E 1655 PDF

 

Designation: E 1655 – 00

Standard Practices for

Infrared Multivariate Quantitative Analysis1 This standard is issued under the fixed designation E 1655; the number immediately following the designation indicates the year of  original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A superscript supersc ript epsilon (e) indicates an editorial change since the last revision or reapproval.

1. Scope Scope

respect respe ct to the handling of outli outliers. ers. Surrogat Surrogatee methods methods may indicate indica te tha thatt the they y make make use of the mat mathem hemati atics cs descri described bed herein her ein,, but the they y sho should uld not claim claim to fol follow low the procedur procedures es described herein. 1.7   This sta standa ndard rd does not purpo purport rt to addre address ss all of the sa safe fety ty conc concer erns ns,, if any any, as asso soci ciat ated ed with with its its us use. e. It is the the responsibility of the user of this standard to establish appro priate safety and health practices and determine the applicability of regulatory limitations prior to use.

1.1 These These pra practi ctices ces cov cover er a gui guide de for the multiv multivari ariate ate calibration of infrared spectrometers used in determining the physical or chemical characteristics of materials. These practices are applicable to analyses conducted in the near infrared (NIR) spectral region (roughly 780 to 2500 nm) through the mid inf infrar rared ed (MI (MIR) R) spectr spectral al reg region ion (ro (rough ughly ly 400 4000 0 to 400 cm−1). NOTE   1—While the practices described herein deal specifically with mid- and near-infrared analysis, much of the mathematical and procedural detail contained herein is also applicable for multivariate multivariate quantitativ quantitativee analysis done using other forms of spectroscopy. The user is cautioned that typical and best practices for multivariate quantitative analysis using other forms of spectroscopy may differ from practices described herein for midand near-infrared spectroscopies.

2. Referenced Referenced Documents Documents 2.1   ASTM Standards: D 1265 1265 Pra Practi ctice ce for Sampli Sampling ng Liq Liquifi uified ed Pet Petrol roleum eum (LP (LP)) 2 Gases (Manual Method) D 4057 4057 Pra Practi ctice ce for Manual Manual Sampli Sampling ng of Pet Petrol roleum eum and 3 Petroleum Products D 4177 Pract Practice ice for Automatic Automatic Sampling of Petr Petroleum oleum and 3 Petroleum Products D 4855 Pract Practices ices for Comparing Comparing Te Test st Methods4 D 6122 Pract Practice ice for Valida Validation tion of Mult Multivari ivariate ate Process Process Infrared Spectrophotometers5 D 6299 Practice for Applying Statistic Statistical al Quality Assurance Techniques Techniques to Evaluate Analytical Measurement System Performance5 D 6300 Pract Practice ice for Deter Determina mination tion of Preci Precision sion and Bias Data for Use in Test Methods for Petroleum Products and Lubricants6 E 131 Ter Terminology minology Relating to Molecular Spectroscopy7 E 168 Practices for General T Techniques echniques of Infrared Quanti7 tative Analysis

1.2 Procedure Proceduress for collecting collecting and trea treating ting data for devel developoping IR calibrations are outlined. Definitions, terms, and calibration brat ion technique techniquess are described. described. Criteria for vali validati dating ng the performance of the calibration model are described. 1.3 The implementat implementation ion of thes thesee pract practices ices requir requiree that the IR spe spectr ctrome ometer ter has bee been n instal installed led in com compli plianc ancee with with the manufacturer’s specifications. In addition, it assumes that, at the times times of cal calibr ibrati ation on and of valida validatio tion, n, the analyz analyzer er is operating at the conditions specified by the manufacturer. 1.4 These These pra practi ctices ces cover cover tec techni hnique quess tha thatt are rou routin tinely ely applie app lied d in the nea nearr and mid infrar infrared ed spe spectr ctral al reg region ionss for quantitative analysis. The practices outlined cover the general cases cas es for coa coarse rse solids solids,, fine gro ground und sol solids ids,, and liq liquid uids. s. All techni technique quess covere covered d requir requiree the use of a com comput puter er for data collection and analysis. 1.5 These practices practices provide a quest questionna ionnaire ire against which multivariate calibrations can be examined to determine if they conform to the requirements defined herein. 1.6 For some multivariate spectroscopic spectroscopic analyses, interferences and matrix effects are sufficiently small that it is possible to calibrate calibrate using mixt mixtures ures that contain subst substanti antially ally fewer chemical components than the samples that will ultimately be analyzed. While these surrogate methods generally make use of the multivariate mathematics described herein, they do not confor con form m to proced procedure uress des descri cribed bed herein herein,, spe specifi cifical cally ly with with

E 275 Pract Practice ice for Descr Describing ibing and Meas Measurin uring g Perf Performa ormance nce of Ultraviolet, Visible, and Near Infrared Spectrophotometers7 E 334 Prac Practic ticee for Gen Genera erall Techniq echniques ues of Inf Infrar rared ed Microanalysis7 E 456 Ter Terminology minology Relating to Quality and Statistics8 E 691 Pract Practice ice for Conducting Conducting an Inte Interlab rlaborato oratory ry Study to 8 Determine the Precision of a Test Method E 932 Pract Practice ice for Descr Describing ibing and Meas Measurin uring g Perf Performa ormance nce 2

 Annual  Annual 4  Annual 5  Annual 6  Annual 7  Annual

3 1

The These se practic practices es are und under er the jurisdi jurisdictio ction n of ASTM Committ Committee ee E13 on Molecular Spectro Molecular Spectroscopy scopy and are the direct responsibility responsibility of Subcommittee Subcommittee E13.11 E13.11 on Chemometrics. Current edition approved Sept. 10, 2000. Published November 2000. Originally

Book Book Book Book Book Book

of of of of of of

ASTM ASTM ASTM ASTM ASTM ASTM

Standards Standards,, Vol 05.01. Standards Standards,, Vol 05.02. Standards Standards,, Vol 07.02. Standards Standards,, Vol 05.04 Standards Standards,, Vol 05.03. Standards Standards,, Vol 03.06.

8

 Annual Book of ASTM Standards Standards,, Vol 14.02.

published publi shed as E 1655 – 97. Last previous previous edition edition E 1655 – 99.

Copyright © ASTM, 100 Barr Harbor Drive, West Conshohocken, PA 19428-2959, United States.

1

 

E 1655 of Dispersive Infrared Spectrometers7 E 1421 Practice for Describing Describing and Measuring Measuring Performance of Four Fourier ier Tra Transfor nsform m Infr Infrared ared (FT (FT-IR) -IR) Spect Spectrome rometers ters:: Level Zero and Level One Tests7 E 1866 1866 Guide for Esta Establis blishing hing Spect Spectropho rophotome tometer ter Perfo Perforrmance Tests7 E 1944 Practice for Describing Describing and Measuring Measuring Performance of Fourier Transform Near-Infrared (FT-NIR) Spectrometers: Level Zero and Level One Tests 7

nents than the samples which will ultimately be analyzed. 3.2.12   surrogate method ,  n —a standard test method that is based on a surrogate calibration. 3.2.13   validation samples—a set of samples used in validating the model. Validation samples are not part of the set of  calibrati cali bration on samp samples. les. Refer Reference ence compo component nent conce concentra ntration tion or property values are known (measured by reference method), and are compared to those estimated using the model. 4. Summar Summary y of Practices Practices

3. Terminology

4.1 Mul Multiv tivari ariate ate mat mathem hemati atics cs is app applie lied d to cor correl relate ate the absorbanc absorb ances es mea measur sured ed for a set of cal calibr ibrati ation on sam sample pless to reference component concentrations or property values for the set of samples. The resultant multivariate calibration model is applie app lied d to the analysis analysis of spe spectr ctraa of unk unknow nown n sam sample pless to provide an estimate of the component concentration or property values for the unknown sample. 4.2 Multilinear Multilinear regre regression ssion (MLR) (MLR),, princ principal ipal compo components nents regression (PCR), and partial least squares (PLS) are examples of mult multivar ivariate iate math mathemat ematical ical tech technique niquess that are comm commonly only used use d for the dev develo elopme pment nt of the cal calibr ibrati ation on mod model. el. Oth Other er mathem mat hemati atical cal tec techni hnique quess are als also o use used, d, but ma may y not det detect ect outliers, and may not be validated by the procedure described in these practices. 4.3 Stat Statisti istical cal tests are applied to detec detectt outl outliers iers during the developmen devel opmentt of the cali calibrati bration on model model.. Outli Outliers ers inclu include de high leverage samples (samples whose spectra contribute a statisticall ca lly y si sign gnifi ifica cant nt fr frac acti tion on of on onee or mo more re of th thee sp spec ectr tral al variables varia bles used in the model), and sampl samples es whose reference reference values are inconsistent with the model. 4.4 Vali alidat dation ion of the cal calibr ibrati ation on mod model el is per perfor formed med by using usi ng the model to ana analyz lyzee a set of val valida idatio tion n sam sample pless and statistically comparing the estimates for the validation samples to reference values measured for these samples, so as to test for bias in the model and for agreement of the model with the reference method. 4.5 Stat Statisti istical cal tests are appli applied ed to detec detectt when values estimated using the model represent extrapolation of the calibration. 4.6 Stat Statisti istical cal expre expression ssionss for calcu calculati lating ng the repea repeatabil tability ity of the infrared analysis and the expected agreement between the infrared analysis and the reference method are given.

3.1   Definitions—For terminology related to molecular spectroscopic methods, refer to Terminology E 131. For terminology relat relating ing to quali quality ty and stat statisti istics, cs, refe referr to Termi erminolo nology gy E 456. 3.2   Definitions of Terms Specific to This Standard: 3.2.1   analysis—in the context of this practice , the process of  applying the calibration model to an absorption spectrum so as to estimate a component concentration value or property. 3.2.2   calibration—a process used to create a model relating two types of measured data. In the context of this practice, a process for creating a model that relates component concentrations or properties to absorbance spectra for a set of known reference samples. 3.2.3   calibration model—the mathematical expression that relates component concentrations or properties to absorbances for a set of reference samples. 3.2.4   calibration calibration sampl samples es—th —thee set of ref refere erence nce sam sample pless used for crea creating ting a calib calibrati ration on model model.. Refer Reference ence component component concen con centra tratio tion n or pro proper perty ty val values ues are kno known wn (me (measu asured red by reference method) for the calibration samples and correlated to the absorbance spectra during the calibration. 3.2.5   estimate—the value for a component concentration or property prope rty obtained by appl applying ying the cali calibrat bration ion model for the analysis of an absorption spectrum. 3.2.6   model validation—the process of testing a calibration model to determine bias between the estimates from the model and the reference method, and to test the expected agreement betwee bet ween n est estima imates tes mad madee wit with h the mod model el and the ref refere erence nce method. 3.2.7   multivariate calibration—a pr proc oces esss fo forr cr crea eati ting ng a model that relates component concentrations or properties to the absorbances of a set of known reference samples at more than one wavelength or frequency. 3.2.8  reference method —the —the analytical method that is used to estimate the reference component concentration or property value which is used in the calibration and validation procedures. 3.2.9   reference reference values—the comp component onent conce concentra ntration tionss or property values for the calibration or validation samples which are measured by the reference analytical method. 3.2.10   spectr spectrometer/ ometer/spectr spectrophotom ophotometer eter qualification qualific ation, n—th —thee pro proced cedure uress by whi which ch a use userr dem demons onstra trates tes tha thatt the performa perf ormance nce of a speci specific fic spect spectrome rometer/ ter/spect spectroph rophotom otometer eter is adequa ade quate te to con conduc ductt a mul multi tivar variat iatee ana analys lysis is so as to obt obtain ain

5. Signifi Significance cance and and Use 5.1 These practices practices can be used to estab establish lish the validity validity of  the results obtained by an infrared (IR) spectrometer at the time the calibration is developed. The ongoing validation of estimates mat es pro produc duced ed by ana analys lysis is of unk unknow nown n sam sample pless usi using ng the calibrati cali bration on model should be cover covered ed sepa separatel rately y (see for example, Practice D 6122). 5.2 The These se pra practi ctices ces are int intend ended ed for all use users rs of inf infrar rared ed spectrosco spect roscopy py.. Near Near-inf -infrare rared d spect spectrosco roscopy py is widel widely y used for quantitative analysis. Many of the general principles described in these practices relate to the common modern practices of  near-infrared spectroscopic analysis. While sampling methods

precision consistent with that specified in the method. 3.2.11   surrogate calibration,   n—a multivariate calibration thatt is dev tha develo eloped ped usi using ng a cal calibr ibrati ation on set whi which ch con consis sists ts of  mixtures which contain substantially fewer chemical compo-

and instrumentation may differ, the general calibration methodologies are equally applicable to mid-infrared spectroscopy. New tec techni hnique quess are und under er stu study dy tha thatt may enh enhanc ancee tho those se discussed within these practices. Users will find these practices 2

 

E 1655 to be applicable to basic aspects of the technique, to include sample samp le sele selectio ction n and prepa preparati ration, on, inst instrumen rumentt opera operation tion,, and data interpretation. 5.3 The calibration calibration procedures procedures define the range over which measurements are valid and demonstrate whether or not the sensitivity and linearity of the analysis outputs are adequate for providing provi ding meaningful meaningful esti estimate matess of the speci specific fic physi physical cal or chemical characteristics of the types of materials for which the calibration is developed.

monitored monito red for con contin tinued ued acc accura uracy cy and pre precis cision ion.. Sim Simult ultaaneously, the instrument performance must be monitored so as to trace any deterioration in performance to either the calibration model itself or to a failure in the instrumentation performance. Procedures for verifying the performance of the analysis are only outlined in Section 22 but are covered in detail in Practice D 6122. The use of this method requires that a model quality control material be established at the time the model is developed. The model QC material is discussed in Section 22.

6. Overv Overview iew of Multiva Multivariate riate Calibration Calibration

For pra practi ctices ces to com compar paree ref refere erence nce me metho thods ds and ana analyz lyzer er methods, refer to Practices D 4855. 6.1.8   Trans —Transferable erable calibrations Transfer fer of Cali Calibrati brations ons—Transf are equations that can be transferred from the original instrument, where calibration data were collected, to other instruments where the calibrations are to be used to predict samples for routine analysis. In order for a calibration to be transferable it must perform prediction after transfer without a significant decrease in performance, as indicated by established statistical tests. tes ts. In add additi ition, on, sta statis tistic tical al tes tests ts tha thatt are use used d to det detect ect extrap ext rapola olatio tion n of the mod model el mus mustt be pre preser served ved dur during ing the transfer. Bias or slope adjustments, or both, are to be made afterr tran afte transfer sfer only when stat statisti isticall cally y warr warranted anted.. Cali Calibrati bration on transfer, that is sometimes referred to as instrument standardization, is discussed in Section 21.

6.1 The practice of infrared multivariate multivariate quantitative quantitative analysis involves the following steps: 6.1.1   Selecting the Calibration Set —This —This set is also termed the training set or spectral library set. This set is to represent all of the chemical and physical variation normally encountered for routine analysis for the desired application. Selection of the calibration set is discussed in Section 17, after the statistical terms ter ms nec necess essary ary to defi define ne the sel select ection ion cri criter teria ia hav havee bee been n defined. 6.1.2   Determi Determinatio nation n of Conce Concentra ntrations tions or Pro Propert perties, ies, or   Both, for Calibration Samples—Th —Thee che chemic mical al or phy physic sical al properties, or both, of samples in the calibration set must be accurately and precisely measured by the reference method in order to accurately calibrate the infrared model for prediction of the unkno unknown wn samp samples. les. Reference Reference meas measurem urements ents are discussed in Section 9. 6.1.3  The Collection of Infrared Spectra—The collection of  optica opt icall dat dataa mus mustt be per perfor forme med d wit with h car caree so as to pre presen sentt calibrat cali bration ion samp samples, les, vali validati dation on sampl samples, es, and predi predictio ction n (unknown) samples for analysis in an alike manner. Variation in sample presentation technique among calibration, validation, and prediction samples will introduce variation and error which has not been modeled within the calibration. Infrared instrumentat men tation ion is dis discus cussed sed in Sec Sectio tion n 7 and inf infrar rared ed spe spectr ctral al measurements in Section 8. 6.1.4   Calculatin Calculating g the Mathe Mathematic matical al Model—The calcu calculalation tio n of mat mathem hemati atical cal (ca (calib librat ration ion)) mod models els may inv involv olvee a variety of data treatments and calibration algorithms. The more common com mon lin linear ear tec techni hnique quess are dis discus cussed sed in Sec Sectio tion n 12. A

7. Infrared Infrared Instrumentation Instrumentation 7.1 A complete complete description description of all applicable applicable types of infr infraared instrumentat instrumentation ion is beyon beyond d the scope of these practices. practices. Only a general outline is given here. 7.2 The IR inst instrume rumentat ntation ion is comp comprise rised d of two cate categorie gories, s, including incl uding inst instrume ruments nts that acqui acquire re conti continuous nuous spec spectral tral data over wavel wavelength ength or freq frequency uency range rangess (spe (spectrop ctrophoto hotomete meters), rs), and tho those se tha thatt onl only y exa examin minee one or sev severa erall dis discre crete te wav waveelengths or frequencies (photometers). 7.2.1 Photo Photomete meters rs may have one or a seri series es of wavel wavelengt ength h filters filt ers and a sin single gle detector detector.. The These se filt filters ers are mounted mounted on a turret wheel so that the individual wavelengths are presented to a sing single le dete detector ctor seque sequentia ntially lly.. Conti Continuous nuously ly varia variable ble filte filters rs may also be used in this fashion. These filters, either linear or circular, are moved past a slit to scan the wavelength being

variety variet y of sta stati tisti stical cal tec techni hnique quess are use used d to eva evalua luate te and optimize the model. These techniques are described in Section 15. Statistics used to detect outliers in the calibration set are covered in Section 16. 6.1.5   Valida Validation tion of the Calib Calibrati ration on Mode Modell—V —Validation alidation of  the efficacy of a specific calibration model (equation) requires that the model be applied for the analysis of a separate set of  test (validation) samples, and that the values predicted for these test samples be statistically compared to values obtained by the refe re fere renc ncee me meth thod od.. Th Thee st stat atis isti tica call te test stss to be ap appl plie ied d fo forr validation of the model are discussed in Section 18. 6.1.6   Ap Appl plic icat atio ion n of th thee Mo Mode dell fo forr th thee An Anal alys ysis is of  Unknowns—The mathematical model is applied to the spectra of unknown samples to estimate component concentrations or property values, or both, (see Section 13). Outlier statistics are

measured. Alternatively, photometers may have several monochromatic chrom atic light sour sources, ces, such as ligh light-em t-emitti itting ng diode diodes, s, that sequentially turn on and off. 7.3 Spect Spectropho rophotome tometers ters can be clas classified sified,, base based d upon the procedure by which light is separated into component wavelengths. lengt hs. Disp Dispersiv ersivee instr instrument umentss gener generally ally use a dif diffrac fraction tion grating to spatially disperse light into a continuum of wavelengths. In scanning-grating systems, the grating is rotated so thatt onl tha only y a nar narrow row band of wav wavele elengt ngths hs is tra transm nsmit itted ted to a single detector at any given time. Dispersion can occur before the sample (pre-dispersed) or after the sample (post-dispersed). 7.3.1 7.3. 1 Spectrop Spectrophoto hotomet meters ers are als also o avai availabl lablee wher wheree the wavelengt wavel ength h sele selectio ction n is accom accomplis plished hed witho without ut movi moving ng parts parts,, using a photodiode array detector. detector. Post-dispersion is utilized. A grating can again provide this function, although other meth-

used to detect when the analysis involves extrapolation of the model (see Section 16). 6.1.7   Routine Analysis and Monitoring—Once the efficacy of calibration equations is established, the equations must be

ods, such as a linear variable filter (LVF) accomplish the same purpose (a LVF is a multilayer filter that has variable thickness along its length, such that different wavelengths are transmitted at different positions). The photodiode array detector is used to 3

 

E 1655 acquire a continuous spectrum over wavelength without mechanical motion. The array detector is a compact aggregate of  up to several thousand individual photodiode detectors. Each photod pho todiod iodee is loc locate ated d in a dif differ ferent ent spe spectr ctral al reg region ion of the disper dis persed sed light beam and detects detects a uni unique que range of wav waveelengths. 7.3.2 7.3 .2 The aco acoust usto-o o-opti ptical cal tun tunabl ablee filt filter er is a con conti tinuo nuous us variant of the fixed filter photometer with no moving optical partss for wavel part wavelengt ength h selec selection. tion. A bire birefrig frigent ent cryst crystal al (for ex-

extraneou extran eouss noi noise se wit withi hin n the spe spectr ctral al si signa gnal. l. Sca Scanni nning/  ng/  interfero inte rferomete meter-ba r-based sed syst systems ems also allo allow w great greater er wavel wavelengt ength/  h/  freque fre quency ncy pre precis cision ion bet betwee ween n ins instru trumen ments ts due to int intern ernal al wavelength/frequency standardization techniques, and the possibiliti sibi lities es of comp computer uter-gene -generated rated spect spectral ral corr correctio ections. ns. For example, scanning instruments have received approval for complex matrices, such as animal feed and forages (1, forages  (1, 2). 2).9 7.6 Descr Descripti iptions ons of inst instrume rumentat ntation ion designs related to Refs (1) and (1)  and (2)  (2) are  are found in Refs (3) Refs  (3) and  and (4)  (4).. Other instrumentation

ample, tellurium oxide) is used, in which acoustic waves at a selected frequency are applied to select the wavelength band of  light transmitted through the crystal. Variations in the acoustic frequency cause the crystal lattice spacing to change, that in turn, tur n, cau causes ses the cry crysta stall to act as a var variab iable le tra transm nsmiss ission ion diffraction grating for one wavelength (that is, a Bragg diffractor). A single detector is used to analyze the signal. 7.3.3 An addit additional ional category of spec spectroph trophotom otometer eterss uses mathematical transformations to convert modulated light signals into spectral data. The most well-known example is the Fourier transform, that when applied to infrared (IR) is known as FT-IR. Light is divided into two beams whose relative paths are varied by use of a moving optical element (for example, either a moving mirror, or a moving wedge of a high refractive index ind ex mat materi erial) al).. The bea beams ms are rec recomb ombine ined d to pro produc ducee an

similar in performance to that described in these references is acceptable for all near-infrared techniques described in these practices. 7.7 For information information describing describing the meas measureme urement nt of perf perforormance of ultraviolet, visible, and near infrared spectrophotometers, refer to Practice E 275. For information describing the measurem meas urement ent of perf performa ormance nce of dispe dispersive rsive infr infrared ared spec spectrotrophotometers, refer to Practice E 932. For information describing the measurement performance of Fourier Transform midinfrar inf rared ed spe spectr ctroph ophoto otome meter ters, s, ref refer er to Pra Practi ctice ce E 142 1421. 1. For information describing the measurement performance of Fourier Transform near-infrared spectrophotometers, refer to Practice E 1944. For spectrophotometers to which these practice do not apply, refer to Guide E 1866.

interference interfer ence pattern that cont contains ains all of the wavelengths wavelengths of  interest. The interference pattern is mathematically converted into spectral data using the Fourier transform. The FT method can operate in the mid-IR and near-IR spectral regions. The FT instruments use a single detector. 7.3.4 A secon second d type of tran transform sformatio ation n spec spectroph trophotom otometer eter uses the Hadamard transformation. Light is initially dispersed with a grating. Light then passes through a mask mounted on or adjacent to a single detector. The mask generates a series of  patt pa tter erns ns.. Fo Forr ex exam ampl ple, e, th thes esee pa patt tter erns ns ma may y be fo form rmed ed by electronically opening and shutting various locations, such as in a liquid crystal display, or by moving an aperture or slit through the beam. These modulations alter the energy distribution incident upon the detector. A mathematical transformation is then used to convert the signal into spectral information.

8.1 Multiv Multivari ariate ate cal calibr ibrati ations ons are bas based ed on Bee Beer’s r’s Law Law,, namely, the absorbance of a homogeneous sample containing an absorbing substance is linearly proportional to the concentration of the absorbing species. The absorbance of a sample is defined as the logarithm to the base ten of the reciprocal of the transmittance, (T ). ).

8. Infra Infrared red Spectral Measurements Measurements

 A 5 log10~1/ T  T !

The transmittance,   T , is defined as the ratio of radiant power transmitted by the sample to the radiant power incident on the sample. 8.1.1 For measurement measurementss condu conducted cted by refle reflectanc ctance, e, the reflectance,  R , is sometimes substituted for the transmittance   T . The reflectan reflectance ce is defi defined ned as the ratio ratio of the radiant radiant power reflected by the sample to the radiant power incident on the sample.

7.4 Infra Infrared red inst instrume ruments nts used in mult multivar ivariate iate cali calibrat brations ions should be ins should instal talled led and ope operat rated ed in acc accord ordanc ancee wit with h the instructions of the instrument manufacturer. Where applicable, the performance of the instrument should be tested at the time the calibration is conducted using procedures defined in the appropriate ASTM practice (see 2.1). The performance of the instrument should be monitored on a periodic basis using the same proc procedure edures. s. The moni monitori toring ng proce procedure dure shoul should d detec detectt changes in the performance of the instrument (relative to that seen during collection collection of the calibration calibration spectra) that would affect the estimation made with the calibration model. 7.5 For most infra infrared red quant quantitat itative ive appli applicatio cations ns invol involving ving complex matrices, it is a general consensus that scanning-type instruments (either dispersive or interferometer based) provide the greatest performance, due to the stability and reproducibil-

NOTE   2—The relationship  A  = log10(1/  R) is not a definition, but rather an app approxi roximat mation ion des designe igned d to line lineari arize ze the rel relati ationsh onship ip bet betwee ween n the measured reflectance,   R, and the concentration of the absorbing species. For some appli applicati cations, ons, othe otherr linea lineariza rization tion func functions tions (for exam example, ple, Kubelka-Munk) may be more appropriate (5) appropriate  (5)..

8.1.2 For most types of instr instrument umentatio ation, n, the radi radiant ant power incident on the sample cannot be measured directly. Instead, a reference (background) measurement of the radiant power is made without the sample being present in the light beam. NOTE  3—To avoid confusion, the reference measurement of the radiant power will be referred to as a background measurement, and the word refe re fere renc ncee wil willl onl only y be use used d to re refe ferr to me meas asur urem emen ents ts ma made de by the refere ref erence nce method against against whic which h the inf infrar rared ed is to be cal calibra ibrated ted.. (Se (Seee Section Sectio n 9.)

ity of mode modern rn inst instrume rumentat ntation ion and to the greater amount of  spectral data provided for computer interpretation. These data allow for greater calibration flexibility and additional options for selections of spectral areas less sensitive to band shifts and

9 The boldface numbers in parentheses refer to a list of references at the end of  the text.

4

 

E 1655 8.1 8.1.3 .3 A me measu asurem rement ent is the then n con conduc ducted ted wit with h the sam sample ple presen pre sent, t, and the rat ratio, io,   T , is cal calcul culate ated. d. The bac backgr kgroun ound d measurement may be conducted in a variety of ways depending on the application and the instrumentation. The sample and its holder may be physically removed from the light beam and a background backg round measurement measurement made on the “empty beam”. The samp sa mple le ho hold lder er (c (cel ell) l) ma may y be em empt ptie ied, d, an and d a ba back ckgr grou ound nd measurement may be taken through the “empty cell”.

acquisition (scan speed). A detailed description of the spectral acquisition parameters and their effect on multivariate calibrations is beyond the scope of these practices. However, it is essential that all adjustable parameters that control the collection and computation of spectral data be maintained constant for the collection of spectra of calibration samples, validation samples, and unknown samples for which estimates are to be made. 8.5 For definitions definitions and furth further er description description of gener general al infra-

NOTE  4—For optically thin cells, care may be necessary to avoid optical interferences resulting from multiple internal reflections within the cell. For very thick cells, differences in the refractive index between the sample and the emp empty ty cel celll may change change prop propert erties ies of the optical optical syst system, em, for example, shift focal points.

red quant quantitat itative ive meas measurem urement ent techn technique iques, s, refe referr to Pract Practice ice E 168 168.. For a des descri cripti ption on of gen genera erall tec techni hnique quess of infrar infrared ed microanalysis, refer to Practice E 334. 9. Refer Reference ence Method and Refer Reference ence Values Values

8.1.4 The sample sample holder (cell) may be filled with a liqui liquid d that has minimal absorption in the spectral range of interest, and the background measurement may be taken through the “background liquid.” Alternatively Alternatively,, the light beam may be split or alt altern ernate ately ly pas passed sed thr throug ough h the sam sample ple and thr throug ough h an “empty beam,” an “empty cell,” or a “background liquid.” For reflectance measurements, the reflectance of a material having minimal absorbance in the region of interest is generally used as the background measurement. 8.1.5 The particular particular backg background round referencing referencing scheme that is used may vary among instruments, and among applications.

9.1 Infrared spectroscopy requires requires calibration to determine the proportionality relationship between the signals measured and the component concentrations or properties that are to be estimate esti mated. d. Durin During g the calib calibrati ration, on, spect spectra ra are meas measured ured for samples for which these reference values are known, and the relationship between the sample absorbances and the reference values is determined. The proportionality relationship is then applie app lied d to the spe spectr ctraa of unk unknow nown n sam sample pless to est estima imate te the concentration or property values for the sample. 9.2 For sim simple ple mix mixtur tures es con contai tainin ning g onl only y a few che chemic mical al components, it is generally possible to prepare mixtures that can serve as standards for the multivariate calibration of an infrared analysis. Because of potential interferences among the absorbances of the components, it is not sufficient to vary the concentration of only some of the mixture components, even when analyses for only one component are being developed. Instead, all components should be varied over a range representative of that expected for future unknown samples that are to be analyzed. Since infrared measurements are conducted on a fixed volume of sample (for example, a fixed cell pathlength), it is preferable that concentration reference values be expressed in volumetric terms, for example, in volume percentage, grams per millilitre, moles per cubic centimetre, and so forth. Developing mult multivari ivariate ate cali calibrat brations ions for refer reference ence conc concentra entrations tions expressed in other terms (for example, weight percentage) can

The same background referencing scheme must be employed for the mea measur sureme ement nt of all spe spectr ctraa of cal calibr ibrati ation on sam sample ples, s, validation samples, and unknown samples to be analyzed. 8.2 Tradi Traditio tional nally ly,, a sam sample ple is man manual ually ly bro brough ughtt to the instrument and placed in a suitable optical container (a cell or cuvette with windows that transmit in the region of interest). Altern Alt ernati ativel vely y, tra transf nsfer er pip pipes es can con contin tinuou uously sly flow liq liquid uid thro th roug ugh h an op opti tica call ce cell ll in th thee in inst stru rume ment nt fo forr co cont ntin inuo uous us analys ana lysis. is. With opt optica icall fibe fibers, rs, the sam sample ple can be ana analyz lyzed ed remote rem otely ly fro from m the ins instru trumen ment. t. Lig Light ht is sen sentt to the sam sample ple through an optical fiber or fibers and returned to the instrument by means of another fiber or group of fibers. Instruments have been developed that use single fibers to transmit and receive the light, light, as wel welll as those using bundles bundles of fibe fibers rs for this purpose. Detectors and light sources external to the instrument can alsoFor be used, in regions which case only one fiber or bundle is needed. spectral where transmitting fibers do not exist, the same function can be performed over limited distances using appropriate optical transfer optics.

8.3 Altho Although ugh most mult multivar ivariate iate calibrations calibrations for liqui liquids ds involve the direct measurement of transmitted light, alternative sampling samp ling techn technologi ologies es (for exam example, ple, atte attenuate nuated d tota totall reflec reflec-tance) can also be employed. Transmittance measurements can be employed for some types of solids (for example, polymer films), whereas other solids (for example, powdered solids) are more commonly measured by diffuse reflectance techniques.

lead to models that are linear approximations to what is really a nonlinear relationship and can lead to less accurate estimates of the concentrations. 9.3 For com comple plex x mix mixtur tures, es, suc such h as tho those se obt obtain ained ed fro from m petrochemical processes, preparation of reference standards is generally gener ally impractical, impractical, and the mult multivari ivariate ate cali calibrati bration on of an infrared analysis must typically be performed on actual process samples. In this case, the reference values used to calibrate the infrared analysis are obtained by a reference analytical method. The accuracy of a component concentration or property value estimated by a multivariate infrared analysis is highly dependent on the accuracy and precision of the reference values used in the calibration. The expected agreement between the infrared estimated values and those obtained from a single reference measurement can never exceed the repeatability of the refer-

8.4 For most infr infrared ared instrumentat instrumentation, ion, a vari variety ety of adjus adjusttable par able parame ameter terss are ava availa ilable ble to con contro troll the col collec lectio tion n and computation of the spectral data. These parameters control, for instance, the optical and digital resolution, and the rate of data

ence met ence method hod,, sin since, ce, eve even n if the inf infrar rared ed est estima imated ted the tru truee value, the measurement of agreement is limited by the precision sio n of the ref refere erence nce val values ues.. Kno Knowle wledge dge of the pre precis cision ion (repea (re peatab tabili ility) ty) of the ref refere erence nce met method hod is cri critic tical al in the

NOTE  5—If the instrument uses predispersion of the light, some caution must be exercised to avoid introducing ambient light into the system at the sample position, since such light may be detected, giving rise to erroneous absorbance measurements.

5

 

E 1655 development of an infrared multivariate calibration. The precision of the reference data used in developing a model, and the accuracy of the model can be improved by averaging repeated reference measurements.

9.6 Refer Reference ence methods methods that are not ASTM methods can be used for the multivariate calibration of infrared analyses, but in this case, it is the responsibility of the method developer to establish the precision of the reference method using procedures similar to those detailed in Practice E 691, in the  Manual  for Determining Precisi Precision on for ASTM Methods on Petrole Petroleum um Products and Lubricants10 and in Practice D 6300. 9.7 When multiple reference reference measurements are made on an individual calibration or validation sample, a Dixon’s Test (see

NOTE  6—If the reference values used to calibrate a multivariate infrared analysiss are generated analysi generated in a sing single le labo laborat ratory ory,, it is ess essent ential ial that the measurement process used to generate these values be monitored for bias and precision using suitable quality assurance assurance proced procedures ures (see for example, Practice D 6299. If primary standards are not available to allow the bi bias as of the re refe fere renc nce meas me asur urem emen entt pr proc oces esssin to be es esta tabli blishe shed, d,crossit is recommended that thee laboratory participate an interlaboratory check program as a means of demonstrating accuracy. NOTE   7—Sa 7—Sample mpless like hydro hydrocarb carbons ons from petr petroche ochemica micall proc process ess streams can degrade with time unless careful sampling and sample storage procedures proced ures are followed. It is critical that the compos composition ition of sample sampless taken for laboratory or at-line infrared analysis, or for laboratory measurement sureme nt of the reference reference data be representative representative of the process at the time the samples are taken, and that composition is maintained during storage and transport of the samples either to the analyzer or to the laboratory. Samplin Sam pling g shou should ld be done in acc accord ordanc ancee with methods methods like Practice Practicess D 126 1265 5 and D 405 4057, 7, or Pra Practi ctice ce D 4177 4177,, whic whicheve heverr are app applic licable able.. Whenev Whe never er poss possible ible,, sam sample ple stor storage age for ext extende ended d time periods periods is not recommended because of the likelihood of samples degrading with time in spite of sampling precautions taken. Degradation of samples can cause changes in the spectra measured by the analyzer and thus in the values estima est imated ted,, and in the pro proper perty ty or qua quality lity measured measured by the referenc referencee method.

A1.1) should be applied to the values to determine if all of the reference values came from the same population, or if one or more of the values is suspect and should be rejected. 10. Simple Procedur Proceduree to Develo Develop p a Feasib Feasibility ility Calibration 10.1 10 .1 For ne new w ap appl plic icat atio ions ns,, it is ge gene nera rall lly y no nott kno known wn whether an adequate IR multivariate model can be developed. In this case, feasibility studies can be performed to determine if th ther eree is a re rela lati tion onsh ship ip be betw twee een n th thee IR sp spec ectr traa an and d th thee compon com ponent ent/pr /prope operty rty of int intere erest, st, and whe whethe therr a mod model el of  adequate adequ ate precision precision coul could d possi possibly bly be built built.. If the feasibility feasibility calibration is successful, then it can be expanded and validated. A feasibility calibration involves the following steps: 10.1.1 Appro Approximat ximately ely 30 to 50 samp samples les are coll collected ected covering the entire range for the constituent/property of interest. Care should be exer exercised cised to avoid intercorrel intercorrelation ationss among major constituents unless such intercorrelations always exist in the materials being analyzed. The range in the concentration/  property should be preferably five times, but not less than three time ti mes, s, th thee st stan anda dard rd de devi viat atio ion n of th thee re repr prod oduc ucib ibil ilit ity y (reproducibility/2.77) of the reference analysis. 10.1.2 10. 1.2 When col collec lectin ting g spe spectr ctral al dat dataa on the these se sam sample ples, s, variations varia tions in part particle icle size, samp sample le prese presentati ntation, on, and proc process ess conditions which are expected during analysis must be reproduced.. Mult duced Multiple iple spectra of the same sampl samplee unde underr dif differe ferent nt conditions can be employed if such variations in conditions are anticipated during analysis. 10.1.3 Refe Reference rence analyses analyses on these samples samples are condu conducted cted using usi ng the acc accept epted ed ref refere erence nce me metho thod. d. If the range for the compon com ponent ent/pr /prope operty rty is not at lea least st five tim times es the sta standa ndard rd deviation of the reproducibility for the reference analysis, then   replicate analyses should be conducted on each sample such r  replicate that the =r  times  times the range is preferably five times, but at least three times, the standard deviation of the reference analysis. 10.1.4 A calibrati calibration on model is devel developed oped using one or more of the mathematical techniques described in Sections 11 and 12. The cali calibrati bration on model is prefe preferably rably tested using cros crosssvalida val idatio tion n met method hodss suc such h as SEC SECV V or PRE PRESS SS (se (seee 15. 15.3.6 3.6). ). Other statistics can also be used to judge the overall quality of  the calibration. 10.1.5 If the SECV value obtained obtained from the cross validation validation suggests that a model of adequate precision can be built, then additional samples are collected to round out the calibration set, and to serve as a validation set, spectra of these samples are collec col lected ted,, a fina finall mod model el is dev develo eloped ped,, and val valida idated ted as de-

9.4 If the reference reference method used to obta obtain in reference reference values for th for thee mu mult ltiv ivar aria iate te ca cali libr brat atio ion n is an es esta tabl blis ished hed AST ASTM M method, meth od, then repeatabilit repeatability y and repr reproduci oducibilit bility y data are included clu ded in the method. method. In thi thiss cas case, e, it is onl only y nec necess essary ary to demonstrate that the reference measurement is being practiced in accordance with the procedure described in the method, and that the repeatability obtained is statistically comparable to that published in the method. Data from established quality control procedures can be used to demonstrate that the repeatability of  the reference method is within ASTM specifications. If such data is not available, then repeatability data should be collected on at least three of the samples that are to be used in the calibration. These samples should be chosen to span the range of values over which the calibration is to be developed, one sample sam ple having having a ref refere erence nce value in the bottom bottom thi third rd of the range, one sample having a value in the middle third of the range, and one sample having a value in the upper third of the range. At least six reference measurements should be made on each sample. The standard deviation among the measurements should be calculated and compared to that expected based on the published repeatability.10 9.5 If the refe reference rence method method to be used for the multivariat multivariatee calibration is an established established ASTM method, and the samples to be used in the calibration have been analyzed by a cooperative testing test ing progr program am (for example, example, octan octanee value valuess obta obtained ined from recognize reco gnized d excha exchange nge group groups), s), then the reference reference valu values es obtained by the cooperative testing program can be used directly, and the stan standard dard deviations deviations estab establishe lished d by the coope cooperati rative ve testing program can be used as the estimate of the precision of  the reference data.

scribed in Sections 13, 14, and 15. 10

 Manual on Determining Precisi Precision on Data for ASTM Methods on Petrole Petroleum um Products and Lubricants, Lubricants, Available from ASTM Headquarters. Request Research Report RR: D02-1007.

11. Data Preprocessing Preprocessing 11.1 Various types of data preprocessing algorithms can be 6

 

E 1655 applie lied d to the spectral spectral data pri prior or to the developm development ent of a app multivariate calibration model. For example, numerical derivatives of the spectra may be calculated using digital filtering algorithms to remove varying baselines. Such filtering generally all y cau causes ses a sig signifi nifican cantt dec decrea rease se in the spe spectr ctral al sig signal nal-to -to-noise. Digital filters may also be employed to smooth data, improv imp roving ing sig signal nal to noi noise se at the exp expens ensee of res resolu olutio tion. n. A complete description of all possible preprocessing methods is beyond the scope of these practices. For the purpose of these

12.1.2 The technique should be capable capable of providing statisstatistics sui tics suitab table le for ide identi ntifyi fying ng if sam sample pless bei being ng ana analyz lyzed ed are outside the range for which the model was developed; that is, when the estimated values represent extrapolation of the model (see 16.3).

practices, preprocessing of the spectral data can be used if it produces a model which has acceptable precision and which passes the validation test described in Section 21. In addition, any spectral preprocessing method must be automated so as to provide provi de an exact exactly ly repr reproduci oducible ble result, and must be appl applied ied consistently to all calibration spectra, validation spectra, and to spectra of unknowns which are to be analyzed. 11.2 11. 2 One type of prepr preproces ocessing sing requires special special ment mention. ion. Mean-centering refers to a procedure in which the average of  the calibration spectra (average absorption over the calibration spectra as a function of wavelength or frequency) is calculated and subtracted from the spectra of the individual calibration samples prior to the development of the model. The average reference value among the calibration samples is also calculated, and subtracted from the individual reference values for

letters indicate matrix or vector dimensions.

th the e ca cali libr brat atio ion n data. samp sa mple s. spectral Thee mo Th mode dell reference is th then en value buil bu iltt data on are thee th mean-centered Ifles. the and mean-centered prior to the development of the model, then: 11.2.1 11. 2.1 When an unkno unknown wn sample is analy analyzed, zed, the avera average ge spectrum for the calibration site must be subtracted from the spectrum of the unknown prior to applying the mean-centered model, and the average reference value for the calibration set must be added to the estimate from the mean-centered model to obtain the final estimate; and 11.2. 11 .2.2 2 The deg degree reess of fre freedo edom m use used d in cal calcul culati ating ng the standa sta ndard rd err error or of cal calibr ibrati ation on mus mustt be dim dimini inishe shed d by one to accoun acc ountt for the deg degree ree of fre freedo edom m use used d in cal calcul culati ating ng the average (see 15.2).

is a vector of dimension  n  by 1, that is the difference between the reference values  values   y  and their estimates, y estimates,  yˆ , where:

12. Multiva Multivariate riate Calibration Calibration Mathematics

where  p  is the least square estimate of the prediction vector  p  p.. It should be noted that, in applying Eq 1-4, it is assumed that

NOTE   8—In the fol followi lowing ng der deriva ivation tions, s, mat matric rices es are ind indica icated ted usin using g boldface capital letters, vectors are indicated using boldface lowercase letters lett ers,, and sca scalar larss are indi indicat cated ed usin using g lowe lowerca rcase se let letter ters. s. Vect ectors ors are column vectors, and their transposes are row vectors. Italicized lowercase

12.1.3 All line linear, ar, multivariat multivariatee tech technique niquess are desi designed gned to solve the same generic problem. If   n  calibra  calibration tion spect spectra ra are measured at f  discrete  discrete wavelengths (or frequencies), then X then  X,, the spectral data matrix, is defined as an   f   by  n  matrix containing the spectra spectra (or some fun functi ction on of the spectra spectra pro produc duced ed by preprocessing, as described in Section 9) as columns. Similarly y   is a vec vector tor of dim dimens ension ion   n   by 1 cont containi aining ng the reference reference valuess for the calib value calibrati ration on sampl samples. es. The objec objectt of the linear, linear, multivariate modeling is to calculate a prediction vector   p   of  dimension   f  by   by 1 that solves Eq 1: y 5 Xtp 1 e

 

(1)

t

where   X is th thee tr tran ansp spos osee of th thee ma matr trix ix   X   obtaine obtained d by interchanging the rows and columns of  X.  X . The error vector, e vector,  e,,

yˆ  5 Xtp

 

(2)

12.1.4 The estimation estimation of the prediction prediction vector p vector p is  is generally calculated so as to minimize the sum of squares of the errors, ete 5 ?? e2 ?? 5 ~y – Xtp!t~y – Xtp!

 

(3)

Since   X  is generally not a square matrix, it cannot be directly Since  invert inv erted ed to sol solve ve Eq 3. Ins Instea tead, d, the pseudo pseudo or gen genera eraliz lized ed + inverse of   X, X , is calculated as: X1y 5 ~XXt!21Xy 5 p

12.1 Multivariate mathematical mathematical techniques techniques are used to relate relate the absorbances measured for a set of   calibration samples   to the refe reference rence values (prop (property erty or comp component onent conce concentra ntration tion values) obtained for this set of samples from a reference test. The object is to establish a multivariate  calibration model  that can be applied to the spectra of future, unknown, samples to estimate values (property or component concentration values). Only Onl y lin linear ear mul multiv tivari ariate ate tec techni hnique quess are des descri cribed bed in the these se practices; that is, it is assumed that the property or component concentration values can be modeled as a linear function of the sample absorptions. Various nonlinear multivariate techniques have been developed, but have generally not been as widely used as the following linear techniques. These practices are not intended to compare or contrast among these techniques. For the purpose of these practices, the suitability of any specific mathematical technique should be judged only on the follow-

 

(4)

the errors in the spectral data in  X  are negligible compared to the errors errors in the referen reference ce dat data, a, and that there is a lin linear ear relationship between the component concentration or property and the spectral data. If either of these assumptions is incorrect, then the linear models derived here will not yield an optimal estimate of  p.  p . 12.1.5 In calculatin calculating g the least square solution solution in Eq 4, it is assumed that the individual error values in   e   (see Eq 1) are normally distributed with common variance. This will be true if each of the individual reference values in   y   represents the result of a single reference measurement, and if the repeatability of the reference method is constant over the range of values in   y. If the values in   y   represent averages of more than one reference method determination, then the least square expression in Eq 4 is not applicable. If  r i  reference values  y i1,  y i2,  y i3, . . .   yir   are measured for calibration sample  i , then a weighted regres reg ressio sion n can be emp employ loyed. ed. If  If    R   is a dia diagon gonal al mat matrix rix of  dimension   n   by   n   containing containing the   r iva valu lues es fo forr ea each ch of th thee calibration samples, then the weighted regression is given by:

ing two criteria: 12.1.1 12. 1.1 The tec techni hnique que sho should uld be cap capabl ablee of pro produc ducing ing a calibration model that can be validated as described in Section 18; and 7

 

E 1655 =R y¯  5 =RXtp 1 e

 

~XRXt!21XRy¯  5 p

 

reference values should still be included in the y the  y  vector if they are available. The use of the average values will lead to better estimates of the regression coeffici coefficients, ents, but the model produced willl not be the least squ wil square aress min minim imum. um. Sta Standa ndard rd err errors ors of  calibrati cali bration on calc calculat ulated ed by the software will gener generally ally not be meaningful in these cases since they are not expressed relative to a single reference measurement. Standard errors of calibration should be recalculated using the procedure described in Section 11.

(5) (6)

where =R indicates the diagonal matrix containing the square roots of the  r ivalues, and y and y¯ is the vector containing the averages of th thee   r i   refere reference nce val values ues for eac each h sam sample ple.. If ave averag rages es of  multip mul tiple le ref refere erence nce val values ues are use used d in   y   and a we weig ight hted ed regression is used, special care must be taken to add back the variance removed by calculating the average reference values (see Sec (see Sectio tion n 11 11)) so tha thatt the statist statistics ics for the model model can be compared to those for a single reference value determination. The specific method in which the weighting is applied depends on the specific multivariate mathematics that are employed.

12.2.4 12. 2.4 The choice choice of the number number of wav wavele elengt ngths hs (or frequencies), k , to use in multilinear regression is a critical factor in the model development. If too few wavelengths are used, a less precise model will be developed. If too many wavelengths are use used, d, col coline ineari arity ty amo among ng the abs absorp orptio tion n val values ues at the these se wavele wav elengt ngths hs may lea lead d to an uns unstab table le mod model. el. The opt optimu imum m number of wavelengths (or frequencies) for a model is related to the number of spectrally distinguishable components in the calibration spectra (see Section 15) and can generally only be dete de term rmin ined ed by tr tria iall an and d er erro rorr. As a ru rule le,, th thee nu numb mber er of  wavelengt wavel engths hs (or frequencies) frequencies) used must be lar large ge enoug enough h to produce a model with the desired precision, but small enough to produce a stable model that passes validation. 12.2.5 The choice of speci specific fic wavelengths wavelengths (or freq frequenci uencies) es) to include in a multilinear regression model is also a critical

12.1.6 For most cases, if the calibration calibration spectra are are collected over an extended wavelength (or frequency) range, the number of individual absorption values per spectrum,  f , will exceed the number numb er of cali calibrat bration ion spec spectra, tra,   n. In this cas case, e, the matrice matricess (XXt)   and   (XRXt)   are rank deficient and cannot be directly inverted. inve rted. Even in cases where   f   <   n, colin colineari earity ty amon among g the calibration spectra can cause   (XXt)   and   (XRXt)   to be nearly singular (to have a determinant that is near zero), and the direct use of Eq 4 and Eq 6 can produce an unstable model, that is, a model for which changes on the order of the spectral noise level produce significant changes in the estimated values. In order to solve Eq 4 and Eq 6, it is therefore necessary to reduce the dimen dimensiona sionality lity of  of    X   so th that at a st stab able le in inve vers rsee ca can n be calculated. The various linear, mathematical techniques used for multivariate calibration are different means of reducing the dimensionality dimensionalit y of  X so  X  so as to be able to calculate stable inverses of   (XXt)   and and (XRX  (XRXt)   and the estimate   p. 12.2  Multilinear Regression Analysis : 12.2.1 In mult multiline ilinear ar regression regression (MLR (MLR), ), a speci specific fic numb number er of wavelengths (or frequencies),  k , are chosen such that  k   3k/n should be eliminated from the calibration set in the development of the model.

16.2.1 If  x 16.2.1  x   is a spectral vector (dimension  f  by 1) and   X   is the matrix of calibration spectra (of dimension  n  by  f ), ), then the leverage statistic is defined as: t

t 1

h 5 x ~XX ! x

 

NOTE   17—If the leverage statistic is scaled as described in (25) in (25),, an  f  test  test can be employed for outlier detection.

(61)

16.3.3 If cali calibrati bration on spect spectra ra with   h   >3k/n   are elim eliminat inated ed from fro m the calibra calibratio tion n set set,, and the mod model el is reb rebuil uilt, t, it is not uncommon for additional spectra with  h  >3 k/n  to be identified for the new model. This occurrence is most likely if removal of  samples reduces   k , but can also be caused merely by scaling changes chang es to the multivariat multivariatee spac spacee indu induced ced by chan changes ges in   n. When Whe n rep repeti etitiv tivee app applic licati ation on of the 3k/n   rule con contin tinues ues to identi ide ntify fy out outlie liers, rs, the out outlie lierr tes testt is sai said d to “sn “snowb owball all.” .” If  “snowballing” occurs, it may indicate some problem with the struct str ucture ure of the spectral spectral data set set.. The variable variable space of the model should be examined for unusual distributions or clusterings. 16.3.3.1 16.3.3 .1 If the following following sequence occurs occurs during the devel devel-opment of a model, the 3k/n  outlier test can be relaxed: (1) a first model is built on an initial calibration set, ( 2) calibration spectra with  h  >3 k/n are eliminated from the calibration set, (3) a second model using the same number,  k , variables is built on the subset of cali calibrat bration ion spectra, spectra, and ( 4) cali calibrat bration ion spect spectra ra with   h   >3k/n  are identified for the second model. The second model should be used providing that no calibration samples have   h  greater than 0.5. 16.3.3 16. 3.3.2 .2 If (1) a first model is built on an initial calibration set, (2) calibration spectra with  h  >3 k/n are eliminated from the calibration set, and (3) a second model using fewer variables is built on the subset of calibration spectra, the 3k/n  outlier test should not automatically be relaxed. Instead, the first model should be rebuilt using the lower number of variables and the sequence in 16.3.3.1 should be applied to the new model. 16.3 16 .3.4 .4 A se seco cond nd ty type pe of ou outl tlie ierr is on onee fo forr wh whic ich h th thee estimated value yˆ dif differs fers by a statistically significant amount from the value from the reference method, y. Such outliers can be detected based on studentized residuals. If   ei  is the difference between the estimated es timated value yˆ i  and the reference value y i for the   ith sample in the calibration set, and   hi  is the leverage statistic for that sample, the studentized residuals for the   ith sample are given by:

 x and  X in 16.2.2 For a mean-centered mean-centered calibration, calibration, x  and X  in Eq 61 are ¯   respectively. replaced by  by   x   −   x¯   and and X  X   −   X 16.2.3 If a weigh weighted ted regression regression is used, the expr expressio ession n for the leverage statistic becomes: h 5 xt ~XRXt!1x

 

(62)

16.2.4 In MLR, MLR, if  m is  m  is the vector (dimension  k  by 1) of the selected absorbance values obtained from a spectral vector   x, and   M   is the matrix matrix of sel select ected ed abs absorb orbanc ancee val values ues for the calibration samples, then the leverage statistic is defined as: h 5 mt ~MMt!21m

 

(63)

16.2.5 Simi 16.2.5 Similar larly ly,, if a wei weight ghted ed reg regres ressio sion n is use used, d, the expression for the leverage statistic becomes: t

t 21

h 5 m ~MRM !

m

 

(64)

16.2.6 In PCR and PLS, the leverage statisti statisticc for a sample with spectrum x spectrum  x is  is obtained by substituting the decompositions for PCR, or for PLS, into Eq 61. The statistic is expressed as: h 5 sts

 

(65)

NOTE  16—If the scores from the PCR or PLS model are not normalized, then the form of Eq 65 becomes  D 2=  s t (StS)−1s

16.2.7 If a wei 16.2.7 weight ghted ed PCR or PLS reg regres ressio sion n is used, the expression for the leverage statistic becomes h 5 st ~StRS!21s

 

(66)

16.3   Outlier Detection During Calibration: 16.3.1 16. 3.1 Two typ types es of out outlie liers rs can be ide identi ntified fied during during the calibration procedures. The first type of outlier is a sample that represents an extreme composition relative to the remainder of  the calibration set. These samples have very high leverage on the regression results; that is, they are largely responsible for the determination of at least one of the regression coefficient values. Generally, there is insufficient data in the calibration set to sta statis tistic ticall ally y det determ ermine ine the acc accura uracy cy of ref refere erence nce val values ues associated with these high leverage samples. Their inclusion in the calibration may lead to erroneous estimations of similar samples if the reference value for the high leverage sample is

ti 5

in er erro rorr. Th Thee se seco cond nd ty type pe of ou outl tlie ierr is on onee fo forr wh whic ich h th thee estimated value differs from the reference value by a statistically significant amount. Such outliers indicate either an error in the reference measurement or a failure of the model.

ei

SEC

12h

 =

 

(67)

16.3.4.1 The studentized 16.3.4.1 studentized residuals residuals should be norm normally ally distributed trib uted with comm common on vari variance. ance. The stud studentiz entized ed resi residuals duals value can be compared to a t distribution value for   n − k   (or 14

 

E 1655 n − k − 1  if mean centered) degrees of freedom, to determine the probability that the error in the estimate fits the expected distribution. If not, the sample should be considered an outlier. A more detai detailed led disc discussi ussion on of stude studentiz ntized ed resi residuals duals can be found in Refs 26–27 Refs  26–27.. 16.3 16 .3.5 .5 If a sa samp mple le is id iden enti tifie fied d as an ou outl tlie ierr ba base sed d on studentized residuals or other similar tests, then the reference value may be in error. When possible, the reference test should be re repe peat ated ed to de dete term rmin inee a co corr rrec ectt va valu luee fo forr th thee sa samp mple le

comparing an estimate of the unknown spectrum derived from the model to the measured spectrum of the unknown. 16.4 16 .4.4 .4.1 .1 For PC PCR, R, an es esti tima mate te of th thee sp spec ectr trum um of th thee unknown can be calculated as: xt 5   sˆ t(Lt

(68)

where the  the   sˆ  is the vector of scores. Similarly for PLS: xt 5   sˆ tLt

(69)

where the s the  sˆ  is the vector of scores. The difference between the estimated spectrum and the actual spectrum can be calculated as:

(multiple tests are recommended). If the reference value is not in error, then the large studentized residuals may indicate a basic bas ic fai failur luree in the mod model. el. For est estima imatio tion n of com compon ponent ent concentrations, there may be sufficient spectral interferences to preclude accurate estimation of the component for this class of  samples. For property estimation, some component that has a significant effect on the property may not be detected. Removing in g ou outl tlie iers rs of th this is ty type pe wi with thou outt ev evid iden ence ce of er erro rorr in th thee reference refe rence value shoul should d be avoid avoided ed whene whenever ver possi possible, ble, sinc sincee these samples may provide the only indication that the model is not applicable to a certain class of materials. 16.4  Interpolation and Extrapolation of the Model During  Analysis: 16.4.1 The spectra spectra of the calibrati calibration on samples samples define a set of  variables that are used in the calibration of the multivariate

r 5 x 2 x

 

(70)

16.4.4.2 The root mean squa 16.4.4.2 square re spectral residuals residuals (RMSSR) (RMSSR) for the spectrum can then be calculated as: RMSSR 5

Œ 

rtr  f 

 

(71)

NOTE   19—Some commercial commercial softwar softwaree packa packages ges may calculate other statistics related to RMSSR, or may call RMSSR by some other name. The model developer should verify what statistics are used in the software to indicate how well the model fits a spectrum being analyzed. The RMSSR is intended as an example of how such a calculation can be done. Other similar statistics can be used.

16.4.5 The RMSSR values values can be calculated calculated for each of the calibration samples. One of the calibration samples will exhibit a maximum RMSSR, RMSSRmax. Assuming that outliers have been bee n rem remove oved d pri prior or to the dev develo elopme pment nt of the cal calibr ibrati ation on model,, RMSSRmax   can be use model used d to cal calcul culate ate a cut cutof offf abo above ve which RMSSR values for unknown spectra are to be taken as evidence of extrapolation of the model. 16.4.6 In general, general, the RMSSR RMSSRmax cannot be used directly to set the cutoff for indicating extrapolation. For PCR and PLS models mod els,, som somee of the spe spectr ctral al noi noise se cha charac racter terist istics ics of the calibrati cali bration on spec spectra tra are alway alwayss inco incorpora rporated ted into the spect spectral ral variables. The RMSSR values calculated for spectra used in the calibrati cali bration on will thus gener generally ally be lower than corr correspon esponding ding values calculated for spectra of the same samples which are not used in the model development. For estimating a suitable cutoff  RMSSR value to serve as an indication of extrapolation, the following procedure is recommended. 16.4.6.1 Replicate spectral spectral measurements measurements (at least seven) of  several sever al (at least three) of the cali calibrat bration ion samples should be made. The replicate measurements should include all steps in the meas measurem urement ent proc procedure edure (for exam example, ple, backg background round spectrum collection, loading of the sample, and measurement of the spectrum). 16.4.6 16. 4.6.2 .2 One spectru spectrum m fr from om the set is to be use used d in the development of the calibration model. The RMSSR values for the spectra used in the calibration are calculated. The RMSSRcal  ( i) is the value for the spectrum of Sample   i. 16.4.6.3 The remaining replicate replicate spectra are analyzed analyzed using the calibration model, and RMSSR values are calculated and averaged for each sample. The RMSSRanal   (i) is the average RMSSR for the replicate spectrum of Sample   i.

model. If, when unknown samples are analyzed, the variables calculated from the spectrum of the unknown sample lie within the range of the variables variables for the cali calibrat bration, ion, the esti estimate mated d value for the unknown sample is obtained by interpolation of  the model. If the variables for the unknown sample are outside the range of the variables in the calibration model, the estimate represents an extrapolation of the model. 16.4.2 Two type typess of extr extrapola apolation tion are possi possible. ble. First, the sample may contain the same components as the calibration samples, but at concentration ranges that are outside the ranges in the calibration set. Second, the sample may contain components that were not present in the calibration samples. 16.4.3 The leverage leverage statistic, statistic,  h , provides a useful indication of the first type of extrapolation. For the calibration set, one sample will have a maximum leverage statistic,   hmax. This is the most extreme sample in the calibration set, in that, it is the farthest from the center of the space defined by the spectral variables. If the leverage statistic for an unknown sample is greater grea ter than   hmax, the then n the estimate estimate for the sam sample ple cle clearl arly y represents an extrapolation of the model. Providing that outliers have been eliminated during the calibration, the distribution of  h  h  should be representative of the calibration model, and  h max can be used as an indication of extrapolation. NOTE  18—Comparison of the spectral variables for an unknown against the range of each spectr spectral al variable in the calibra calibration tion model could be done, and extrapolation of any single variable could be taken as extrapolation of  the model. The use of the leverage statistic as an indicator of extrapolation may not detect certain spectra which are slight extrapolations on one or more spectral variables; variables; however however,, signific significant ant extrapolation extrapolation of any one variab var iable le wil willl res result ult in a hig high h lev levera erage ge stat statisti istic, c, and thus dete detectio ction n of  extrapolation. Use of individual variables in tests for extrapolation is not recommended since it can unduly restrict the range of samples to which the model is applicable.

16.4.6.4 16.4.6 .4 The ratios ratios of the RMSSR values values from the analyses to those from the calibration are calculated and averaged, and RMSSRmax   is mul multip tiplie lied d by the average average rat ratio io to obt obtain ain the cutoff:

16.4.4 16. 4.4 The sec second ond typ typee of ext extrap rapola olatio tion n of the mod model, el, namely, the presence of a new component, can be detected by 15

 

E 1655

F

RMSSRlimit 5 (

RMSSRanal~i! RMSSRmax RMSSRcal~i!

G

 

sample spectra. A maximum NND value is determined. This value val ue rep repres resent entss the lar larges gestt dis distan tance ce bet betwee ween n cal calibr ibrati ation on sample spectra. 16.4.8.5 16.4.8 .5 Durin During g analy analysis, sis, the NND value is calculated calculated for the unk unknow nown n sam sample ple spe spectr ctrum um rel relati ative ve to the cal calibr ibrati ation on spectra. If the calculated value is greater than the maximum NND fro from m 16. 16.5.3 5.3,, the then n the min minim imum um dis distan tance ce bet betwee ween n the process sample spectrum and the calibration spectra is greater than the largest distance between calibration sample spectra,

(72)

16.4.6.5 If the RMSSR value for an unkno 16.4.6.5 unknown wn sample being analyzed exceeds RMSSRlimit, then the analysis of the sample represents an extrapolation of the model. 16.4.7 Statistics Statistics comparable comparable to RMSSR cannot be calcu calcu-lated for multiple linear regression. The MLR is thus incapable of det detect ecting ing the sec second ond typ typee of ext extrap rapola olatio tion, n, nam namely ely,, the presence of a new component that was not in the calibration sample sam ples. s. Car Caree sho should uld be exe exerci rcised sed whe when n app applyi lying ng MLR in systems where the calibration set used in the development of  the MLR model may not represent the total range of sample compositions that will be encountered during analyses. In such cases, MLR should be supplemented with other techniques to determine if the sample being analyzed falls within the scope of the calibration. calibration. For exam example, ple, outlier statistics statistics from PCR models developed on the same calibration set could be used for this purpose.

the unknown sample spectrum falls within a sparsely populated region of the calibration space. Such samples are referred to as Nearest Neighbor Outliers. 17. Selec Selection tion of Calibration Samples Samples 17.1 For the development development of a mult multivar ivariate iate model, model, an idea ideall calibration sample set will: 17.1.1 17. 1.1 Contai Contain n sam sample pless whi which ch pro provid videe exa exampl mples es of all chemical components which are expected to be present in the samples which are to be analyzed using the model, thereby ensuring that analyses involve interpolation of the model; 17.1.2 Conta Contain in samples for which the range of vari variation ation in the conce concentra ntrations tions of the chem chemical ical components components excee exceeds ds the rang ra ngee of va vari riat atio ion n ex expe pect cted ed fo forr sa samp mple less wh whic ich h ar aree to be analyz ana lyzed ed usi using ng the mod model, el, the thereb reby y ens ensuri uring ng tha thatt ana analys lyses es

NOTE   20—For PLS models, residuals residuals calculations calculations such as RMSSR are not al not alwa ways ys a use usefu full ind indic icat ator or of out outlie liers rs.. If If,, du durin ring g ca cali libr brati ation on,, a significant percentage of the spectral( X -block) -block) variance due to signal is not used in the model, then the model residuals used to calculate RMSSRcal may contain significant contributions due to calibration sample component absorptions. In such cases, RMSSRlimit  values calculated on the basis of  cal such RMSSR may beintoo large tobeing detect model extrapolation due to new chemical values components samples analyzed. Thee pr Th proc oced edur uree de desc scri ribe bed d in 15 15.3 .3.3 .3 ca can n be use used d to est estim imat atee th thee percen per centage tage of the tota totall   X -block - block variance variance that is due to sign signal. al. If the varian var iance ce incl include uded d in the mod model el is sign signific ificant antly ly les lesss tha than n the signal variance, then the modeler may wish to supplement the PLS model with a PCR model built on the same data. RMSSR statistics from the PCR model are then used for outlier detection. The number of variables used in the PCR model should be sufficient to account for the signal variance.

involve ofples the 17.1.3 17. 1.3interpolation Contai Con tain n sam sample s model; for whi which ch the con concen centra tratio tions ns of  chemical components are uniformly distributed over their total range of variation; 17.1.4 Contain a suffi sufficient cient number of samples to statistically statistically define the relationships between the spectral variables and the component concentrations or properties to be modeled. 17.2 For simple mixtures, mixtures, calibration samples samples can generally be prepared to meet the criteria above. For complex mixtures, obtaining an ideal calibration set is difficult, if not impossible. The statistical tests that are used to detect outliers guard against non-ideal non-i deal calibration calibration sets sets.. The RMSSR values detect when sample sam pless bei being ng ana analyz lyzed ed con contai tain n com compon ponent entss tha thatt are not repres rep resent ented ed in the cal calibr ibrati ation on set (vi (viola olatio tion n of cri criter terion ion 1 above). above ). Lever Leverage age stati statistic sticss dete detect ct when samples being analyzed are outside the concentration ranges represented in the calibr cal ibrati ation on set (vi (viola olati tion on of cri criter terion ion 2). Out Outlie lierr det detect ection ion during model development identifies components for which the range of concentrations in the calibration set is not uniform (violation of criterion 3). 17.3 The number of samples that that are required to calibrate calibrate an infrared multivariate model (see 17.1.4) depends on the complexity of the samples being analyzed. If the samples to be analyzed contain only a few components that vary in concentration, then there will be a small number of spectral variables, and a relatively small calibration set is adequate to define the relationship between the variables and the concentrations or proper pro pertie ties. s. If a lar larger ger num number ber of com compon ponent entss var vary y in the samples to be analyzed, then a larger number of calibration samples is required for the model development. Determining whether or not a set of calibration samples is adequate can only

16.4.8   Neare Nearest st Neig Neighbor hbor Dist Distance ance—If the cal calibr ibrati ation on sample sam ple spe spectr ctraa for form m mul multip tiple le clu cluste sters rs wit within hin the var variab iable le space, the spectrum of the unknown being analyzed can have a  D2 less than  D2max yet fall into a relatively unpopulated portion of the calibration space. In this case, the sample being analyzed contains the same components as the calibration samples (since the sample is not a RMSSR outlier), but at combinations that are not represented in the calibration set. The spectrum of the unknow unk nown n doe doess not bel belong ong to any of the calibrat calibration ion sample sample spectra clusters, and the results produced by application of the mode mo dell ma may y be in inva vali lid. d. Un Unde derr th thes esee ci circ rcum umst stan ance ces, s, it is desirable to employ a Nearest Neighbor Distance test to detect unknow unk nown n sam sample pless tha thatt fal falll wit within hin voi voids ds in the cal calibr ibrati ation on space. 16.4.8.1 16.4.8 .1 Neare Nearest st Neigh Neighbor bor Dista Distance, nce, NND, meas measures ures the distance dist ance betwe between en the spec spectrum trum being anal analyzed, yzed,   x, and individual spectra in the calibration set,  set,   xi. NND 5 min@~x 2 xi!t  ~XXt !21 ~x 2 xi!#

 

(73)

16.4.8.2 16.4.8 .2 For MLR, NND is calculated calculated as NND 5 min@~m 2 mi !t  ~MMt !21~m 2 mi !#

 

(74)

16.4.8.3 For PCR and PLS (with orthogonal 16.4.8.3 orthogonal scores), scores), NND is calculated as NND 5 min@~s 2 si!t  ~s 2 si!#

 

be done after a model is developed and an estimate of the number of spectral variables required for the model is made. 17.4 17. 4 If a mul multiv tivari ariate ate mod model el is dev develo eloped ped usi using ng thr three ee or fewer few er var variab iables les,, the then n the cal calibr ibrati ation on set sho should uld con contai tain n a

(75)

16.4.8.4 16.4.8 .4 NND valu values es are calc calculate ulated d for all the calibration calibration 16

 

E 1655 minimum of 24 samples after elimination of outliers. 17.5 17. 5 If a mul multiv tivari ariate ate mod model el is dev develo eloped ped usi using ng   k   (>3) variables, then the calibration set should contain a minimum of  6k  spectra   spectra after elimination of outliers. If the model is mean centered, a minimum of 6(k  +   + 1) spectra should remain.

for which the model was developed; that is, the span and the standard deviation of the range of concentrations or property values for the validation samples should be at least 95 % of the span and the standard deviation of the range of concentrations or pro proper perty ty val values ues in the mod model, el, and the concentr concentrati ation on or property values for the validation samples should be distributed as uniformly as possible across the range; and 18.2.3.2 18.2.3 .2 Span the range of spect spectral ral variables variables for which the model was developed; that is, if the range of a spectral variable

NOTE   21—6k   is ch chose osen n to ensure ensure at lea least st 20 df in the mo mode dell fo forr statist sta tistica icall tes testing ting,, and to ens ensure ure that the there re is an ade adequa quate te num number ber of  samples to define the relationship between the spectral variables and the concentration concen tration or proper property ty values.

17.6 17. 6 For som somee spe spectr ctrosc oscopi opicc ana analys lyses, es, it is pos possib sible le to calibratee usin calibrat using g gravi gravimetr metrical ically ly or volum volumetri etricall cally y prepa prepared red mixtures mixt ures which conta contain in sign significan ificantly tly fewer comp component onentss than the sam sample pless whi which ch wil willl ult ultima imatel tely y be ana analyz lyzed. ed. For the these se surrogate surr ogate methods, the outl outlier ier stat statisti istics cs descr described ibed here herein in are not strictly appropriate since all actual samples are by definition tio n out outli liers ers rel relati ative ve to the sim simpli plified fied cal calibr ibrati ations ons.. Thu Thus, s, surrogate surr ogate methods canno cannott stri strictly ctly fulfill the requ requirem irements ents of  this practice. Surrogate methods should, however, follow the requireme requi rements nts described described here herein in for the numbe numberr and range of  calibration samples.

in the cal calibr ibrati ation on mod model el is fro from m   a   to   b, and the sta standa ndard rd deviat dev iation ion of the spe spectr ctral al var variab iable le is   c, th then en th thee sp spec ectr tral al variables estimated for the validation samples should cover at least 95 % of the range from  a  to  b , and should be distributed as uniformly as possible across the range such that the standard deviation in the spectral variables estimated for the validation samples will be at least 95 % of   c. 18.2.4 Determination of of whether a validation set set is adequate willl gen wil genera erally lly require require tha thatt the set be ana analyz lyzed ed so tha thatt the spectr spe ctral al var variab iables les for the set can be det determ ermine ined. d. Sam Sample pless whose analyses are extrapolations of the model should not be included in the validation set. If the validation set does not meet the criteria in 18.2.3.1 and 18.2.3.2, additional validation samples should be taken. 18.3   Vali Validat dation ion Spe Spectr ctra a Mea Measur sureme ement nt and Anal Analysi ysiss—

18. Validati alidation on of a Multivariate Model 18.1 Validation of an infrared multivariate model is accomplished by applying the model for the analysis of a set of   v vali validati on samples, les, statisti stat isticall cally yerence comparing compa ring the estimate mates s for dation these the se samp sample sam ples s and to kno known wn refere ref nce val values ues. . Vesti alidat ali dation ion requ re quir ires es th thor orou ough gh te test stin ing g of th thee mo mode dell to en ensu sure re th that at it performs up to the expectations derived from the calibration set statistics. 18.2   Validation Sample Set : 18.2.1 For the vali validati dation on of a multivariate multivariate model, model, an ideal validation sample set will: 18.2.1 18. 2.1.1 .1 Contain Contain sam sample pless tha thatt pro provid videe exa exampl mples es of all chemical components which are expected to be present in the samples which are to be analyzed using the model; 18.2.1.2 Contain samples for which which the range of variation in in the concentrations of the chemical components is comparable to the range of variation expected for samples that are to be analyzed using the model: 18.2.1.3 18.2.1 .3 Conta Contain in sampl samples es for which the concentratio concentrations ns of  chemical components are uniformly distributed over their total range of variation; and 18.2.1.4 18.2.1 .4 Conta Contain in a suf suffficie icient nt number of samp samples les to stat statisti isti-cally test the relationships between the spectral variables and the component concentrations or properties that were modeled. 18.2.2 For simp simple le mixt mixtures ures,, vali validatio dation n sampl samples es can generally be prepared to meet the criteria in 18.2.1.1-18.2.1.4. For complex mixtures, obtaining an ideal validation set is difficult if not impossible. 18.2.3 The number of samples needed needed to vali validate date an infrared mul multiv tivari ariate ate mo model del dep depend endss on the com comple plexit xity y of the model. Only samples whose analyses are found to be interpolations of the model should be used in the validation procedure. If five or fewer spectral variables are used in the model, then a minimum of 20 interpolation samples is recommended. If  k  >

Spectra of validation samples should be collected using exactly the same pro proced cedure uress as wer weree use used d to col collec lectt spe spectr ctraa of the calibra cali bration tion sam samples ples.. Refe Referenc rencee valu values es for the vali validati dation on samples should be obtained using the same reference method as was use used d for the calibrat calibration ion sam sample ples. s. Spe Spectr ctraa sho should uld be analyzed using the multivariate model to produce estimates of  the component concentrations or properties, and the statistics described in Sections 18 and 19 should be calculated. 18.4   Validation Error : 18.4.1 18. 4.1 If   v   (a vec vector tor of dim dimens ension ionss   v   by on one) e) ar aree th thee estimates obtained by analysis of the spectra of the  v  validation samples, and   v  are the corresponding values measured by the reference method, then the validation error,  error,   e  is given by: e 5 v 2 v

 

(76)

18.4.2 If multiple reference reference values are available available for some of  the val valida idatio tion n sam sample ples, s, the then n the ave averag ragee of the ind indivi ividua duall reference refe rence measuremen measurements ts can be used in   v, and the variance removed by calculating the averages should be calculated using Eq 52. 18.5  Variance of the Validation Error —The —The variance of the error of the validation measurements is calculated as: v

r i

VARv 5 etRe 1 s2avg 5   (

(

i 5 1  j 5 1

~vij 2   vi! 2

(77)

where   s2avg  is zero and   R  is an identity matrix if individual reference measurements are used in  v . 18.6  Standard Error of Validation: 18.6.1 The standard standard error of vali validatio dation n (SEV) is give given n by: v

r i

( ( SEV  5

5 spectral variables are used in the model, then a minimum of  4k  interpolation   interpolation samples should be used in the validation. In addition, the validation samples should: 18.2.3.1 Span the range of concentrations concentrations or property property values

v VAR d  5

Œ 

v

i 5 1  j 5 1

! 

~vij 2 vi! 2 (78)

v

(

i51

r i

d v  is the total number of reference values available for all  v 17

 

E 1655 SEC 3   =1 1 D to yˆ +  t   3   SEC 3   =1 1 D . If more than 5 % of the reference values fall outside this range, then the confidence limit estimates based on SEC are questionable, and further testing is required to demonstrate the agreement between the model and the reference method. 2

valida idati tion on sam sample ples. s. SEV is the sta standa ndard rd dev deviat iation ion in the val differe dif ferences nces between refe reference rence and IR esti estimate mated d valu values es for samples in the validation set. The standard error of validation is sometimes referred to as a standard error of prediction. A bias corrected version of this statistic has also been defined as the standard error of performance. To avoid confusion between two terms that are both abbreviated SEP, the use of SEV is preferred in these practices. 18.6.2 Stude Studentiz ntized ed residuals residuals test testing ing can be appl applied ied to the

18.10.2 18.10 .2 An alternative alternative method can be used to demo demonstra nstrate te agreement between the model and the reference method. This altern alt ernati ative ve met method hod is pre prefer ferred red whe when n the pre precis cision ion of the reference method is not constant across the range of reference values used in the calibration, but can be applied even when the precis pre cision ion is con consta stant. nt. If R(yi) is the reproduc reproducibi ibilit lity y of the reference method at level yi, then the percentage of reference values for which:

estimates of the validation set to detect possible errors in the reference values. 18.7   Validation Bias—The average bias for the estimation of the validation set, e¯ v, is calculated as: v

v

(

e¯ v 5

 j 5 1

ij

5

d v

r i

( ( ~v

r iei

yˆ i 2 R ~   yˆ i ! , yij ,   yˆ i 1 R ~  yˆ i!

2   vi!

i 5 1  j 5 1

(79)

v

(

where  r i  is 1 if individual reference values were used, or is the number num ber of ref refere erence nce val values ues tha thatt wer weree ave averag raged ed for the   ith validation sample if averages are used.  d v  is the total number of  reference values used in the calculation.

SDV 5

Œ 

i

i

2

v

d v 2 1

 

!  v

1 s2avg

5

r i

( ( @~v i 5 1  j 5 1

ij

instrument is properly calibrated. This instrument qualification procedure typically involves the analysis of gravimetrically or volumetri volum etricall cally y prepa prepared red mixt mixtures ures that cont contain ain sign significan ificantly tly fewer components than the samples which will ultimately be analyzed. There is no a priori relationship between the standard error err or tha thatt is cal calcul culate ated d fro from m thi thiss pro proced cedure ure and the err error or expected during application of the model to actual samples. To avoid avo id con confus fusion ion,, it is rec recomm ommend ended ed tha thatt the pro proced cedure ure be referred to as a spectrometer/spectrophotometer qualification, not validation. Additionally, it is recommended that the standard error calculated from this procedure be referred to as a Standard Error of Qualification (SEQsurrogate ), not as a Standard Error of Validation.

2 vi! 2   e¯ v# 2

v

~ ( r i ! 2 1 i51

(80)

where   r i  is 1 and   s2avg  is 0 if individual reference measurements are used in calculating  calculating   yˆ . 18.9   Significance of Validation Bias: 18.9.1 18. 9.1 A   t    test test is us used ed to de dete term rmin inee if th thee va vali lida dati tion on estimate esti matess show a stat statisti isticall cally y signi significant ficant bias. A   t   value value is calculated as: t  5

| e¯ v| =d v SDV 

 

(82)

18.11 18.1 1 For mult multivari ivariate ate analy analyses ses empl employing oying surrogate surrogate cali cali-brations, a procedure similar to that described here for validation is often performed for the purpose of verifying that the

18.8   Standard stan-Standard Devi Deviation ation of Valida alidation tion Err Errors ors—The stan dard deviation of the validation errors, SDV, is calculated as v

 

is cal calcul culate ated. d. If 95 % or mor moree of the referenc referencee val values ues fall within this interval, then estimates produced with the multivariate IR model agree with those produced by the reference method as well as a second laboratory repeating the reference measurement would agree.

r i

i51

( r ~e 2   e¯ ! i51

2

(81)

19. Prec Precision ision of Infra Infrared red Estimated Values Values

The  t  value   value of is compared   values from Tab Table le A1.3 for d v  degrees freedom. to critical  t  values

19.1 19. 1 The pre precis cision ion of val values ues est estim imate ated d fro from m an inf infrar rared ed multivariate model is calculated from repeated spectral measurements. The number of samples for which repeat measurement me ntss is ma made de sh shou ould ld be at le leas astt eq equa uall to th thee nu numb mber er of  variables used in the model, and never less than three. The samples used for repeat spectral measurements should span at least 95 % of the rang rangee of conce concentrat ntration ion or prope property rty values used in the model. When possible, samples should be selected to ens ensure ure tha thatt som somee var variat iation ion on eac each h spe spectr ctral al var variab iable le is exhibited among the samples. At least six spectra should be collected for each sample. The spectra should be analyzed and values estimated. The average estimate for each sample should be calculated, and the standard deviation among the estimates should be obtained. If  y  y ij  is the estimate for the  j th spectrum of  r i  total spectra for the  i th sample, then the average estimate for

18.9.2 18. 9.2 If the   t   value is less than the critical   t  value, then analyses based on the multivariate model are expected to give essentially the same average result as measurements conducted by the reference method, provided that the analysis represents an interpolation of the model. 18.9.3 If the the  t  value calculated is greater than the tabulated t  value,   value, there is a 95 % probability that the estimate from the multivariate model will not give the same average results as the reference method. Validity of the multivariate model is then suspec sus pect. t. Fur Furthe therr inv invest estiga igatio tion n of the mod model el is req requir uired ed to resolve the probable bias that is indicated. 18.10  Validation of Agreement Between Model and Reference Method : 18.10. 18. 10.1 1 The con confide fidence nce lim limits its on the est estima imates tes for the validation samples should be calculated, and a determination should be made as to whether the individual reference values for the valid validation ation samples lie withi within n the rang rangee from yˆ −  t 3

this sample is:

r i

(

 yˆ  ˆ i 5

18

 j 5 1

r i

 yˆ  ˆ ij

(83)

 

E 1655 standard devi deviatio ation n of the repli replicate cate estimates estimates is 19.1.1 The standard calculated as:

 Œ  r i

si 5

( ~ yˆ  2   y¯ ¯  !  j 5 1 ij

20.2   Sampling Related Errors—Table 2 lists errors arising from sampling problems and possible solutions to these problems  (28)  (28).. 20.3  Sources of Calibration Error —Table —Table 3 lists sources of  error in the development of the calibration model and possible ways to minimize these errors. 20.4  Analysis Errors—Table 4 lists factors that can contribute to errors in the estimated values for unknown samples and possible ways to minimize such errors.

2

i

 

r i 2 1

(84)

19.2 A  x 2 value is calculated using the standard deviation values calculated in Eq 81: 2.3026

2

x 5

r i

2

2

5 1 r i  log si ! c   ~r  log  log s 2  i (

 

(85)

21. Wavelen avelength gth (Frequency) (Frequency) Sensitivity of a Multiva Multivariate riate Model

where: t 

r  5   ( r i

 

i51

s5 c511

Œ 

1 t  r is2i r  i ( 51

1

 S (

3~ z 2 1!

 z

21.1 Wavelength stability of spectrometers spectrometers is often a critical factor in the performance factor performance of a mult multivar ivariate iate calibration calibration.. The estimation of the sensitivity of a multivariate model to changes in the wavelength scale provides a useful parameter against which instrument performance can be judged. The wavelength sens se nsit itiv ivit ity y of a mo mode dell ca can n be ro roug ughl hly y es esti tima mate ted d by th thee following procedure: 21.1.1 Iden Identify tify the samples in the cali calibrati bration on set that reprerepresent the extreme values of each of the spectral variables; 21.1.2 If the spectra are are collected with a digital digital resolution resolution of  D, then shift each spectrum by + D  and by − D. 21.1.3 21. 1.3 Ana Analyz lyzee the shi shifte fted d spe spectr ctra, a, usi using ng the cal calibr ibrati ation on

(86)

(87)

D

1 1 2 r  r  i51 i

 

(88)

and   z  is the number of samples for which replicate measurements were made. 19.3 The   x2 value calculated in Eq 85 is compared with a 19.3 critical value from a chi-squared table (see Table A1.4) for  t  − 1 degrees of freedom. If the calculated x2 value is less than the critical critic al val value, ue, the then n all of the var varian iances ces for the rep replic licate ated d measurements belong to the same population, and the average variance calculated in Eq 87 can be used as a measure of the repeatability of the infrared measurement. The infrared analysis is expected to have a repeatability on the order of t 3   =2 s¯ . 19.4 If the calc calculate ulated d   x2 value is greater than the critical chi-squared value, then the repeatability of the infrared estimatee may var mat vary y wit with h sam sample ple composit composition ion.. In thi thiss cas case, e, the infrared analysis is expected to have a repeatability that is no worse than t 3   =2   3 smax, where  s max  is the maximum  s i value for the replicate measurements.

model, calculateand the change in the estimates between the +D  and and −D  spectra, 21.1.4 21. 1.4 Ide Identi ntify fy the spe spectr ctrum um sho showin wing g the lar larges gestt cha change nge upon shifting. If the estimates are   yˆ +D   and   yˆ −D   respectively, then the wavelength (frequency) sensitivity of the model can be estimated as: 0.1 3 D 3 SEC/ ~  yˆ 1D 2   yˆ 2D!

 

(89)

21.2 The value calculate calculated d in Eq 89 is the wavelength wavelength shift that, tha t, in the wor worst st cas casee (th (thee mos mostt sen sensit sitive ive spe spectr ctrum) um) wil willl produce a change in the estimate that is on the order of 5 % of  the standard error of calibration. NOTE   22—The wavelength wavelength sensitivity sensitivity of a model calculated in Section

20. Major Sources Sources of Calibr Calibration ation and Analysis Analysis Error TABLE TA BLE 2 Sampl Sampling ing Related Errors

20.1  General Sources of Error in Spectral Measurements—

Sampling Error

T able abl e 1the listtspect lis some som e pos possib sible le source sou rces s ofpoten error err ortial that tha t can occur during duri ng spectral ral measureme measu rement nt and potential solution solutions s for these problems.

Nonhomogeneity of sample

TABLE TA BLE 1 Gener General al Sources of Error in Spect Spectral ral Measurements Measurements Source of Spectral Error Poor instr instrument ument perfo performanc rmance e

Opticall pol Optica polari arizat zation ion eff effect ects s Variabl Va riable e sampl sample e prese presentatio ntation n

Conduct instr Conduct instrument ument perfor performance mance tests regularly to monitor changes in instrument performance Analyze QC (Quality Check) sample to determine if instrument performance changes affect analysis Determine linear response range for instrument Choose pathlengths to keep bands of interest in range Use dep depola olariz rizing ing ele elemen ments ts Improve Improv e sampl sample e prese presentati ntation on metho methods ds

Optical component contamination

Investigate commercially available sample presentation equipment Inspect windows, etc., for contamination and clean as necessary

Absorbance exceeds linear response range

Physical variation in solid samples

Possible Solution

Chemical varia Chemical variation tion in sample sam ple wit with h tim time e

Bubbles in liquid samples

19

Possible Solution Improve mixing guidelines or grinding procedures, or both For solids, average replicate repacks For solids, rotate sample cups Measure multiple aliquots from large sample volume Improved sample mixing during sample preparation Diffuse light before it strikes the sample using a light diffusing plate Pulverize Pulve rize sample to parti particle cle size of less than 40 µm (NIR) or 2 µm (MIR) Average multiple repacks of each sample Rotate sample or average five sample measurements Freeze-dry Freez e-dry sampl sample e for stora storage ge and measu measuremen rementt Immedi Imm ediate ate dat data a col collec lectio tion n and ana analys lysis is fol follow lowing ing sample preparation Identification of kinetics of chemical change and avoidance of rapidly changing spectral regions Check pressure requirements for single-phase sample Check flow properties properties of cell for sample introduction

 

E 1655 TABLE TA BLE 3 Source Sources s of Calibr Calibration ation Error S ou ou rc rc e o f Ca Call ib ib ra ra titio n Err rro or Spectroscopy insensitive to component/property being modeled Inadequate sampling of population in calibration set Outlier samples within calibration set

Refe Re fere renc nce e da data ta er erro rors rs

Non-Beer’s Law relationship (Nonlinearity due to component interactions) (Nonlinearity due to instrument response) Sensitivity to baseline shifts, etc. Tran Tr ansc scri ript ptio ion n er erro rors rs

developed. Instrument standardization can also involve actual adjustment of the instrument hardware to achieve such agreement.. Instr ment Instrument ument standardizat standardization ion is one mean meanss of achie achieving ving calibration transfer transfer.. 22.3 Calibration transfer or instrument instrument standardization standardization may be required when maintenance is done to an instrument if such maintenance produces a change in the spectral response large enough eno ugh to cha change nge the val values ues est estima imated ted by the cal calibr ibrati ation on model. The calibration can be thought of as being transferred

Po ss ss ib ib le le So lu lu titi on on Try alternative spectral region Redefine requirement in terms of measurable components/properties Review criteria for calibration set selection Use sample selection techniques for selecting calibration set (29) set  (29) Employ outlier detection algorithms Eliminate spectral outliers or find additional examples

from one instrument (before maintenance) to a second instrument (after maintenance). 22.4 When a calibration transfer or instrument instrument standardization procedure is developed, it is necessary to demonstrate that the performa performance nce of the model is not degraded degraded dur during ing the transfer. To demonstrate that a calibration transfer or instrument standardization procedure preserves the performance of a model, it is necessary to validate the model as described in Section 18. Each calibration transfer or instrument standardization procedure must be tested at least once by performing a full validation of the transferred model. Once the success of a particula parti cularr calib calibrati ration on trans transfer fer or inst instrumen rumentt stand standardiz ardization ation proced pro cedure ure has bee been n dem demons onstra trated ted for a par partic ticula ularr typ typee of  inst in stru rume ment nt,, th then en qu qual alit ity y co cont ntro roll sa samp mple less ca can n be us used ed to evaluate additional transfers and standardizations.

Eliminate reference data outliers or remeasure Anal An alyz yze e bl blin ind d re repl plic icat ates es to te test st pr prec ecis isio ion n Correct procedural errors, improve analytical procedures Check and recalibrate reagents, equipment, etc. (30) etc.  (30) Develop multiple calibrations over smaller concentration ranges Check dynamic range of instrument, Try shorter pathlengths Preprocessing of data to minimize effects of baseline Two pe peop ople le cr cros osss-ch chec eck k or on one e pe pers rson on triple- check all handscribed data

TABLE TA BLE 4 Analy Analysis sis Errors Sources of Analysis Error Poor cal Poor calibr ibrati ation on mod model el

Possible Solution Vali alidat date e cal calibr ibrati ation on mod model el on rep repres resent entati ative ve validation set Check performa performance nce of instrument/m instrument/model odel with with QC samples Diagnose instrument problems with instrument performance tests Vali alidat date e cal calibr ibrati ation on tra transf nsfer er and ins instru trumen mentt standardization procedures Select calibrations with lowest noise, wave length shift sensitivity, and offset sensitivity Employ Empl oy outlier outlier statistics statistics to test test that sample sample is interpolation of model

23. Calibr Calibration ation Quality Control Control

22.1 Cal 22.1 Calibr ibrati ation on tra transf nsfer er ref refers ers to a pro proces cesss by whi which ch a calibration model is developed using data from one spectrometer, is possibly modified, and is applied for the analysis of  spectra spect ra colle collected cted on a secon second d spec spectrom trometer eter.. The cali calibrat bration ion transfer may require that spectral data for a common sample or sample sam pless be col collec lected ted on bot both h ins instru trumen ments, ts, and tha thatt som somee transfer function be developed and applied to the spectra or the model. A complete description of calibration transfer methodologies is beyond the scope of these practices.

23.1 When an IR, mult multivari ivariate, ate, analysis analysis is used to estimate estimate component concentrations or properties, or both, it is desirable to per period iodica ically lly tes testt the ana analys lysis is (in (instr strume ument nt and mod model) el) to ensure that the performance of the analysis is unchanged. To perform such tests, it is sometimes necessary to choose one or more quality control samples that will be used for this purpose. A complete discussion of methods used to validate the performance of an IR analyzer is beyond the scope of these practices. Thee us Th user er is re refe ferr rred ed to Pr Prac acti tice ce D 61 6122 22 wh whic ich h di disc scus usse sess validatio vali dation n of IR anal analyzers yzers for hydr hydrocarb ocarbon on analy analysis, sis, and to Refs   30   and   31   which which dis discus cusss met method hodss tha thatt hav havee gai gained ned acceptance within the agricultural community. 23.2 Control samples (materials (materials for which reference reference values havee bee hav been n mea measur sured ed usi using ng the ref refere erence nce met method hod)) can be employed to monitor the performance of the analysis, provided that the analyses of the control samples involve interpolation of  the model. The IR estimated values for the control samples are compared compa red to the refer reference ence values using established established ASTM 32). These tests procedures or alternative statistical tests  (30, 32). willl gen wil genera erally lly req requir uiree tha thatt the IR est estim imate ated d val values ues and the refere ref erence nce val values ues agr agree ee to wit within hin the con confide fidence nce int interv ervals als defined in 15.3. Since the confidence limits are based on SEC, and since SEC is often dominated by the error in the reference measur mea surem ement ent,, the these se pro proced cedure uress may not pro provid videe the mos mostt sensit sen sitive ive ind indica icatio tion n of cha change ngess in the per perfor forma mance nce of the analysis. analy sis. Alternative Alternatively ly,, quali quality ty contr control ol (QC) samp samples les can be employed. 23.3 23. 3 Qua Qualit lity y con contro troll (QC (QC)) sam sample pless are use used d to mon monito itorr changes chang es in the perf performa ormance nce of an anal analysis ysis (instrument (instrument and

22.2 22. 2 Instrume Instrument nt sta standa ndardi rdizat zation ion is a pro proces cesss whe where re the spectra collected on a second instrument are mathematically adjusted in an attempt to match the spectra that would have been collected on the instrument on which the calibration was

model), after the analysis has been validated. Quality control materials should be identified at the time the model is developed based on the following criteria: 23.3.1 23. 3.1 QC mat materi erials als mus mustt be che chemi mical cally ly and phy physic sicall ally y

Poor instrumen instrumentt performance performance

Poorr cal Poo calibr ibrati ation on tra transf nsfer er

Sample Sampl e outside outside model range range

12 will depend on a variety of factors, including the optical and digital resolution of the instrument relative to the bandwidths of the sample being measured. Calculation of a wavelength sensitivity is done to provide a useful diagnostic for analyses conducted on the same type of analyzer. The wavelength stability of the analyzer can be compared to the value in Eq 83 as a means of monitoring the performance of the analyzer. Because the value in Eq 83 is depend dependent ent on specific instrumental instrumental parameters, parameters, it should genera gen erally lly not be used to com compar paree the suitabilit suitability y of ana analyz lyzers ers for a particular application.

22. Calibration Transfer Transfer and Instrument Standardization

20

 

E 1655 compatible with materials being analyzed, so as to not introduce contaminants into the samples being analyzed, and not to cause safety problems. 23.3.2 QC materials must be chemically chemically stable when stored and sam sample pled. d. If mix mixtur tures es are use used, d, the com compos positi ition on of the mixtur mix turee mus mustt be kno known wn and met method hodss for rep reprod roduci ucing ng the mixture must be established. 23.3.3 The spectra spectra of the QC material must be compatible compatible with the model. Absorption bands for the QC material should

23.6 The use of bia 23.6 biass and slope adjustme adjustments nts to imp improv rovee calibration or prediction statistics for IR multivariate models is generally not recommended. Prediction errors requiring continued tinue d bias and slop slopee corr correctio ections ns indi indicate cate drift in refe referenc rencee method meth od or chang changes es in the inst instrume rument nt photo photometr metric ic or wavelength len gth sta stabil bility ity.. If a cal calibr ibrati ation on mod model el fai fails ls dur during ing the QC monitoring step, the performance of the instrument should be evaluated using the appropriate ASTM instrument performance test, and any instrument problem that is identified should be

not exc exceed eed the lin linear ear res respon ponse se ran range ge of the ins instru trumen mentt in regions used in the calibration model. The spectra of the QC materi mat erial al sho should uld be as sim simila ilarr as pos possib sible le to spe spectr ctraa of the calibration samples. However, analysis of the QC sample can be an extrapolation of the model. 23.4 Spectral data on the the QC material is collected during the same time period that spectra of the calibration and validation samples are collected. The QC material should be treated in exactly the same fashion as other samples so that variations in the spectra are representative of the variations which will occur during dur ing the col collec lectio tion n of spe spectr ctraa for unk unknow nowns. ns. Sep Separa arate te samples should be used for each measurement. A minimum of  20 spectra should be collected.

corrected correc ted.. If con contro troll sam sample pless are use used, d, che checks cks sho should uld be performed on the reference method to ensure that reference valuess are corr value correct. ect. If inst instrume rument nt main maintenan tenance ce is perf performe ormed, d, calibration transfer or instrument standardization procedures, or both, should be followed to reestablish the calibration. 24. Model Updating Updating 24.1 It ma 24.1 may y so some meti time mess be de desi sira rabl blee to ad add d ad addi diti tion onal al calibration samples to an existing model to increase the range of applicability of the model. The new calibration samples may contai con tain n the sam samee com compon ponent entss as the ori origin ginal al cal calibr ibrati ation on samples but at more extreme concentrations, or new components not present in the original calibration samples. The new calibration samples may fill voids in the original calibration space.

NOTE  23—If the QC spectra are collected over too short a time interval, the variation variation see seen n in the spectra spectra will be sma smaller ller than tha thatt typi typical cally ly encountered in application of the model to unknowns, and QC limits set based on these spectra will be excessively tight.

When a model is updated, updated , the matrix  matrix X  containing the24.1.1 original calibration spectra is augmented with  the spectra of  the additional calibration samples, and the vector y vector  y  containing the property or composition values for the calibration samples is augme augmented nted with the values for the addi additiona tionall cali calibrat bration ion samples. 24.1.2 Outlier procedures procedures described in 16.3 must be applied applied to updated models in the same way they are applied to new models. Thus, if additional samples are being added to increase the span of the calibration, it may be necessary to add several samples of each new type to avoid the added samples being rejected as outliers. 24.2 24 .2 When When a ca cali libr brat atio ion n mo mode dell is up upda date ted, d, it mu must st be revalidat reval idated. ed. The requ requirem irements ents for vali validati dation on samp samples les for an update upd ated d mod model el are the sam samee as for the ori origin ginal al mod model el (se (seee Section 18). The spectra used to validate the original model can be us used ed to va vali lida date te th thee up upda date ted d mo mode del, l, bu butt th they ey mu must st be supplemented to cover an adequate range as described in 18.2. The percentage of new samples added to the validation set for the updated model must be at least as large as the percentage of  new samples added to the calibration set.

23.4.1 The spectra for the QC mate material rial are analyzed analyzed using the calibration model, and the average value, y¯ qc  is calculated: q

(  yˆ ˆ  i51

i

y¯ qc 5

q

 

(90)

where q  is the number of spectra collected for the QC material. The standard deviation in the estimated values,   sqc, is calculated as

Œ  q

( ~  yˆ i 2   y¯ qc!2

sqc 5

i51

q21

 

(91)

23.4 23 .4.1. .1.1 1 Di Dixo xon’ n’ss te test st ca can n be ap appl plie ied d to th thee in indi divi vidu dual al estimated values to identify outliers in the calculations in Eq 90 and Eq 91. 23.5 23. 5 The QC mat materi erial al is ana analyz lyzed ed per period iodica ically lly whe when n the anal an alys ysis is (i (ins nstr trum umen entt an and d mo mode del) l) is in us usee fo forr an anal alyz yzin ing g unknowns. The QC material is treated exactly the same as an unknown sample being estimated. The estimated value for the QC ma mate teri rial al is co comp mpar ared ed to yqc. Th Thee es esti tima mate ted d va valu luee is expected to be within the range from y qc  −  t  3 sqc  to yqc +  t  3 sqc 95 % of the time, where  t  is  is the studentized t  value   value for q − 1 df and the 95 % confidence level. 23.5.1 If the analysis of the QC mate material rial is an inter interpolat polation ion of the model, then   sqc  should be consistent with the repeatabil ab ilit ity y of th thee IR an anal alys ysis is as de defin fined ed in Se Sect ctio ion n 19 19.. If th thee analysis of the QC material is an extrapolation of the model, then   sqc   may be somewhat higher than the   si   calculated calculated in

25. Multiva Multivariate riate Calibration Calibration Quest Questionnair ionnairee 25.1 The following following questionnaire questionnaire is designed to assi assist st the user in determining if a multivariate calibration conforms to the requirements set forth in these practices. 25.1.1 If all of the following following questions questions in 25.1.3 25.1.3-25.1 -25.1.7 .7 are answered in the affirmative, then the calibration can be said to have been developed and validated according to E 1655. 25.1.2 If any of the following following questions in 25.1.3-25.1.7 are answered in the negative, then the calibration can not be said to

Section 19. However, since the control limits are still based on the rep repeat eatabi abilit lity y of the spe spectr ctral al mea measur sureme ement nt and do not depend on the reference method, they are expected generally to be tighter than those derived from control samples.

have been developed and validated according to E 1655. If the calibration method is MLR, PCR or PLS-1, the calibration may be said to have been developed using mathematical techniques described in E 1655. ASTM methods that reference E 1655 21

 

E 1655 should not claim calibration or validation via E 1655 unless all of the following questions would have been answered in the affirmative for the procedures followed during the collection of  round robin data on which the method is based. 25.1.3 The following following quest questions ions apply to the mathematica mathematicall methodology used in the calibration: 25.1.3.1 25.1.3 .1 Was the mathematica mathematicall techn technique ique used in the calibration MLR, PCR or PLS-1? (Sections 12 and 13) 25.1.3.2 25.1.3 .2 Did the cali calibrati bration on meth methodolo odology gy incl include ude the capa-

25.1.5.3 Was the number of validation samples greater than 4k  if   if the model was not mean centered, or greater than 4(k  +  + 1) if the model was mean centered? (18.2.3)

bility of detecting high leverage outliers using a statistic such as the leverage statistic,   h? (16.2) 25.1.3.3 25.1.3 .3 Did the anal analysis ysis methodology methodology include the capab capabilility to det detect ect outlier outlierss via a sta statis tistic tic such as tho those se bas based ed on spectral residuals? (16.4.4-16.4.7) 25.1.4 25. 1.4 The fol follow lowing ing que questi stions ons app apply ly to the cal calibr ibrati ation on model where  n  is the number of samples in the calibration set, and  k  is the number of variables (MLR wavelengths, Principal Components, or PLS latent variables) in the model. 25.1.4.1 25.1.4 .1 Was n >6k  if   if the model is not mean centered, or  n  > 6(k  + 1) if the model is mean centered? (17.5) 25.1.4.2 Was the number of samples in the calibration set at least 24? (17.4) 25.1.5 The following following questions questions apply to the valid validation ation of  the model:

95 % of the results for the validation samples fall within 6   t·SEC·   =1 1 h   of th thee re refe fere renc ncee va valu lues es wh wher eree t is th thee Studentized   t   value for   n−k   degrees of freedom ( n−k −1 − 1 for mean mea n cen center tered ed mod models els), ), and   h   is the lev levera erage ge sta statis tistic tic?? (18.10.1)

25.1.5.1 25.1.5 .1 Was a sepa separate rate set of vali validatio dation n sampl samples es used to test the calibration? (18.2) 25.1.5.2 Were validation spectra which were outliers outliers based on either leverage (Mahalanobis Distance) or spectral residuals excluded from the validation set? (18.2.3)

26. Keywo Keywords rds 26.1 infrared analysis; molecular molecular spectroscopy; spectroscopy; multivariate multivariate analysis; quantitative analysis

25.1.5.4 Was the numb 25.1.5.4 number er of vali validatio dation n samples at least 20? (18.2.3) 25.1.5.5 Did the validation 25.1.5.5 validation samples samples span 95 % of the range of the calibration samples? (18.2.3.1) 25.1.5 25. 1.5.6 .6 If SEC is the Standard Standard Error of Cal Calibr ibrati ation, on, do

25.1.5.7 Do the validation results results show a statistically statistically insignificant bias? (18.9.1) 25.1.6 Was the precision of the model determined using t $ k   $   3 tes testt sam sample pless and   r   $   6 repl replicat icatee meas measurem urements ents per sample? (Section 19) 25.1.7 If the calibration and analysis analysis methodology includes preprocess prepr ocessing ing or post postproce processin ssing, g, are these calc calculati ulations ons per per-formed automatically? (Sections 11 and 14)

22

 

E 1655 ANNEXES (Mandatory Information) A1. ST STAT ATISTICAL ISTICAL TREATMENT TREATMENT TABLE TA BLE A1.1 Critic Critical al Values for Rejection of a Disco Discordant rdant Measurement (31)

A1.1 Dixon’ Dixon’ss Test Test Functio Functions ns for Reject Rejection ion of Outlier Outlierss A1.1.1 A1. 1.1forThis testt pro tes provid vides es a sim simple ple and highly hig hly efffici ef icient ent method determining whether all data obtained came from the same population (with unknown mean and standard deviation) and if one or more of the data points are suspect and should be rejected. A1.1.2 In applying this test the numb number er of dete determin rminatio ations ns ( N ) are tabulated in increasing order of magnitude and designated as  X 1,   X 2,   X 3, . . .   X n. A1.1.3 The values at the extremes extremes of the tabulatio tabulation n   X 1   and  X n  are tested in turn in accordance with the number of values in the tabulation.

 

Statistic r 10

 

r 11

 

r 21 21

 

r 22

 

A1.2 Sel A1.2 Select ect the proper proper expressi expression on shown as fol follow lowss in accordance with the number ( N ) of the values in the tabulation and the upper or lower limit to be tested: Outliers Under Test For N  For  N  =  = 3 to 7

 

X 1

r    5 r 

 2 X    ~X 2  2  X 1!  2 X   X 1! ~X n  2

8 to 10

A1.3 A1. 3

 

X n 

r    5 r 

 2 X   X 1!   ~X 2  2 r    5 r  ~X  ~n  2 1  2  1!! 2  X 1 !

r    5 r 

  ~X 3  2  2 X   X 1! r    5 r  ~X  ~n  2 1  2  1!! 2  X 1 !

r    5 r 

11 to 13

14 to 30

 

r    5 r 

 2 X   X 1!   ~X 3  2 ~X  ~n  2 2  2  2!! 2  X 1 !

r    5 r 

 2 X  ~X n  2  X ~n  2 1  2  1!!! ~X n  2  2 X   X 1!

N 

 

a  = 0.05

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

 

a  = 0.01

0.941 0.765 0.642 0.560 0.507 0.554 0.512 0.477 0.576 0.546 0.521 0.546 0.525 0.507 0.490 0.475 0.462 0.450 0.440 0.430 0.421 0.413 0.406

0.988 0.889 0.780 0.698 0.637 0.683 0.635 0.597 0.679 0.642 0.615 0.641 0.616 0.595 0.577 0.561 0.547 0.535 0.524 0.514 0.505 0.497 0.489

 2 X  ~X n  2  X ~n  2 1  2  1!!!  2 X   X 2! ~X n  2

the historical standard deviation of the test method. Therefore the  sample  standard deviation may be less reliable (because of  these random fluctuations) than the  historical  standard deviation tio n in det determ ermini ining ng the con confide fidence nce lim limits its of an ave averag ragee of  results of several determinations.

 2 X  ~X n  2  X ~n  2 2  2  2!!! ~X n  2  2 X   X 1!

A1.6.1 If the histo historical rical standard standard deviation deviation is unkno unknown, wn, the sample standard deviation may be substituted for it in using the nomograph nomog raph and then then multiply multiplying ing the value value found on the 95 % CL scale by the factor given as follows for the number of 

 2 X  ~X n  2  X ~n  2 2  2  2!!!  2 X   X 3! ~X n  2

Substi Sub stitut tutee the appropr appropriat iatee values values in the equati equation on

selected, calculate “r ” and compare the value obtained to the  r  value in Table A1.1 for the appropriate sample size ( N ). ).

results in the average to obtain reliable 95 % confidence limits. No. of Results Factor No. of Results Factor

A1.4 Rejec Rejectt the value value if the calcul calculated ated “r ”is ”is greater than the tabulated value.

3 2.20 8 1.21

4 1.62 10 1.15

5 1.42 15 1.09

6 1.31 25 1.05

7 1.25 35 1.04

A1.7   To Find the Number of Determinations Needed in an  Average to Give Specific Confidence Limits—Lay a straight Average edge across the nomograph so that its edge passes through the point on the right scale corresponding to the standard deviation for the test and through the desired point on the confidence limit scale. Read the number of determinations required from the left scale.

A1.5 His A1.5 Histor torica icall sta standa ndard rd deviatio deviation n as used in Fig. A1.1 means the standard deviation of a test method. It is established by averaging the standard deviations of many samples tested by many laboratories. The samples should cover the range of  usefulness of the test method and should include materials of  diverse composition if the latter has any effect on the reproducibility of results. A1.6   Sampl merely ly the stan standard dard Samplee Stand Standar ard d Devia Deviation tion   is mere deviat dev iation ion computed computed from the data obt obtain ained ed by a gro group up of 

A1.8   To Find the Confidence Limits of an Average Average—Using the number of determinations in the average, lay a straightedge from this point on the left scale through the point on the right

laborator labora tories ies tes testin ting g the sam samee sam sample ple usi using ng the sam samee tes testt method. Obviously it may be much lower or much higher than

scale corresponding to the standard deviation. Read the confidence limits from the intermediate scale.

23

 

E 1655 TABLE A1.2   F -Distribution: -Distribution: Degrees of Freedom for Numerator 1

2

3

4

5

6

7

8

9

10

12 12

15 15

20 20

1 2 3 4 5 6 7 8 9

161 18.5 10.1 7.71 6.61 5.99 5.59 5.32 5.12

200 19.0 9.55 6.94 5.79 5.14 4.74 4.46 4.26

216 19.2 9.28 6.59 5.41 4.76 4.35 4.07 3.86

225 19.2 9.12 6.39 5.19 4.53 4.12 3.54 3.63

230 19.3 9.01 6.26 5.06 4.39 3.97 3.69 3.48

234 19.3 8.94 6.16 4.95 4.28 3.87 3.58 3.37

237 19.4 8.87 6.09 4.88 4.21 3.79 3.50 3.29

239 19.4 8.85 6.04 4.81 4.15 3.73 3.44 3.23

241 19.4 8.81 6.00 4.77 4.10 3.68 3.39 3.18

242 19.4 8.79 5.96 4.74 4.06 3.64 3.35 3.14

244 19.4 8.74 5.91 4.68 4.00 3.57 3.28 3.07

246 19.4 8.70 5.86 4.62 3.94 3.51 3.22 3.01

248 19.4 8.66 5.80 4.56 3.87 3.44 3.15 2.94

10 11 12 13 14 15 16 17 18 19 20 `

4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38 4.35 3.84

4.10 3.98 3.89 3.81 3.74 3.68 3.63 3.59 3.55 3.52 3.49 3.00

3.70 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.16 3.13 3.10 2.50

3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90 2.87 2.37

3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81 2.77 2.74 2.71 2.21

3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63 2.60 2.10

3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54 2.51 2.01

3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 1.94

3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 1.88

2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 1.83

2.91 2.79 2.69 2.60 2.53 2.48 2.42 2.38 2.34 2.31 2.28 1.75

2.85 2.72 2.62 2.53 2.46 2.40 2.35 2.31 2.27 2.23 2.20 1.67

2.77 2.55 2.54 2.46 2.39 2.33 2.28 2.23 2.19 2.16 2.12 1.57

 

TABLE TA BLE A1.3 Ta Table ble of  t  at 5 % Probability Level  

Degrees of Freedom

t 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086

2

 x

TABLE A1.4 Critical TABLE Values NOTE   1— x2 values for (t  −   − 1) degrees of freedom and 95 % confidence level. (t  −  − 1) 1 2 3 4 5

 

x2

(t  −  − 1)

3.84 5.99 7.81 9.49 11.07

6 7 8 9 10

 

x2

(t  −  − 1)

12.59 14.07 15.51 16.92 18.31

11 12 13 14 15

 

24

x2

(t  −  − 1)

19.68 21.03 22.36 23.68 25.00

16 17 18 19 20

 

x2 26.30 27.59 28.87 30.14 31.41

 

E 1655

FIG. A1.1 Nomograph for Number of Determinations to Obtain Desired Confidence Limits

STATI ATISTICAL STICAL TESTS COMMON TO NIRS METHODS (18, 19) (SUPPLEMENTAL (SUPPLEMENTAL INFORMATION) INFORMATION) A2. ST

A2.1 Common Symbols Symbols

represent dimensions of vectors and matrices. Italicized sub-

A2.1.1 Throughout Throughout thes thesee prac practices tices,, lowe lowercase rcase letters are used to represent scalar quantities. Lower case  bold  bold letters  letters are used to represent vectors, and upper case   BOLD  letter  letterss are used use d to rep repres resent ent mat matric rices. es. Ita Italic licize ized d let letter terss are use used d to

scripts are sample, wavelength indices. For example: Scalar ar refere reference nce value value for the   ith sample.  yi   = Scal

25

 

E 1655 between the actual values for the data points and the predicted or est estima imated ted val values ues for the these se poi points nts are exp explai lained ned by the calibrati cali bration on equat equation ion (mat (mathema hematica ticall mode model), l), and 50 % is not explained. Squared values approaching 1.0 are attempted when developing calibrations. R-squared can be estimated using a simple method as outlined as follows. A2.3.2 A2. 3.2.1 .1 The R2 is determined using the equation:

 yˆ  ˆ i   = The estima estimated ted y-valu y-valuee for   ith sa samp mple le ba base sed d on a regression model.  y¯  ¯    = The mean y value value for all samp samples. les. y   = Vector of refer reference ence value valuess for for   n  samples.  xi   = Spect Spectral ral vect vector or of of length length  f  for   for the   ith sample. X   = Matr Matrix ix of spect spectra, ra, the n  rows of  X contain  X  contain the spectra of length  f   for   n  samples. Numberr of sample sampless used in a calibra calibration tion model. model. n   = Numbe Numb mber er of freque frequenc ncie iess or wa wave vele leng ngth thss us used ed in a  f    = Nu

n

( R 2 5 1 2 i 5 1n

calibration model. k    = Number Number of variab variables les used used in a calibr calibratio ation n model. model. r    = Numbe Numberr of replic replicate ate measur measuremen ements ts on a sampl sample. e. (   = Cap Capita itall sig sigma ma rep repres resent entss sum summat mation ion of all values values within parentheses. 2 R = Coeff Coefficient icient of multiple determination determination (R-squared). R   = The simple correlation coef coeffic ficient ient for a linear regression for any set of data points; this is equal to the square root of the R-squared value. b0   = Th Thee bi bias as or yy-in inte terc rcep eptt va valu luee fo forr an any y ca cali libr brat atio ion n function fit to x, y data. For bias-corrected standard error calculations the bias is equal to the difference between the average reference analytical values and the IR predicted values.

(

i51

A2.2.1 Sum of squa squares res for regression: regression:

A2.2.2 A2. 2.2

(A2.1)

Sum of squar squares es for resi residua dual: l: n

SSres 5   ( ~  yˆ i 2 yi! 2

(A2.2)

i51

A2.2.3

Mean squar squaree for for regres regression sion:: n

MSreg 5

A2.2.4

( ~  yˆ i 2   y¯ ! 2 i51 k  2 1

 

(A2.3)

Mean squar squaree for for resi residual: dual: n

MS

5

( ~  yˆ i 2 yi! 2 i51

A2.2.5 A2. 2.5

 

(A2.4)

n 2 k  2 1

reg

F5

Tota otall sum of of square squares: s: n

SStot 5   ( ~yi 2   y¯ ! 2 i51

~ yi 2   y¯ !  / ~n 2 1!

reg 5 SS SS tot

(A2.6)

sion equ sion equati ation on dec decrea reases ses,, F ten tends ds to inc increa rease. se. Del Deleti eting ng an unimportant wavelength from an equation will cause the F for regression to increase. A2.3.3.2 A2.3.3 .2 The F-statistic F-statistic can also be useful in reco recognizi gnizing ng suspec sus pected ted out outlie liers rs wit within hin a cal calibr ibrati ation on sam sample ple set set;; if the F-value decreases when a sample is deleted, the sample was not an outlier. This situation is the result of the sample not affecting the overall fit of the calibration line to the data while at th thee sa same me ti time me de decr crea easi sing ng th thee nu numb mber er of sa samp mple le (n). Conversely, if deleting a single sample increases the overall F for regression, the sample is considered a suspected outlier. F is defined as the mean square for regression divided by the mean square for residual (see statistical terms in A1.2). A2.3.3 A2. 3.3.3 .3 The F for the reg regres ressio sion n is det determ ermine ined d by the equation:

n i51

2

A2.3.2 A2.3 .2.2 .2 If sR  is the standard deviation of the errors in the reference method measurement, and  s Y  is the standard deviation in the reference values used in the calibration (a measure of the range spanned by the reference data), then R2 values that exceed 1 − sR2 / sY2 are probable indications of overfitting of  the data. A2.3.3   F-Test Statistic for the Regression : A2.3.3.1 A2.3.3 .1 This statistic statistic is also termed termed F for regression, regression, or t-squared. F increases as the equation begins to model, or fit, more of the variation variation with within in the data. With R-squared R-squared held consta con stant, nt, the F val value ue inc increa reases ses as the number number of sam sample pless increases. As the number wavelengths used within the regres-

A2.2 Statis Statistical tical Terms Terms

SSreg 5   ( ~  yˆ i 2   y¯ ! 2

~yi 2   yˆ i! / ~n 2 k  2 1!

R 2~n 2 k  2 1! 2

~1 2 R !k 

MSreg

5 MS

(A2.7)

res

A2.3.4   Student’s t-Value t-Value (For a Regression): Regression): A2.3.4.1 A2.3.4 .1 This statistic statistic is equivalent equivalent to the F stat statisti isticc in the determination of the correlation between  between   X   and and   y  data. It can be us used ed to de dete term rmin inee wh whet ethe herr th ther eree is a tr true ue co corr rrel elat atio ion n betwee bet ween n an IR est estima imated ted val value ue and the pri primar mary y che chemi mical cal analysis for that sample. It is used to test the hypothesis that the correlation really exists and has not happened only by chance. A lar large ge t val value ue (ge (gener nerall ally y gre greate aterr tha than n ten ten)) ind indica icates tes a rea reall (statistically significant) correlation between  between   X   and and   y. A2.3.4.2 A2.3.4 .2 The t for regression regression is calculated calculated as:

(A2.5)

A2.3 Test Statistics Statistics A2.3.1 The statistics statistics discussed discussed as follo follows ws have most commonly been applied to MLR models. The statistics assume that the data has bee been n mea mean n cen center tered ed in dev develo elopin ping g the mod model. el. Similar statistics can be derived for PCR and PLS models, and for models that are not mean centered. A2.3.2  Coeffıcient of Multiple Determination The coefficient of multiple determination is also termed the R-squared statistic, or total explained variation. This statistic allows determination of the amount of variation in the data that

t5

R=n 2 k  2 1 1 2 R2

=

is adequately modeled by the calibration equation as a total fraction of 1.0. Thus R2 = 1.00 indicates the calibration equation ti on mo mode dels ls 10 100 0 % of th thee va vari riat atio ion n wi with thin in th thee da data ta.. An R2 = 0.50 indicates that 50 % of the variation in the differences

 

(A2.8)

A2.3.5   Pa Part rtia iall F or tt-Sq Squa uarred Tes estt for for a Re Regr gres essi sion on Coeffıcient: A2.3 A2 .3.5 .5.1 .1 This This te test st in indi dica cate tess wh whet ethe herr th thee addi additi tion on of a 26

 

E 1655 particul icular ar wavel wavelengt ength h (inde (independe pendent nt vari variable able)) and its corr correepart sponding regression coefficient coefficient (multiplier) adds any significant improvement to an equation’s ability to model the data (including the remaining unexplained variation). Small F or t values indicate no real improvement is given by adding the wavelength into the equation. A2.3.5.2 A2.3.5 .2 If several several wavelengths wavelengths (variables (variables)) have low t or F values (less than 10 or 100, respectively), it may be necessary to de dele lete te ea each ch of th thee su susp spec ectt wa wave vele leng ngth ths, s, si sing ngly ly or in

of all regression coefficients. The larger the value, the greater is the sensitivity to particle size differences between samples or to the isotropic (mirror-like) scattering properties of samples. The offset sensitivity is used to compare two or more equations for the their ir “bl “blind indnes ness” s” to of offse fsett var variat iation ion bet betwee ween n sam sample ples. s. Equations with large offset sensitivities indicate that particle size variations within a data set may cause wide variations in the analytical result. A2.3.8.2 A2.3.8 .2 The ISV is calculated calculated as:

combination, combinati on, to deter determine mine which wavel wavelength engthss are the most critical for predicting constituent values. In the case where an important wavelength is masked by intercorrelation with another wavelength, a sharp increase in the partial F will occur when an unimportant wavelength is deleted and where there is no lon longer ger hig high h int interc ercorr orrela elatio tion n bet betwee ween n the var variab iables les sti still ll within the regression equation. A2.3.5.3 The t-statistic is sometimes sometimes referred to as the ratio ratio of the actual regression coefficient for a particular wavelength to the standard deviation of that coefficient. The partial F value described is equal to this t value squared; note that the t value calculated this way retains the sign of the coefficient, whereas all F values are positive. A2.3.5.4 A2.3.5 .4 The partial F for a regre regression ssion coeff coefficien icientt is calculated as: SSres ~all variables except one! 2 SSres ~all variables!   MSres ~all variables!

k 

ISV 5   ( bi i51

k 

IRV IR V 5 (

i51

A2.3.6.2 A2.3.6 .2

 

 j 5 1

 

Œ 

A2.3.1 A2. 3.10.2 0.2

(A2.10)

Œ 

ni

( (

SEL 5

( ~yi 2   yˆ i! 2 b0!2

(A2.15)

SEL is give given n by: by: n

n

i51

(A2.14)

r 

y¯ i 5   ( yij

The bias bias corrected corrected standar standard d error is is calculate calculated d as:

SEc 5

=b2i

A2.3.10   Stand Standard ard Error Error of the Laboratory Laboratory (SEL) for Reference Chemical Methods: A2.3.10.1 The SEL can be determined by using one or more more samples properly aliquoted and analyzed in replicate by one or more laboratories. The average analytical value for the replicates on a single sample is determined as:

(A2.9)

n

1 ~yi 2   yˆ i! n  i ( 51

(A2.13)

A2.3.9   Random Variation Variation Sensitivity: Sensitivity: A2.3.9.1 A2.3.9 .1 This statistic statistic is also termed the index of random variation (IRV). Random variation sensitivity is calculated as the sum of the squares of the values of all regression coefficients. The larger the value, the greater the sensitivity to factors such as: poor wavel wavelength ength precision, precision, temp temperatu erature re varia variations tions within wit hin sam sample pless and ins instru trumen ment, t, and ele electr ctroni onicc noi noise. se. The higher the value, the less likely the equation can be transferred successfully to other instruments. A2.3.9.2 A2.3.9 .2 The IRV is calculated calculated using the expression: expression:

A2.3.6   The Bias Corrected Standard Error : A2.3.6.1 Bias corrected standard error error measurements allow the chara characteri cterizati zation on of the varia variance nce attr attribut ibutable able to rando random m unexplained error within. The bias value, b0, is calculated as the mea mean n dif differ ferenc encee bet betwee ween n ref refere erence nce and IR est estim imate ated d values: b0 5

 

i 5 1  j 5 1

~yij 2   y¯ i!2

n~r i 2 1!

 

(A2.16)

(A2.11)

where the  i  index represents different samples and the   j  index

Similar bias corrected values can be calculated for SECV. A2.3.7   Standard Deviation of Repeatability (SDR) : A2.3.7.1 SDR is also referred to as as the standard deviation deviation of  difference (SDD) or standard error of difference for replicate measurements (SD replicates). The SDR is calculated to allow accurate estimation of the variation in an analytical method due to both sampling, sample presentation, and analysis errors. The SDR can be used as a measure of precision for the reference analytical method. A2.3.7.2 A2.3.7 .2 The SDR is calc calculat ulated ed using:

different measurements on the same sample. A2.3.10.3 A2.3.1 0.3 This can apply whether the repli replicates cates were performed in a single laboratory or whether a collaborative study was undertaken at multiple laboratories. Additional techniques for planning collaborative tests can be found in Ref  20 Ref  20.. Some care ca re mu must st be ta take ken n in ap appl plyi ying ng Eq Eq.. 2. 2.3. 3.16 16.. If al alll of th thee analytical results are from a single analyst in a single laboratory, then the repeatability of the analysis is defined as =2 t(n   (r  − 1), 95 %) SEL, where t( n   (r  − 1), 95 %) is the Student’s   t  value   value for the 95 % confidence level and   n   (r   − 1) degrees of freedom. If the analytical results are from multiple analys ana lysts ts and lab labora orator tories ies,, the sam samee cal calcul culati ation on yie yields lds the reproducibility of the analysis. For many analytical tests, SEL may vary with the magnitude of y. SEL values calculated for samples having different y¯ i  can be compared by an F-test to

n21

Œ 

 

r 

( ~ y j 2  y¯ ¯ j  !2

SDR 5

 j 5 1

r  2 1

 

(A2.12)

A2.3.8   Offs Offset et Sensitivity: Sensitivity: A2.3.8 A2. 3.8.1 .1 Also ter terme med d sys system temati aticc var variat iation ion or ind index ex of  systematic variation (ISV), offset sensitivity is equal to the sum

determine if the SEL valu determine values es show a stat statisti isticall cally y signi significant ficant variation as a function of y¯ i.

27

 

E 1655 REFERENCES (1) Association of Official Official Analytical Chemists, AOAC Offıcial Methods of   Analysis, Method 989.03, 1990, pp. 74–76. (2)   Journal of the Association of Of Offıci fıcial al Analytic Analytical al Chemis Chemists ts, Vol 71, 1988, p. 1162. (3) Landa, I.,  Review of Scientific Instruments , Vol 50, 1979, pp. 34–40. (4) Landa, I., and Norris, K. H.,   Applied Spectroscopy, Vol 23, 1979, pp. 105–107. (5) Kortüm, Kortüm, G.,   Reflectance Spectroscopy, Springe Springer-V r-Verlag, New York, NY, 1969, p. 111. (6) Honigs, D. E., Freelin, J. M., Hieftje, G. M., and Hirschfeld, T. B.,  Applied Spectroscopy, Vol 37, No. 6, 1983, pp. 491–497. (7) Hrushka, W., “Data Analysis: Wavelength Selection Techniques,” in  Near Infrared Technology in the Agricultural and Food Industries, P. Williams Willi ams and K. Norris Norris,, Eds., American American Association Association of Cereal Chemists, St. Paul, MN, 1987. (8) Mark, H.,  Applied Spectroscopy, Vol 42, No. 8, 1988, pp. 1427–1440. (9) Brown, P. J.,  Journal of Chemometrics, Vol 6, 1992, pp. 151–161. (10 10)) Fred Fredric ricks, ks, P. M., Osb Osborn, orn, P. R., and Swi Swinke nkels, ls, P. R.,   Analytical Chemistry, Vol 57, 1985, pp. 1947–1950. (11 11)) Kenne Kennedy dy,, W. J., and Gen Gentle tle,, J. E.,   Statistical Computi Computing ng, Marce Marcell Dekker, New York, NY, 1980. (12 12)) Allen, D. M.,  Tec  Technical hnical Report Number 23, University of Kentucky Department Depar tment of Statisti Statistics, cs, August 1981. (13 13)) Lindberg, W., Persson, J., and Wold, S.,  Analytical Chemistry, Vol 55, 1983, p. 643.

(18 18)) Manne, R.,  Chemometrics and Intelligent Laboratory Systems , Vol 2, 1987, p. 187. (19 19)) Helland, I. S.,   Communications in Statistics (Simulation and Com putation) , Vol 17, 1988, p. 581. (20 20)) Helland, I. S.,  Scandinavian Journal of Statistics, Vol 17, 1990, p. 97. (21 21)) Drap Draper er,, N. R., and Smi Smith, th, A.,   Applied Regr Regression ession Analysi Analysiss, John Wiley and Sons, New York, NY, 1981. (22 orkman man,, J., “NI “NIR R Spe Spectr ctrosco oscopy py Cal Calibr ibratio ation n Bas Basics ics,” ,” in   Near  22)) Work  Infrared A nalysis, Burns, D., and Ciurczak, E., eds., Marcel-Dekker, Inc., New York, NY, 1992, pp. 247–280. (23 23)) Mark, Mark, H., and Workman, Workman, J.,   Statistics in Spectr Spectroscopy oscopy, Academic Press, Boston, MA, 1991. (24 24)) Hoaglin, D.C., Welsch, R.E.  Amer. Statist.  1978, 32, 17. (25 25)) Whitfield, Whitfield, R.G., Gerge Gerger, r, M.E., and Sharp, R.L.,  Applied Spectr Spectrososcopy, Vol 41, 1987, pp. 1204–1213. (26 26)) Geladi, P., and Kowalski, B. R., Analytica Chimica Acta, 185, 1986, pp. 1–17. (27 27)) Miller, R.,  Simultaneous Inference, 2nd ed., Springer, New York, NY, 1981. (28 28)) Mark, H., and Workman, J.,   Analytical Chemistry, Vol 58, 1986, p. 1454. (29 29)) Honigs, D. E., Hieftje, G. M., Mark, H. L., and Hirschfeld, T. B.,  Analytical Chemistry, Vol 57, 1985, p. 2299. (30 30)) Youden, W. J.,   Statistic Statistical al Manu Manual al of the Asso Associa ciation tion of Of Offı fıcial cial  Analytical Chemists, AOAC, Arlington, VA, 1979. (31 31)) Martens, H., and Naes, T., In Williams, P., and Norris, K., Eds.,  Near   Infrared Technology in the Agricultural and Food Industries, American Association of Cereal Chemists, St. Paul, MN, 1987, pp. 57–87. (32 32)) Hald, Hald, A.,   Statistical Statistical Theory with Enginee Engineering ring Applica Applications tions, Joh John n Wiley and Sons, New York, NY, 1952. (33 33)) Dixon, W. J.,   Biometrics, Vol 9, 1953, pp. 74–89.

(14 A.,York, and Naes, T.,   Multiva Multivariate riate Calibrat Calibration ion, John Wiley 14)) Martens, and Sons,H. New NY, 1989. (15 15)) Geladi, P., and Kowalski, B. R.,   Journal of Chemometrics, Vol 1, 1986, pp. 1 and 18. (16 16)) Haaland, D. M., and Thomas, E. V.,  Analytical Chemistry, Vol 60, 1988, pp. 1193– 1193–1202. 1202. (17 17)) Wold, S., Ruhe, A., Wold, H., and Dunn, W. J.,   SIAM Journal of  Sciencee and Statisti Scienc Statistical cal Computa Computations tions, Vol 5, 1984, p. 735.

The American Society for Testing Testing and Materials takes no position respecting the validity of any patent rights asserted in connection  with any item mentioned in this standard. Users of this standard are expressly advised that determination of the validity of any such  patent rights, and the risk of infringement of such rights, are entirely their own responsibility. This standard is subject to revision at any time by the responsible technical committee and must be reviewed every five years and  if not revised, either reapproved or withdrawn. Your comments are invited either for revision of this standard or for additional standards  and should be addressed to ASTM Headquarters. Your comments will receive careful consideration at a meeting of the responsible  technical committee, which you may attend. If you feel that your comments have not received a fair hearing you should make your  views known to the ASTM Committee on Standards, at the address shown below. This standard is copyrighted by ASTM, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States. Individual reprints (single or multiple copies) of this standard may be obtained by contacting ASTM at the above address or at  610-832-9585 (phone), 610-832-9555 (fax), or [email protected] (e-mail); or through the ASTM website (www.astm.org).

28