Method Validation of Analytical Procedures
|
|
- Emma Thornton
- 5 years ago
- Views:
Transcription
1 32 Revew Artcle Method Valdaton of Analytcal Procedures Prakash Chanda Gupta QC Executve, Natonal Healthcare Pvt. Ltd., Nepal ABSTRACT After the development of an analytcal procedure, t s must mportant to assure that the procedure wll consstently produce the ntended a precse result wth hgh degree of accuracy. The method should gve a specfc result that may not be affected by external matters. Ths creates a requrement to valdate the analytcal procedures. The valdaton procedures conssts of some characterstcs parameters that makes the method acceptable wth addton of statstcal tools. Keywords: Valdaton, Analytcal Procedure, Accuracy, Precson, Robustness INTRODUCTION Valdaton of an analytcal procedure s the process by whch t s establshed, by laboratory studes, that the performance characterstcs of the procedure meet the requrements for the ntended analytcal applcatons. [1] Method valdaton provdes an assurance of relablty durng normal use, and s sometme referred to as the process for provdng documented evdence that the method does what t s ntended to do. The man objectve of the valdaton s to demonstrate that the analytcal method s sutable for ts ntended purpose, s accurate, specfc and precse over the specfed range that an analyte wll be analyzed. Analytcal Method Valdaton s to be performed for new analyss methods or for current methods when any changes are made to the procedure, composton of the drug product and synthess of the drugs substances. Common types of analytcal procedure that can be valdated [2] Identfcaton tests; Quanttatve tests for mpurtes content; Lmt tests for the control of mpurtes; Quanttatve tests of the actve moety n samples of drug substance or drug product or other selected component(s) n the drug product. Typcal valdaton characterstcs whch should be consdered are lsted below: [1] Accuracy Precson Specfcty Detecton Lmt Quanttaton Lmt Lnearty Range Robustness The valdaton characterstcs are to be evaluated on the bass of the type of analytcal procedures. Table 1: Evaluaton of Valdaton Characterstcs Characterstcs Type of Analytcal Procedures Identfcaton Impurtes Quanttatve Tests Quanttatve Lmt How to cte ths artcle: PC Gupta; Method Valdaton of Analytcal Procedures; PharmaTutor; 2015; 3(1); 32-39
2 33 Accuracy Not evaluated Evaluated Not evaluated Evaluated Precson Not evaluated Evaluated Not evaluated Evaluated Specfcty Evaluated Evaluated Evaluated Evaluated Detecton Lmt Not evaluated Not evaluated Evaluated Not evaluated Quanttaton Lmt Not evaluated Evaluated Not evaluated Not evaluated Lnearty Not evaluated Evaluated Not evaluated Evaluated Range Not evaluated Evaluated Not evaluated Evaluated METHODS AND TERMINOLOGY Accuracy The accuracy of an analytcal method s the closeness of the test results obtaned by that method to the true value. [3] Ths s sometmes termed trueness. It s recommended that accuracy should be determned usng a mnmum of nne determnatons over a mnmum of the three concentraton levels, coverng the specfed range (3 concentratons/3 replcates each of total analytcal procedures). [4] It s measured as the percent of analyte recovered by assay. The recovery can be determned by the equaton: Recovery = Analytcal Result x 100% True Value The recovery should be n the range of Control lmt. The followng method can be appled for calculatng the Upper Control Lmt (UCL) and Lower Control Lmt (LCL). The method nvolves the movng range, whch s defned as the absolute dfference between two consecutve measurements ( x x 1 ). These movng range are averaged ( MR ) and used n the followng formulae: [5] UCL and MR x 3 d 2 LCL MR x 3 d 2 Where, x s the ndvdual analytcal result, x s the sample mean, and 2 d s a constant commonly used for ths type of chart and s based on the number of observatons assocated wth the movng range calculaton. Where n = 2 (two consecutve measurements), as here, d 2 = Precson The precson of the analytcal method descrbes the closeness of repeated ndvdual measures of analyte. [6] The precson of an analytcal procedure s usually expressed as the standard devaton or relatve standard devaton (coeffcent of varaton) of a seres of measurements. It s ndcated by Relatve Standard Devaton, RSD, whch s determned by the equaton: 1/ 2 n 2 ( ) 100 x x 1 (%) n RSD x n 1 Where x s an ndvdual measurement n a set of n measurement and x s the arthmetc mean of the set. Generally, the RSD should not be more than 2%.
3 34 Repeatablty Repeatablty refers to the use of the analytcal procedure wthn a laboratory over a short perod of tme usng the same analyst wth the same equpment. [3] Repeatablty should be assessed usng a mnmum of nne determnatons coverng the specfed range for the procedure (.e., three concentratons and three replcates of each concentraton or usng a mnmum of sx determnatons at 100% of the test concentraton). [4] assay and mpurty tests by chromatographc procedures, specfcty can be demonstrated by the resoluton of the two components whch elute closest to each other. [9] It s not always possble to demonstrate that an analytcal procedure s specfc for a partcular analyte (complete dscrmnaton). In ths case a combnaton of two or more analytcal procedures s recommended to acheve the necessary level of dscrmnaton. Reproducblty Reproducblty expresses the precson between laboratores (collaboratve studes, usually appled to standardsaton of methodology). Reproducblty s usually demonstrated by means of an nter-laboratory tral. [7] Intermedate Precson Intermedate precson s the results from wthn lab varatons due to random events such as dfferent days, dfferent analysts, dfferent equpment, etc. [8] The standard devaton, relatve standard devaton (coeffcent of varaton) and confdence nterval should be reported for each type of precson nvestgated. Specfcty Specfcty s the ablty to measure accurately and specfcally the analyte of nterest n the presence of other components that may be expected to be present n the sample matrx such as mpurtes, degradaton products and matrx components. It must be demonstrated that the analytcal method s unaffected by the presence of spked materals (mpurtes and/or excpents). In case of dentfcaton tests, the method should be able to dscrmnate between compounds of closely related structures whch are lkely to be present. Smlarly, n case of Lnearty Lnearty s the ablty of the method to elct test results that are drectly, or by a welldefned mathematcal transformaton, proportonal to analyte concentraton wthn a gven range. [10] It should be establshed ntally by vsual examnaton of a plot of sgnals as a functon of analyte concentraton of content. If there appears to be a lnear relatonshp, test results should be establshed by approprate statstcal methods. Data from the regresson lne provde mathematcal estmates of the degree of lnearty. The correlaton coeffcent, y-ntercept, and the slope of the regresson lne should be submtted. It s recommended to have a mnmum of fve concentraton levels, along wth certan mnmum specfed ranges. For assay, the mnmum specfed range s from 80% -120% of the target concentraton. [11] Regresson lne, y ax b Where, a s the slope of regresson lne and b s the y - ntercept. Here, x may represent analyte concentraton and y may represent the sgnal responses. Correlaton Coeffcent, r n ( x x)( y x) ( x y)( y y) 1/ 2
4 35 Where x s an ndvdual measurement n a set of n measurement and x s the arthmetc mean y of the set, s an ndvdual measurement n a set of n measurement and y s the arthmetc mean of the set. DETECTION LIMIT AND QUANTITATION LIMIT The Detecton Lmt s defned as the lowest concentraton of an analyte n a sample that can be detected, not quantfed. The Quanttaton Lmt s the lowest concentraton of an analyte n a sample that can be determned wth acceptable precson and accuracy under the stated operatonal condtons of the analytcal procedures. [12] Some of the approaches to determne the Detecton Lmt and Quanttaton Lmt are: [13] the determnaton of Detecton Lmt and relably quantfed for the determnaton of Quanttaton Lmt. A sgnal-to-nose rato between 3 or 2:1 s generally consdered acceptable for estmatng the detecton lmt and A typcal sgnal-to-nose rato s 10:1 s consdered for establshng the quanttaton lmt. c. Standard Devaton of the response and the Slope. The Detecton Lmt may be expressed as: DL 3.3 / s The Quanttaton Lmt may be expressed as: QL 10 / s Where, s standard devaton of the response and s s slope of the lnearty curve. a. Vsual Evaluaton Vsual evaluaton may be used for nonnstrumental methods. For non-nstrumental procedures, the detecton lmt s generally determned by the analyss of samples wth known concentratons of analyte and by establshng the mnmum level at whch the analyte can be relably detected. And the quanttaton lmt s generally determned by the analyss of samples wth known concentratons of analyte and by establshng the mnmum level at whch the analyte can be determned wth acceptable accuracy and precson. Vsual Evaluaton approach may also be used wth nstrumental methods. b. Sgnal to Nose Ths approach can only be appled to analytcal procedures that exhbt baselne nose. Determnaton of the sgnal-to-nose rato s performed by comparng measured sgnals from samples wth known low concentratons of analyte wth those of blank samples and establshng the mnmum concentraton at whch the analyte can be relably detected for The method used for determnng the detecton lmt and the quanttaton lmt should be presented. If DL and QL are determned based on vsual evaluaton or based on sgnal to nose rato, the presentaton of the relevant chromatograms s consdered acceptable for justfcaton. Range The range of an analytcal procedure s the nterval between the upper and lower levels of analyte (ncludng these levels) that have been demonstrated to be determned wth a sutable level of precson, accuracy, and lnearty usng the procedure as wrtten. The range s normally expressed n the same unts as test results (e.g., percent) obtaned by the analytcal procedure. [10] The followng mnmum specfed ranges should be consdered: [14] For Assay of a Drug Substance (or a drug product) the range should be from 80% to 120% of the test concentraton.
5 36 For Determnaton of an Impurty: from 50% to 120% of the acceptance crteron. For Content Unformty: a mnmum of 70% to 130% of the test concentraton, unless a wder or more approprate range based on the nature of the dosage form (e.g., metered-dose nhalers) s justfed. For Dssoluton Testng: ±20% over the specfed range (e.g., f the acceptance crtera for a controlledrelease product cover a regon from 20%, after 1 hour, and up to 90%, after 24 hours, the valdated range would be 0% to 110% of the label clam). Robustness The robustness of an analytcal procedure s a measure of ts capacty to reman unaffected by small but delberate varatons n procedural parameters lsted n the procedure documentaton and provdes and ndcaton of ts sutablty durng normal usage. Robustness may be determned durng development of the analytcal procedure. [15] If measurements are susceptble to varatons n analytcal condtons, the analytcal condtons should be sutably controlled or a precautonary statement should be ncluded n the procedure. One consequence of the evaluaton of robustness should be that a seres of system sutablty parameters (e.g., resoluton test) s establshed to ensure that the valdty of the analytcal procedure s mantaned whenever used. [16] Examples of typcal varatons are: Stablty of analytcal solutons; Extracton tme. Dfferent columns (dfferent lots and/or supplers); Temperature; Flow rate. In the case of gas-chromatography, examples of typcal varatons are: Dfferent columns (dfferent lots and/or supplers); Temperature; Flow rate. System Sutablty Testng System sutablty testng s an ntegral part of many analytcal procedures. The tests are based on the concept that the equpment, electroncs, analytcal operatons and samples to be analyzed consttute an ntegral system that can be evaluated as such. System sutablty test parameters to be establshed for a partcular procedure depend on the type of procedure beng valdated. They are especally mportant n the case of chromatographc procedures. [16] INTERPRETATION AND TREATMENT OF VARIATION OF ANALYTICAL DATA Analytcal procedures are developed and valdated to ensure the qualty of drug products. The analytcal data can be treated and nterpreted for the scentfc acceptance. The statstcal tools that may be helpful n the nterpretaton of analytcal data are descrbed. Many descrptve statstcs, such as the mean and standard devaton, are n common use. Other statstcal tools, such as calculatng confdence nterval, outler tests, etc. can be performed usng several dfferent, scentfcally vald approaches. In the case of lqud chromatography, examples of typcal varatons are: Influence of varatons of ph n a moble phase; Influence of varatons n moble phase composton; 1. Confdence Interval: A confdence nterval for the mean may be consdered n the nterpretaton of data. Such ntervals are calculated from several data ponts usng the sample mean (x) and sample
6 37 standard devaton (s) accordng to the be performed on the same sample, f possble, formula: [17] or on a new sample. [17] x t s x t / 2, n1, / 2, n1 n s n n whch t / 2, n1 s a statstcal number dependent upon the sample sze (n), the number of degrees of freedom ( n 1), and the desred confdence level ( 1 ). Its values are obtaned from publshed tables of the Student t-dstrbuton. A confdence nterval provdes lmts around the expermentally determned value of the mean wthn whch the true value les wth a gven value of probablty, usually 95%. 2. Outlyng Results: Occasonally, observed analytcal results are very dfferent from those expected. Aberrant, anomalous, contamnated, dscordant, spurous, suspcous or wld observatons; and flyers, rogues, and mavercks are properly called outlyng results. Lke all laboratory results, these outlers must be documented, nterpreted, and managed. Such results may be accurate measurements of the entty beng measured, but are very dfferent from what s expected. Alternatvely, due to an error n the analytcal system, the results may not be typcal, even though the entty beng measured s typcal. When an outlyng result s obtaned, systematc laboratory and process nvestgatons of the result are conducted to determne f an assgnable cause for the result can be establshed. Factors to be consdered when nvestgatng an outlyng result nclude but are not lmted to human error, nstrumentaton error, calculaton error, and product or component defcency. If an assgnable cause that s not related to a product or component defcency can be dentfed, then retestng may When used approprately, outler tests are valuable tools for pharmaceutcal laboratores. Several tests exst for detectng outlers such as the Extreme Studentzed Devate (ESD) Test, Dxon's Test, and Hampel's Rule. Choosng the approprate outler test wll depend on the sample sze and dstrbutonal assumptons. Many of these tests (e.g., the ESD Test) requre the assumpton that the data generated by the laboratory on the test results can be thought of as a random sample from a populaton that s normally dstrbuted, possbly after transformaton. 3. Generalzed Extreme Studentzed Devate (ESD) Test Ths s a modfed verson of the ESD Test that allows for testng up to a prevously specfed number, r, of outlers from a normally dstrbuted populaton. Let r equal 1, and n equal 10. Normalze each result by subtractng the mean from each value and dvdng ths dfference by the standard devaton. Take the absolute value of these results, select the maxmum value R, and compare t to a 1 prevously specfed tabled crtcal value 1 based on the selected sgnfcance level (for example, 5%). If the the maxmum value s larger than the tabled crtcal value, t s dentfed as beng nconsstent wth the remanng data. If the maxmum value s less than the tabled crtcal value, there s not an outler. Sources for -values are ncluded n many statstcal textbooks. CONCLUSION Method Valdaton s an mportant analytcal tool to ensure the accuracy and specfcty of
7 38 the analytcal procedures wth a precse agreement. Ths process determnes the detecton and quanttaton lmt for the estmaton of drug components. The valdaton procedures are performed along wth the system sutablty. Some statstcal tools are also used to nterpret the analytcal results of the valdaton characterstcs. The valdaton of analytcal methods not only requres the performance of characterstcs parameter but also the statstcal treatments of the analytcal data. The acceptance of the varaton of the analytcal data s determned by these treatments. REFERENCES 1. Valdaton of Compendal Procedures <1225>, The Unted States Pharmacopea, 32th Rev., and The Natonal Formulary, 27th Rev., Rockvlle, MD: The Unted States Pharmacopeal Conventon Inc., 2009; I: Valdaton of Analytcal Procedures: Text and Methodology Q2(R1), ICH Harmonsed Trpartte Pharmaceutcals For Human Use, 2005; Valdaton of Compendal Procedures <1225>, The Unted States Pharmacopea, 32th Rev., and The Natonal Formulary, 27th Rev., Rockvlle, MD: The Unted States Pharmacopeal Conventon Inc., 2009; I: Valdaton of Analytcal Procedures: Text and Methodology Q2(R1), ICH Harmonsed Trpartte Pharmaceutcals For Human Use, 2005; Analytcal Data-Interpretaton and Treatment <1010>, The Unted States Pharmacopea, 32th Rev., and The Natonal Formulary, 27th Rev., Rockvlle, MD: The Unted States Pharmacopeal Conventon Inc., 2009; I: Gudelne on boanalytcal method valdaton, European Medcnes Agency, London, UK, 2011; I: Valdaton of Analytcal Procedures SC III F, Brtsh Pharmacopea, Brtsh Pharmacopea Commsson, Valdaton of Analytcal Procedures: Text and Methodology Q2(R1), ICH Harmonsed Trpartte Pharmaceutcals For Human Use, 2005; Valdaton of Analytcal Procedures: Text and Methodology Q2(R1), ICH Harmonsed Trpartte Pharmaceutcals For Human Use, 2005; Valdaton of Compendal Procedures <1225>, The Unted States Pharmacopea, 32th Rev., and The Natonal Formulary, 27th Rev., Rockvlle, MD: The Unted States Pharmacopeal Conventon Inc., 2009; I: Valdaton of Analytcal Procedures: Text and Methodology Q2(R1), ICH Harmonsed Trpartte Pharmaceutcals For Human Use, 2005; Valdaton of Compendal Procedures <1225>, The Unted States Pharmacopea, 32th Rev., and The Natonal Formulary, 27th Rev., Rockvlle, MD: The Unted States Pharmacopeal Conventon Inc., 2009; I: Valdaton of Analytcal Procedures: Text and Methodology Q2(R1), ICH Harmonsed Trpartte
8 39 Pharmaceutcals For Human Use, 2005; Valdaton of Analytcal Procedures: Text and Methodology Q2(R1), ICH Harmonsed Trpartte Pharmaceutcals For Human Use, 2005; Valdaton of Compendal Procedures <1225>, The Unted States Pharmacopea, 32th Rev., and The Natonal Formulary, 27th Rev., Rockvlle, MD: The Unted States Pharmacopeal Conventon Inc., 2009; I: Valdaton of Analytcal Procedures: Text and Methodology Q2(R1), ICH Harmonsed Trpartte Pharmaceutcals For Human Use, 2005; Analytcal Data-Interpretaton and Treatment <1010>, The Unted States Pharmacopea, 32th Rev., and The Natonal Formulary, 27th Rev., Rockvlle, MD: The Unted States Pharmacopeal Conventon Inc., 2009; I: 399.
Chapter 13: Multiple Regression
Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to
More informationStatistics Chapter 4
Statstcs Chapter 4 "There are three knds of les: les, damned les, and statstcs." Benjamn Dsrael, 1895 (Brtsh statesman) Gaussan Dstrbuton, 4-1 If a measurement s repeated many tmes a statstcal treatment
More informationStatistics for Economics & Business
Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable
More informationDepartment of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6
Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationComparison of Regression Lines
STATGRAPHICS Rev. 9/13/2013 Comparson of Regresson Lnes Summary... 1 Data Input... 3 Analyss Summary... 4 Plot of Ftted Model... 6 Condtonal Sums of Squares... 6 Analyss Optons... 7 Forecasts... 8 Confdence
More information2016 Wiley. Study Session 2: Ethical and Professional Standards Application
6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton
More informationAnswers Problem Set 2 Chem 314A Williamsen Spring 2000
Answers Problem Set Chem 314A Wllamsen Sprng 000 1) Gve me the followng crtcal values from the statstcal tables. a) z-statstc,-sded test, 99.7% confdence lmt ±3 b) t-statstc (Case I), 1-sded test, 95%
More informationCorrelation and Regression. Correlation 9.1. Correlation. Chapter 9
Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,
More informationECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More informationCopyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for P Charts. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. Control Lmts for P Charts Copyrght 2017 by Taylor Enterprses, Inc., All Rghts Reserved. Control Lmts for P Charts Dr. Wayne A. Taylor Abstract: P charts are used for count data
More information1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands
Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of
More informationUncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
More informationBasic Business Statistics, 10/e
Chapter 13 13-1 Basc Busness Statstcs 11 th Edton Chapter 13 Smple Lnear Regresson Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc. Chap 13-1 Learnng Objectves In ths chapter, you learn: How to use regresson
More informationThis column is a continuation of our previous column
Comparson of Goodness of Ft Statstcs for Lnear Regresson, Part II The authors contnue ther dscusson of the correlaton coeffcent n developng a calbraton for quanttatve analyss. Jerome Workman Jr. and Howard
More informationx i1 =1 for all i (the constant ).
Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by
More informationSystematic Error Illustration of Bias. Sources of Systematic Errors. Effects of Systematic Errors 9/23/2009. Instrument Errors Method Errors Personal
9/3/009 Sstematc Error Illustraton of Bas Sources of Sstematc Errors Instrument Errors Method Errors Personal Prejudce Preconceved noton of true value umber bas Prefer 0/5 Small over large Even over odd
More informationx = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationChapter 3 Describing Data Using Numerical Measures
Chapter 3 Student Lecture Notes 3-1 Chapter 3 Descrbng Data Usng Numercal Measures Fall 2006 Fundamentals of Busness Statstcs 1 Chapter Goals To establsh the usefulness of summary measures of data. The
More informationStatistical Evaluation of WATFLOOD
tatstcal Evaluaton of WATFLD By: Angela MacLean, Dept. of Cvl & Envronmental Engneerng, Unversty of Waterloo, n. ctober, 005 The statstcs program assocated wth WATFLD uses spl.csv fle that s produced wth
More informationStatistics for Business and Economics
Statstcs for Busness and Economcs Chapter 11 Smple Regresson Copyrght 010 Pearson Educaton, Inc. Publshng as Prentce Hall Ch. 11-1 11.1 Overvew of Lnear Models n An equaton can be ft to show the best lnear
More informationNegative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationCopyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for U Charts. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts Copyrght 207 by Taylor Enterprses, Inc., All Rghts Reserved. Adjusted Control Lmts for U Charts Dr. Wayne A. Taylor Abstract: U charts are used
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationEconomics 130. Lecture 4 Simple Linear Regression Continued
Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationU-Pb Geochronology Practical: Background
U-Pb Geochronology Practcal: Background Basc Concepts: accuracy: measure of the dfference between an expermental measurement and the true value precson: measure of the reproducblty of the expermental result
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased
More informationChemometrics In Spectroscopy Limitations in Analytical Accuracy, Part I: Horwitz s Trumpet
18 Spectroscopy 1(9) September 006 Chemometrcs In Spectroscopy Lmtatons n Analytcal Accuracy, Part I: Horwtz s Trumpet Two techncal papers recognzed as sgnfcant early contrbutons n the dscusson of the
More informationChapter 14 Simple Linear Regression
Chapter 4 Smple Lnear Regresson Chapter 4 - Smple Lnear Regresson Manageral decsons often are based on the relatonshp between two or more varables. Regresson analss can be used to develop an equaton showng
More information4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected
More informationSTAT 511 FINAL EXAM NAME Spring 2001
STAT 5 FINAL EXAM NAME Sprng Instructons: Ths s a closed book exam. No notes or books are allowed. ou may use a calculator but you are not allowed to store notes or formulas n the calculator. Please wrte
More informationLecture 6: Introduction to Linear Regression
Lecture 6: Introducton to Lnear Regresson An Manchakul amancha@jhsph.edu 24 Aprl 27 Lnear regresson: man dea Lnear regresson can be used to study an outcome as a lnear functon of a predctor Example: 6
More informationCorrelation and Regression
Correlaton and Regresson otes prepared by Pamela Peterson Drake Index Basc terms and concepts... Smple regresson...5 Multple Regresson...3 Regresson termnology...0 Regresson formulas... Basc terms and
More informationLecture 16 Statistical Analysis in Biomaterials Research (Part II)
3.051J/0.340J 1 Lecture 16 Statstcal Analyss n Bomaterals Research (Part II) C. F Dstrbuton Allows comparson of varablty of behavor between populatons usng test of hypothess: σ x = σ x amed for Brtsh statstcan
More informationChemometrics. Unit 2: Regression Analysis
Chemometrcs Unt : Regresson Analyss The problem of predctng the average value of one varable n terms of the known values of other varables s the problem of regresson. In carryng out a regresson analyss
More informationFREQUENCY DISTRIBUTIONS Page 1 of The idea of a frequency distribution for sets of observations will be introduced,
FREQUENCY DISTRIBUTIONS Page 1 of 6 I. Introducton 1. The dea of a frequency dstrbuton for sets of observatons wll be ntroduced, together wth some of the mechancs for constructng dstrbutons of data. Then
More informationLearning Objectives for Chapter 11
Chapter : Lnear Regresson and Correlaton Methods Hldebrand, Ott and Gray Basc Statstcal Ideas for Managers Second Edton Learnng Objectves for Chapter Usng the scatterplot n regresson analyss Usng the method
More informationSIMPLE LINEAR REGRESSION
Smple Lnear Regresson and Correlaton Introducton Prevousl, our attenton has been focused on one varable whch we desgnated b x. Frequentl, t s desrable to learn somethng about the relatonshp between two
More informationx yi In chapter 14, we want to perform inference (i.e. calculate confidence intervals and perform tests of significance) in this setting.
The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson Introducton In chapter 3 we used a least-squares regresson lne (LSRL) to represent a lnear relatonshp etween two quanttatve explanator
More informationJanuary Examinations 2015
24/5 Canddates Only January Examnatons 25 DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR STUDENT CANDIDATE NO.. Department Module Code Module Ttle Exam Duraton (n words)
More informationSupplementary Notes for Chapter 9 Mixture Thermodynamics
Supplementary Notes for Chapter 9 Mxture Thermodynamcs Key ponts Nne major topcs of Chapter 9 are revewed below: 1. Notaton and operatonal equatons for mxtures 2. PVTN EOSs for mxtures 3. General effects
More informationRegulation No. 117 (Tyres rolling noise and wet grip adhesion) Proposal for amendments to ECE/TRANS/WP.29/GRB/2010/3
Transmtted by the expert from France Informal Document No. GRB-51-14 (67 th GRB, 15 17 February 2010, agenda tem 7) Regulaton No. 117 (Tyres rollng nose and wet grp adheson) Proposal for amendments to
More informationChapter 8 Indicator Variables
Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n
More informationChapter 15 Student Lecture Notes 15-1
Chapter 15 Student Lecture Notes 15-1 Basc Busness Statstcs (9 th Edton) Chapter 15 Multple Regresson Model Buldng 004 Prentce-Hall, Inc. Chap 15-1 Chapter Topcs The Quadratc Regresson Model Usng Transformatons
More informationStatistics II Final Exam 26/6/18
Statstcs II Fnal Exam 26/6/18 Academc Year 2017/18 Solutons Exam duraton: 2 h 30 mn 1. (3 ponts) A town hall s conductng a study to determne the amount of leftover food produced by the restaurants n the
More informationHere is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y)
Secton 1.5 Correlaton In the prevous sectons, we looked at regresson and the value r was a measurement of how much of the varaton n y can be attrbuted to the lnear relatonshp between y and x. In ths secton,
More information/ n ) are compared. The logic is: if the two
STAT C141, Sprng 2005 Lecture 13 Two sample tests One sample tests: examples of goodness of ft tests, where we are testng whether our data supports predctons. Two sample tests: called as tests of ndependence
More informationChapter 9: Statistical Inference and the Relationship between Two Variables
Chapter 9: Statstcal Inference and the Relatonshp between Two Varables Key Words The Regresson Model The Sample Regresson Equaton The Pearson Correlaton Coeffcent Learnng Outcomes After studyng ths chapter,
More informationis the calculated value of the dependent variable at point i. The best parameters have values that minimize the squares of the errors
Multple Lnear and Polynomal Regresson wth Statstcal Analyss Gven a set of data of measured (or observed) values of a dependent varable: y versus n ndependent varables x 1, x, x n, multple lnear regresson
More informationAn identification algorithm of model kinetic parameters of the interfacial layer growth in fiber composites
IOP Conference Seres: Materals Scence and Engneerng PAPER OPE ACCESS An dentfcaton algorthm of model knetc parameters of the nterfacal layer growth n fber compostes o cte ths artcle: V Zubov et al 216
More informationDETERMINATION OF UNCERTAINTY ASSOCIATED WITH QUANTIZATION ERRORS USING THE BAYESIAN APPROACH
Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata Proceedngs, XVII IMEKO World Congress, June 7, 3, Dubrovn, Croata TC XVII IMEKO World Congress Metrology n the 3rd Mllennum June 7, 3,
More informationInterval Estimation in the Classical Normal Linear Regression Model. 1. Introduction
ECONOMICS 35* -- NOTE 7 ECON 35* -- NOTE 7 Interval Estmaton n the Classcal Normal Lnear Regresson Model Ths note outlnes the basc elements of nterval estmaton n the Classcal Normal Lnear Regresson Model
More informationSee Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition)
Count Data Models See Book Chapter 11 2 nd Edton (Chapter 10 1 st Edton) Count data consst of non-negatve nteger values Examples: number of drver route changes per week, the number of trp departure changes
More informationDO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR. Introductory Econometrics 1 hour 30 minutes
25/6 Canddates Only January Examnatons 26 Student Number: Desk Number:...... DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR Department Module Code Module Ttle Exam Duraton
More informationIntroduction to Regression
Introducton to Regresson Dr Tom Ilvento Department of Food and Resource Economcs Overvew The last part of the course wll focus on Regresson Analyss Ths s one of the more powerful statstcal technques Provdes
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationPredictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore
Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models
More informationLOGIT ANALYSIS. A.K. VASISHT Indian Agricultural Statistics Research Institute, Library Avenue, New Delhi
LOGIT ANALYSIS A.K. VASISHT Indan Agrcultural Statstcs Research Insttute, Lbrary Avenue, New Delh-0 02 amtvassht@asr.res.n. Introducton In dummy regresson varable models, t s assumed mplctly that the dependent
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed
More informationChapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
More informationSTAT 3008 Applied Regression Analysis
STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,
More informationLecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding
Recall: man dea of lnear regresson Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 8 Lnear regresson can be used to study an
More informationLecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 212. Chapters 14, 15 & 16. Professor Ahmadi, Ph.D. Department of Management
Lecture Notes for STATISTICAL METHODS FOR BUSINESS II BMGT 1 Chapters 14, 15 & 16 Professor Ahmad, Ph.D. Department of Management Revsed August 005 Chapter 14 Formulas Smple Lnear Regresson Model: y =
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours
UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x
More informationMultiple Contrasts (Simulation)
Chapter 590 Multple Contrasts (Smulaton) Introducton Ths procedure uses smulaton to analyze the power and sgnfcance level of two multple-comparson procedures that perform two-sded hypothess tests of contrasts
More informationLecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding
Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 008 Recall: man dea of lnear regresson Lnear regresson can be used to study
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationSTATISTICS QUESTIONS. Step by Step Solutions.
STATISTICS QUESTIONS Step by Step Solutons www.mathcracker.com 9//016 Problem 1: A researcher s nterested n the effects of famly sze on delnquency for a group of offenders and examnes famles wth one to
More informationLaboratory 1c: Method of Least Squares
Lab 1c, Least Squares Laboratory 1c: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly
More informationCathy Walker March 5, 2010
Cathy Walker March 5, 010 Part : Problem Set 1. What s the level of measurement for the followng varables? a) SAT scores b) Number of tests or quzzes n statstcal course c) Acres of land devoted to corn
More informationDepartment of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution
Department of Statstcs Unversty of Toronto STA35HS / HS Desgn and Analyss of Experments Term Test - Wnter - Soluton February, Last Name: Frst Name: Student Number: Instructons: Tme: hours. Ads: a non-programmable
More information7.1. Single classification analysis of variance (ANOVA) Why not use multiple 2-sample 2. When to use ANOVA
Sngle classfcaton analyss of varance (ANOVA) When to use ANOVA ANOVA models and parttonng sums of squares ANOVA: hypothess testng ANOVA: assumptons A non-parametrc alternatve: Kruskal-Walls ANOVA Power
More informationOF THE IN FIRE RESISTANCE TESTING
SUMMARISED REPORT OF THE EGOLF ROUND-ROBIN NR. TC 9- IN FIRE RESISTANCE TESTING Prepared by Faben Dumont Qualty Manager Fre testng laboratory Unversty of Lège on behalf of EGOLF CONTENTS SCOPE... PART
More informationa. (All your answers should be in the letter!
Econ 301 Blkent Unversty Taskn Econometrcs Department of Economcs Md Term Exam I November 8, 015 Name For each hypothess testng n the exam complete the followng steps: Indcate the test statstc, ts crtcal
More informationRegression. The Simple Linear Regression Model
Regresson Smple Lnear Regresson Model Least Squares Method Coeffcent of Determnaton Model Assumptons Testng for Sgnfcance Usng the Estmated Regresson Equaton for Estmaton and Predcton Resdual Analss: Valdatng
More informationCode_Aster. Identification of the model of Weibull
Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : /8 Responsable : PARROT Aurore Clé : R70209 Révson : Identfcaton of the model of Webull Summary One tackles here the problem of the
More informationA LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS. Dr. Derald E. Wentzien, Wesley College, (302) ,
A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS Dr. Derald E. Wentzen, Wesley College, (302) 736-2574, wentzde@wesley.edu ABSTRACT A lnear programmng model s developed and used to compare
More informationAnalytical Chemistry Calibration Curve Handout
I. Quck-and Drty Excel Tutoral Analytcal Chemstry Calbraton Curve Handout For those of you wth lttle experence wth Excel, I ve provded some key technques that should help you use the program both for problem
More informationSimulated Power of the Discrete Cramér-von Mises Goodness-of-Fit Tests
Smulated of the Cramér-von Mses Goodness-of-Ft Tests Steele, M., Chaselng, J. and 3 Hurst, C. School of Mathematcal and Physcal Scences, James Cook Unversty, Australan School of Envronmental Studes, Grffth
More informationNEW ASTERISKS IN VERSION 2.0 OF ACTIVEPI
NEW ASTERISKS IN VERSION 2.0 OF ACTIVEPI ASTERISK ADDED ON LESSON PAGE 3-1 after the second sentence under Clncal Trals Effcacy versus Effectveness versus Effcency The apprasal of a new or exstng healthcare
More informationComparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method
Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method
More informationThe Ordinary Least Squares (OLS) Estimator
The Ordnary Least Squares (OLS) Estmator 1 Regresson Analyss Regresson Analyss: a statstcal technque for nvestgatng and modelng the relatonshp between varables. Applcatons: Engneerng, the physcal and chemcal
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 14 Multiple Regression Models
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 14 Multple Regresson Models 1999 Prentce-Hall, Inc. Chap. 14-1 Chapter Topcs The Multple Regresson Model Contrbuton of Indvdual Independent Varables
More informationProblem of Estimation. Ordinary Least Squares (OLS) Ordinary Least Squares Method. Basic Econometrics in Transportation. Bivariate Regression Analysis
1/60 Problem of Estmaton Basc Econometrcs n Transportaton Bvarate Regresson Analyss Amr Samm Cvl Engneerng Department Sharf Unversty of Technology Ordnary Least Squares (OLS) Maxmum Lkelhood (ML) Generally,
More informationThe SAS program I used to obtain the analyses for my answers is given below.
Homework 1 Answer sheet Page 1 The SAS program I used to obtan the analyses for my answers s gven below. dm'log;clear;output;clear'; *************************************************************; *** EXST7034
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More informationInvestigation of Uncertainty Sources in the External Dosimetry Laboratory
Investgaton of Uncertanty Sources n the External Dosmetry Laboratory Specfcaton.1.1. Analyss of uncertantes Methods for calculatng the overall uncertanty from ndvdual measured uncertantes are gven n the
More informationChapter 6. Supplemental Text Material
Chapter 6. Supplemental Text Materal S6-. actor Effect Estmates are Least Squares Estmates We have gven heurstc or ntutve explanatons of how the estmates of the factor effects are obtaned n the textboo.
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationEcon107 Applied Econometrics Topic 9: Heteroskedasticity (Studenmund, Chapter 10)
I. Defnton and Problems Econ7 Appled Econometrcs Topc 9: Heteroskedastcty (Studenmund, Chapter ) We now relax another classcal assumpton. Ths s a problem that arses often wth cross sectons of ndvduals,
More informationECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE)
ECONOMETRICS - FINAL EXAM, 3rd YEAR (GECO & GADE) June 7, 016 15:30 Frst famly name: Name: DNI/ID: Moble: Second famly Name: GECO/GADE: Instructor: E-mal: Queston 1 A B C Blank Queston A B C Blank Queston
More informationLecture 3 Stat102, Spring 2007
Lecture 3 Stat0, Sprng 007 Chapter 3. 3.: Introducton to regresson analyss Lnear regresson as a descrptve technque The least-squares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture
More information