Preparation of Calibration Curves

Transcription

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2 Preparato of Calbrato Curves A Gude to Best Practce September 3 Cotact Pot: Lz Prchard Tel: Prepared by: Vck Barwck Approved by: Date: The work descrbed ths report was supported uder cotract wth the Departmet of Trade ad Idustry as part of the Natoal Measuremet System Vald Aalytcal Measuremet (VAM) Programme Mlestoe Referece: KT/.3 LGC/VAM/3/3 LGC Lmted 3

3 Cotets. Itroducto. The Calbrato Process. Plag the epermets. Makg the measuremets 3.3 Plottg the results 4.3. Evaluatg the scatter plot 5.4 Carryg out regresso aalyss 6.4. Assumptos 7.4. Carryg out regresso aalyss usg software 7.5 Evaluatg the results of the regresso aalyss 8.5. Plot of the resduals 8.5. Regresso statstcs 9.6 Usg the calbrato fucto to estmate values for test samples 4.7 Estmatg the ucertaty predcted cocetratos 4.8 Stadard error of predcto worked eample 6 3. Coclusos 8 Apped : Protocol ad results sheet 9 Apped : Eample set of results 5 Apped 3: Lear regresso equatos 7 LGC/VAM/3/3 Page

4 . Itroducto Istrumet calbrato s a essetal stage most measuremet procedures. It s a set of operatos that establsh the relatoshp betwee the output of the measuremet system (e.g., the respose of a strumet) ad the accepted values of the calbrato stadards (e.g., the amout of aalyte preset). A large umber of aalytcal methods requre the calbrato of a strumet. Ths typcally volves the preparato of a set of stadards cotag a kow amout of the aalyte of terest, measurg the strumet respose for each stadard ad establshg the relatoshp betwee the strumet respose ad aalyte cocetrato. Ths relatoshp s the used to trasform measuremets made o test samples to estmates of the amout of aalyte preset, as show Fgure. Istrumet respose y r Cocetrato /mg L - Fgure : Typcal calbrato curve As calbrato s such a commo ad mportat step aalytcal methods, t s essetal that aalysts have a good uderstadg of how to set up calbrato epermets ad how to evaluate the results obtaed. Durg August a bechmarkg eercse was udertake, whch volved the preparato ad aalyss of calbrato stadards ad a test sample usg UV spectrophotometry. The am of the eercse was to vestgate the ucertates assocated wth the costructo of a calbrato curve, ad wth usg the calbrato curve to determe the cocetrato of a ukow compoud a aqueous soluto. I addto, t was hoped to detfy ay commo problems ecoutered by aalysts udertakg calbrato epermets. Members of the Evrometal Measuremet Trag Network (EMTN) ad the SOCSA Aalytcal Network Group (SANG) were vted to partcpate the eercse. Fve members of EMTN, s members of SANG ad three orgasatos who are members of both EMTN ad SANG submtted results. Some partcpats submtted results from more tha oe aalyst, gvg 9 sets of results total. Full detals of the protocol ad results sheet crculated to the laboratores ca be foud Apped. Apped cotas a deal set of results from the bechmarkg eercse to llustrate how the report should be preseted. The results of the bechmarkg eercse were terestg. Although the eercse tally appeared relatvely straghtforward, a umber of mstakes carryg out the epermets ad aalysg the data were detfed. Sce a umber of the mstakes occurred more tha oe laboratory, t s lkely that other laboratores carryg out smlar eercses may make the same errors. LGC/VAM/3/3 Page

5 The am of ths gude s to hghlght good practce settg up calbrato epermets, ad to epla how the results should be evaluated. The gude focuses o calbrato epermets where the relatoshp betwee respose ad cocetrato s epected to be lear, although may of the prcples of good practce descrbed ca be appled to o-lear systems. Wth software packages such as Ecel, t easy to geerate a large umber of statstcs. The gude also eplas the meag ad correct terpretato of some of the statstcal terms commoly assocated wth calbrato.. The Calbrato Process There are a umber of stages the process of calbratg a aalytcal strumet. These are summarsed below: Pla the epermets; Make measuremets; Plot the results; Carry out statstcal (regresso) aalyss o the data to obta the calbrato fucto; Evaluate the results of the regresso aalyss; Use the calbrato fucto to estmate values for test samples; Estmate the ucertaty assocated wth the values obtaed for test samples. The gude cosders each of these steps tur.. Plag the epermets The ssues a aalyst eeds to cosder whe plag a calbrato study are as follows: The umber of calbrato stadards; The cocetrato of each of the calbrato stadards; The umber of replcates at each cocetrato; Preparato of the calbrato stadards; Oe of the frst questos aalysts ofte ask s, How may epermets do I eed to do?. Due to tme ad other costras, ths ofte traslates as, What s the absolute mmum I ca do?. Whe thkg about a calbrato epermet, ths relates to the umber of calbrato stadards that eed to be aalysed, ad the amout of replcato at each calbrato level. For a tal assessmet of the calbrato fucto, as part of method valdato for eample, stadards wth at least seve dfferet cocetratos (cludg a blak) should be prepared. The stadard cocetratos should cover, at least, the rage of cocetratos ecoutered durg the aalyss of test samples ad be evely spaced across the rage (see Secto.3.). Ideally, the calbrato rage should be establshed so that the majorty of the test sample cocetratos fall towards the cetre of the rage. As dscussed Secto.7, ths s the area of the calbrato rage where the ucertaty assocated wth predcted cocetratos s at ts mmum. It s also useful to make at least duplcate measuremets at each cocetrato level, partcularly at the method valdato stage, as t allows the precso of the calbrato process to be evaluated at each cocetrato level. The replcates should deally be depedet makg replcate measuremets o the same calbrato stadard gves oly partal formato about the calbrato varablty, as t oly covers the precso of the strumet used to make the measuremets, ad does ot clude the preparato of the stadards. Page LGC/VAM/3/3

6 Havg decded o the umber ad cocetratos of the calbrato stadards, the aalyst eeds to cosder how best to prepare them. Frstly, the source of the materal used to prepare the stadards (.e., the referece materal used) requres careful cosderato. The ucertaty assocated wth the calbrato stage of ay method wll be lmted by the ucertaty assocated wth the values of the stadards used to perform the calbrato the ucertaty a result ca ever be less tha the ucertaty the stadard(s) used. Typcally, calbrato solutos are prepared from a pure substace wth a kow purty value or a soluto of a substace wth a kow cocetrato. The ucertaty assocated wth the property value (.e., the purty or the cocetrato) eeds to be cosdered to esure that t s ft for purpose. The matr used to prepare the stadards also requres careful cosderato. Is t suffcet to prepare the stadards a pure solvet, or does the matr eed to be closely matched to that of the test samples? Ths wll deped o the ature of the strumet beg used to aalyse the samples ad stadards ad ts sestvty to compoets the sample other tha the target aalyte. The accuracy of some methods ca be mproved by addg a sutable teral stadard to both calbrato stadards ad test samples ad basg the regresso o the rato of the aalyte respose to that of the teral stadard. The use of a teral stadard corrects for small varatos the operatg codtos. Ideally the stadards should be depedet,.e., they should ot be prepared from a commo stock soluto. Ay error the preparato of the stock soluto wll propagate through the other stadards leadg to a bas the calbrato. A procedure sometmes used the preparato of calbrato stadards s to prepare the most cocetrated stadard ad the dlute t by, say, 5%, to obta the et stadard. Ths stadard s the dluted by 5% ad so o. Ths procedure s ot recommeded as, addto to the lack of depedece, the stadard cocetratos wll ot be evely spaced across the cocetrato rage leadg to the problem of leverage (see Secto.3.). Costructo of a calbrato curve usg seve calbrato stadards every tme a batch of samples s aalysed ca be tme-cosumg ad epesve. If t has bee establshed durg method valdato that the calbrato fucto s lear the t may be possble to use a smplfed calbrato procedure whe the method s used routely, for eample usg fewer calbrato stadards wth oly a sgle replcate at each level. A sgle pot calbrato s a fast way of checkg the calbrato of a system whe there s o doubt about the learty of the calbrato fucto ad the system s ubased (.e., the tercept s ot sgfcatly dfferet from zero, see Secto.5.). The cocetrato of the stadard should be equal to or greater tha the mamum cocetrato lkely to be foud test samples. If there s o doubt about the learty of the calbrato fucto, but there s a kow bas (.e., a o-zero tercept), a two pot calbrato may be used. I ths case, two calbrato stadards are prepared wth cocetratos that ecompass the lkely rage of cocetratos for test samples. Where there s some doubt about the learty of the calbrato fucto over the etre rage of terest, or the stablty of the measuremet system over tme, the bracketg techque may be useful. A prelmary estmate of the aalyte cocetrato the test sample s obtaed. Two calbrato stadards are the prepared at levels that bracket the sample cocetrato as closely as possble. Ths approach s tme cosumg but mmses ay errors due to olearty.. Makg the measuremets It s good practce to aalyse the stadards a radom order, rather tha a set sequece of, for eample, the lowest to the hghest cocetrato. All equpmet used a aalytcal method, from volumetrc glassware to HPLC systems must be ft for ther teded purpose. It s good scece to be able to demostrate that strumets are ft for purpose. Equpmet qualfcato (EQ) s a formal process that LGC/VAM/3/3 Page 3

7 provdes documeted evdece that a strumet s ft for ts teded purpose ad kept a state of mateace ad calbrato cosstet wth ts use. Ideally the strumet used to make measuremets o the stadards ad samples should have goe through the EQ [,, 3] process..3 Plottg the results It s always good practce to plot data before carryg out ay statstcal aalyss. I the case of regresso ths s essetal, as some of the statstcs geerated ca be msleadg f cosdered solato (see secto.5). Ay data sets of equal sze ca be plotted agast each other o a dagram to see f a relatoshp (a correlato) ests betwee them (Fgure ). Istrumet respose y-as as Cocetrato /mg L - Fgure : Scatter plot of strumet respose data versus cocetrato The horzotal as s defed as the -as ad the vertcal as as the y-as. Whe plottg data from a calbrato epermet, the coveto s to plot the strumet respose data o the y-as ad the values for the stadards o the -as. Ths s because the statstcs used the regresso aalyss assume that the errors the values o the -as are sgfcat compared wth those o the y-as. I the case of calbrato data, the assumpto s that the errors the strumet respose values (due to radom varato) are greater tha those the values assged to the stadards. I most cases ths s ot a ureasoable assumpto. The values plotted o the y-as are sometmes referred to as the depedet varable, because ther values deped o the magtude of the other varable. For eample, the strumet respose wll obvously be depedet o the cocetrato of the aalyte preset the stadards. Coversely, the data plotted o the -as are referred to as the depedet varable. P. Bedso ad M. Sarget, J. Accred. Qual. Assur., 996,, P. Bedso ad D. Rudd, J. Accred. Qual. Assur., 999, 4, D. G. Holcombe ad M. C. Boardma, J. Accred. Qual. Assur.,, 6, Page 4 LGC/VAM/3/3

8 .3. Evaluatg the scatter plot The plot of the data should be spected for possble outlers ad pots of fluece. I geeral, a outler s a result whch s sgfcatly dfferet from the rest of the data set. I the case of calbrato, a outler would appear as a pot whch s well removed from the other calbratos pots. A pot of fluece s a calbrato pot whch has a dsproportoate effect o the posto of the regresso le. A pot of fluece may be a outler, but may also be caused by poor epermetal desg (see secto.). Pots of fluece ca have oe of two effects o a calbrato le leverage or bas. Leverage 4 pot wth hgh leverage 8 Istrumet respose Istrumet respose Cocetrato /mg L - Cocetrato /mg L - a) Leverage due to uequal dstrbuto of calbrato levels b) Leverage due to the presece of a outler Fgure 3: Pots of fluece leverage A outler at the etremes of the calbrato rage wll chage the posto of the calbrato le by tltg t upwards or dowwards (see Fgure 3b). The pot s sad to have a hgh degree of leverage. Leverage ca be a problem f oe or two of the calbrato pots are a log way from the others alog the -as (see Fgure 3a). These pots wll have a hgh degree of leverage, eve f they are ot outlers. I other words, a relatvely small error the measured respose wll have a sgfcat effect o the posto of the regresso le. Ths stuato arses whe calbrato stadards are prepared by sequetal dluto of solutos (e.g., 3 mg L -, 6 mg L -, 8 mg L -, 4 mg L -, mg L -, mg L -, as llustrated Fgure 3a). Leverage affects both the gradet ad tercept of the le wth the y-as. LGC/VAM/3/3 Page 5

9 Bas 8 Istrumet respose Cocetrato /mg L - Fgure 4: Pots of fluece bas A outler the mddle of the calbrato rage (see Fgure 4) wll shft the regresso le up or dow. The outler s a pot of fluece as t has troduced a bas to the posto of the le. The gradet of the le wll be appromately correct but the tercept wll be wrog..4 Carryg out regresso aalyss I relato to strumet calbrato, the am of lear regresso s to establsh the equato that best descrbes the lear relatoshp betwee strumet respose (y) ad aalyte level (). The relatoshp s descrbed by the equato of the le,.e., y m + c, where m s the gradet of the le ad c s ts tercept wth the y-as. Lear regresso establshes the values of m ad c whch best descrbe the relatoshp betwee the data sets. The equatos for calculatg m ad c are gve Apped 3, but ther values are most readly obtaed usg a sutable software package. Note that regresso of y o (as s usually doe a calbrato study) s ot the same as the regresso of o y. Ths s because the procedures used lear regresso assume that all the errors are the y values ad that the errors the values are sgfcat. Ths s a reasoable assumpto for may aalytcal methods as t s possble to prepare stadards where the ucertaty the cocetrato s sgfcat compared wth the radom varablty of the aalytcal strumet. It s therefore essetal to esure that the strumet respose data ad the stadard cocetratos are correctly assged. Uderstadg the prcples of lear regresso requres a uderstadg of resduals. A resdual s the dfferece betwee a observed y value, ad the y value calculated usg the equato of the ftted le (see Fgure 5). The resduals gve a dcato of how well the le fts the data. I Fgure 5, the sum of the squared resduals for the poorly fttg dashed le wll be much larger tha for the sold best-ft le. It ca be show that the le that gves the smallest sum of the squared resduals best represets the lear relatoshp betwee the ad y varables. Software for lear regresso smply calculates the values for m ad c that mmse the sum of the squared resduals. For ths reaso, ths type of regresso s ofte referred to as, least squares regresso. Page 6 LGC/VAM/3/3

10 8 Istrumet respose Observed Estmated Resdual Cocetrato /mg L - Fgure 5: Least squares lear regresso calculatg the best straght le.4. Assumptos Basc least squares lear regresso reles o a umber of assumptos. A best-ft le wll oly be obtaed whe the assumptos are met. Sgfcat volato of ay of the assumptos wll usually requre specal treatmet whch s outsde the scope of ths gude. The frst assumpto was metoed Secto.4, that s that the error the values should be sgfcat compared wth that of the y values. I addto, the error assocated wth the y values must be ormally dstrbuted. Normalty s hard to test for statstcally wth oly small data sets. If there s doubt about the ormalty t may be suffcet to replace sgle y values wth averages of three or more for each value of, as mea values ted to be ormally dstrbuted eve where dvdual results are ot. The magtude of the error the y values should also be costat across the rage of terest,.e. the stadard devato should be costat. Smple least squares regresso gves equal weght to all pots ths wll ot be approprate f some pots are much less precse tha others. I may chemcal measuremet systems the stadard devato creases wth cocetrato,.e., t s the relatve stadard devato that remas appromately costat rather tha the stadard devato. The geeral soluto to ths problem s to use weghted regresso, whch takes accout of the varablty the y values. [4] Both the ad y data must be cotuous valued,.e., ot restrcted to tegers, sgfcatly trucated or categorsed (e.g., sample umbers, days of the week, etc.). Ths assumpto should be met the case of strumet calbrato as both the strumet respose ad the cocetratos of the stadards ca, theory, take ay value o a cotuous scale..4. Carryg out regresso aalyss usg software The equatos for carryg out a lear regresso are gve Apped 3, but regresso s usually carred out usg software suppled wth the strumet or packages such as Ecel. May software packages allow a regresso aalyss to be carred out wthout frst plottg the data, however t s good practce to produce a plot before carryg out the statstcal aalyss (see Sectos.3 ad.5.). If the opto s avalable, t s also useful to obta a plot of the resduals (see Secto.5.). 4 Statstcs ad chemometrcs for aalytcal chemstry, J. N. Mller ad J. C. Mller, 4 th Edto, Pretce Hall,, ISBN LGC/VAM/3/3 Page 7

11 Most software also allows the tercept wth the y-as to be set to zero whe carryg out the regresso. Ths opto should ot be selected uless t has bee proved that the tercept s cosstetly ot sgfcatly dfferet from zero (see Secto.5.). Fally, esure that the ad y data have bee correctly assged (see Secto.4)..5 Evaluatg the results of the regresso aalyss Usg software to carry out the lear regresso wll result a umber of dfferet statstcal parameters ad possbly (depedg o the software used) a table ad/or plot of the resduals. The meag ad terpretato of each of the commo outputs from a regresso aalyss s dscussed below..5. Plot of the resduals Obtag a plot of the resduals s strogly recommeded as t ca hghlght problems wth the calbrato data that may ot be mmedately obvous from a smple scatter plot of the data. The costructo of a resdual plot s llustrated Fgure 6. 8 Istrumet respose Resdual Cocetrato /mg L - Fgure 6: The resduals plot Page 8 LGC/VAM/3/3

12 Resdual Cocetrato /mg L - Resdual Cocetrato /mg L - 9 a) Ideal - radom dstrbuto of resduals about zero b) Stadard devato creases wth cocetrato Resdual Resdual Cocetrato /mg L - - Cocetrato /mg L - c) Curved respose d) Itercept correctly set to zero Fgure 7: Eamples of resduals plots Fgure 7a shows a deal resdual plot. The resduals are scattered appromately radomly aroud zero, ad there s o tred the spread of resduals wth cocetrato. Fgure 7b shows the patter of resduals that s obtaed f the stadard devato of the strumet respose creases wth aalyte cocetrato. Fgure 7c llustrates a typcal resdual plot that s obtaed whe a straght le s ftted through data that are o-lear. Fally, Fgure 7d shows a possble patter of resduals whe the regresso le has bee correctly ftted through zero (see Sectos.4. ad.5.). Fgures b) to d) should all cause cocer as the patter of the resduals s clearly ot radom..5. Regresso statstcs A typcal output from a regresso aalyss s show Fgure 8. The output show s from Ecel, but smlar formato s obtaed from other software. The dfferet parts of the output are descrbed more detal the followg sectos. LGC/VAM/3/3 Page 9

13 Regresso Statstcs Multple R R Square Adjusted R Square Stadard Error.5646 Observatos 6 ANOVA df SS MS F Sgfcace F Regresso Resdual Total 5.9 Coeffcets Stadard Error t Stat P-value Lower 95% Upper 95% Itercept X Varable Fgure 8: Typcal output from a regresso aalyss usg Ecel The correlato coeffcet, r The correlato coeffcet, r (ad the related parameters r ad adjusted r ) s a measure of the stregth of the degree of correlato betwee the y ad values. I Ecel output t s descrbed as Multple R. r ca take ay value betwee + ad ; the closer t s to, the stroger the correlato. The correlato coeffcet s oe of the statstcs commoly used aalytcal measuremet. Ufortuately, t s easly (ad frequetly) msterpreted. The r value s easly msterpreted because: correlato ad learty are oly loosely related. The coeffcet r s a measure of correlato ot a measure of learty; t s relatvely easy to geerate data wth apparetly good correlato. However, a plot of the data may well reveal that the data would be usatsfactory for the purposes of calbrato (see Fgure 9); for predctos made from the calbrato curve to have small ucertates, r eeds to be very close to (see Secto.7); A low r value does ot ecessarly mea that there s o correlato. There could be a relatoshp betwee the y ad values, but ot a lear oe (see Fgure 9). For these reasos, t s essetal to plot calbrato data, ad ot just rely o the statstcs, whe assessg the ftess-for-purpose of a calbrato curve. Fgure 9 shows some eamples of how the correlato coeffcet ca be msleadg. Page LGC/VAM/3/3

14 8 r r a) No-learty b) Relatoshp other tha lear 6 r.989 r c) A outler causg bas d) A outler causg leverage 8 r e) Poor epermetal desg Fgure 9: Iterpretg the correlato coeffcet Fgure 9a shows a case where the relatoshp s clearly ot lear across the etre rage of values. I Fgure 9b, the r value of zero dcates that there s o lear correlato. However, there s clearly a sgfcat o-lear relatoshp. Fgures 9c ad 9d show the effect of dvdual outlers. I c) the tercept of the le ftted wth the outler preset (broke le) wll be correct compared to the le ftted wth the outler removed (sold le). I d) both the gradet ad the tercept wll be correct. I Fgure 9e, the r value of.998 dcates a strog correlato. However, the two groups of pots are dstct ad ether shows ay LGC/VAM/3/3 Page

15 sgfcat correlato. Although there are data pots, the dstrbuto of the pots dcates that ths s effect a two-pot calbrato (whch wll always gve r ). The questo whch aalysts ofte ask s, How close to does the correlato coeffcet have to be for a good calbrato curve?. What the aalyst s really after s a calbrato le that wll result a satsfactory level of ucertaty the values predcted from the ftted le. The partcular value of r that shows a statstcally sgfcat correlato betwee y ad depeds o the umber of data pots used to calculate t. Fgure shows the value of r that would dcate statstcally sgfcat correlato for dfferet umbers of data pots. Absolute values of r wth the shaded area dcate a statstcally sgfcat correlato, at the 95% cofdece level. Remember that statstcal sgfcace oly dcates some evdece for correlato. It does ot ecessarly mea that the data would be approprate for calbrato. For eample, wth data pots a value of r.6 would be statstcally sgfcat. However, t s hghly ulkely that a calbrato curve wth a correlato coeffcet of.6 would be of ay use, as the ucertates assocated wth predcted values obtaed from such a le would be prohbtvely large (see secto.7) r No. of pots (, y ) Fgure : Statstcally sgfcat values of r (shaded area) at the 95% cofdece level The parameters related to r are r ad adjusted r. r s ofte used to descrbe the fracto of the total varace the data whch s cotrbuted by the le that has bee ftted. Ideally, f there s a good lear relato, the majorty of varablty ca be accouted for by the ftted le. r should therefore be close to. The adjusted r value s terpreted the same way as r but s always lower. It s useful for assessg the effect of addg addtoal terms to the equato of the ftted le (e.g., f a quadratc ft s used stead of a lear ft). The r value always creases o the addto of a etra term to the equato, but ths does ot mea that the eteded equato s ecessarly a better ft of the data. The adjusted r value s more useful such cases as t takes accout of the reducto the degrees of freedom whch occurs each tme a addtoal term s added to the equato of the le (see secto below o the resdual stadard devato), ad therefore does ot automatcally crease o addto of etra terms. Ths guards agast overfttg, whch occurs whe the equato ftted has more terms tha ca be supported by the amout of data avalable (.e., there are suffcet degrees of freedom). Page LGC/VAM/3/3

16 Resdual stadard devato (or stadard error) The resdual stadard devato (also kow as the resdual stadard error) s a statstcal measure of the devato of the data from the ftted regresso le. It s calculated usg Eq. : s () r ( y yˆ ) Eq. where y s the observed value of y for a gve value of, ŷ s the value of y predcted by the equato of the calbrato le for a gve value of, ad s the umber of calbrato pots. ANOVA table Software such as Ecel produces a aalyss of varace (ANOVA) table for the regresso. The sum of squares terms (SS) represet dfferet sources of varablty the calbrato data. Fgure llustrates the org of these terms. The regresso term represets the varablty the data that ca be accouted for by the ftted regresso le. Ideally ths should be large; f there s a good lear relatoshp, the ftted le wll descrbe the majorty of the varablty respose wth cocetrato. The resdual term s the sum of the squared resduals (see secto.4). Ths value should be small compared to the regresso sum of squares terms because f the regresso le fts the data well, the resduals wll be small. df SS MS F Sgfcace F Regresso Resdual Total Respose Cocetrato /mg L - Resdual Cocetrato /mg L - Fgure : Org of sum of squares terms regresso aalyss Each mea square (MS) term s smply the sum of squares term dvded by ts degrees of freedom. The F value s the rato of the regresso MS term to the resdual MS term. Ideally ths rato should be very large; f there s a good lear relatoshp the regresso MS term wll be much greater tha the resdual MS term (see Fgure 8). The sgfcace F value represets the probablty of obtag the results the ANOVA table f there s o correlato betwee y ad values,.e., obtag the results by chace. A small value dcates that the results were ulkely to have happeed by chace, dcatg that t s hghly LGC/VAM/3/3 Page 3

17 lkely that there s a strog relatoshp betwee the y ad values. For a calbrato curve to be of ay use the sgfcace F value should be etremely small (see Fgure 8). Ths value s also kow as the p-value. Regresso coeffcets The fal secto of the regresso output show Fgure 8 gves formato about the regresso coeffcets m (the gradet of the le) ad c (the tercept of the le wth the y- as). The frst colum of umbers gves the values of the coeffcets. I Ecel, the gradet s descrbed as X Varable. The et colum gves the stadard errors (also kow as the stadard devatos) for each coeffcet. These values gve a dcato of the rages wth whch the values for the gradet ad tercept could le. Related to these values are the lower ad upper cofdece lmts for the gradet ad tercept (fal two colums of the table). These represet the etremes of the values that the gradet ad tercept could take, at the chose level of cofdece (usually 95%). The equatos for calculatg these values are gve Apped 3. The t-stat ad p-value relate to the sgfcace of the coeffcets,.e. whether or ot they are statstcally sgfcatly dfferet from zero. I a calbrato epermet we would epect the gradet of the le to be very sgfcatly dfferet from zero. The t-value should therefore be a large umber (for a calbrato wth 7 data pots the t-value should be much greater tha.6, the -taled Studet t value for 5 degrees of freedom at the 95% cofdece level) ad the p-value should be small (much less tha.5 f the regresso aalyss has bee carred out at the 95% cofdece level). Typcal values are show Fgure 8. Ideally, we would lke the calbrato le to pass through the org. If ths s the case the the tercept should ot be sgfcatly dfferet from zero. I the regresso output we would epect to see a small value for t (less tha.6 for a calbrato wth 7 data pots) ad a p-value greater tha.5 (for regresso at the 95% cofdece level). Whether the calbrato le ca reasoably be assumed to pass through zero ca also be judged by spectg the cofdece terval for the tercept. If ths spas zero, the the tercept s ot statstcally dfferet from zero, as the eample show Fgure 8..6 Usg the calbrato fucto to estmate values for test samples If, after plottg the data ad eamg the regresso statstcs, the calbrato data are judged to be satsfactory the calbrato equato (.e., the gradet ad the tercept) ca be used to estmate the cocetrato of the aalyte test samples. Ths requres each sample to be aalysed oe or more tmes, uder the same codtos that the calbrato stadards were measured. It s also useful to obta a estmate of the ucertaty assocated wth the predcted cocetrato values for test samples. Ths s descrbed Secto.7..7 Estmatg the ucertaty predcted cocetratos Fgure llustrates the cofdece terval for the regresso le. The terval s represeted by the curved les o ether sde of the regresso le ad gves a dcato of the rage wth whch the true le mght le. Note that the cofdece terval s arrowest ear the cetre (the pot, y ) ad less certa ear the etremes. Page 4 LGC/VAM/3/3

18 8 7 Istrumet respose Cocetrato /mg L - Fgure : 95% cofdece terval for the le I addto, t s possble to calculate a cofdece terval for values predcted usg the calbrato fucto. Ths s sometmes referred to as the stadard error of predcto ad s llustrated Fgure 3. The predcto terval gves a estmate of the ucertaty assocated wth predcted values of. 8 y Istrumet respose predcto terval pred Cocetrato /mg L - Fgure 3: Predcto terval The predcto terval s s calculated usg Eq. : s () r ( y y) s + + m N m o ( ) Eq. LGC/VAM/3/3 Page 5

19 Where: s(r) s the resdual stadard devato (see Eq. ) s the umber of pared calbrato pots (,y ) m N yo y s the calculated best-ft gradet of the calbrato curve s the umber of repeat measuremets made o the sample (ths ca vary from sample to sample ad ca equal ) s the mea of N repeat measuremets of y for the sample s the mea of the y values for the calbrato stadards s a value o the -as s the mea of the values A cofdece terval s obtaed by multplyg s by the -taled Studet t value for the approprate level of cofdece ad - degrees of freedom..8 Stadard error of predcto worked eample Table ad Fgure 4 show a set of calbrato data whch wll be used to llustrate the calculato of a predcto terval. Cocetrato /mg L - Absorbace Table : Calbrato data for stadard error of predcto eample Absorbace y r Cocetrato /mg L - Fgure 4: Plot of data for stadard error of predcto eample Page 6 LGC/VAM/3/3

20 The data requred to calculate a predcto terval are show Table. Coc Absorbace y ( ) Predcted yˆ m + c Resduals y ˆ y Resduals ( y y ) ˆ y ( ) ( y yˆ ) Table : Data requred to calculate a predcto terval Usg Eq. the resdual stadard devato s calculated as: 8.37 () 5 s r Applyg Eq., the predcto terval for a sample whch gves a strumet respose of.87, s: ( ) - s mg L Note that a sgle measuremet s made o the sample so N. pred mg L.54 - Epressed as a % of pred, s.53%. At the 95% cofdece level, the -taled Studet t value for 5 degrees of freedom s.57. The 95% cofdece terval for pred s mg L - (.4%). LGC/VAM/3/3 Page 7

21 The ucertaty predcted values ca be reduced by creasg the umber of replcate measuremets (N) made o the test sample. Table 3 shows how s chages as N s creased. 3. Coclusos N s /mg L - Ucertaty /%relatve Table 3: Stadard error of predcto for dfferet values of N The bechmarkg eercse hghlghted a umber of problems assocated wth carryg out strumet calbrato. Some commo ptfalls ecoutered calbrato studes clude: the cocetrato rage covered by the calbrato stadards does ot adequately cover the rage of cocetratos ecoutered for test samples; the cocetratos of the calbrato stadards are ot evely spaced across the calbrato rage; the ucertaty assocated wth the cocetratos of the calbrato stadards s too large (e.g., approprate glassware s used to prepare the stadards, the materal used to prepare the stadards s ot of a approprate purty); the wrog regresso s carred out (.e., regresso of o y rather tha y o ); the calbrato le s ftted through zero whe the tercept s, fact, sgfcatly dfferet from zero; strumet software s used to carry out the regresso ad automatcally calculate the cocetrato of test samples but a plot of the calbrato data s ot obtaed; the resdual stadard devato s used as a estmate of the ucertaty predcted cocetrato values, rather tha carryg out the full stadard error of predcto calculato; the performace of the strumet used to make the measuremets s ot wth specfcato. Followg the steps lsted below should avod these problems: pla the calbrato study so that the cocetrato rage of terest s covered ad the cocetratos of the calbrato stadards are evely dstrbuted across the rage; clude a stadard wth zero aalyte cocetrato (.e., a blak); esure that approprate materals ad apparatus are used to prepare the calbrato stadards; esure that the strumet used to make the measuremets s ft for purpose (.e., carry out equpmet qualfcato); plot ad eame the results; use valdated software to perform the lear regresso; Page 8 LGC/VAM/3/3

22 do ot set the tercept to zero uless there s evdece that the tercept s ot statstcally dfferet from zero; plot ad eame the resduals; calculate the ucertaty (predcto terval) for test sample cocetratos predcted usg the calbrato le. LGC/VAM/3/3 Page 9

23

24 Backgroud Apped : Protocol ad results sheet Partcpatg orgasatos were suppled wth a sold sample of a photographc chemcal, Z, ad a waste water sample, S, cotag Z at a ukow cocetrato. The partcpats were requred to use compoud Z to prepare a set of calbrato solutos, costruct a calbrato curve ad the use the curve to predct the cocetrato of Z soluto S. Each aalyst takg part was asked to repeat the eercse three tmes. Partcpats were gve the opto of calculatg the stadard error of predcto for oe of ther estmates of the cocetrato of S. A Ecel spreadsheet for eterg the results of the study was suppled to each partcpat. LABORATORY PROTOCOL Costructo of a Calbrato Curve ad the Determato of the Cocetrato of a Substace Water by UV Aalyss The am of ths eercse s to vestgate the ucertates assocated wth the costructo of a calbrato curve, ad wth the determato of the cocetrato of a ukow soluto usg the calbrato curve. I addto, by requestg partcpats to repeat the eercse trplcate, t wll be possble to evaluate the effect of ay homogeety the materal used to prepare the calbrato stadards. Partcpats wll be suppled wth a sold sample of a photographc chemcal, Z, ad a waste water sample (S) cotag Z at a ukow cocetrato. Partcpats are requred to use the sold sample to prepare calbrato solutos, costruct a calbrato curve ad the use the curve to determe the cocetrato of Z soluto S. Procedure Test Prepare a stock soluto (soluto A) wth a cocetrato of mg L - by weghg out 5 mg of the sold sample, trasferrg to a 5 ml volumetrc flask ad dlutg to the mark wth de-osed water. Through approprate dlutos of soluto A, prepare 7 calbrato solutos as follows: Cocetrato /mg L - Dluto of soluto A requred.5 5 ml dluted to ml ml dluted to ml 8 (a) ml dluted to 5 ml 8 (b) ml dluted to 5 ml 8 (c) ml dluted to 5 ml 5 5 ml dluted to ml.5 5 ml dluted to ml Calculate the actual cocetratos of the calbrato solutos, takg to accout the amout of materal used (W). Measure the absorbace of each calbrato soluto at 36 m ad produce a plot of absorbace vs. actual cocetrato. Carry out a lear regresso aalyss to determe the equato of the relatoshp betwee absorbace ad cocetrato (.e., y m + c). LGC/VAM/3/3 Page

25 Measure the absorbace of soluto S at 36 m. Use the calbrato fucto to calculate the cocetrato of Z S mg L -. Test Repeat Test, ecept for the replcato of the preparato of the 8 mg L - stadard. Therefore, oly 5 calbrato stadards are prepared as show the table: Test 3 As Test. Reportg results Cocetrato /mg L - Dluto of soluto A requred.5 5 ml dluted to ml ml dluted to ml 8 ml dluted to 5 ml 5 5 ml dluted to ml.5 5 ml dluted to ml For each test, please report the followg formato the spreadsheet suppled: Laboratory ame Aalyst ame Date of aalyss Wavelegth used Cell path legth Amout of sold sample used to prepare the stock soluto Cocetratos of the calbrato solutos Absorbace readg for each calbrato soluto Correlato coeffcet for the calbrato curve Equato of the calbrato curve (.e., gradet ad tercept of the le) Absorbace readg for soluto S Cocetrato of Z soluto S Further commets please clude a bref descrpto of the equpmet used, the temperature at whch the measuremets were made, the order whch the solutos were aalysed ad ay devatos from the protocol. I addto, please clude a copy of the calbrato graph from each test, ad ay addtoal statstcal formato relatg to the determato of the equato of the calbrato le (e.g., output from software used to carry out the aalyss, resdual plots). Optoal For at least oe of the tests, use the stadard error of predcto equato to calculate the stadard error assocated wth the cocetrato determed for Z soluto S. Page LGC/VAM/3/3

26 Evaluato of Results The results suppled by each aalyst were evaluated agast the followg crtera: Carryg out the three tests as specfed by the protocol, partcular: preparato of three solutos wth a cocetrato of 8 mg L - test, ot makg repeat measuremets o the same soluto; preparg depedet sets of calbrato solutos for each test; measurg the absorbace of the soluto S for each test. Correct calculato of the cocetratos of the calbrato stadards; Correct regresso aalyss ad correct reportg of the correlato coeffcet, r, the gradet ad tercept of the ftted le; Learty of the calbrato curve; Correct calculato of the cocetrato of S; Correct calculato of the stadard error of predcto f reported. For each set of results, the mea ad stadard devato of three estmates of the cocetrato of S was calculated, alog wth the stadard devato of the three absorbaces reported for the 8 mg L - stadards test. LGC/VAM/3/3 Page 3

27 Lab ame Ecel Results Sheet (Test ) Aalyst ame Date of aalyss Wavelegth used Path legth of cell Calbrato Results Amout of stadard used W (mg) Target cocetrato (mg L - ) (a) 8 (b) 8 (c).5 Actual cocetrato (mg L - ) Absorbace Correlato coeffcet (r ) for calbrato curve Equato of calbrato curve (y m + c ) Gradet (m) Itercept (c) Cocetrato of Z soluto S Absorbace Cocetrato (mg L - ) Stadard error (mg L - ) (optoal) Further commets (e.g., strumet used, type of cell, temperature, order of aalyss, dluto scheme f dfferet from protocol) Page 4 LGC/VAM/3/3

28 Apped : Eample set of results Test Lab ame Aalyst ame Date of aalyss Wavelegth used Path legth of cell Calbrato Results Amout of stadard used W (mg) 36 m. cm 5.6 mg Target cocetrato (mg L - ) Actual cocetrato (mg L - ) Absorbace (a) (b) (c) Correlato coeffcet (r) for calbrato curve.9999 Equato of calbrato curve (y m + c) Gradet (m) Itercept (c) Cocetrato of Z soluto S Absorbace Cocetrato (mg L - ) Stadard error (mg L - ) (optoal) Further commets (e.g., strumet used, type of cell, temperature, order of aalyss, dluto scheme f dfferet from protocol) Phlps 88 UV/VIS strumet. Absorbace recorded at 5 C All volumetrc solutos prepared at C. Grade A volumetrc glassware used. Stadard solutos aalysed radom order. Sample soluto aalysed last. The same cm glass cuvette used for all measuremets ad always placed the cell holder the same way roud. LGC/VAM/3/3 Page 5

29 Absorbace y r Cocetrato /mg L -.4. Resdual Cocetrato /mg L - Fgure 5: Scatter plot ad resdual plot for typcal bechmarkg data set (test ) Page 6 LGC/VAM/3/3

30 LGC/VAM/3/3 Page 7 Apped 3: Lear regresso equatos Parameter Equato Gradet of the least squares le, m ( )( ) { } ( ) y y m Itercept of the least squares le, c m y c Correlato coeffcet, r ( )( ) { } ( ) ( ) y y y y r Resdual stadard devato, s(r) () ( ) ˆ y y r s Stadard devato (error) of the gradet, s m () ( ) m r s s Stadard devato (error) of the tercept, s c () ( ) c r s s Cofdece terval of the gradet, c m m m ts c Cofdece terval of the tercept, c c c c ts c Stadard devato of the regresso le, s L () ( ) ( ) + L r s s Cofdece terval for the regresso le, c L L L ts c Predcto terval for predcted values of, s () ( ) ( ) + + o m y y N m r s s Cofdece terval for predcted values of, c ts c value o the -as N umber of repeated measuremets made o the test soluto y observed value o the y-as y mea of N repeat measuremets of y for the test soluto mea of values ŷ predcted value of y for a gve value y mea of y values t -taled Studet s t value for - degrees of freedom umber of calbrato pots