total A A reag total A A r eag

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1 hapter 5 Standardzng nalytcal Methods hapter Overvew 5 nalytcal Standards 5B albratng the Sgnal (S total ) 5 Determnng the Senstvty (k ) 5D Lnear Regresson and albraton urves 5E ompensatng for the Reagent Blank (S reag ) 5F Usng Excel and R for a Regresson nalyss 5G Key Terms 5H hapter Summary 5I Problems 5J Solutons to Practce Exercses The mercan hemcal Socety s ommttee on Envronmental Improvement defnes standardzaton as the process of determnng the relatonshp between the sgnal and the amount of analyte n a sample. 1 In hapter 3 we defned ths relatonshp as S k n + S or S k + S total reag total r eag where S total s the sgnal, n s the moles of analyte, s the analyte s concentraton, k s the method s senstvty for the analyte, and S reag s the contrbuton to S total from sources other than the sample. To standardze a method we must determne values for k and S reag. Strateges for accomplshng ths are the subject of ths chapter. 1 S ommttee on Envronmental Improvement Gudelnes for Data cquston and Data Qualty Evaluaton n Envronmental hemstry, nal. hem. 1980, 5, Source URL: ttrbuted to [Davd Harvey] Page 1 of 56

2 154 nalytcal hemstry.0 See hapter 9 for a thorough dscusson of ttrmetrc methods of analyss. 5 nalytcal Standards To standardze an analytcal method we use standards contanng known amounts of analyte. The accuracy of a standardzaton, therefore, depends on the qualty of the reagents and glassware used to prepare these standards. For example, n an acd base ttraton the stochometry of the acd base reacton defnes the relatonshp between the moles of analyte and the moles of ttrant. In turn, the moles of ttrant s the product of the ttrant s concentraton and the volume of ttrant needed to reach the equvalence pont. The accuracy of a ttrmetrc analyss, therefore, can be no better than the accuracy to whch we know the ttrant s concentraton. 5.1 Prmary and Secondary Standards The base NaOH s an example of a secondary standard. ommercally avalable NaOH contans mpurtes of Nal, Na O 3, and Na SO 4, and readly absorbs H O from the atmosphere. To determne the concentraton of NaOH n a soluton, t s ttrated aganst a prmary standard weak acd, such as potassum hydrogen phthalate, KH 8 H 4 O 4. We dvde analytcal standards nto two categores: prmary standards and secondary standards. prmary standard s a reagent for whch we can dspense an accurately known amount of analyte. For example, a g sample of K r O 7 contans moles of K r O 7. If we place ths sample n a 50-mL volumetrc flask and dlute to volume, the concentraton of the resultng soluton s M. prmary standard must have a known stochometry, a known purty (or assay), and t must be stable durng long-term storage. Because of the dffculty n establshng the degree of hydraton, even after dryng, a hydrated reagent usually s not a prmary standard. Reagents that do not meet these crtera are secondary standards. The concentraton of a secondary standard must be determned relatve to a prmary standard. Lsts of acceptable prmary standards are avalable. ppendx 8 provdes examples of some common prmary standards. 5. Other Reagents Preparng a standard often requres addtonal reagents that are not prmary standards or secondary standards. Preparng a standard soluton, for example, requres a sutable solvent, and addtonal reagents may be need to adjust the standard s matrx. These solvents and reagents are potental sources of addtonal analyte, whch, f not accounted for, produce a determnate error n the standardzaton. If avalable, reagent grade chemcals conformng to standards set by the mercan hemcal Socety should be used. 3 The label on the bottle of a reagent grade chemcal (Fgure 5.1) lsts ether the lmts for specfc mpurtes, or provdes an assay for the mpurtes. We can mprove the qualty of a reagent grade chemcal by purfyng t, or by conductng a more accurate assay. s dscussed later n the chapter, we (a) Smth, B. W.; Parsons, M. L. J. hem. Educ. 1973, 50, ; (b) Moody, J. R.; Greenburg, P. R.; Pratt, K. W.; Rans, T.. nal. hem. 1988, 60, ommttee on nalytcal Reagents, Reagent hemcals, 8th ed., mercan hemcal Socety: Washngton, D.., Source URL: ttrbuted to [Davd Harvey] Page of 56

3 hapter 5 Standardzng nalytcal Methods 155 can correct for contrbutons to S total from reagents used n an analyss by ncludng an approprate blank determnaton n the analytcal procedure. 5.3 Preparng Standard Solutons It s often necessary to prepare a seres of standards, each wth a dfferent concentraton of analyte. We can prepare these standards n two ways. If the range of concentratons s lmted to one or two orders of magntude, then each soluton s best prepared by transferrng a known mass or volume of the pure standard to a volumetrc flask and dlutng to volume. When workng wth larger ranges of concentraton, partcularly those extendng over more than three orders of magntude, standards are best prepared by a seral dluton from a sngle stock soluton. In a seral dluton we prepare the most concentrated standard and then dlute a porton of t to prepare the next most concentrated standard. Next, we dlute a porton of the second standard to prepare a thrd standard, contnung ths process untl all we have prepared all of our standards. Seral dlutons must be prepared wth extra care because an error n preparng one standard s passed on to all succeedng standards. (a) (b) Fgure 5.1 Examples of typcal packagng labels for reagent grade chemcals. Label (a) provdes the manufacturer s assay for the reagent, NaBr. Note that potassum s flagged wth an astersk (*) because ts assay exceeds the lmts establshed by the mercan hemcal Socety (S). Label (b) does not provde an assay for mpurtes, but ndcates that the reagent meets S specfcatons. n assay for the reagent, NaHO 3 s provded. Source URL: ttrbuted to [Davd Harvey] Page 3 of 56

4 156 nalytcal hemstry.0 See Secton D.1 to revew how an electronc balance works. albratng a balance s mportant, but t does not elmnate all sources of determnate error n measurng mass. See ppendx 9 for a dscusson of correctng for the buoyancy of ar. 5B albratng the Sgnal (S total ) The accuracy of our determnaton of k and S reag depends on how accurately we can measure the sgnal, S total. We measure sgnals usng equpment, such as glassware and balances, and nstrumentaton, such as spectrophotometers and ph meters. To mnmze determnate errors affectng the sgnal, we frst calbrate our equpment and nstrumentaton. We accomplsh the calbraton by measurng S total for a standard wth a known response of S, adjustng S total untl S total S Here are two examples of how we calbrate sgnals. Other examples are provded n later chapters focusng on specfc analytcal methods. When the sgnal s a measurement of mass, we determne S total usng an analytcal balance. To calbrate the balance s sgnal we use a reference weght that meets standards establshed by a governng agency, such as the Natonal Insttute for Standards and Technology or the mercan Socety for Testng and Materals. n electronc balance often ncludes an nternal calbraton weght for routne calbratons, as well as programs for calbratng wth external weghts. In ether case, the balance automatcally adjusts S total to match S. We also must calbrate our nstruments. For example, we can evaluate a spectrophotometer s accuracy by measurng the absorbance of a carefully prepared soluton of mg/l K r O 7 n M H SO 4, usng M H SO 4 as a reagent blank. 4 n absorbance of ± absorbance unts at a wavelength of nm ndcates that the spectrometer s sgnal s properly calbrated. Be sure to read and carefully follow the calbraton nstructons provded wth any nstrument you use. 5 Determnng the Senstvty (k ) To standardze an analytcal method we also must determne the value of k n equaton 5.1 or equaton 5.. S k n + S total reag 5.1 S k + S total reag 5. In prncple, t should be possble to derve the value of k for any analytcal method by consderng the chemcal and physcal processes generatng the sgnal. Unfortunately, such calculatons are not feasble when we lack a suffcently developed theoretcal model of the physcal processes, or are not useful because of nondeal chemcal behavor. In such stuatons we must determne the value of k by analyzng one or more standard solutons, each contanng a known amount of analyte. In ths secton we consder 4 Ebel, S. Fresenus J. nal. hem. 199, 34, 769. Source URL: ttrbuted to [Davd Harvey] Page 4 of 56

5 hapter 5 Standardzng nalytcal Methods 157 several approaches for determnng the value of k. For smplcty we wll assume that S reag has been accounted for by a proper reagent blank, allowng us to replace S total n equaton 5.1 and equaton 5. wth the analyte s sgnal, S. S S k n 5.3 k Sngle-Pont versus Multple-Pont Standardzatons The smplest way to determne the value of k n equaton 5.4 s by a sngle-pont standardzaton n whch we measure the sgnal for a standard, S, contanng a known concentraton of analyte,. Substtutng these values nto equaton 5.4 k S 5.5 gves the value for k. Havng determned the value for k, we can calculate the concentraton of analyte n any sample by measurng ts sgnal, S samp, and calculatng usng equaton 5.6. Ssamp 5.6 k sngle-pont standardzaton s the least desrable method for standardzng a method. There are at least two reasons for ths. Frst, any error n our determnaton of k carres over nto our calculaton of. Second, our expermental value for k s for a sngle concentraton of analyte. Extendng ths value of k to other concentratons of analyte requres us to assume a lnear relatonshp between the sgnal and the analyte s concentraton, an assumpton that often s not true. 5 Fgure 5. shows how assumng a constant value of k may lead to a determnate error n the analyte s concentraton. Despte these lmtatons, sngle-pont standardzatons fnd routne use when the expected range for the analyte s concentratons s small. Under these condtons t s often safe to assume that k s constant (although you should verfy ths assumpton expermentally). Ths s the case, for example, n clncal labs where many automated analyzers use only a sngle standard. The preferred approach to standardzng a method s to prepare a seres of standards, each contanng the analyte at a dfferent concentraton. Standards are chosen such that they bracket the expected range for the analyte s concentraton. multple-pont standardzaton should nclude at least three standards, although more are preferable. plot of S versus Equaton 5.3 and equaton 5.4 are essentally dentcal, dfferng only n whether we choose to express the amount of analyte n moles or as a concentraton. For the remander of ths chapter we wll lmt our treatment to equaton 5.4. You can extend ths treatment to equaton 5.3 by replacng wth n. Lnear regresson, whch also s known as the method of least squares, s one such algorthm. Its use s covered n Secton 5D. 5 ardone, M. J.; Palmero, P. J.; Sybrandt, L. B. nal. hem. 1980, 5, Source URL: ttrbuted to [Davd Harvey] Page 5 of 56

6 158 nalytcal hemstry.0 assumed relatonshp actual relatonshp S samp Fgure 5. Example showng how a sngle-pont standardzaton leads to a determnate error n an analyte s reported concentraton f we ncorrectly assume that the value of k s constant. S ( ) reported ( ) actual s known as a calbraton curve. The exact standardzaton, or calbraton relatonshp s determned by an approprate curve-fttng algorthm. There are at least two advantages to a multple-pont standardzaton. Frst, although a determnate error n one standard ntroduces a determnate error nto the analyss, ts effect s mnmzed by the remanng standards. Second, by measurng the sgnal for several concentratons of analyte we no longer must assume that the value of k s ndependent of the analyte s concentraton. onstructng a calbraton curve smlar to the actual relatonshp n Fgure 5., s possble. ppendng the adjectve external to the noun standard mght strke you as odd at ths pont, as t seems reasonable to assume that standards and samples must be analyzed separately. s you wll soon learn, however, we can add standards to our samples and analyze them smultaneously. 5. External Standards The most common method of standardzaton uses one or more external standards, each contanng a known concentraton of analyte. We call them external because we prepare and analyze the standards separate from the samples. SINGLE EXTERNL STNDRD quanttatve determnaton usng a sngle external standard was descrbed at the begnnng of ths secton, wth k gven by equaton 5.5. fter determnng the value of k, the concentraton of analyte,, s calculated usng equaton 5.6. Example 5.1 spectrophotometrc method for the quanttatve analyss of Pb + n blood yelds an S of for a sngle standard whose concentraton of lead s 1.75 ppb What s the concentraton of Pb + n a sample of blood for whch S samp s 0.361? Source URL: ttrbuted to [Davd Harvey] Page 6 of 56

7 hapter 5 Standardzng nalytcal Methods 159 SOLUTION Equaton 5.5 allows us to calculate the value of k for ths method usng the data for the standard. k S ppb 175. ppb -1 Havng determned the value of k, the concentraton of Pb + n the sample of blood s calculated usng equaton 5.6. Ssamp ppb -1 k ppb MULTIPLE EXTERNL STNDRDS Fgure 5.3 shows a typcal multple-pont external standardzaton. The volumetrc flask on the left s a reagent blank and the remanng volumetrc flasks contan ncreasng concentratons of u +. Shown below the volumetrc flasks s the resultng calbraton curve. Because ths s the most common method of standardzaton the resultng relatonshp s called a normal calbraton curve. When a calbraton curve s a straght-lne, as t s n Fgure 5.3, the slope of the lne gves the value of k. Ths s the most desrable stuaton snce the method s senstvty remans constant throughout the analyte s concentraton range. When the calbraton curve s not a straght-lne, the S (M) Fgure 5.3 Shown at the top s a reagent blank (far left) and a set of fve external standards for u + wth concentratons ncreasng from left to rght. Shown below the external standards s the resultng normal calbraton curve. The absorbance of each standard, S, s shown by the flled crcles. Source URL: ttrbuted to [Davd Harvey] Page 7 of 56

8 160 nalytcal hemstry.0 method s senstvty s a functon of the analyte s concentraton. In Fgure 5., for example, the value of k s greatest when the analyte s concentraton s small and decreases contnuously for hgher concentratons of analyte. The value of k at any pont along the calbraton curve n Fgure 5. s gven by the slope at that pont. In ether case, the calbraton curve provdes a means for relatng S samp to the analyte s concentraton. Example 5. second spectrophotometrc method for the quanttatve analyss of Pb + n blood has a normal calbraton curve for whch S -1 ( ppb ) What s the concentraton of Pb + n a sample of blood f S samp s 0.397? SOLUTION To determne the concentraton of Pb + n the sample of blood we replace S n the calbraton equaton wth S samp and solve for. S samp ppb ppb ppb It s worth notng that the calbraton equaton n ths problem ncludes an extra term that does not appear n equaton 5.6. Ideally we expect the calbraton curve to have a sgnal of zero when s zero. Ths s the purpose of usng a reagent blank to correct the measured sgnal. The extra term of n our calbraton equaton results from the uncertanty n measurng the sgnal for the reagent blank and the standards. The one-pont standardzaton n ths exercse uses data from the thrd volumetrc flask n Fgure 5.3. Practce Exercse 5.1 Fgure 5.3 shows a normal calbraton curve for the quanttatve analyss of u +. The equaton for the calbraton curve s S 9.59 M What s the concentraton of u + n a sample whose absorbance, S samp, s 0.114? ompare your answer to a one-pont standardzaton where a standard of M u + gves a sgnal of lck here to revew your answer to ths exercse. n external standardzaton allows us to analyze a seres of samples usng a sngle calbraton curve. Ths s an mportant advantage when we have many samples to analyze. Not surprsngly, many of the most common quanttatve analytcal methods use an external standardzaton. There s a serous lmtaton, however, to an external standardzaton. When we determne the value of k usng equaton 5.5, the analyte s pres- Source URL: ttrbuted to [Davd Harvey] Page 8 of 56

9 hapter 5 Standardzng nalytcal Methods 161 S samp standard s matrx sample s matrx ( ) reported ( ) actual Fgure 5.4 albraton curves for an analyte n the standard s matrx and n the sample s matrx. If the matrx affects the value of k, as s the case here, then we ntroduce a determnate error nto our analyss f we use a normal calbraton curve. ent n the external standard s matrx, whch usually s a much smpler matrx than that of our samples. When usng an external standardzaton we assume that the matrx does not affect the value of k. If ths s not true, then we ntroduce a proportonal determnate error nto our analyss. Ths s not the case n Fgure 5.4, for nstance, where we show calbraton curves for the analyte n the sample s matrx and n the standard s matrx. In ths example, a calbraton curve usng external standards results n a negatve determnate error. If we expect that matrx effects are mportant, then we try to match the standard s matrx to that of the sample. Ths s known as matrx matchng. If we are unsure of the sample s matrx, then we must show that matrx effects are neglgble, or use an alternatve method of standardzaton. Both approaches are dscussed n the followng secton. The matrx for the external standards n Fgure 5.3, for example, s dlute ammona, whch s added because the u(nh 3 ) 4 + complex absorbs more strongly than u +. If we fal to add the same amount of ammona to our samples, then we wll ntroduce a proportonal determnate error nto our analyss. 5.3 Standard ddtons We can avod the complcaton of matchng the matrx of the standards to the matrx of the sample by conductng the standardzaton n the sample. Ths s known as the method of standard addtons. SINGLE STNDRD DDITION The smplest verson of a standard addton s shown n Fgure 5.5. Frst we add a porton of the sample, V o, to a volumetrc flask, dlute t to volume, V f, and measure ts sgnal, S samp. Next, we add a second dentcal porton of sample to an equvalent volumetrc flask along wth a spke, V, of an external standard whose concentraton s. fter dlutng the spked sample to the same fnal volume, we measure ts sgnal, S spke. The followng two equatons relate S samp and S spke to the concentraton of analyte,, n the orgnal sample. Source URL: ttrbuted to [Davd Harvey] Page 9 of 56

10 16 nalytcal hemstry.0 add V o of add V of Fgure 5.5 Illustraton showng the method of standard addtons. The volumetrc flask on the left contans a porton of the sample, V o, and the volumetrc flask on the rght contans an dentcal porton of the sample and a spke, V, of a standard soluton of the analyte. Both flasks are dluted to the same fnal volume, V f. The concentraton of analyte n each flask s shown at the bottom of the fgure where s the analyte s concentraton n the orgnal sample and s the concentraton of analyte n the external standard. oncentraton of nalyte V V dlute to V f o f V V o V V f f The ratos V o /V f and V /V f account for the dluton of the sample and the standard, respectvely. S k V o 5.7 V samp f V V o S k spke V V 5.8 f f s long as V s small relatve to V o, the effect of the standard s matrx on the sample s matrx s nsgnfcant. Under these condtons the value of k s the same n equaton 5.7 and equaton 5.8. Solvng both equatons for k and equatng gves S samp V V o f V V o f S spke + whch we can solve for the concentraton of analyte,, n the orgnal sample. Example 5.3 thrd spectrophotometrc method for the quanttatve analyss of Pb + n blood yelds an S samp of when a 1.00 ml sample of blood s dluted to 5.00 ml. second 1.00 ml sample of blood s spked wth 1.00 μl of a 1560-ppb Pb + external standard and dluted to 5.00 ml, yeldng an V V f 5.9 Source URL: ttrbuted to [Davd Harvey] Page 10 of 56

11 hapter 5 Standardzng nalytcal Methods 163 S spke of What s the concentraton of Pb + n the orgnal sample of blood? SOLUTION We begn by makng approprate substtutons nto equaton 5.9 and solvng for. Note that all volumes must be n the same unts; thus, we frst covert V from 1.00 μl to ml ml ml ml ml ppb ml 500. ml ppb ppb ppb 1.33 ppb The concentraton of Pb + n the orgnal sample of blood s 1.33 ppb. It also s possble to make a standard addton drectly to the sample, measurng the sgnal both before and after the spke (Fgure 5.6). In ths case the fnal volume after the standard addton s V o + V and equaton 5.7, equaton 5.8, and equaton 5.9 become add V of V o V o oncentraton of nalyte V o V o V V o V V Fgure 5.6 Illustraton showng an alternatve form of the method of standard addtons. In ths case we add a spke of the external standard drectly to the sample wthout any further adjust n the volume. Source URL: ttrbuted to [Davd Harvey] Page 11 of 56

12 164 nalytcal hemstry.0 S k samp V V o S k spke V V V V 5.10 o o S samp V o V o + V S spke + V o V + V 5.11 Example 5.4 fourth spectrophotometrc method for the quanttatve analyss of Pb + n blood yelds an S samp of 0.71 for a 5.00 ml sample of blood. fter spkng the blood sample wth 5.00 μl of a 1560-ppb Pb + external standard, an S spke of s measured. What s the concentraton of Pb + n the orgnal sample of blood. V o + V 5.00 ml ml ml SOLUTION To determne the concentraton of Pb + n the orgnal sample of blood, we make approprate substtutons nto equaton 5.11 and solve for ml ppb ml ml ml ppb ppb ppb The concentraton of Pb + n the orgnal sample of blood s 1.33 ppb. MULTIPLE STNDRD DDITIONS We can adapt the sngle-pont standard addton nto a multple-pont standard addton by preparng a seres of samples contanng ncreasng amounts of the external standard. Fgure 5.7 shows two ways to plot a standard addton calbraton curve based on equaton 5.8. In Fgure 5.7a we plot S spke aganst the volume of the spkes, V. If k s constant, then the calbraton curve s a straght-lne. It s easy to show that the x-ntercept s equvalent to V o /. Source URL: ttrbuted to [Davd Harvey] Page 1 of 56

13 hapter 5 Standardzng nalytcal Methods 165 (a) y-ntercept k V o V f S spke slope k V f 0.10 (b) S spke V x-ntercept - (ml) V o y-ntercept k V o V f x-ntercept - V o V f V Vf slope k (mg/l) Fgure 5.7 Shown at the top s a set of sx standard addtons for the determnaton of Mn +. The flask on the left s a 5.00 ml sample dluted to ml. The remanng flasks contan 5.00 ml of sample and, from left to rght, 1.00,.00, 3.00, 4.00, and 5.00 ml of an external standard of mg/l Mn +. Shown below are two ways to plot the standard addtons calbraton curve. The absorbance for each standard addton, S spke, s shown by the flled crcles. Example 5.5 Begnnng wth equaton 5.8 show that the equatons n Fgure 5.7a for the slope, the y-ntercept, and the x-ntercept are correct. SOLUTION We begn by rewrtng equaton 5.8 as S spke kv k o + V V V whch s n the form of the equaton for a straght-lne f Y y-ntercept + slope X f Source URL: ttrbuted to [Davd Harvey] Page 13 of 56

14 166 nalytcal hemstry.0 where Y s S spke and X s V. The slope of the lne, therefore, s k /V f and the y-ntercept s k V o /V f. The x-ntercept s the value of X when Y s zero, or kv k o 0 + x-ntercept V V f f x-ntercept kv o V V f k V f o Practce Exercse 5. Begnnng wth equaton 5.8 show that the equatons n Fgure 5.7b for the slope, the y-ntercept, and the x-ntercept are correct. lck here to revew your answer to ths exercse. Because we know the volume of the orgnal sample, V o, and the concentraton of the external standard,, we can calculate the analyte s concentratons from the x-ntercept of a multple-pont standard addtons. Example 5.6 ffth spectrophotometrc method for the quanttatve analyss of Pb + n blood uses a multple-pont standard addton based on equaton 5.8. The orgnal blood sample has a volume of 1.00 ml and the standard used for spkng the sample has a concentraton of 1560 ppb Pb +. ll samples were dluted to 5.00 ml before measurng the sgnal. calbraton curve of S spke versus V has the followng equaton S spke ml 1 V What s the concentraton of Pb + n the orgnal sample of blood. SOLUTION To fnd the x-ntercept we set S spke equal to zero ml 1 V Solvng for V, we obtan a value of ml for the x-ntercept. Substtutng the x-nterecpt s value nto the equaton from Fgure 5.7a 4 V 100. ml o ml 1560 ppb and solvng for gves the concentraton of Pb + n the blood sample as 1.33 ppb. Source URL: ttrbuted to [Davd Harvey] Page 14 of 56

15 hapter 5 Standardzng nalytcal Methods 167 Practce Exercse 5.3 Fgure 5.7 shows a standard addtons calbraton curve for the quanttatve analyss of Mn +. Each soluton contans 5.00 ml of the orgnal sample and ether 0, 1.00,.00, 3.00, 4.00, or 5.00 ml of a mg/l external standard of Mn +. ll standard addton samples were dluted to ml before readng the absorbance. The equaton for the calbraton curve n Fgure 5.7a s S V What s the concentraton of Mn + n ths sample? ompare your answer to the data n Fgure 5.7b, for whch the calbraton curve s S (V /V f ) lck here to revew your answer to ths exercse. Snce we construct a standard addtons calbraton curve n the sample, we can not use the calbraton equaton for other samples. Each sample, therefore, requres ts own standard addtons calbraton curve. Ths s a serous drawback f you have many samples. For example, suppose you need to analyze 10 samples usng a three-pont calbraton curve. For a normal calbraton curve you need to analyze only 13 solutons (three standards and ten samples). If you use the method of standard addtons, however, you must analyze 30 solutons (each of the ten samples must be analyzed three tmes, once before spkng and after each of two spkes). USING STNDRD DDITION TO IDENTIFY MTRIX EFFETS We can use the method of standard addtons to valdate an external standardzaton when matrx matchng s not feasble. Frst, we prepare a normal calbraton curve of S versus and determne the value of k from ts slope. Next, we prepare a standard addtons calbraton curve usng equaton 5.8, plottng the data as shown n Fgure 5.7b. The slope of ths standard addtons calbraton curve provdes an ndependent determnaton of k. If there s no sgnfcant dfference between the two values of k, then we can gnore the dfference between the sample s matrx and that of the external standards. When the values of k are sgnfcantly dfferent, then usng a normal calbraton curve ntroduces a proportonal determnate error. 5.4 Internal Standards To successfully use an external standardzaton or the method of standard addtons, we must be able to treat dentcally all samples and standards. When ths s not possble, the accuracy and precson of our standardzaton may suffer. For example, f our analyte s n a volatle solvent, then ts concentraton ncreases when we lose solvent to evaporaton. Suppose we Source URL: ttrbuted to [Davd Harvey] Page 15 of 56

16 168 nalytcal hemstry.0 have a sample and a standard wth dentcal concentratons of analyte and dentcal sgnals. If both experence the same proportonal loss of solvent then ther respectve concentratons of analyte and sgnals contnue to be dentcal. In effect, we can gnore evaporaton f the samples and standards experence an equvalent loss of solvent. If an dentcal standard and sample lose dfferent amounts of solvent, however, then ther respectve concentratons and sgnals wll no longer be equal. In ths case a smple external standardzaton or standard addton s not possble. We can stll complete a standardzaton f we reference the analyte s sgnal to a sgnal from another speces that we add to all samples and standards. The speces, whch we call an nternal standard, must be dfferent than the analyte. Because the analyte and the nternal standard n any sample or standard receve the same treatment, the rato of ther sgnals s unaffected by any lack of reproducblty n the procedure. If a soluton contans an analyte of concentraton, and an nternal standard of concentraton, IS, then the sgnals due to the analyte, S, and the nternal standard, S IS, are S k S k IS IS IS where k and k IS are the senstvtes for the analyte and nternal standard. Takng the rato of the two sgnals gves the fundamental equaton for an nternal standardzaton. S S IS k K 5.1 k IS Because K s a rato of the analyte s senstvty and the nternal standard s senstvty, t s not necessary to ndependently determne values for ether k or k IS. IS IS SINGLE INTERNL STNDRD In a sngle-pont nternal standardzaton, we prepare a sngle standard contanng the analyte and the nternal standard, and use t to determne the value of K n equaton 5.1. K IS S S Havng standardzed the method, the analyte s concentraton s gven by S IS K S IS IS samp 5.13 Source URL: ttrbuted to [Davd Harvey] Page 16 of 56

17 hapter 5 Standardzng nalytcal Methods 169 Example 5.7 sxth spectrophotometrc method for the quanttatve analyss of Pb + n blood uses u + as an nternal standard. standard contanng 1.75 ppb Pb + and.5 ppb u + yelds a rato of (S /S IS ) of.37. sample of blood s spked wth the same concentraton of u +, gvng a sgnal rato, (S /S IS ) samp, of Determne the concentraton of Pb + n the sample of blood. SOLUTION Equaton 5.13 allows us to calculate the value of K usng the data for the standard IS K S. 5 ppb u ppb u SIS ppb Pb ppb Pb The concentraton of Pb +, therefore, s S IS K SIS samp. 5 ppb u ppb u ppb u 3.05 ppb Pb MULTIPLE INTERNL STNDRDS sngle-pont nternal standardzaton has the same lmtatons as a snglepont normal calbraton. To construct an nternal standard calbraton curve we prepare a seres of standards, each contanng the same concentraton of nternal standard and a dfferent concentratons of analyte. Under these condtons a calbraton curve of (S /S IS ) versus s lnear wth a slope of K/ IS. Example 5.8 seventh spectrophotometrc method for the quanttatve analyss of Pb + n blood gves a lnear nternal standards calbraton curve for whch S S IS 1 (. 11 ppb ) lthough the usual practce s to prepare the standards so that each contans an dentcal amount of the nternal standard, ths s not a requrement. What s the ppb Pb + n a sample of blood f (S /S IS ) samp s.80? SOLUTION To determne the concentraton of Pb + n the sample of blood we replace (S /S IS ) n the calbraton equaton wth (S /S IS ) samp and solve for. Source URL: ttrbuted to [Davd Harvey] Page 17 of 56

18 170 nalytcal hemstry.0 S S IS samp.11 ppb ppb ppb The concentraton of Pb + n the sample of blood s 1.33 ppb. In some crcumstances t s not possble to prepare the standards so that each contans the same concentraton of nternal standard. Ths s the case, for example, when preparng samples by mass nstead of volume. We can stll prepare a calbraton curve, however, by plottng (S /S IS ) versus / IS, gvng a lnear calbraton curve wth a slope of K. 5D Lnear Regresson and albraton urves In a sngle-pont external standardzaton we determne the value of k by measurng the sgnal for a sngle standard contanng a known concentraton of analyte. Usng ths value of k and the sgnal for our sample, we then calculate the concentraton of analyte n our sample (see Example 5.1). Wth only a sngle determnaton of k, a quanttatve analyss usng a sngle-pont external standardzaton s straghtforward. multple-pont standardzaton presents a more dffcult problem. onsder the data n Table 5.1 for a multple-pont external standardzaton. What s our best estmate of the relatonshp between S and? It s temptng to treat ths data as fve separate sngle-pont standardzatons, determnng k for each standard, and reportng the mean value. Despte t smplcty, ths s not an approprate way to treat a multple-pont standardzaton. So why s t napproprate to calculate an average value for k as done n Table 5.1? In a sngle-pont standardzaton we assume that our reagent blank (the frst row n Table 5.1) corrects for all constant sources of determnate error. If ths s not the case, then the value of k from a sngle-pont standardzaton has a determnate error. Table 5. demonstrates how an Table 5.1 Data for a Hypothetcal Multple-Pont External Standardzaton (arbtrary unts) S (arbtrary unts) k S / mean value for k 1.5 Source URL: ttrbuted to [Davd Harvey] Page 18 of 56

19 hapter 5 Standardzng nalytcal Methods 171 Table 5. Effect of a onstant Determnate Error on the Value of k From a Sngle- Pont Standardzaton S k S / (S ) e k (S ) e / (wthout constant error) (actual) (wth constant error) (apparent) mean k (true) 1.00 mean k (apparent) 1.3 uncorrected constant error affects our determnaton of k. The frst three columns show the concentraton of analyte n the standards,, the sgnal wthout any source of constant error, S, and the actual value of k for fve standards. s we expect, the value of k s the same for each standard. In the fourth column we add a constant determnate error of to the sgnals, (S ) e. The last column contans the correspondng apparent values of k. Note that we obtan a dfferent value of k for each standard and that all of the apparent k values are greater than the true value. How do we fnd the best estmate for the relatonshp between the sgnal and the concentraton of analyte n a multple-pont standardzaton? Fgure 5.8 shows the data n Table 5.1 plotted as a normal calbraton curve. lthough the data certanly appear to fall along a straght lne, the actual calbraton curve s not ntutvely obvous. The process of mathematcally determnng the best equaton for the calbraton curve s called lnear regresson S Fgure 5.8 Normal calbraton curve for the hypothetcal multple-pont external standardzaton n Table 5.1. Source URL: ttrbuted to [Davd Harvey] Page 19 of 56

20 17 nalytcal hemstry.0 5D.1 Lnear Regresson of Straght Lne albraton urves When a calbraton curve s a straght-lne, we represent t usng the followng mathematcal equaton y β + β x where y s the sgnal, S, and x s the analyte s concentraton,. The constants β 0 and β 1 are, respectvely, the calbraton curve s expected y-ntercept and ts expected slope. Because of uncertanty n our measurements, the best we can do s to estmate values for β 0 and β 1, whch we represent as b 0 and b 1. The goal of a lnear regresson analyss s to determne the best estmates for b 0 and b 1. How we do ths depends on the uncertanty n our measurements. 5D. Unweghted Lnear Regresson wth Errors n y The most common approach to completng a lnear regresson for equaton 5.14 makes three assumptons: (1) that any dfference between our expermental data and the calculated regresson lne s the result of ndetermnate errors affectng y, () that ndetermnate errors affectng y are normally dstrbuted, and (3) that the ndetermnate errors n y are ndependent of the value of x. Because we assume that the ndetermnate errors are the same for all standards, each standard contrbutes equally n estmatng the slope and the y-ntercept. For ths reason the result s consdered an unweghted lnear regresson. The second assumpton s generally true because of the central lmt theorem, whch we consdered n hapter 4. The valdty of the two remanng assumptons s less obvous and you should evaluate them before acceptng the results of a lnear regresson. In partcular the frst assumpton s always suspect snce there wll certanly be some ndetermnate errors affectng the values of x. When preparng a calbraton curve, however, t s not unusual for the uncertanty n the sgnal, S, to be sgnfcantly larger than that for the concentraton of analyte n the standards. In such crcumstances the frst assumpton s usually reasonable. HOW LINER REGRESSION WORKS To understand the logc of an lnear regresson consder the example shown n Fgure 5.9, whch shows three data ponts and two possble straght-lnes that mght reasonably explan the data. How do we decde how well these straght-lnes fts the data, and how do we determne the best straghtlne? Let s focus on the sold lne n Fgure 5.9. The equaton for ths lne s ŷ b 0 + bx Source URL: ttrbuted to [Davd Harvey] Page 0 of 56

21 hapter 5 Standardzng nalytcal Methods 173 Fgure 5.9 Illustraton showng three data ponts and two possble straght-lnes that mght explan the data. The goal of a lnear regresson s to fnd the mathematcal model, n ths case a straght-lne, that best explans the data. where b 0 and b 1 are our estmates for the y-ntercept and the slope, and ŷ s our predcton for the expermental value of y for any value of x. Because we assume that all uncertanty s the result of ndetermnate errors affectng y, the dfference between y and ŷ for each data pont s the resdual error, r, n the our mathematcal model for a partcular value of x. r ( y yˆ ) If you are readng ths aloud, you pronounce ŷ as y-hat. Fgure 5.10 shows the resdual errors for the three data ponts. The smaller the total resdual error, R, whch we defne as R ( y yˆ ) 5.16 the better the ft between the straght-lne and the data. In a lnear regresson analyss, we seek values of b 0 and b 1 that gve the smallest total resdual error. ŷ 3 ŷ b bx 0 1 The reason for squarng the ndvdual resdual errors s to prevent postve resdual error from cancelng out negatve resdual errors. You have seen ths before n the equatons for the sample and populaton standard devatons. You also can see from ths equaton why a lnear regresson s sometmes called the method of least squares. r ( y yˆ ) y r ( y yˆ ) ŷ 1 ŷ r ( y yˆ ) y 3 y 1 Fgure 5.10 Illustraton showng the evaluaton of a lnear regresson n whch we assume that all uncertanty s the result of ndetermnate errors affectng y. The ponts n blue, y, are the orgnal data and the ponts n red, ŷ, are the predcted values from the regresson equaton, ŷ b 0 + bx 1.The smaller the total resdual error (equaton 5.16), the better the ft of the straght-lne to the data. Source URL: ttrbuted to [Davd Harvey] Page 1 of 56

22 174 nalytcal hemstry.0 FINDING THE SLOPE ND Y-INTEREPT lthough we wll not formally develop the mathematcal equatons for a lnear regresson analyss, you can fnd the dervatons n many standard statstcal texts. 6 The resultng equaton for the slope, b 1, s b 1 n x y x y n x x and the equaton for the y-ntercept, b 0, s 5.17 y b x 1 b n lthough equaton 5.17 and equaton 5.18 appear formdable, t s only necessary to evaluate the followng four summatons x y xy x See Secton 5F n ths chapter for detals on completng a lnear regresson analyss usng Excel and R. Equatons 5.17 and 5.18 are wrtten n terms of the general varables x and y. s you work through ths example, remember that x corresponds to, and that y corresponds to S. Many calculators, spreadsheets, and other statstcal software packages are capable of performng a lnear regresson analyss based on ths model. To save tme and to avod tedous calculatons, learn how to use one of these tools. For llustratve purposes the necessary calculatons are shown n detal n the followng example. Example 5.9 Usng the data from Table 5.1, determne the relatonshp between S and usng an unweghted lnear regresson. SOLUTION We begn by settng up a table to help us organze the calculaton. x y x y x ddng the values n each column gves x y xy x See, for example, Draper, N. R.; Smth, H. ppled Regresson nalyss, 3rd ed.; Wley: New York, Source URL: ttrbuted to [Davd Harvey] Page of 56

23 hapter 5 Standardzng nalytcal Methods 175 Substtutng these values nto equaton 5.17 and equaton 5.18, we fnd that the slope and the y-ntercept are ( b 1. ) (.. ) ( ) ( ) ( ) b The relatonshp between the sgnal and the analyte, therefore, s S For now we keep two decmal places to match the number of decmal places n the sgnal. The resultng calbraton curve s shown n Fgure UNERTINTY IN THE REGRESSION NLYSIS s shown n Fgure 5.11, because of ndetermnate error affectng our sgnal, the regresson lne may not pass through the exact center of each data pont. The cumulatve devaton of our data from the regresson lne that s, the total resdual error s proportonal to the uncertanty n the regresson. We call ths uncertanty the standard devaton about the regresson, s r, whch s equal to s r ( y y ˆ ) n 5.19 Dd you notce the smlarty between the standard devaton about the regresson (equaton 5.19) and the standard devaton for a sample (equaton 4.1)? S Fgure 5.11 albraton curve for the data n Table 5.1 and Example 5.9. Source URL: ttrbuted to [Davd Harvey] Page 3 of 56

24 176 nalytcal hemstry.0 where y s the th expermental value, and ŷ s the correspondng value predcted by the regresson lne n equaton Note that the denomnator of equaton 5.19 ndcates that our regresson analyss has n degrees of freedom we lose two degree of freedom because we use two parameters, the slope and the y-ntercept, to calculate ŷ. more useful representaton of the uncertanty n our regresson s to consder the effect of ndetermnate errors on the slope, b 1, and the y- ntercept, b 0, whch we express as standard devatons. s b 1 ns n x x r sr ( x x ) 5.0 s b 0 s x r n x x s r x ( ) n x x 5.1 We use these standard devatons to establsh confdence ntervals for the expected slope, β 1, and the expected y-ntercept, β 0 β 1 b 1 ± ts b 5. 1 β 0 b 0 ± ts b You mght contrast ths wth equaton 4.1 for the confdence nterval around a sample s mean value. s you work through ths example, remember that x corresponds to, and that y corresponds to S. where we select t for a sgnfcance level of α and for n degrees of freedom. Note that equaton 5. and equaton 5.3 do not contan a factor of ( n) 1 because the confdence nterval s based on a sngle regresson lne. gan, many calculators, spreadsheets, and computer software packages provde the standard devatons and confdence ntervals for the slope and y-ntercept. Example 5.10 llustrates the calculatons. Example 5.10 alculate the 95% confdence ntervals for the slope and y-ntercept from Example 5.9. SOLUTION We begn by calculatng the standard devaton about the regresson. To do ths we must calculate the predcted sgnals, ŷ, usng the slope and y-ntercept from Example 5.9, and the squares of the resdual error, ( y yˆ ). Usng the last standard as an example, we fnd that the predcted sgnal s yˆ b + bx ( ) and that the square of the resdual error s Source URL: ttrbuted to [Davd Harvey] Page 4 of 56

25 hapter 5 Standardzng nalytcal Methods 177 ( y yˆ ) ( ) The followng table dsplays the results for all sx solutons. x y ŷ ( y yˆ ) ddng together the data n the last column gves the numerator of equaton 5.19 as The standard devaton about the regresson, therefore, s s r Next we calculate the standard devatons for the slope and the y-ntercept usng equaton 5.0 and equaton 5.1. The values for the summaton terms are from n Example 5.9. s b 1 ns n x x r 6 ( ) ( ) ( 1 550) s b 0 s x r ( ) n x x ( ) ( ) Fnally, the 95% confdence ntervals (α 0.05, 4 degrees of freedom) for the slope and y-ntercept are You can fnd values for t n ppendx 4. b ± ts b ± ( ) ±. 7 β b ± ts b 009. ± ( ) 0. ± 08. β The standard devaton about the regresson, s r, suggests that the sgnal, S, s precse to one decmal place. For ths reason we report the slope and the y-ntercept to a sngle decmal place. Source URL: ttrbuted to [Davd Harvey] Page 5 of 56

26 178 nalytcal hemstry.0 MINIMIZING UNERTINTY IN LIBRTION URVES To mnmze the uncertanty n a calbraton curve s slope and y-ntercept, you should evenly space your standards over a wde range of analyte concentratons. close examnaton of equaton 5.0 and equaton 5.1 wll help you apprecate why ths s true. The denomnators of both equatons nclude the term ( x x). The larger the value of ths term whch you accomplsh by ncreasng the range of x around ts mean value the smaller the standard devatons n the slope and the y-ntercept. Furthermore, to mnmze the uncertanty n the y-ntercept, t also helps to decrease the value of the term x n equaton 5.1, whch you accomplsh by ncludng standards for lower concentratons of the analyte. OBTINING THE NLYTE S ONENTRTION FROM REGRESSION EQUTION Once we have our regresson equaton, t s easy to determne the concentraton of analyte n a sample. When usng a normal calbraton curve, for example, we measure the sgnal for our sample, S samp, and calculate the analyte s concentraton,, usng the regresson equaton. S b b samp Equaton 5.5 s wrtten n terms of a calbraton experment. more general form of the equaton, wrtten n terms of x and y, s gven here. s x s 1 1 r + + b m n 1 ( Y y) ( b) ( x x) 1 close examnaton of equaton 5.5 should convnce you that the uncertanty n s smallest when the sample s average sgnal, S, s equal to the average samp sgnal for the standards, S. When practcal, you should plan your calbraton curve so that S samp falls n the mddle of the calbraton curve. What s less obvous s how to report a confdence nterval for that expresses the uncertanty n our analyss. To calculate a confdence nterval we need to know the standard devaton n the analyte s concentraton, s, whch s gven by the followng equaton s sr b m n 1 ( Ssamp S ) ( ) ( ) b where m s the number of replcate used to establsh the sample s average sgnal ( S samp ), n s the number of calbraton standards, S s the average sgnal for the calbraton standards, and and are the ndvdual and mean concentratons for the calbraton standards. 7 Knowng the value of s, the confdence nterval for the analyte s concentraton s μ ± ts where μ s the expected value of n the absence of determnate errors, and wth the value of t based on the desred level of confdence and n degrees of freedom. 7 (a) Mller, J. N. nalyst 1991, 116, 3 14; (b) Sharaf, M..; Illman, D. L.; Kowalsk, B. R. hemometrcs, Wley-Interscence: New York, 1986, pp ; (c) nalytcal Methods ommttee Uncertantes n concentratons estmated from calbraton experments, M Techncal Bref, March 006 ( Source URL: ttrbuted to [Davd Harvey] Page 6 of 56

27 hapter 5 Standardzng nalytcal Methods 179 Example 5.11 Three replcate analyses for a sample contanng an unknown concentraton of analyte, yeld values for S samp of 9.3, 9.16 and Usng the results from Example 5.9 and Example 5.10, determne the analyte s concentraton,, and ts 95% confdence nterval. SOLUTION The average sgnal, S samp, s 9.33, whch, usng equaton 5.4 and the slope and the y-ntercept from Example 5.9, gves the analyte s concentraton as S b samp b To calculate the standard devaton for the analyte s concentraton we must ( ) determne the values for S and. The former s just the average sgnal for the calbraton standards, whch, usng the data n Table ( ) 5.1, s alculatng looks formdable, but we can smplfy ts calculaton by recognzng that ths sum of squares term s the numerator n a standard devaton equaton; thus, ( s n ) ( ) 1 ( ) where s s the standard devaton for the concentraton of analyte n the calbraton standards. Usng the data n Table 5.1 we fnd that s s and ( ) ( ) ( 6 1) Substtutng known values nto equaton 5.5 gves s ( ) ( ) Fnally, the 95% confdence nterval for 4 degrees of freedom s ± ts 0. 41± ( ) 0. 41± μ Fgure 5.1 shows the calbraton curve wth curves showng the 95% confdence nterval for. You can fnd values for t n ppendx 4. Source URL: ttrbuted to [Davd Harvey] Page 7 of 56

28 180 nalytcal hemstry S 30 Fgure 5.1 Example of a normal calbraton curve wth a supermposed confdence nterval for the analyte s concentraton. The ponts n blue are the orgnal data from Table 5.1. The black lne s the normal calbraton curve as determned n Example 5.9. The red lnes show the 95% confdence nterval for assumng a sngle determnaton of S samp Practce Exercse 5.4 Fgure 5.3 shows a normal calbraton curve for the quanttatve analyss of u +. The data for the calbraton curve are shown here. [u + ] (M) bsorbance omplete a lnear regresson analyss for ths calbraton data, reportng the calbraton equaton and the 95% confdence nterval for the slope and the y-ntercept. If three replcate samples gve an S samp of 0.114, what s the concentraton of analyte n the sample and ts 95% confdence nterval? lck here to revew your answer to ths exercse. In a standard addton we determne the analyte s concentraton by extrapolatng the calbraton curve to the x-ntercept. In ths case the value of s and the standard devaton n s b x-ntercept 0 b 1 Source URL: ttrbuted to [Davd Harvey] Page 8 of 56

29 hapter 5 Standardzng nalytcal Methods 181 s sr 1 + b n 1 ( S ) ( ) ( ) b 1 where n s the number of standard addtons (ncludng the sample wth no added standard), and S s the average sgnal for the n standards. Because we determne the analyte s concentraton by extrapolaton, rather than by nterpolaton, s for the method of standard addtons generally s larger than for a normal calbraton curve. EVLUTING LINER REGRESSION MODEL You should never accept the result of a lnear regresson analyss wthout evaluatng the valdty of the your model. Perhaps the smplest way to evaluate a regresson analyss s to examne the resdual errors. s we saw earler, the resdual error for a sngle calbraton standard, r, s r ( y yˆ ) If your regresson model s vald, then the resdual errors should be randomly dstrbuted about an average resdual error of zero, wth no apparent trend toward ether smaller or larger resdual errors (Fgure 5.13a). Trends such as those shown n Fgure 5.13b and Fgure 5.13c provde evdence that at least one of the model s assumptons s ncorrect. For example, a trend toward larger resdual errors at hgher concentratons, as shown n Fgure 5.13b, suggests that the ndetermnate errors affectng the sgnal are not ndependent of the analyte s concentraton. In Fgure 5.13c, the resdual (a) (b) (c) resdual error resdual error resdual error Fgure 5.13 Plot of the resdual error n the sgnal, S, as a functon of the concentraton of analyte, for an unweghted straght-lne regresson model. The red lne shows a resdual error of zero. The dstrbuton of the resdual error n (a) ndcates that the unweghted lnear regresson model s approprate. The ncrease n the resdual errors n (b) for hgher concentratons of analyte, suggest that a weghted straght-lne regresson s more approprate. For (c), the curved pattern to the resduals suggests that a straght-lne model s napproprate; lnear regresson usng a quadratc model mght produce a better ft. Source URL: ttrbuted to [Davd Harvey] Page 9 of 56

30 18 nalytcal hemstry.0 Practce Exercse 5.5 Usng your results from Practce Exercse 5.4, construct a resdual plot and explan ts sgnfcance. lck here to revew your answer to ths exercse. errors are not random, suggestng that the data can not be modeled wth a straght-lne relatonshp. Regresson methods for these two cases are dscussed n the followng sectons. 5D.3 Weghted Lnear Regresson wth Errors n y Our treatment of lnear regresson to ths pont assumes that ndetermnate errors affectng y are ndependent of the value of x. If ths assumpton s false, as s the case for the data n Fgure 5.13b, then we must nclude the varance for each value of y nto our determnaton of the y-ntercept, b o, and the slope, b 1 ; thus b 1 b 0 wy b wx n n w x y w x w y 5.7 n w x w x Ths s the same data used n Example 5.9 wth addtonal nformaton about the standard devatons n the sgnal. where w s a weghtng factor that accounts for the varance n y ns ( ) y w 5.8 s ( y ) and s y s the standard devaton for y. In a weghted lnear regresson, each xy-par s contrbuton to the regresson lne s nversely proportonal to the precson of y that s, the more precse the value of y, the greater ts contrbuton to the regresson. Example 5.1 Shown here are data for an external standardzaton n whch s s the standard devaton for three replcate determnaton of the sgnal. (arbtrary unts) S (arbtrary unts) s Source URL: ttrbuted to [Davd Harvey] Page 30 of 56

31 hapter 5 Standardzng nalytcal Methods Determne the calbraton curve s equaton usng a weghted lnear regresson. s you work through ths example, remember that x corresponds to, and that y corresponds to S. SOLUTION We begn by settng up a table to ad n calculatng the weghtng factors. x y s y ( s y ) w ddng together the values n the forth column gves s a check on your calculatons, the sum of the ndvdual weghts must equal the number of calbraton standards, n. The sum of the entres n the last column s , so all s well. ( s y ) whch we use to calculate the ndvdual weghts n the last column. fter calculatng the ndvdual weghts, we use a second table to ad n calculatng the four summaton terms n equaton 5.6 and equaton 5.7. x y w w x w y w x w x y ddng the values n the last four columns gves wx wx wy wxy Substtutng these values nto the equaton 5.6 and equaton 5.7 gves the estmated slope and estmated y-ntercept as Source URL: ttrbuted to [Davd Harvey] Page 31 of 56

32 184 nalytcal hemstry.0 ( b 1. ) (.. ) ( ) ( ) ( ) b The calbraton equaton s S Fgure 5.14 shows the calbraton curve for the weghted regresson and the calbraton curve for the unweghted regresson n Example 5.9. lthough the two calbraton curves are very smlar, there are slght dfferences n the slope and n the y-ntercept. Most notably, the y-ntercept for the weghted lnear regresson s closer to the expected value of zero. Because the standard devaton for the sgnal, S, s smaller for smaller concentratons of analyte,, a weghted lnear regresson gves more emphass to these standards, allowng for a better estmate of the y-ntercept. Equatons for calculatng confdence ntervals for the slope, the y-ntercept, and the concentraton of analyte when usng a weghted lnear regresson are not as easy to defne as for an unweghted lnear regresson. 8 The confdence nterval for the analyte s concentraton, however, s at ts 8 Bonate, P. J. nal. hem. 1993, 65, weghted lnear regresson unweghted lnear regresson S Fgure 5.14 comparson of unweghted and weghted normal calbraton curves. See Example 5.9 for detals of the unweghted lnear regresson and Example 5.1 for detals of the weghted lnear regresson. Source URL: ttrbuted to [Davd Harvey] Page 3 of 56

33 hapter 5 Standardzng nalytcal Methods 185 optmum value when the analyte s sgnal s near the weghted centrod, y c, of the calbraton curve. y 1 wx n c 5D.4 Weghted Lnear Regresson wth Errors n Both x and y If we remove our assumpton that the ndetermnate errors affectng a calbraton curve exst only n the sgnal (y), then we also must factor nto the regresson model the ndetermnate errors affectng the analyte s concentraton n the calbraton standards (x). The soluton for the resultng regresson lne s computatonally more nvolved than that for ether the unweghted or weghted regresson lnes. 9 lthough we wll not consder the detals n ths textbook, you should be aware that neglectng the presence of ndetermnate errors n x can bas the results of a lnear regresson. See Fgure 5. for an example of a calbraton curve that devates from a straghtlne for hgher concentratons of analyte. 5D.5 urvlnear and Multvarate Regresson straght-lne regresson model, despte ts apparent complexty, s the smplest functonal relatonshp between two varables. What do we do f our calbraton curve s curvlnear that s, f t s a curved-lne nstead of a straght-lne? One approach s to try transformng the data nto a straghtlne. Logarthms, exponentals, recprocals, square roots, and trgonometrc functons have been used n ths way. plot of log(y) versus x s a typcal example. Such transformatons are not wthout complcatons. Perhaps the most obvous complcaton s that data wth a unform varance n y wll not mantan that unform varance after the transformaton. nother approach to developng a lnear regresson model s to ft a polynomal equaton to the data, such as y a + bx + cx. You can use lnear regresson to calculate the parameters a, b, and c, although the equatons are dfferent than those for the lnear regresson of a straght lne. 10 If you cannot ft your data usng a sngle polynomal equaton, t may be possble to ft separate polynomal equatons to short segments of the calbraton curve. The result s a sngle contnuous calbraton curve known as a splne functon. The regresson models n ths chapter apply only to functons contanng a sngle ndependent varable, such as a sgnal that depends upon the analyte s concentraton. In the presence of an nterferent, however, the sgnal may depend on the concentratons of both the analyte and the nterferent It s worth notng that n mathematcs, the term lnear does not mean a straghtlne. lnear functon may contan many addtve terms, but each term can have one and only one adjustable parameter. The functon y ax + bx s lnear, but the functon y ax b s nonlnear. Ths s why you can use lnear regresson to ft a polynomal equaton to your data. Sometmes t s possble to transform a nonlnear functon. For example, takng the log of both sdes of the nonlnear functon shown above gves a lnear functon. log(y) log(a) + blog(x) 9 See, for example, nalytcal Methods ommttee, Fttng a lnear functonal relatonshp to data wth error on both varable, M Techncal Bref, March, 00 ( 10 For detals about curvlnear regresson, see (a) Sharaf, M..; Illman, D. L.; Kowalsk, B. R. hemometrcs, Wley-Interscence: New York, 1986; (b) Demng, S. N.; Morgan, S. L. Expermental Desgn: hemometrc pproach, Elsever: msterdam, Source URL: ttrbuted to [Davd Harvey] Page 33 of 56

34 186 nalytcal hemstry.0 heck out the ddtonal Resources at the end of the textbook for more nformaton about lnear regresson wth errors n both varables, curvlnear regresson, and multvarate regresson. S k + k + S I I reag where k I s the nterferent s senstvty and I s the nterferent s concentraton. Multvarate calbraton curves can be prepared usng standards that contan known amounts of both the analyte and the nterferent, and modeled usng multvarate regresson. 11 5E Blank orrectons Thus far n our dscusson of strateges for standardzng analytcal methods, we have assumed the use of a sutable reagent blank to correct for sgnals arsng from sources other than the analyte. We dd not, however ask an mportant queston What consttutes an approprate reagent blank? Surprsngly, the answer s not mmedately obvous. In one study, approxmately 00 analytcal chemsts were asked to evaluate a data set consstng of a normal calbraton curve, a separate analyte-free blank, and three samples of dfferent sze but drawn from the same source. 1 The frst two columns n Table 5.3 shows a seres of external standards and ther correspondng sgnals. The normal calbraton curve for the data s S W where the y-ntercept of s the calbraton blank. separate reagent blank gves the sgnal for an analyte-free sample. In workng up ths data, the analytcal chemsts used at least four dfferent approaches for correctng sgnals: (a) gnorng both the calbraton blank, B, and the reagent blank, RB, whch clearly s ncorrect; (b) usng the calbraton blank only; (c) usng the reagent blank only; and (d) usng both the calbraton blank and the reagent blank. Table 5.4 shows the equa- 11 Beebe, K. R.; Kowalsk, B. R. nal. hem. 1987, 59, ardone, M. J. nal. hem. 1986, 58, Table 5.3 Data Used to Study the Blank n an nalytcal Method W S Sample Number W samp S samp reagent blank albraton equaton: S W W : weght of analyte used to prepare the external standard; dluted to volume, V. W samp : weght of sample used to prepare sample; dluted to volume, V. Source URL: ttrbuted to [Davd Harvey] Page 34 of 56

35 hapter 5 Standardzng nalytcal Methods 187 Table 5.4 Equatons and Resultng oncentratons of nalyte for Dfferent pproaches to orrectng for the Blank oncentraton of nalyte n... pproach for orrectng Sgnal Equaton Sample 1 Sample Sample 3 gnore calbraton and reagent blank W S samp W kw samp samp use calbraton blank only W W samp S samp kw B samp use reagent blank only W W samp S samp kw RB samp use both calbraton and reagent blank W W samp S samp B RB kw samp use total Youden blank W W samp S samp kw TYB samp concentraton of analyte; W weght of analyte; W samp weght of sample; k slope of calbraton curve (0.075 see Table 5.3); B calbraton blank (0.15 see Table 5.3); RB reagent blank (0.100 see Table 5.3); TYB total Youden blank (0.185 see text) tons for calculatng the analyte s concentraton usng each approach, along wth the resultng concentraton for the analyte n each sample. That all four methods gve a dfferent result for the analyte s concentraton underscores the mportance of choosng a proper blank, but does not tell us whch blank s correct. Because all four methods fal to predct the same concentraton of analyte for each sample, none of these blank correctons properly accounts for an underlyng constant source of determnate error. To correct for a constant method error, a blank must account for sgnals from any reagents and solvents used n the analyss, as well as any bas resultng from nteractons between the analyte and the sample s matrx. Both the calbraton blank and the reagent blank compensate for sgnals from reagents and solvents. ny dfference n ther values s due to ndetermnate errors n preparng and analyzng the standards. Unfortunately, nether a calbraton blank nor a reagent blank can correct for a bas resultng from an nteracton between the analyte and the sample s matrx. To be effectve, the blank must nclude both the sample s matrx and the analyte and, consequently, must be determned usng the sample tself. One approach s to measure the sgnal for samples of dfferent sze, and to determne the regresson lne for a plot of S samp versus the Because we are consderng a matrx effect of sorts, you mght thnk that the method of standard addtons s one way to overcome ths problem. lthough the method of standard addtons can compensate for proportonal determnate errors, t cannot correct for a constant determnate error; see Ellson, S. L. R.; Thompson, M. T. Standard addtons: myth and realty, nalyst, 008, 133, Source URL: ttrbuted to [Davd Harvey] Page 35 of 56

36 188 nalytcal hemstry.0 amount of sample. The resultng y-ntercept gves the sgnal n the absence of sample, and s known as the total youden blank. 13 Ths s the true blank correcton. The regresson lne for the three samples n Table 5.3 s S samp W samp gvng a true blank correcton of s shown by the last row of Table 5.4, usng ths value to correct S samp gves dentcal values for the concentraton of analyte n all three samples. The use of the total Youden blank s not common n analytcal work, wth most chemsts relyng on a calbraton blank when usng a calbraton curve, and a reagent blank when usng a sngle-pont standardzaton. s long we can gnore any constant bas due to nteractons between the analyte and the sample s matrx, whch s often the case, the accuracy of an analytcal method wll not suffer. It s a good dea, however, to check for constant sources of error before relyng on ether a calbraton blank or a reagent blank. 5F Usng Excel and R for a Regresson nalyss lthough the calculatons n ths chapter are relatvely straghtforward consstng, as they do, mostly of summatons t can be qute tedous to work through problems usng nothng more than a calculator. Both Excel and R nclude functons for completng a lnear regresson analyss and for vsually evaluatng the resultng model. 5F.1 Excel B 1 S Fgure 5.15 Porton of a spreadsheet contanng data from Example 5.9 ( ; S S ). Let s use Excel to ft the followng straght-lne model to the data n Example 5.9. y β + β 0 1 Enter the data nto a spreadsheet, as shown n Fgure Dependng upon your needs, there are many ways that you can use Excel to complete a lnear regresson analyss. We wll consder three approaches here. USE EXEL S BUILT-IN FUNTIONS If all you need are values for the slope, β 1, and the y-ntercept, β 0, you can use the followng functons: ntercept(known_y s, known_x s) x slope(known_y s, known_x s) where known_y s s the range of cells contanng the sgnals (y), and known_x s s the range of cells contanng the concentratons (x). For example, clckng on an empty cell and enterng 13 ardone, M. J. nal. hem. 1986, 58, Source URL: ttrbuted to [Davd Harvey] Page 36 of 56

37 hapter 5 Standardzng nalytcal Methods 189 slope(b:b7, :7) returns Excel s exact calculaton for the slope ( ). USE EXEL S DT NLYSIS TOOLS To obtan the slope and the y-ntercept, along wth addtonal statstcal detals, you can use the data analyss tools n the nalyss ToolPak. The ToolPak s not a standard part of Excel s nstllaton. To see f you have access to the nalyss ToolPak on your computer, select Tools from the menu bar and look for the Data nalyss... opton. If you do not see Data nalyss..., select dd-ns... from the Tools menu. heck the box for the nalyss ToolPak and clck on OK to nstall them. Select Data nalyss... from the Tools menu, whch opens the Data nalyss wndow. Scroll through the wndow, select Regresson from the avalable optons, and press OK. Place the cursor n the box for Input Y range and then clck and drag over cells B1:B7. Place the cursor n the box for Input X range and clck and drag over cells 1:7. Because cells 1 and B1 contan labels, check the box for Labels. Select the rado button for Output range and clck on any empty cell; ths s where Excel wll place the results. lckng OK generates the nformaton shown n Fgure There are three parts to Excel s summary of a regresson analyss. t the top of Fgure 5.16 s a table of Regresson Statstcs. The standard error s the standard devaton about the regresson, s r. lso of nterest s the value for Multple R, whch s the model s correlaton coeffcent, r, a term wth whch you may already by famlar. The correlaton coeffcent s a measure of the extent to whch the regresson model explans the varaton n y. Values of r range from 1 to +1. The closer the correlaton coeffcent s to ±1, the better the model s at explanng the data. correlaton coeffcent of 0 means that there s no relatonshp between x and y. In developng the calculatons for lnear regresson, we dd not consder the correlaton coeffcent. There Once you nstall the nalyss ToolPak, t wll contnue to load each tme you launch Excel. Includng labels s a good dea. Excel s summary output uses the x-axs label to dentfy the slope. SUMMRY OUTPUT Regresson Statstcs Multple R R Square djusted R Square Standard Error Observatons 6 NOV df SS MS F Sgnfcance F Regresson E-08 Resdual Total oeffcents Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept E Fgure 5.16 Output from Excel s Regresson command n the nalyss ToolPak. See the text for a dscusson of how to nterpret the nformaton n these tables. Source URL: ttrbuted to [Davd Harvey] Page 37 of 56

38 190 nalytcal hemstry.0 y r x Fgure 5.17 Example of fttng a straght-lne to curvlnear data. See Secton 4F. and Secton 4F.3 for a revew of the F-test. See Secton 4F.1 for a revew of the t-test. s a reason for ths. For most straght-lne calbraton curves the correlaton coeffcent wll be very close to +1, typcally 0.99 or better. There s a tendency, however, to put too much fath n the correlaton coeffcent s sgnfcance, and to assume that an r greater than 0.99 means the lnear regresson model s approprate. Fgure 5.17 provdes a counterexample. lthough the regresson lne has a correlaton coeffcent of 0.993, the data clearly shows evdence of beng curvlnear. The take-home lesson here s: don t fall n love wth the correlaton coeffcent! The second table n Fgure 5.16 s enttled NOV, whch stands for analyss of varance. We wll take a closer look at NOV n hapter 14. For now, t s suffcent to understand that ths part of Excel s summary provdes nformaton on whether the lnear regresson model explans a sgnfcant porton of the varaton n the values of y. The value for F s the result of an F-test of the followng null and alternatve hypotheses. H 0 : regresson model does not explan the varaton n y H : regresson model does explan the varaton n y The value n the column for Sgnfcance F s the probablty for retanng the null hypothess. In ths example, the probablty s %, suggestng that there s strong evdence for acceptng the regresson model. s s the case wth the correlaton coeffcent, a small value for the probablty s a lkely outcome for any calbraton curve, even when the model s napproprate. The probablty for retanng the null hypothess for the data n Fgure 5.17, for example, s %. The thrd table n Fgure 5.16 provdes a summary of the model tself. The values for the model s coeffcents the slope, β 1, and the y-ntercept, β 0 are dentfed as ntercept and wth your label for the x-axs data, whch n ths example s. The standard devatons for the coeffcents, s b 0 and s b, are n the column labeled Standard error. The column t Stat and the 1 column P-value are for the followng t-tests. slope H 0 : β 1 0, H : β 1 0 y-ntercept H 0 : β 0 0, H : β 0 0 The results of these t-tests provde convncng evdence that the slope s not zero, but no evdence that the y-ntercept sgnfcantly dffers from zero. lso shown are the 95% confdence ntervals for the slope and the y-ntercept (lower 95% and upper 95%). PROGRM THE FORMULS YOURSELF thrd approach to completng a regresson analyss s to program a spreadsheet usng Excel s bult-n formula for a summaton sum(frst cell:last cell) and ts ablty to parse mathematcal equatons. The resultng spreadsheet s shown n Fgure Source URL: ttrbuted to [Davd Harvey] Page 38 of 56

39 hapter 5 Standardzng nalytcal Methods 191 B D E F 1 x y xy x^ n *B ^ slope (F1*8-8*B8)/(F1*D8-8^) *B3 3^ y-nt (B8-F*8)/F *B4 4^ *B5 5^ *B6 6^ *B7 7^ 8 9 sum(:7) sum(b:b7) sum(:7) sum(d:d7) <--sums Fgure 5.18 Spreadsheet showng the formulas for calculatng the slope and the y-ntercept for the data n Example 5.9. The cells wth the shadng contan formulas that you must enter. Enter the formulas n cells 3 to 7, and cells D3 to D7. Next, enter the formulas for cells 9 to D9. Fnally, enter the formulas n cells F and F3. When you enter a formula, Excel replaces t wth the resultng calculaton. The values n these cells should agree wth the results n Example 5.9. You can smplfy the enterng of formulas by copyng and pastng. For example, enter the formula n cell. Select Edt: opy, clck and drag your cursor over cells 3 to 7, and select Edt: Paste. Excel automatcally updates the cell referencng. USING EXEL TO VISULIZE THE REGRESSION MODEL You can use Excel to examne your data and the regresson lne. Begn by plottng the data. Organze your data n two columns, placng the x values n the left-most column. lck and drag over the data and select Insert: hart... from the man menu. Ths launches Excel s hart Wzard. Select xy-chart, choosng the opton wthout lnes connectng the ponts. lck on Next and work your way through the screens, talorng the plot to meet your needs. To add a regresson lne to the chart, clck on the chart and select hart: dd Trendlne... from the man men. Pck the straght-lne model and clck OK to add the lne to your chart. By default, Excel dsplays the regresson lne from your frst pont to your last pont. Fgure 5.19 shows the result for the data n Fgure y-axs 30 0 Excel s default optons for xy-charts do not make for partcularly attractve scentfc fgures. For example, Excel automatcally adds grd lnes parallel to the x-axs, whch s a common practce n busness charts. You can deselect them usng the Grd lnes tab n the hart Wzard. Excel also defaults to a gray background. To remove ths, just double-clck on the chart s background and select none n the resultng pop-up wndow x-axs Fgure 5.19 Example of an Excel scatterplot showng the data and a regresson lne. Source URL: ttrbuted to [Davd Harvey] Page 39 of 56

40 19 nalytcal hemstry Resduals Fgure 5.0 Example of Excel s plot of a regresson model s resdual errors Excel also wll create a plot of the regresson model s resdual errors. To create the plot, buld the regresson model usng the nalyss ToolPak, as descrbed earler. lckng on the opton for Resdual plots creates the plot shown n Fgure 5.0. Practce Exercse 5.6 Use Excel to complete the regresson analyss n Practce Exercse 5.4. lck here to revew your answer to ths exercse. LIMITTIONS TO USING EXEL FOR REGRESSION NLYSIS Excel s bggest lmtaton for a regresson analyss s that t does not provde a functon for calculatng the uncertanty when predctng values of x. In terms of ths chapter, Excel can not calculate the uncertanty for the analyte s concentraton,, gven the sgnal for a sample, S samp. nother lmtaton s that Excel does not nclude a bult-n functon for a weghted lnear regresson. You can, however, program a spreadsheet to handle these calculatons. 5F. R Let s use Excel to ft the followng straght-lne model to the data n Example 5.9. y β + β 0 1 x ENTERING DT ND RETING THE REGRESSION MODEL To begn, create objects contanng the concentraton of the standards and ther correspondng sgnals. > conc c(0, 0.1, 0., 0.3, 0.4, 0.5) > sgnal c(0, 1.36, 4.83, 35.91, 48.79, 60.4) The command for creatng a straght-lne lnear regresson model s s you mght guess, lm s short for lnear model. lm(y ~ x) Source URL: ttrbuted to [Davd Harvey] Page 40 of 56

41 hapter 5 Standardzng nalytcal Methods 193 where y and x are the objects contanng our data. To access the results of the regresson analyss, we assgn them to an object usng the followng command > model lm(sgnal ~ conc) where model s the name we assgn to the object. You can choose any name for the object contanng the results of the regresson analyss. EVLUTING THE LINER REGRESSION MODEL To evaluate the results of a lnear regresson we need to examne the data and the regresson lne, and to revew a statstcal summary of the model. To examne our data and the regresson lne, we use the plot command, whch takes the followng general form plot(x, y, optonal arguments to control style) where x and y are objects contanng our data, and the ablne command ablne(object, optonal arguments to control style) where object s the object contanng the results of the lnear regresson. Enterng the commands > plot(conc, sgnal, pch 19, col blue, cex ) > ablne(model, col red ) creates the plot shown n Fgure 5.1. To revew a statstcal summary of the regresson model, we use the summary command. > summary(model) The resultng output, shown n Fgure 5., contans three sectons. The frst secton of R s summary of the regresson model lsts the resdual errors. To examne a plot of the resdual errors, use the command sgnal > plot(model, whch1) conc The name ablne comes from the followng common form for wrtng the equaton of a straght-lne. y a + bx The reason for ncludng the argument whch1 s not mmedately obvous. When you use R s plot command on an object created by the lm command, the default s to create four charts summarzng the model s sutablty. The frst of these charts s the resdual plot; thus, whch1 lmts the output to ths plot. Fgure 5.1 Example of a regresson plot n R showng the data and the regresson lne. You can customze your plot by adjustng the plot command s optonal arguments. The argument pch controls the symbol used for plottng ponts, the argument col allows you to select a color for the ponts or the lne, and the argument cex sets the sze for the ponts. You can use the command help(plot) to learn more about the optons for plottng data n R. Source URL: ttrbuted to [Davd Harvey] Page 41 of 56

42 194 nalytcal hemstry.0 > modellm(sgnal~conc) > summary(model) all: lm(formula sgnal ~ conc) Resduals: Fgure 5. The summary of R s regresson analyss. See the text for a dscusson of how to nterpret the nformaton n the output s three sectons. oeffcents: Estmate Std. Error t value Pr(> t ) (Intercept) conc e-08 *** --- Sgnf. codes: 0 *** ** 0.01 * Resdual standard error: on 4 degrees of freedom Multple R-Squared: , djusted R-squared: F-statstc: 1.568e+04 on 1 and 4 DF, p-value:.441e-08 See Secton 4F.1 for a revew of the t-test. whch produces the result shown n Fgure 5.3. Note that R plots the resduals aganst the predcted (ftted) values of y nstead of aganst the known values of x. The choce of how to plot the resduals s not crtcal, as you can see by comparng Fgure 5.3 to Fgure 5.0. The lne n Fgure 5.3 s a smoothed ft of the resduals. The second secton of Fgure 5. provdes the model s coeffcents the slope, β 1, and the y-ntercept, β 0 along wth ther respectve standard devatons (Std. Error). The column t value and the column Pr(> t ) are for the followng t-tests. slope H 0 : β 1 0, H : β 1 0 y-ntercept H 0 : β 0 0, H : β 0 0 The results of these t-tests provde convncng evdence that the slope s not zero, but no evdence that the y-ntercept sgnfcantly dffers from zero. Resduals Resduals vs Ftted Fgure 5.3 Example showng R s plot of a regresson model s resdual error Ftted values lm(sgnal ~ conc) Source URL: ttrbuted to [Davd Harvey] Page 4 of 56

43 hapter 5 Standardzng nalytcal Methods 195 The last secton of the regresson summary provdes the standard devaton about the regresson (resdual standard error), the square of the correlaton coeffcent (multple R-squared), and the result of an F-test on the model s ablty to explan the varaton n the y values. For a dscusson of the correlaton coeffcent and the F-test of a regresson model, as well as ther lmtatons, refer to the secton on usng Excel s data analyss tools. See Secton 4F. and Secton 4F.3 for a revew of the F-test. PREDITING THE UNERTINTY IN GIVEN S SMP Unlke Excel, R ncludes a command for predctng the uncertanty n an analyte s concentraton,, gven the sgnal for a sample, S samp. Ths command s not part of R s standard nstallaton. To use the command you need to nstall the chemal package by enterng the followng command (note: you wll need an nternet connecton to download the package). > nstall.packages( chemal ) fter nstallng the package, you wll need to load the functons nto R usng the followng command. (note: you wll need to do ths step each tme you begn a new R sesson as the package does not automatcally load when you start R). > lbrary( chemal ) The command for predctng the uncertanty n s nverse.predct, whch takes the followng form for an unweghted lnear regresson nverse.predct(object, newdata, alpha value) where object s the object contanng the regresson model s results, newdata s an object contanng values for S samp, and value s the numercal value for the sgnfcance level. Let s use ths command to complete Example Frst, we create an object contanng the values of S samp > sample c(9.3, 9.16, 9.51) and then we complete the computaton usng the followng command > nverse.predct(model, sample, alpha 0.05) producng the result shown n Fgure 5.4. The analyte s concentraton,, s gven by the value $Predcton, and ts standard devaton, s, s shown as $`Standard Error`. The value for $onfdence s the confdence nterval, ±ts, for the analyte s concentraton, and $`onfdence Lmts` provdes the lower lmt and upper lmt for the confdence nterval for. You need to nstall a package once, but you need to load the package each tme you plan to use t. There are ways to confgure R so that t automatcally loads certan packages; see n Introducton to R for more nformaton (clck here to vew a PDF verson of ths document). USING R FOR WEIGHTED LINER REGRESSION R s command for an unweghted lnear regresson also allows for a weghted lnear regresson by ncludng an addtonal argument, weghts, whose value s an object contanng the weghts. lm(y ~ x, weghts object) Source URL: ttrbuted to [Davd Harvey] Page 43 of 56

44 196 nalytcal hemstry.0 > nverse.predct(model, sample, alpha 0.05) $Predcton [1] $`Standard Error` [1] $onfdence [1] Fgure 5.4 Output from R s command for predctng the analyte s concentraton,, from the sample s sgnal, S samp. $`onfdence Lmts` [1] You may have notced that ths way of defnng weghts s dfferent than that shown n equaton 5.8. In dervng equatons for a weghted lnear regresson, you can choose to normalze the sum of the weghts to equal the number of ponts, or you can choose not to the algorthm n R does not normalze the weghts. Practce Exercse 5.7 Use Excel to complete the regresson analyss n Practce Exercse 5.4. lck here to revew your answer to ths exercse. Let s use ths command to complete Example 5.1. Frst, we need to create an object contanng the weghts, whch n R are the recprocals of the standard devatons n y, (s y ). Usng the data from Example 5.1, we enter > syc(0.0, 0.0, 0.07, 0.13, 0., 0.33) > w1/sy^ to create the object contanng the weghts. The commands > modelw lm(sgnal ~ conc, weghts w) > summary(modelw) generate the output shown n Fgure 5.5. ny dfference between the results shown here and the results shown n Example 5.1 are the result of round-off errors n our earler calculatons. > modelwlm(sgnal~conc, weghts w) > summary(modelw) all: lm(formula sgnal ~ conc, weghts w) Resduals: Fgure 5.5 The summary of R s regresson analyss for a weghted lnear regresson. The types of nformaton shown here s dentcal to that for the unweghted lnear regresson n Fgure 5.. oeffcents: Estmate Std. Error t value Pr(> t ) (Intercept) conc e-08 *** --- Sgnf. codes: 0 *** ** 0.01 * Resdual standard error: on 4 degrees of freedom Multple R-Squared: , djusted R-squared: F-statstc: 1.717e+04 on 1 and 4 DF, p-value:.034e-08 Source URL: ttrbuted to [Davd Harvey] Page 44 of 56

45 hapter 5 Standardzng nalytcal Methods 197 5G Key Terms external standard nternal standard lnear regresson matrx matchng method of standard addtons multple-pont standardzaton normal calbraton curve prmary standard reagent grade resdual error secondary standard seral dluton sngle-pont standardzaton unweghted lnear regresson standard devaton about the regresson weghted lnear regresson total Youden blank s you revew ths chapter, try to defne a key term n your own words. heck your answer by clckng on the key term, whch wll take you to the page where t was frst ntroduced. lckng on the key term there, wll brng you back to ths page so that you can contnue wth another key term. 5H hapter Summary In a quanttatve analyss we measure a sgnal, S total, and calculate the amount of analyte, n or, usng one of the followng equatons. S k n + S total reag S k + S total reag To obtan an accurate result we must elmnate determnate errors affectng the sgnal, S total, the method s senstvty, k, and the sgnal due to the reagents, S reag. To ensure that we accurately measure S total, we calbrate our equpment and nstruments. To calbrate a balance, for example, we a standard weght of known mass. The manufacturer of an nstrument usually suggests approprate calbraton standards and calbraton methods. To standardze an analytcal method we determne ts senstvty. There are several standardzaton strateges, ncludng external standards, the method of standard addton and nternal standards. The most common strategy s a multple-pont external standardzaton, resultng n a normal calbraton curve. We use the method of standard addtons, n whch known amounts of analyte are added to the sample, when the sample s matrx complcates the analyss. When t s dffcult to reproducbly handle samples and standards, we may choose to add an nternal standard. Sngle-pont standardzatons are common, but are subject to greater uncertanty. Whenever possble, a multple-pont standardzaton s preferred, wth results dsplayed as a calbraton curve. lnear regresson analyss can provde an equaton for the standardzaton. reagent blank corrects for any contrbuton to the sgnal from the reagents used n the analyss. The most common reagent blank s one n whch an analyte-free sample s taken through the analyss. When a smple reagent blank does not compensate for all constant sources of determnate error, other types of blanks, such as the total Youden blank, can be used. Source URL: ttrbuted to [Davd Harvey] Page 45 of 56

46 198 nalytcal hemstry.0 5I Problems 1. Descrbe how you would use a seral dluton to prepare 100 ml each of a seres of standards wth concentratons of , , , and M from a M stock soluton. alculate the uncertanty for each soluton usng a propagaton of uncertanty, and compare to the uncertanty f you were to prepare each soluton by a sngle dluton of the stock soluton. You wll fnd tolerances for dfferent types of volumetrc glassware and dgtal ppets n Table 4. and Table 4.3. ssume that the uncertanty n the stock soluton s molarty s ± Three replcate determnatons of S total for a standard soluton that s 10.0 ppm n analyte gve values of 0.163, 0.157, and (arbtrary unts). The sgnal for the reagent blank s alculate the concentraton of analyte n a sample wth a sgnal of g sample contanng an analyte s transferred to a 50-mL volumetrc flask and dluted to volume. When a ml alquot of the resultng soluton s dluted to 5.00 ml t gves sgnal of 0.35 (arbtrary unts). second mL porton of the soluton s spked wth ml of a 1.00-ppm standard soluton of the analyte and dluted to 5.00 ml. The sgnal for the spked sample s alculate the weght percent of analyte n the orgnal sample ml sample contanng an analyte gves a sgnal of 11.5 (arbtrary unts). second 50 ml alquot of the sample, whch s spked wth 1.00 ml of a 10.0-ppm standard soluton of the analyte, gves a sgnal of 3.1. What s the analyte s concentraton n the orgnal sample? 5. n approprate standard addtons calbraton curve based on equaton 5.10 places S spke (V o + V ) on the y-axs and V on the x-axs. learly explan why you can not plot S spke on the y-axs and [V / (V o + V )] on the x-axs. In addton, derve equatons for the slope and y-ntercept, and explan how you can determne the amount of analyte n a sample from the calbraton curve. 6. standard sample contans 10.0 mg/l of analyte and 15.0 mg/l of nternal standard. nalyss of the sample gves sgnals for the analyte and nternal standard of and 0.33 (arbtrary unts), respectvely. Suffcent nternal standard s added to a sample to make ts concentraton 15.0 mg/l nalyss of the sample yelds sgnals for the analyte and nternal standard of 0.74 and 0.198, respectvely. Report the analyte s concentraton n the sample. Source URL: ttrbuted to [Davd Harvey] Page 46 of 56

47 hapter 5 Standardzng nalytcal Methods 199 (a) Sgnal Sgnal (b) Sgnal Sgnal (c) Sgnal Sgnal Fgure 5.6 albraton curves to accompany Problem For each of the par of calbraton curves shown n Fgure 5.6, select the calbraton curve usng the more approprate set of standards. Brefly explan the reasons for your selectons. The scales for the x-axs and y-axs are the same for each par. 8. The followng data are for a seres of external standards of d + buffered to a ph of [d + ] (nm) S total (n) Wojcechowsk, M.; Balcerzak, J. nal. hm. cta 1991, 49, Source URL: ttrbuted to [Davd Harvey] Page 47 of 56

48 00 nalytcal hemstry.0 (a) Use a lnear regresson to determne the standardzaton relatonshp and report confdence ntervals for the slope and the y-ntercept. (b) onstruct a plot of the resduals and comment on ther sgnfcance. t a ph of 3.7 the followng data were recorded for the same set of external standards. [d + ] (nm) S total (n) (c) How much more or less senstve s ths method at the lower ph? (d) sngle sample s buffered to a ph of 3.7 and analyzed for cadmum, yeldng a sgnal of Report the concentraton of d + n the sample and ts 95% confdence nterval. 9. To determne the concentraton of analyte n a sample, a standard addtons was performed mL porton of sample was analyzed and then successve 0.10-mL spkes of a mg/L standard of the analyte were added, analyzng after each spke. The followng table shows the results of ths analyss. V spke (ml) S total (arbtrary unts) onstruct an approprate standard addtons calbraton curve and use a lnear regresson analyss to determne the concentraton of analyte n the orgnal sample and ts 95% confdence nterval. 10. Troost and Olavsesn nvestgated the applcaton of an nternal standardzaton to the quanttatve analyss of polynuclear aromatc hydrocarbons. 15 The followng results were obtaned for the analyss of phenanthrene usng sotopcally labeled phenanthrene as an nternal standard. Each soluton was analyzed twce. / IS S /S IS (a) Determne the standardzaton relatonshp usng a lnear regresson, and report confdence ntervals for the slope and the y-ntercept. verage the replcate sgnals for each standard before completng the lnear regresson analyss. (b) Based on your results explan why the authors concluded that the nternal standardzaton was napproprate. 15 Troost, J. R.; Olavesen, E. Y. nal. hem. 1996, 68, Source URL: ttrbuted to [Davd Harvey] Page 48 of 56

49 hapter 5 Standardzng nalytcal Methods In hapter 4 we used a pared t-test to compare two analytcal methods used to ndependently analyze a seres of samples of varable composton. n alternatve approach s to plot the results for one method versus the results for the other method. If the two methods yeld dentcal results, then the plot should have an expected slope, β 1, of 1.00 and an expected y-ntercept, β 0, of 0.0. We can use a t-test to compare the slope and the y-ntercept from a lnear regresson to the expected values. The approprate test statstc for the y-ntercept s found by rearrangng equaton 5.3. b t β 0 0 b 0 exp s s b b 0 0 Rearrangng equaton 5. gves the test statstc for the slope. t exp β b. b s s b b 1 1 Reevaluate the data n problem 5 from hapter 4 usng the same sgnfcance level as n the orgnal problem. lthough ths s a common approach for comparng two analytcal methods, t does volate one of the requrements for an unweghted lnear regresson that ndetermnate errors affect y only. Because ndetermnate errors affect both analytcal methods, the result of unweghted lnear regresson s based. More specfcally, the regresson underestmates the slope, b 1, and overestmates the y-ntercept, b 0. We can mnmze the effect of ths bas by placng the more precse analytcal method on the x-axs, by usng more samples to ncrease the degrees of freedom, and by usng samples that unformly cover the range of concentratons. For more nformaton, see Mller, J..; Mller, J. N. Statstcs for nalytcal hemstry, 3rd ed. Ells Horwood PTR Prentce-Hall: New York, lternatve approaches are found n Hartman,.; Smeyers-Verbeke, J.; Pennnckx, W.; Massart, D. L. nal. hm. cta 1997, 338, 19 40, and Zwanzger, H. W.; Sârbu,. nal. hem. 1998, 70, onsder the followng three data sets, each contanng value of y for the same values of x. Data Set 1 Data Set Data Set 3 x y 1 y y (a) n unweghted lnear regresson analyss for the three data sets gves nearly dentcal results. To three sgnfcant fgures, each data set has a slope of and a y-ntercept of The standard devatons n the slope and the y-ntercept are and 1.15 for each Source URL: ttrbuted to [Davd Harvey] Page 49 of 56

50 0 nalytcal hemstry.0 data set. ll three standard devatons about the regresson are 1.4, and all three data regresson lnes have a correlaton coeffcents of Based on these results for a lnear regresson analyss, comment on the smlarty of the data sets. (b) omplete a lnear regresson analyss for each data set and verfy that the results from part (a) are correct. onstruct a resdual plot for each data set. Do these plots change your concluson from part (a)? Explan. (c) Plot each data set along wth the regresson lne and comment on your results. (d) Data set 3 appears to contan an outler. Remove ths apparent outler and reanalyze the data usng a lnear regresson. omment on your result. (e) Brefly comment on the mportance of vsually examnng your data. 13. Fanke and co-workers evaluated a standard addtons method for a voltammetrc determnaton of Tl. 16 summary of ther results s tabulated n the followng table. ppm Tl added Instrument Response (μ) Use a weghted lnear regresson to determne the standardzaton relatonshp for ths data. 5J Solutons to Practce Exercses Practce Exercse 5.1 Substtutng the sample s absorbance nto the calbraton equaton and solvng for gve S samp M M For the one-pont standardzaton, we frst solve for k 16 Franke, J. P.; de Zeeuw, R..; Hakkert, R. nal. hem. 1978, 50, Source URL: ttrbuted to [Davd Harvey] Page 50 of 56

51 hapter 5 Standardzng nalytcal Methods 03 k S M M 1 and then use ths value of k to solve for. Ssamp k M 3 M When usng multple standards, the ndetermnate errors affectng the sgnal for one standard are partally compensated for by the ndetermnate errors affectng the other standards. The standard selected for the onepont standardzaton has a sgnal that s smaller than that predcted by the regresson equaton, whch underestmates k and overestmates. lck here to return to the chapter. Practce Exercse 5. We begn wth equaton 5.8 V S k V spke o f V V f rewrtng t as 0 kv V o k V V f f whch s n the form of the lnear equaton Y y-ntercept + slope X where Y s S spke and X s V /V f. The slope of the lne, therefore, s k, and the y-ntercept s k V o /V f. The x-ntercept s the value of X when Y s zero, or kv o 0 + k { x-ntercept} V f x-ntercept kv o V V f k V f o lck here to return to the chapter. Practce Exercse 5.3 Usng the calbraton equaton from Fgure 5.7a, we fnd that the x-ntercept s Source URL: ttrbuted to [Davd Harvey] Page 51 of 56

52 04 nalytcal hemstry.0 x-ntercept ml ml Pluggng ths nto the equaton for the x-ntercept and solvng for gves the concentraton of Mn + as ml x-ntercept ml mg/ L 696. mg/l For Fgure 7b, the x-ntercept s x-ntercept ml ml and the concentraton of Mn + s ml x-ntercept ml mg/l L lck here to return to the chapter. Practce Exercse 5.4 We begn by settng up a table to help us organze the calculaton. x y x y x ddng the values n each column gves x y xy x Substtutng these values nto equaton 5.17 and equaton 5.18, we fnd that the slope and the y-ntercept are (. ) (. ) (. ) 4 ( ) ( ) 3 b Source URL: ttrbuted to [Davd Harvey] Page 5 of 56

53 hapter 5 Standardzng nalytcal Methods ( ) b The regresson equaton s S To calculate the 95% confdence ntervals, we frst need to determne the standard devaton about the regresson. The followng table wll help us organze the calculaton. x y ŷ ( y yˆ ) ddng together the data n the last column gves the numerator of equaton 5.19 as The standard devaton about the regresson, therefore, s s r Next, we need to calculate the standard devatons for the slope and the y-ntercept usng equaton 5.0 and equaton ( ) s b1 4 6 ( ) ( ) 3 4 ( ) ( ) s b0 4 6 ( ) ( ) The 95% confdence ntervals are b ± ts b ± ( ) M -1 ± M -1 β β 0 0 b ± ts b ± {. 78 ( } ± Wth an average S samp of 0.114, the concentraton of analyte,, s S samp b b M 1 3 M Source URL: ttrbuted to [Davd Harvey] Page 53 of 56

54 06 nalytcal hemstry.0 The standard devaton n s s ( ) ( 9. 57) ( ) and the 95% confdence nterval s μ 3 5 ± ts ± {. 78 ( )} M± lck here to return to the chapter. 3 3 Practce Exercse 5.5 To create a resdual plot, we need to calculate the resdual error for each standard. The followng table contans the relevant nformaton. M resdual error Fgure 5.7 Plot of the resdual errors for the data n Practce Exercse 5.5. x y ŷ y yˆ Fgure 5.7 shows a plot of the resultng resdual errors s shown here. The resdual errors appear random and do not show any sgnfcant dependence on the analyte s concentraton. Taken together, these observatons suggest that our regresson model s approprate. lck here to return to the chapter Practce Exercse 5.6 Begn by enterng the data nto an Excel spreadsheet, followng the format shown n Fgure Because Excel s Data nalyss tools provde most of the nformaton we need, we wll use t here. The resultng output, whch s shown n Fgure 5.8, contans the slope and the y-ntercept, along wth ther respectve 95% confdence ntervals. Excel does not provde a functon for calculatng the uncertanty n the analyte s concentraton,, gven the sgnal for a sample, S samp. You must complete these calculatons by hand. Wth an S samp. of 0.114, S b samp M -1 b M 1 Source URL: ttrbuted to [Davd Harvey] Page 54 of 56

55 hapter 5 Standardzng nalytcal Methods 07 SUMMRY OUTPUT Regresson Statstcs Multple R R Square djusted R Sq Standard Error Observatons 6 NOV df SS MS F Sgnfcance F Regresson E-08 Resdual E E-06 Total oeffcents Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept E Fgure 5.8 Excel s summary of the regresson results for Practce Exercse 5.6. The standard devaton n s s ( ) ( 9. 59) ( ) and the 95% confdence nterval s μ 3 5 ± ts ± {. 78 ( )} M± lck here to return to the chapter 3 3 Practce Exercse 5.7 Fgure 5.9 shows an R sesson for ths problem, ncludng loadng the chemal package, creatng objects to hold the values for, S, and Ssamp. Note that for Ssamp, we do not have the actual values for the three replcate measurements. In place of the actual measurements, we just enter the average sgnal three tmes. Ths s okay because the calculaton depends on the average sgnal and the number of replcates, and not on the ndvdual measurements. lck here to return to the chapter M Source URL: ttrbuted to [Davd Harvey] Page 55 of 56

56 08 nalytcal hemstry.0 > lbrary("chemal") > concc(0, 1.55e-3, 3.16e-3, 4.74e-3, 6.34e-3, 7.9e-3) > sgnalc(0, 0.050, 0.093, 0.143, 0.188, 0.36) > modellm(sgnal~conc) > summary(model) all: lm(formula sgnal ~ conc) Resduals: oeffcents: Estmate Std. Error t value Pr(> t ) (Intercept) conc e-08 *** --- Sgnf. codes: 0 *** ** 0.01 * Resdual standard error: on 4 degrees of freedom Multple R-Squared: , djusted R-squared: F-statstc: 9690 on 1 and 4 DF, p-value: 6.386e-08 > sampc(0.114, 0.114, 0.114) > nverse.predct(model,samp,alpha0.05) $Predcton [1] $`Standard Error` [1] e-05 $onfdence [1] $`onfdence Lmts` [1] Fgure 5.9 R sesson for completng Practce Exercse 5.7. Source URL: ttrbuted to [Davd Harvey] Page 56 of 56