the analytical process – more than IQC Make use of quality assurance • Knowledge of instrument • Unequivocal working instructions • Inventory control (reagent batches) • Maintenance • Qualified personnel, etc. Electronic QC Pre-analytics (sample) • Clot • Hemolysis, etc • Sample volume Analytics (instrument & reagent) • Pipette volumes • Wavelength • Light source output, sensor response • Temperature • Kinetic Post-analytics (verification, reports, etc) • Calibration verification • Calculation verification Make use of EQA • Company • National This information is particularly useful for troubleshooting (explained later in more detail). For more information about quality assurance, the reader is referred to the books cited below. References • Stewart CE, Koepke JA. Basic quality assurance practices for clinical laboratories. Philadelphia (USA): J. B. Lippincott Company, 1987. • Garfield FM. Quality assurance principles for analytical laboratories. AOAC International: 1994. • St John A. Critical care testing. Quality assurance. Mannheim: Roche Diagnostics, 2001. • Nilsen CL. Managing the analytical laboratory: plain and simple. Buffalo Grove (IL): Interpharm Press, 1996. Statistics & graphics for the laboratory 11

IQC in the broader context Purpose of IQC We have seen that IQC is an integral part of the analytical process and of the quality system. Therefore, IQC serves two purposes It is primarily useful for the laboratory itself • It monitors analytical performance and indicates when performance deteriorates. • It allows actions to be taken before quality specifications are exceeded It demonstrates to the outside (e. g. physicians, accreditation bodies, health authorities) • That analytical performance was adequate at the time patient results have been reported Summary – IQC in the broader context IQC is an integral part of the • analytical process • quality system IQC is useful for • the laboratory itself • demonstration of performance to the outside IQC is one mosaic stone of the whole quality management/quality assurance process Statistics & graphics for the laboratory 12

IQC and regulation Introduction IQC and regulation Manufacturers • IVD 98/79/EC Directive & CEN-documents • Manufacturer recommendations Laboratories • Belgian Guidelines (Koninklijk Besluit/ Praktijkrichtlijn) • Summary and conclusion Foreword Regulatory requirements for IQC exist for laboratories as well as for manufacturers. For manufacturers, usually, international rules apply (or at least, for the "big regions", such as the United States, Europe, Japan). For laboratories, usually, national rules apply, mostly associated with the rules for external quality assessment. Examples Manufacturer The key elements of the European rules will be presented. Laboratory The key elements of the Belgian rules will be presented. Statistics & graphics for the laboratory 13

IQC and regulation Manufacturers European IVD Directive 98/79/EC 8. Information supplied by the manufacturer (k) information appropriate to users on: - internal quality control including specific validation procedures The IVD directive states that - Manufacturers should give information about appropriate IQC procedures, but - The content of the information is not detailed. Corresponding CEN-standard# (EN 375: 2001, Information by the manufacturer) 5. 15 Internal quality control Suitable procedures for internal quality control shall be given including a means for the user to establish criteria for assessing the validity of the measurement procedure. The CEN standard states that -Suitable procedures for IQC shall be given, but -the content of the information is not detailed. Conclusion European requirements for manufacturers are vague: There should be recommendations for IQC, but the content is left over to the manufacturer. A closer look at manufacturers’ recommendations References • #For more information see: www. cenorm. be Statistics & graphics for the laboratory 14

IQC and regulation Manufacturers’ recommendations Sources of information Manufacturers’ recommendations for IQC can be found in the technical documentation (reagent data sheets; instrument manuals; dedicated brochures). Information from 5 different manufacturers was investigated. Test systems (manufacturer information 2001) Company Clinical Chemistry Immunochemistry Beckman Synchron LX 20 Access Bayer --Advia Centaur Abbott Aeroset Architect Ortho Vitros 700 Vitros Eci Roche Modular Elecsys 2010 General recommendations of manufacturers • 2 to 3 levels • Once per day • No interpretation rules Summary Manufacturers recommend MINIMUM IQC efforts Manufacturers give no recommendations for interpretation of IQC results Statistics & graphics for the laboratory 15

IQC and regulation Laboratories The Belgian guidelines Royal decree (Koninklijk Besluit; KB) Art. 34. § 1. The laboratory director has to organize IQC in all disciplines. § 3. IQC consists of several procedures which allow, before the release of patient results, to detect all significant within- or between-day variations. Art. 35. § 1. The frequency of control measurements has to be such that it can guarantee a clinically acceptable imprecision. This frequency depends on the characteristics of the method and/or the instrument. § 2. The control material, … must be stable within a defined period of time. Different aliquots of the same lot must be homogeneous. § 3. For each new lot, the mean and the SD have to be determined. … IQC materials may, at the same time, not be used as calibrator and control material. Practice guideline (Praktijkrichtlijn) 10. 7. Validation REQUIREMENT • A procedure for internal quality control (for every analyte) • nature of control samples • concentration, location in the run, number and frequency (concentration & location: additional to KB) • control rules used for start • control rules used for acceptance of a run IQC at least at 2 occasions • Control of one and the same test with different instruments • Panic values In essence, the Belgian guidelines: TELL: WHAT to do, but NOT: HOW to do it The Belgian guidelines in a nutshell: • "Suitable" IQC procedures • At least 2 IQC events: start/end • Determine mean & SD • Documentation References 1 Royal Decree from December 3 1999 regarding the authorization of clinical chemical laboratories. Moniteur Belge. December 30, 1999. 2 Implementation document: : Praktijkrichtlijn (Practice guideline): www. iph. fgov. be/Clinbiol/NL/index. htm Statistics & graphics for the laboratory 16

IQC and regulation Regulation Overall summary Regulation gives minimum rules for laboratory IQC • Follow manufacturer • Follow regulation General: minimum frequency, no rules Development of an IQC-policy: TASK of the LABORATORY Requirement: Knowledge of the statistical basis of the analytical process and of IQC. We look at the “analytical paradigm” and the “IQC paradigm” Statistics & graphics for the laboratory 17

Fundamentals of IQC • The analytical paradigm • The IQC paradigm The “analytical-paradigm“ We assume that: Analytical procedures give results (xi) that are independent from other results xi comes from a Gaussian distribution with a mean µ and a standard deviation s Note: An experimentally determined standard deviation (finite number of measurements) is denoted by the symbol s We assume that: Analytical procedures have periods of stable performance. The performance characteristics (mean, standard deviation) of the stable process are known from sufficiently frequent measurements under stable conditions. We assume that: In the course of time, analytical procedures tend to instability: • Measured means deviate from the "true" mean due to the occurrence of systematic error • Measured s is >"true" s due to increased random error The “IQC-paradigm“ We assume that: IQC can detect process deterioration (increased systematic or random error) at a sufficiently early stage • By repeated measurement of the sample • Investigation of the results by statistical methods Statistical methods (control rules) indicate, for example, whether • The actual mean deviates from the "true" mean • The actual s is > than the "true" s Summary We have to learn: Basic statistics • Particularly: the Gaussian distribution The metrological error concept • Systematic error, random error, total error, … Before that, we look at some Practical aspects of IQC … repeated measurement of the sample we look at the sample & other practical aspects of IQC. Statistics & graphics for the laboratory 18

IQC – practical aspects IQC materials • Nature • Target mean & SD/CV – Statistical excursion: uncertainty of mean & SD • Frequency and location Presentation of results Software & Documentation IQC materials Nature Materials for IQC should resemble the actually measured samples as far as possible. Matrix Serum analysis should apply materials with a serum-like matrix Urine analysis should apply materials with an urine-like matrix Whole blood should apply materials based on a whole blood matrix Note 1 Usually, a compromise has to be made between stability and “nativity”. Note 2 Most commercial IQC materials exhibit artificial matrix effects. Therefore, they usually cannot be used for the assessment of trueness. Concentration Analyte concentrations should be medically relevant (e. g. , be in the mid, the upper, and the lower part of the reference range, or near decision points; see www. westgard. com for medical decision levels). Be compatible with the test Be stable & homogeneous (bottle to bottle) Lyophilized samples are preferred for long-term stability. Problem associated with lyophilization: -Reconstitution accuracy (particular important for analytes that require tight control limits; e. g. , Na, Cl) Be available in large batches to allow their use over an extended period of time (e. g. , two years). Be purchased overlapping The new material should be tested for some time together with the old material in order to have continuous experience. This prevents difficulties in problem-solving when they just occur at the moment a new IQC batch is introduced. Statistics & graphics for the laboratory 19

IQC – practical aspects IQC materials Target mean Notes in advance • The target mean of a control material is particularly important. • Control materials should not be used as calibrators. Target means should: • Be test-specific • Have negligible uncertainty • Be provided with sufficient digits (adjusted to the precision of the method) General problems with digits Too few digits may give problems with target uncertainty (rounding problem), calculation of CV, violation of IQC rules & graphical display. Lactate simulation (n = 20): Mean = 1, 6 mmol/l; CV = 1, 8% Red squares: 2 digits; Blue diamonds: 3 digits Too few digits may give problems with target, IQC rules, calculation of CV, & graphic Target setting (mean) • May be done by the laboratory itself • May be part of the control sample Target part of the control sample: CAVE Method dependent assigned values are valid only for homogeneous test-systems (instrument/reagent/-calibrator from the same manufacturer). In case that the laboratory uses, for example, reagent and calibrator from different sources (= heterogeneous test), it might obtain a value that is different from the original one. In that case, the laboratory has to determine the target with its own test procedure. Target setting by the laboratory This is done, for example, by parallel analysis of a new batch of unassigned material over 21 days under stable operation conditions. A closer look Statistics & graphics for the laboratory 20

IQC – practical aspects Target setting (mean & SD) by the laboratory Usually recommended • Measure the control at least 21 times on separate occasions • Calculate the mean and SD Reject outliers (e. g. , values more than 3 SDs from the mean) and recalculate the mean and SD In that time, IQC should indicate no problems (be aware of the limitations of your IQC procedure). Care has to be taken that no bias is introduced. Reflect on the uncertainty of the estimates Calculation of imprecision with undetected shift Be aware that undetected shift (also drift) increases the magnitude of the “stable” CV and introduces a bias. Statistical considerations Reflect on the uncertainty of the estimates Note This part is based on the general principles of the Gaussian statistics (see chapter: Statistical basis of IQC). It is treated here because the statistical uncertainty of the target mean and CV of an IQC sample are often underestimated. Statistical excursion Mean & SD (CV) – Statistical considerations While 21 measurements seem to be quite a burden, one has to realize that the estimate of the mean may have an uncertainty that is relevant for IQC! The formula for the calculation of the CI of the mean is: Note: The term s/ n is called the standard error of the mean (SEM). Statistics & graphics for the laboratory 21

Sampling statistics – Confidence intervals Confidence interval/limits of the mean Relationship confidence interval/confidence limit The confidence interval (mean ± CI) spans from the lower to the higher confidence limit (CL): CI = - CL < mean < + CL • CI = ± t • s/ n • Lower CL = - t • s/ n • Higher CL = + t • s/ n The CI/CL of the mean depends • on the probability level, a • on the sort of tail (1 -/2 -tailed, also called 1 -sided, or 2 -sided) • on n (n, respectively) a, n, and the "sort of tail" determine the magnitude of t • the standard deviation s (also denoted SD in the book) The expression t/ n can be summarized by a factor k. Then, a CL can be calculated as k • SD. A table of k-factors is given below, as well as a graphical presentation. Relationship between confidenc limit and sample size: k-factors for the 2 -sided 95% confidence limit of a mean Statistics & graphics for the laboratory 22

Sampling statistics – Confidence intervals Confidence interval/limits of s (SD) The CI/CL of s (SD) depends • on the probability level, a (1 -sided, or 2 -sided, also called 1 -/2 -tailed) • on n (n, respectively) Calculation Lower CL = SD • [(n-1)/X 20. 025(n-1)]0. 5 Upper CL = SD • [(n-1)/X 20. 975(n-1)]0. 5 A table of factors is given below for the calculation of a lower/upper CL, as well as a graphical presentation. Relationship between confidenc limit and sample size: Factors for the 2 -sided 95% confidence limit of s (SD) Statistics & graphics for the laboratory 23

Statistical excursison Target setting (mean & SD) by the laboratory Statistical considerations Uncertainty of the mean and the SD with n = 21 measurements Estimates of the mean and the SD with 21 measurements, still, have an uncertainty that should not be neglected: • Uncertainty of the mean: ~0, 5 • SDexper • Upper limit of the SD: ~1, 44 • SDexper • Lower limit of the SD: ~0, 77 • SDexper IQC – Manufacturer Peer groups NOTE: Due to the uncertainty of the laboratory mean and CV (SD), consider the participation in: Manufacturer Peer groups Advantages Due to the high number of participants Sample • System specific target means and CVs • Low target uncertainty • Control of sample stability Laboratory • Better IQC-sample • Easier set-up of IQC (more reliable estimates of stable performance) • Easier troubleshooting by direct comparison with “peer” Summary IQC samples with target values IQC samples with system specific target values for mean & CV (SD), that have a negligible uncertainty, are the preferred sample for the laboratory due to -easy IQC set-up & -easier troubleshooting Statistics & graphics for the laboratory 24

Statistical excursison Target mean & SD – Exercise uncertainty Use Sampling. Statistics to get a feeling for the uncertainty of a mean & SD calculated with 21 measurements. Verification of target mean and CV Exercise with the Confidence calculator Is my experimental mean (CV) different from a target mean (CV)? Statistics & graphics for the laboratory 25

IQC – practical aspects Target CV (SD) of the laboratory Some remarks on its meaning for IQC purposes The target CV (= stable imprecision) is the cornerstone of IQC. It deserves special attention. All instabilities (random and systematic) are compared relative to the stable imprecision. Usually, one selects CVa, total for IQC purposes. CVa, total includes variations from • within-run (-day) • between-run (-day) • [calibration] “Target CV” and calibration intervals/tolerance Decide whether/or not to include in the “target CV” the betweencalibration variation • Monthly calibration: ? • Weekly calibration: Perhaps • Daily calibration: Yes, but be aware of the tolerance Know the calibration tolerance • Be aware of “shifts” after calibration • Indicator: the CVtotal/CVwithin ratio Know the lot variation of calibrators/reagents “Target CV” and the CVa, total/CVa, within ratio The CVa, total/CVa, within ratio: a general indicator for test stability CVa, total/CVa, within ratios 2. 5 may indicate the necessity of special attention to quality assurance (e. g. ; new lots). Also, in order to pick up unwanted variations, one may consider to use a CV lower than CVa, total for IQC. General remark It would be desirable to have in-depth information about instrument and test variation. Ideally, a GUM type variance analysis should be available for all important elements (e. g. , calibrator lots, reagent lots). Then, a distinction could be be made which variation one wants to pick up by IQC and which variation is accepted as inherent to the system. Statistics & graphics for the laboratory 26

IQC – practical aspects Checklist for the IQC-sample Nature • Correspond with patient sample • Compatible with the test Concentration: medically relevant • Number of levels Stability (liquid versus lyophilized) Lyophilized • Variation in fill content • Accuracy of reconstitution Target mean • Sufficient digits • System specific • Uncertainty Target SD/CV • Representative for system • Uncertainty Checklist “stable” imprecision Gather as much information as possible Instrument stability (general system robustness), e. g. : • Pipetting • Temperature • Photometer (wavelength/intensity/sensor) Test stability & reproducibility (individual test robustness) • Total/within-day CV (CVa, tot/CVa, w ratio) • Calibration tolerance (within/between lot), -function • Reagent (within/between lot) • Test robustness Statistics & graphics for the laboratory 27

IQC – practical aspects IQC materials – Frequency Minimum frequency Legislation often requires that at least 2 control samples should be measured per analytical run. In consequence, even when only one patient specimen is measured in a run, two controls have to be included. Note 1 The maximum length of a run is 24 hours, or a shift (for hematology, often a maximum run lenght of 8 hours is recommended). Note 2 Control materials should be introduced in regular analytical runs and not be treated separately. Desirable frequency No general rule can be given • The laboratory should adapt the number of control samples to the stability of the system and the control rule it applies. • Realistically, the frequency of control samples may be in the order of …% of the patient samples. • Consider “dummy” measurements before 1 st IQC (system warm-up). IQC materials – Location Random versus regular • Random placement gives a better estimate of the CV • Regular placement makes “administration easier” Small runs Consider to “bracket” the patient sample(s) with 2 controls. Medium runs Place 2 IQC samples before, in the mid, and after patient samples. Long runs Measure IQC samples several times (e. g. , 1 -2% of the samples, dependent on system stability); start and end with IQC. Note: In case that the system is recalibrated, the following measurements have to be considered as a new run. Statistics & graphics for the laboratory 28

IQC – practical aspects IQC materials – Location Block (bracket) versus continuous (see Figure) Consider one performs 4 IQC measurements. Those can be done in block (all at once), or continuously. • Block: more patient samples are between the IQC events, but statistics are stronger. • Continuously: fewer patient samples are between the IQC events, but statistics are weaker in the beginning. See also later Remedial actions “Block” IQC-events More samples between events, but stronger statistics Maximizes chance of assignable cause variability between subgroups Continuous IQC-events Fewer samples between events, weaker statistics in the beginning Maximizes chance of assignable cause variability within subgroups Checklist – Frequency and location • Minimum: 2 samples per run • Desirable: ~1 -2% of patient samples -Make a cost/benefit calculation • Frequency should be related to test stability: requires knowledge of instrument and test • Consider “dummy” measurements before 1 st IQC • Frequency may depend on the control rule • Block: Maximizes chance of assignable cause of variability between subgroups • Continuous: Maximizes chance of assignable cause of variability within subgroups Statistics & graphics for the laboratory 29

IQC – practical aspects Presentation of IQC data In the form of • Tables • Control charts • Histograms (summary data) Example of a control chart The example shows an IQC-chart with a 2. 5 s warning limit and a 3. 5 s action limit. The actual chart to be used in the laboratory will depend on the data system used. It is assumed that the laboratory has computerized IQC! Transforming a result into a point on the chart • Mean: 200 mg/d. L • Standard deviation: 4. 0 mg/d. L • Result for daily control run: 205 mg/d. L • Computation: (205 - 200)/4. 00 = +1. 25, i. e. the point representing the control is plotted at a distance of +1. 25 s from the mean. Alternative The measured values can be listed directly on a chart. However, the y-axis (and the s-limits) must then be constructed with the individual data of a specific control material. Statistics & graphics for the laboratory 30

IQC – practical aspects Documentation of IQC data Some general remarks • IQC data that are stored on electronic media should be printed regularly (e. g. , weekly). • IQC data have to be archieved for an adequate peroid of time, respecting the respective regulations. Content of documents • Identification of the laboratory • Instrument (workplace) • Test • Date • Signature of the operator • Identification of the control material (lot & expiry) • Target & Limits (rules) • Data (individual & summary, e. g. , monthly mean & CV) • Accept/reject boxes • Corrective actions taken • New IQC bottle, reagent (lot & expiry), calibrator (lot & expiry) • Instrument calibration or major maintenance • Instrument function checks (temperature, wavelength, etc. ) Note: Documentation must be able to identify which IQC-data belong to which patient results. Statistics & graphics for the laboratory 31

IQC – practical aspects IQC software Software Checklist Administrative capabilities -Easy set-up and modification -Online (real time) connection with LIS -Full sample & IQC traceability -“Accreditation-conform” documentation -Up-to-date data safety “IQC”-capabilities -Transparent & efficient data presentation -Great variety of rules -Rule selection logic -Automatic release -Automatic “flags” and remedial action (but with open decision logic) Examples • QC Today (IL) • EZ Runs (Westgard) • Unity (Bio. Rad) • Computrol on Line (Sigma) • [System-specific] Statistics & graphics for the laboratory 32

Statistical basis of IQC Introduction Statistics • Basic calculations • Gaussian distribution (standard and cumulated) • Gaussian distribution and %-age of observations • Statistical probabilities • “Rare events” • Summary outside probabilities Control rules • From outside probabilities to control rules • Control rules – An example • Control rules – basic monitoring principles • Selection of control rules: Fundamental problems • Power function graphs • Construction of the power function for 13 s and SE • Construction of the power function for 12 s and RE • Power of control rules and s-limits • Power of control rules and n • Comparison of the power of control rules • Control rules: The problem of false rejection (Pfr) • Circumventing the increase in Pfr with n: control rules with variable limits • The ideal control rule • Magnitude of errors detected by IQC Conclusions from statistical considerations Statistics & graphics for the laboratory 33

Statistical basis of IQC Introduction The chapter addresses: Basic statistical knowledge required for understanding the selection of IQCrules • Main characteristics of the Gaussian (or Normal) distribution, • Power functions for IQC and their importance for the selection of quality control rules, • The problem of false rejections of stable analytical runs. Statistical basis of IQC • Basic calculations • Gaussian distribution (standard and cumulated) • Gaussian distribution and %-age of observations • Statistical probabilities • “Rare events” • Summary outside probabilities Statistics & graphics for the laboratory 34

Statistical basis of IQC Basic calculations REMEMBER • Analytical procedures give results (xi) that are independent from other results • xi comes from a Gaussian distribution with a mean µ and a standard deviation s Calculation of • Mean • Standard deviation • Coefficient of variation (CV) >See: Basic statistics Remark The standard deviation, often, increases with increasing concentration of the analyte (see Figure). the coefficient of variation (CV) is often more convenient for the description of random error (imprecision). Gaussian distribution (standard and cumulated) A Gaussian (= Normal) distribution is characterized by • its mean and • standard deviation (s) To understand the basis of statistical IQC, it is important to memorize the key characteristics of the Gaussian function, in particular, the expected location of single values that constitute the distribution. We look at the percentage of observations that we expect in certain regions of the distribution. REMEMBER • xi comes from a Gaussian distribution with a mean µ and a standard deviation s • If we know the stable mean and SD of an analytical process • We can predict the location of future measurement with a certain probability Statistics & graphics for the laboratory 35

Gaussian distribution Graphical presentation of the Gaussian distribution The Gaussian distribution can be presented • In the normal way: "Bell-shaped" (similar to a histogram) • Cumulated: "S-shaped" • Cumulated & linearized = Normal probability plot EXCEL® template from P Hyltoft Petersen (note: not available in EXCEL ® itself) Gaussian. Distribution (Worksheets "Gauss. Bell"; "Gauss. Cumul") These worksheets use the EXCEL NORMDIST function. The "Print Screens" guide you through their application. The graphs will appear automatically. Statistics & graphics for the laboratory 36

Gaussian distributions – Probabilities IMPORTANT NOTE When data are Gaussian distributed, we can predict the frequencies (or probabilities) of their occurrence within or outside certain distances (s, or z-values) from the mean (see also Figures above). These probabilities are used in parametric statistical calculations. They are listed in tables, but they also can be calculated with EXCEL®. Of particular importance are probabilities that are used in statistical tests (95%, 99% probabilities). 2 -sided and 1 -sided probabilities Statistics distinguish probabilities in 2 -sided & 1 -sided • 2 -sided probabilities: question is A different from B? • 1 -sided probabilities: question(s) is A > B (A < B)? Of practical importance are probabilities "Inside" & "Outside" • Outside probabilities, for example, are important in internal quality control. Statistics & graphics for the laboratory 37

Gaussian distributions – Probabilities at selected s (z) values 1. 65 s INSIDE 1 -sided 95% 2 -sided [90 %] OUTSIDE 1 -sided 5% 2 -sided [10 %] 1. 96 s 97. 5% 95% 2. 5% 5% 2. 0 s 97. 7% 95. 5% 2. 3% 4. 5% 2. 33 s 99% 98% 1. 0% 2. 58 s 99. 5% 99% 0. 5% 1. 0% 3. 0 s 99. 87% 99. 7% 0. 13% 0. 3% 1 -sided probabilities can be expected in the presence of considerable systematic error. At SE RE (SE/RE 1) the probabilities become practically 1 -sided (see Figure) Statistics & graphics for the laboratory 38

Statistical basis of IQC Statistics “Rare events” and outside probabilities We have seen that very few results can be found in certain regions of the Gaussian distribution. For example, it is highly unlikely to find results beyond a distance of 4 s from the mean. In that connection, a convention has been made about what we consider “unlikely” (“rare events”). Convention Values outside ± 1. 96 s (2 -sided view) are deemed “rare events”, they occur in 5% of the cases, only. Values that are found outside ± 1. 96 s, are not by chance. It is assumed a non-statistical reason (e. g. , systematic error) causes values to be found outside ± 1. 96 s. Monitoring “the outsides”: One IQC-principle This gives an indication “whether something happened”. REMARK on “rare events” We called observations that happen in <5% of the cases rare events. At the same time, when we observe them in daily practice, we suspect that their occurrence has non-statistical reasons. BEWARE: Our judgement may be wrong in 5%, 1%, 0. 3%, etc. of the cases! See later: probabilities of false rejection! Statistics & graphics for the laboratory 39

Statistical basis of IQC Checklist – Basic statistics Calculations • Mean • SD • CV Gaussian (normal) distribution • Graphic of the usual & the cumulated distribution • Probabilities within certain distances (s) of the mean • Probabilities outside certain distances (s) of the mean • 1 -sided and 2 -sided probabilities • Important values for s • Convention on “rare events” • Possibility of wrong decisions • The 1 st IQC principle: monitoring the outsides Exercises with EXCEL® Tools > Add-ins > Analysis Tool. Pak & -VBA Installs Data Analysis Tools > Data Analysis > Random number generation Investigate them with > Descriptive Statistics Normal Simulator Statistics & graphics for the laboratory 40

Statistical basis of IQC Control rules From outside probabilities to control rules REMEMBER Monitoring “the outsides”: One IQC-principle This principle leads us to a simple family of IQC-rules. Namely, based on a known Gaussian distribution, we monitor in practice whether we observe an IQC result that falls, for example, out of the ± 3 s limits of that population. In case that happens (note: the probability is less than 0. 3%), we assume that the process became unstable. the 13 s-rule The process is out-of-control when 1 IQC result is outside a distance of ± 3 s from the «true» mean. It is a member of the family : nz • s • n: number of observations • z: certain number of standard deviations of the Gaussian distribution (= standard normal deviate) • s: stable (“true”) standard deviation Basic monitoring principles of IQC-rules • “The outsides” (for example 13 s) • A trend • The location towards the mean (above or below) • A range (difference between results) • The mean (several results) • The imprecision, or variance (several results) Control rules Selection of control rules – Fundamental problems • There are many different control rules • Different control rules have different power for error detection • Different control rules have different probabilities of false rejections The selection of a particular control rule is always a trade-off between error detection and false rejection! Statistics & graphics for the laboratory 41

Statistical basis of IQC Power functions Power function graphs (#) – are useful tools for the selection of control rules The power function graph • indicates the power of a rule for error detection (Ped) • and the probability of false rejection (Pfr). NOTE Separate graphs must be constructed for systematic and random error. For details of the error concept used here, see: Metrological basis of IQC #Hyltoft Petersen P, Ricós C, Stöckl D, Libeer J-C, Baadenhuijsen H, Fraser CG, Thienpont LM. Proposed guidelines for the internal quality control of analytical results in the medical laboratory. Eur J Clin Chem Clin Biochem 1996; 34: 983 -99. The power function graph • The x-axis plots the size of error in multiples of the analytical standard deviation • The y-axis plots the probability of error detection (Ped) (rejecting a run) against the size of error on the x-axis. • The probability of false rejection (Pfr) can be read at the point DRE = 1 or DSE = 0. Statistics & graphics for the laboratory 42

Statistical basis of IQC Power functions Construction of the power function graph for the • 13 s-rule and systematic error (SE) Start with the error-free situation of the stable process (DSE = 0; naturally, the intrinsic RE is present). The first point is observed at DSE = 0; Ped = 0. 3%. However, this corresponds to the false rejection of the rule. Even under stable conditions, 0. 3% of the results will be outside the ± 3 slimit: Pfr = 0. 3%. Example point: Introduce DSE in the direction of +3 s • Transform DSE in fractions of s, e. g. , 2 • Read Ped as the cumulated %-age from - to the respective s, calculated as – 3 s (= limit) +2 s (= DSE): read at s = -1: Ped = 15. 9% Plot the point (2/15. 9) A shift of DSE = 2 will be detected in 16% of the cases, only! Statistics & graphics for the laboratory 43

Statistical basis of IQC Power functions Construction of the power function graph for the • 12 s-rule and random error (RE) Start with the error-free situation of the stable process (DRE = 1; intrinsic random error). The first point is observed at DRE = 1; Ped = 5%. However, this corresponds to the false rejection of the rule. Even under stable conditions, 5% of the results will be outside the ± 2 s-limit: Pfr = 5%. Example point: Introduce DRE • Transform DRE in fractions of s, e. g. , 2 • Read Ped as 2 • cumulated %-age from - to – 2 s (= DRE) from a Gauss function with SD = 2: Ped = 32% Plot the point (2/32) A doubling of RE will be detected in 32% of the cases, only! Statistics & graphics for the laboratory 44

Statistical basis of IQC Power of control rules and s-limit The Figure presents different power function graphs for the family of the 1 z • s control rules. Following observations can be made: • The power of error detection (Ped) increases with decrease of the s-limit • But also • The Pfr rate increases with the decrease of the s-limit Simulation exercise • Shewart tutorial from marquis-soft Statistics & graphics for the laboratory 45

Statistical basis of IQC Power of control rules & n Power function graphs for the n 3 s rule (various numbers of measurement) for Systematic error & Random error Notes - Power increases with n - Pfr increases with n Usually, - Power for detection: RE < SE Comparison of the power of control rules The Figure compares power functions for mean rules with “simple” individual rules (e. g. , the 13 s rule) (#). NOTE • Mean rules, usually, are more powerful than quality control rules based on individual values. #Linnet K. Mean and variance rules are more powerful or selective than quality control rules based on individual values. Eur J Clin Chem Clin Biochem 1991; 29: 417 -24. Statistics & graphics for the laboratory 46

Statistical basis of IQC Power of control rules Comparison of the power of control rules The Figure compares power functions for variance rules with “simple” individual rules (e. g. , the 13 s rule) (#). NOTE • Variance rules, usually, are more selective than quality control rules based on individual values. #Linnet K. Mean and variance rules are more powerful or selective than quality control rules based on individual values. Eur J Clin Chem Clin Biochem 1991; 29: 417 -24. Note Generally one strives for a Ped of …% Statistics & graphics for the laboratory 47

Statistical basis of IQC Pfr of control rules The problem of false rejection (P fr) The Figure • compares. Pfr values for control rules with low (14 s) and high (12 s) Ped. Observation Powerful control rules often have a high Pfr Figure • comparison Pfr values of the 12 s, 12. 5 s and 13 s-rule with measurements ranging from 1 – 20. Observations Pfr increases dramatically with the number of measurements Control rules with a limit >3 s would be desirable to keep the false alarms low and the productivity high. Unfortunately, such rules have a low power for error detection (Ped). Generally: do not consider rules with Pfr > …% Statistics & graphics for the laboratory 48

Statistical basis of IQC Circumventing the increase in Pfr with n The increase of Pfr with can be circumvented by control rules with variable limit, but fixed Pfr. In the Westgard notation, they are designated as X 0. 01: mean rule for SE, Pfr 1% (independent of n). R 0. 01: range rule for RE, Pfr 1% (independent of n). Usually the rules for SE and RE are applied together: X 0. 01/R 0. 01. Power function of the X 0. 01 rule (n=4, Pfr = 1%) For SE: Quite powerful at higher n Statistics & graphics for the laboratory 49

Statistical basis of IQC The ideal control rule would indicate process deterioration only shortly before one exceeds the critical error. Its Pfr is 0 and would be kept shortly before the critical error. • Then, it would jump to Ped of 100%: extreme steepness. A near-ideal control rule The mean-rule (n = 6) with a Pfr of the 3 s-rule (0, 3%) and moved by 0, 5 s is a near ideal control rule. It • Keeps Pfr until 0, 5 s • Has a good steepness: reaches Ped 90% at D SE = 2. Its major disadvantage is the high number of measurements. Statistics & graphics for the laboratory 50

Statistical basis of IQC Magnitude of errors detected by IQC Comparison of the different IQC-power curves learns, realistic errors to be detected by IQC are § D SE = > … § D RE = >… • REstable D RE Statistics & graphics for the laboratory 51

Statistical basis of IQC Conclusions from statistical considerations • Pfr increases dramatically when multiple measurements are performed, or when many analytes are controlled, for example, on a multichannel analyzer. • But, also Ped increases with the number of measurements. • Control rules with a limit >3 s would be desirable to keep the false alarms low and the productivity of the process high: low Pfr. • However, rules with a low Pfr, often have a low power for error detection (Ped). • Multi-rules or mean and variance rules are superior to the single rules. • Control rules for SE are more powerful than those for RE. Conclusion: The selection of a control rule always has to compromise between high power for error detection and low probability of false rejection. Note: Because of the complexity to deal with both random and systematic errors at the same time, separate IQC procedures have to be used for detection of systematic and random error (or multirules). Checklist – Power of control rules • Ped should be 90% • Realistic errors to be detected by IQC are • D SE = >2 • D RE = >3 • REstable • Ped AND Pfr increase with • lower s-limits (2 s > 3 s) • n (note, some rules are connected to the number of materials: multiples of 2, 3 with 2 or 3 materials) • Generally, do not consider rules with Pfr >1% • Ped increases by combination of rules • Ped of mean and variance rules > than single or combined rules • Ped for SE > RE • Pfr at non-zero can be minimized by movement of the power curve • The power curve should have a good steepness Statistics & graphics for the laboratory 52

Statistical basis of IQC simulation tools - www. westgard. com/qctools. html -STT Consulting -www. marquis-soft. com Exercises with the software tools from Marquis We set ARL = 370 (= 3 s limit) ARL & probability: ARL = 1/P Outside 3 s: 0. 27%, P = 0. 0027; ARL = 1/0. 0027 = 370 1. Set SE = 0. 5 s, let run ………………………………………………………………. 2. Set SE = 1. 5 s, let run ………………………………………………………………. 3. Set SE = 1. 5 s, run until red, repeat several times ………………………………………………………………. Statistics & graphics for the laboratory 53