Heteroscedastic controlled calibration model applied to analytical chemistry

Heteroscedastic con trolled calibration mo del applied to analytica l c hemistry Betsab ´ e G. Blas A chic and Mˆ onica C. Sando v al Departamen to de Estat ´ ıstica, Universidade de S˜ ao P aulo, S˜ ao Paulo, Brasil Abstract In chemica l analysis made by lab oratories one has the pr oblem of determining the concen tration of a chemica l element in a sample. In order to tac kle this problem the guide EURACHEM/CIT A C recom- mends the application of the linear calibration mo del, so implicitly assume that th ere is no measurement error in the indep end en t v ari- able X . In this w ork, it is prop osed a new calibration mo del assuming that the indep enden t v a riable is con trolled. This assumption is appro- priate in c hemical analysis wh er e th e pro cess tempting to attain the fixed kno wn v alue X generates an err or and the resu lting v a lue is x , whic h is not an observ able. Ho w ev er, observ ations on its su rrogate X are a v a ilable. A sim ulation stud y is carried out in ord er to v erify some prop erties of the estimators deriv ed for the new mo del and it is also considered the usual calibratio n mo del to compare it with the n ew ap- proac h. Three applications are considered to ve rify the p erformance of the new app r oac h. Keyw ords: linear calibration mo del, con trolled v ariable, measurement error mo del, uncertain ty , c hemical analysis. 1 1 In tro duction The usual calibration mo del [1] is commonly used to estimate the concen- tration X 0 of a c hemical sp ecies in a test sample. It typic ally a ssumes that the indep enden t v ariable is fixed and it is not sub ject to error. How ev er, in applications in analytical chemis try this v ariable is sub ject to error whic h arises f rom the preparation process of a standard solution. In many studies, suc h as [2], [3] and [4], it is a ttempted to consider the uncertainties due to the preparatio n pro cess of the standard solutions b y application of the error propagation la w to the standard error of the estimator of X 0 . W e hav e that the concen tration o f the standard solutio n is pre-fixed by the c hemical analyst and a pro cess is carried out attempting to attain it, this pro cess generates erro rs. Hence, in this case it arises the so called controlled v ariable [5], where the controlled v ariable X is defined by the pre-fixed con- cen tration v alue of t he standard solution whic h is expressed b y the equation X = x + δ , where x is the unobserv ed v ariable and δ is the measuremen t error v ariable. In [6] it was prop osed the so called homoscedastic controlled calibration mo del. This mo del is fo rm ulated in the framew o rk of the usual calibration mo del assuming that the independent v ariable is a con trolled v a r ia ble and the associated measuremen t errors ha ve e qual v a r iances. In [7] and [8], some metho ds to compute the uncertain ties in certain v al- ues obtained through measuremen ts are studied. In [8], the uncertain ties of standard solutions are computed a nd it is o bserv ed that these uncertain ties dep end on the concen tration v alues, so w e can observ e that the usual calibra- tion mo del a nd the homo scedastic controlled calibration mo del seem not to b e the more suitable ones. This problem motiv ates us to study a calibra tion mo del that considers the errors v ariability of the preparation of standard solutions. In this w ork w e propo se a calibration mo del t ha t incorp o rates the errors v ariabilit y arisen from the preparatio n process of the standard solu- tion and w e call it a s the heter osc e dastic c on tr ol le d c ali br ation mo del . This w ork is a contin uation to our previous pap er [6] in whic h it w as undertaken the study of the so-called homoscedastic con trolled calibration mo del whic h assumed e qual v a riance errors. The pap er is orga nized as follow s. In Section 2, we form ulate the het- eroscedastic con trolled calibration model. In Sec tion 3, a sim ulation study to test the new approac h is presen ted. In Section 4, three applications a r e considered whic h show that the prop osed mo de l seems to b e more adequate. Section 5 presen ts our concluding remarks. Finally , w e presen t in App endix A the usual calibration mo del, and in App endix 3 some ta bles sho wing t he results of the sim ulation study . 2 2 The prop os ed mo del Among the relev an t problems in c hemical analys is is the one related to the estimation of the concen tratio n X 0 of a c hemical compo und in a giv en sample. In order to tac kle this problem it is used a statistical calibration mo del, whic h is defined b y a tw o- step pro c ess. This problem has b een considered in [9] and [10]. The first stage of the calibra t io n mo del is given b y dat a p oin ts ( X , Y ) whic h is determined in an exp erimen t where the indep enden t v ariable X is the one that the exp erimen ter selects. F or instance, t he concen trations of the standard solutions that a c hemist prepares are independen t v aria bles since an y concen tration may b e chos en. The dep enden t v ariable Y is a measurable prop ert y of the indep enden t v ariable. F or example, the dep enden t v ariable ma y be the amount of intens ity supplied b y t he plasma spectrometry metho d, since the intens ity depends on the concen tration. In the second stage of the calibration mo del it is prepared a suitable sample related to the unknow n concen tratio n X 0 in order to obtain the mea- suremen ts Y 0 . W e hav e that the standard conce ntration X is fixed b y the analyst and the pro cess of preparation attempting to get it pro duces an error δ , and the unobserv ed quan tit y attained is x . Considering the usual calibra tion mo del defined b y the e quations A.1 and A.2 in the App endix A and the eq uation X = x + δ , w e define the heteroscedas tic con trolled calibration mo del as Y i = α + β x i + ǫ i , i = 1 , 2 · · · , n, (2.1) X i = x i + δ i , i = 1 , 2 · · · , n, (2.2) Y 0 i = α + β X 0 + ǫ i , i = n + 1 , n + 2 , · · · , n + k . (2.3) It is considered the usual calibration mo del assumptions (see App endix A) in addition to the follow ing conditions • δ 1 , δ 2 , · · · , δ n are indep enden t and normally distributed with mean 0. • the v ariances σ 2 δ i , ( i = 1 , · · · , n ) are supp osed to b e kno wn. • δ i , i = 1 , · · · , n and ǫ i , i = 1 , · · · , n + k are indep enden t. Observ e that in the mo del describ ed ab ov e we only consider the case when the v ariances σ 2 δ i , i = 1 , · · · , n a re kno wn. It is a g eneralization of the homoscedastic con trolled calibra t io n mo del discusse d in [6], when it is con- sidered σ 2 δ i = σ 2 δ for a ll i and the know n σ 2 δ case. This new mo del is a lso a generalization of the usual calibration mo del in whic h one tak es δ i = 0 , i = 1 , · · · , n . F or the heteroscedastic con trolled calibration mo del the logarithm of the lik eliho o d function is giv en b y l ( α, β , X 0 , σ 2 ǫ ) ∝ − 1 2 n X i =1 log ( γ i ) − k 2 log ( σ 2 ǫ ) 3 − 1 2   n X i =1 ( Y i − α − β X i ) 2 γ i + n + k X i = n +1 ( Y 0 i − α − β X 0 ) 2 σ 2 ǫ   , (2.4) where γ i = σ 2 ǫ + β 2 σ 2 δ i , i = 1 , · · · , n . Solving ∂ l/∂ α = 0 and ∂ l/∂ X 0 = 0 one can get the maximum lik eliho o d estimator of α and X 0 giv en, respectiv ely , b y ˆ α = ¯ Y − ˆ β ¯ X and ˆ X 0 = ¯ Y 0 − ˆ α ˆ β . (2.5) F rom (2.4) and (2.5), it follo ws that the logarithm of the lik eliho o d function for ( α, β , X 0 , σ 2 ǫ ) can b e writen as l ( α, β , X 0 , σ 2 ǫ ) ∝ − 1 2 n X i =1 log ( γ i ) − k 2 log ( σ 2 ǫ ) (2.6) − 1 2   n X i =1 [( Y i − ¯ Y ) − β ( X i − ¯ X )] 2 γ i + 1 σ 2 ǫ n + k X i = n +1 ( Y 0 i − ¯ Y 0 ) 2   . Making ∂ l/ ∂ β = 0, ∂ l /∂ σ 2 ǫ = 0 in the loga r it hm of the lik eliho o d function (2.6), w e hav e t he f o llo wing equations n X i =1 β σ 2 δ i [ γ i − ( Y i − α − β X i ) 2 ] γ 2 i = n X i =1 X i ( Y i − α − β X i ) γ i (2.7) n X i =1 γ i − ( Y i − α − β X i ) 2 γ 2 i = n + k X i = n +1 ( Y 0 i − ¯ Y 0 ) 2 σ 4 ǫ − k σ 2 ǫ . (2.8 ) The estimates of β and σ 2 ǫ can b e obtained through some iterativ e method that solv es t he equations (2.7) and (2 .8 ). The Fisher exp ected info r mation I ( θ ) = I ( α, β , X 0 , σ 2 ǫ ) is g iven by I ( θ ) =          P n i =1 1 γ i + k σ 2 ǫ P n i =1 X i γ i + k X 0 σ 2 ǫ k β σ 2 ǫ 0 P n i =1 X i γ i + k X 0 σ 2 ǫ P n i =1 X 2 i γ i + 2 β 2 P n i =1 σ 4 δ i γ 2 i + k X 2 0 σ 2 ǫ k β X 0 σ 2 ǫ β P n i =1 σ 2 δ i γ 2 i k β σ 2 ǫ k β X 0 σ 2 ǫ k β 2 σ 2 ǫ 0 0 β P n i =1 σ 2 δ i γ 2 i 0 P n i =1 1 2 γ 2 i + k 2 σ 4 ǫ          When k = q n , q ∈ Q + and n → ∞ , the estimator ˆ θ is appro ximately normally distributed with mean θ and v aria nce I ( θ ) − 1 , thu s t he approximate v ariance to order n − 1 for ˆ X 0 is giv en by V ( ˆ X 0 ) = σ 2 ǫ β 2 " 1 n + 1 k − E 1 nσ 2 ǫ E 2 # , (2.9) where E 1 = − n n X i =1 X 2 0 σ 4 ǫ γ i n X i =1 1 γ 2 i − nk n X i =1 X 2 0 γ i − n n X i =1 X 2 i σ 4 ǫ γ i n X i =1 1 γ 2 i − nk n X i =1 X 2 i γ i 4 − 2 nβ 2 n X i =1 σ 4 δ i σ 4 ǫ γ 2 i n X i =1 1 γ 2 i − 2 nk β 2 n X i =1 σ 4 δ i γ 2 i + 2 nβ 2 σ 4 ǫ " n X i =1 σ 2 δ i γ 2 i # 2 +2 nX 0 σ 4 ǫ n X i =1 X i γ i n X i =1 1 γ 2 i + 2 nk X 0 n X i =1 X i γ i + σ 6 ǫ n X i =1 X 2 i γ i n X i =1 1 γ 2 i n X i =1 1 γ i + k σ 2 ǫ n X i =1 X 2 i γ i n X i =1 1 γ i + 2 β 2 σ 6 ǫ n X i =1 σ 4 δ i γ 2 i n X i =1 1 γ 2 i n X i =1 1 γ i + 2 k β 2 σ 2 ǫ n X i =1 1 γ i n X i =1 σ 4 δ i γ 2 i − 2 β 2 σ 6 ǫ " n X i =1 σ 2 δ i γ 2 i # 2 n X i =1 1 γ i − σ 6 ǫ " n X i =1 X i γ i # 2 n X i =1 1 γ 2 i − k σ 2 ǫ " n X i =1 X i γ i # 2 and E 2 = σ 4 ǫ n X i =1 X 2 i γ i n X i =1 1 γ 2 i n X i =1 1 γ i + k n X i =1 X 2 i γ i n X i =1 1 γ i + 2 σ 4 ǫ β 2 n X i =1 σ 4 δ i γ 2 i n X i =1 1 γ 2 i n X i =1 1 γ i +2 k β 2 n X i =1 σ 4 δ i γ 2 i n X i =1 1 γ i − 2 β 2 σ 4 ǫ " n X i =1 σ 2 δ i γ 2 i # 2 n X i =1 1 γ i − σ 4 ǫ " n X i =1 X i γ i # 2 n X i =1 1 γ 2 i − k " n X i =1 X i γ i # 2 . Note that when σ 2 δ i = 0 , i = 1 , · · · , n, the expression (2.9) is reduc ed to the v aria nce of the usual mo del giv en in (A.5) and when σ 2 δ i = σ 2 δ (for all i ) the expression ( 2.9) is also reduced to the v ariance of the homoscedastic mo del when σ 2 δ is kno wn (see eq. (2 .12) of ref. [6]). In order to construct a confidence in terv al for X 0 w e consider the in terv al (A.7), whe re ˆ V ( ˆ X 0 C ) is the estimated v ariance that follo ws from (2.9). 3 Sim ulation study W e presen t a simulation study to compare the p erforma nce of t he estimators obtained from the heteroscedastic con trolled calibration mo del (Prop osed-M) with the results obtained b y considering the usual mo de l (Usual-M). It w as considered 3000 samples generated from the Propo sed-M. In a ll the samples, the parameters α and β tak e the v alues 0 .1 and 2, resp ectiv ely . The range of v alues for the con tro lled v a r ia ble w as [0,2]. The fixed v alues f or the con trolled v a r iable w ere x 1 = 0 , x i = x i − 1 + 2 / ( n − 1) , i = 2 , · · · , n, and the parameter v alues X 0 w ere 0.01 (extreme inferior v a lue), 0.8 (near to the cen- tral v alue) and 1.9 (extreme sup erior v alue). F o r the first and second stages w e consider the sample of sizes n = 5 , 20 , 100 , 5000 and k = 2 , 20 , 100 , 500, resp ectiv ely . W e consider σ 2 ǫ = 0 . 04 and the maxim um para meter v alues of σ 2 δ as max { σ 2 δ i } n i =1 = 0 .1. W e consider σ 2 δ i = i × 0 . 1 /n for i = 1 , · · · , n . The empirical mean bias is giv en b y P 3000 j =1 ( ˆ X 0 − X 0 ) / 3000 and the em- pirical mean s quared error (MSE) is giv en by P 3000 j =1 ( ˆ X 0 − X 0 ) 2 / 3000. The 5 mean estimated v a r ia nce of ˆ X 0 is giv en b y P 3000 j =1 ˆ V ( ˆ X 0 ) / 3000. The theoreti- cal v ariances of ˆ X 0 is r eferred to the expressions (A.5) and (2.9) ev alua ted on the relev a nt pa r a meter v alues. In Appendix B it is presen ted the sim ulation results. In T a ble 1, w e observ e that, in general, the bias of ˆ X 0 from the usual mo del is smaller than the v a lue supplied b y the propo sed mo del, but related to the MSE w e hav e that the outcome f r o m usual mo de l is g r eat er compared with MSE of the prop osed mo del. Also, we observ e t hat the mean estimated v ariance from t he pr o p osed mo del is closer to the theoretical v ar ia nce as compared to the outcome from the usual mo del. Analyzing T able 2, w e observ e t ha t the amplitude of the pro p osed mo del, in most cases, is smaller when compared with the estimate of the usual mo del. F or all n and X 0 the amplitude fr o m the usual mo del greatly decreases as the size of k increase s, this b e havior is b eing r eflected on the cov ering p ercen tage decreasing to less than 9 5 %. Adopting the correct mo del w e ha v e that when k increases the confidence in terv al amplitude decreases and the co v ering p ercen tage increases approachin g 95%. 6 T able 1 : Em pirical bias and mean squared error, t he mean estimated v ariance and theoretical v a riance of ˆ X 0 . X 0 n k Usual-M Prop osed-M U sual-M Proposed-M Theorical v ariance Bias MSE Bias MSE ˆ V ( ˆ X 0 ) ˆ V ( ˆ X 0 ) V ( ˆ X 0 ) 0.01 5 2 -0.0156 0.0350 -0.0318 0.0334 0.0398 0.0143 0.0257 20 -0.0236 0.0319 -0.0445 0.0278 0.0131 0.0156 0.0211 100 -0.0183 0.0306 - 0.0429 0.0276 0.0084 0.0155 0.0207 20 2 -0.0076 0.0119 -0.0049 0.0100 0.0365 0.0053 0.0097 20 -0.0055 0.0074 -0.0073 0.0055 0.0081 0.0036 0.0051 100 -0.0059 0.0068 - 0.0101 0.0050 0.0036 0.0033 0.0047 100 2 0.0003 0.0063 0.0020 0.0059 0.0315 0.0047 0.0059 20 -0.0023 0.0019 -0.0020 0.0014 0.0046 0.0011 0.0014 100 -0.0014 0.0015 - 0.0013 0.0010 0.0017 0.0007 0.0010 5000 2 0.0008 0.0055 0.0008 0.0055 0.0300 0.0050 0.0050 20 0.0000 0.0005 0.0000 0.0005 0.0030 0.0005 0.0005 100 -0.0003 0.0001 - 0.0004 0.0001 0.0006 0.0001 0.0001 500 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000 0.0000 0.8 5 2 0.0061 0.0193 0.0025 0.0202 0.0254 0.0089 0.0167 20 0.0033 0.0135 -0.0019 0.0139 0.0047 0.0081 0.0122 100 0.0014 0.0132 -0.0037 0.0137 0.0029 0.0078 0.0118 20 2 0.0015 0.0077 0.0015 0.0077 0.0291 0.0042 0.0074 20 0.0016 0.0032 0.0009 0.0032 0.0036 0.0020 0.0029 100 -0.0005 0.0026 - 0.0018 0.0026 0.0012 0.0016 0.0025 100 2 0.0010 0.0055 0.0014 0.0054 0.0299 0.0044 0.0055 20 0.0006 0.0010 0.0007 0.0010 0.0031 0.0008 0.0010 100 -0.0001 0.0006 - 0.0001 0.0006 0.0007 0.0004 0.0006 5000 2 0.0014 0.0051 0.0014 0.0051 0.0300 0.0050 0.0050 20 0.0006 0.0005 0.0006 0.0005 0.0030 0.0005 0.0005 100 0.0001 0.0001 0.0001 0.0001 0.0006 0.0001 0.0001 500 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 1.9 5 2 0.0500 0.0802 0.0582 0.0704 0.0432 0.0275 0.0482 20 0.0314 0.0620 0.0503 0.0562 0.0124 0.0278 0.0435 100 0.0434 0.0645 0.0594 0.0587 0.0085 0.0283 0.0430 20 2 0.0070 0.0213 0.0054 0.0185 0.0351 0.0086 0.0166 20 0.0117 0.0160 0.0127 0.0132 0.0075 0.0067 0.0118 100 0.0099 0.0161 0.0104 0.0130 0.0033 0.0064 0.0114 100 2 0.0016 0.0080 0.0007 0.0076 0.0312 0.0055 0.0074 20 0.0019 0.0035 0.0014 0.0029 0.0043 0.0017 0.0028 100 0.0001 0.0031 0.0009 0.0025 0.0015 0.0013 0.0024 5000 2 -0.0008 0.0051 - 0.0009 0.0051 0.0300 0.0050 0.0050 20 -0.0003 0.0006 -0.0004 0.0006 0.0030 0.0005 0.0005 100 -0.0003 0.0002 - 0.0002 0.0001 0.0006 0.0001 0.0001 500 0.0000 0.0001 0.0000 0.0001 0.0001 0.0000 0.0001 4 Applicatio n In t his section w e illustrate the usefulness of the prop osed mo del by a pply- ing it to the data supplied by the c hemical labo ratory o f the “Instituto de P esquisas T ecnol´ ogicas do Estado de S˜ ao P aulo (IPT)” - Brasil. The o ut- come from the prop osed approac h are also compared with the results from the usual mo del. Our main in terest is to estimate the unkno wn concen tration v alue X 0 of a sample of the c hemical elemen ts suc h as c hromium, cadmium and lead. T able 3 below presen ts the fixed v alues of concentration for the standard solutions with their r elat ed uncertain ty ( u ( X i )) and the corresp onding in- 7 T able 2 : Co v ering p ercen tage (%) and amplitude (A) of the interv a ls with a 95% confidenc e lev el for the parameter X 0 . X 0 n k Us ual-M Prop osed-M % A % A 0.01 5 2 89 0.35 78 0.22 20 78 0.21 89 0.24 100 70 0.17 88 0.24 20 2 100 0.37 74 0.13 20 95 0.17 90 0.12 100 85 0.12 90 0.11 100 2 100 0.35 84 0.13 20 100 0.13 91 0.06 100 96 0.08 90 0.05 5000 2 100 0.34 94 0.14 20 100 0.11 95 0.04 100 100 0.05 94 0.02 500 100 0.02 94 0.01 0.8 5 2 90 0.28 78 0.18 20 73 0.13 87 0.17 100 63 0.10 87 0.17 20 2 100 0.33 73 0.11 20 95 0.12 87 0.09 100 81 0.07 88 0.08 100 2 100 0.34 86 0.12 20 100 0.11 91 0.05 100 97 0.05 89 0.04 5000 2 100 0.34 95 0.14 20 100 0.11 95 0.04 100 100 0.05 95 0.02 500 100 0.02 93 0.01 1.9 5 2 78 0.35 81 0.31 20 59 0.20 86 0.32 100 51 0.17 87 0.32 20 2 98 0.36 78 0.17 20 81 0.17 84 0.16 100 62 0.11 84 0.16 100 2 100 0.34 87 0.14 20 97 0.13 87 0.08 100 83 0.08 84 0.07 5000 2 100 0.34 95 0.14 20 100 0.11 94 0.04 100 100 0.05 93 0.02 500 99 0.02 89 0.01 tensities for the c hromium, cadmium and lead elemen ts. T he uncertain ties considered are computed using the metho d r ecommende d b y the ISOGUM guide (see [11]) and the in tensities are supplied by the plasma sp ectrome- try metho d. This data is referred to the first stage of the heteroscedastic con trolled calibration mo del. Moreo v er, T a ble 4 b elo w presen ts the in tensities of t he sample solutions of ch romium, cadmium a nd lead elemen ts. These data are referred to as the second stage o f the calibration mo del. Observing T ables 3 and 4 w e v erify that the uncertaint y v alues increase with the concen tratio n v alues. W e consider σ 2 δ i = u ( X i ) 2 . The expanded uncertain t y U ( X 0 ) is obtained 8 T able 3: Conce ntration ( mg /g ), uncertaint y( u ( X i )) a nd inte nsity of the stan- dard solutions o f chromium, cadmium and lead elemen ts. Chromium element Cadmium element lead element X i u ( X i ) Intensit y X i u ( X i ) Int ensity X i u ( X i ) Int ensity 0.05 0.00016 6455.900 0.05 0. 00016 4.8973 3 0.05 0.00015 0.9471 0.11 0.00027 13042.933 0.10 0.00027 9.706 0.10 0.00025 1.46833 0.26 0.00040 32621.733 0.25 0.00041 23.41333 0.26 0.00039 3.09033 0.79 0.00122 97364.500 0.73 0.00122 69.73 0.77 0.00117 8.40533 1.05 0.00161 129178.100 1.01 0.00168 96.85667 1.01 0.00155 10.92667 T able 4: In tensit y of the sample solutions o f c hromium, cadmium and lead elemen t s. Chromium element Cadmium element Lead element 10173.6 5.066 1.303 10516.9 5.027 1.290 10352.2 5.085 1.341 m ultiplying the squared ro ot of the estimate of v ariance of ˆ X 0 b y the v a lue 1.96 (see [2] and [8]). W e use the o ptim command from R-pro ject prog r am to estimate the pa- rameters β and σ 2 ǫ on the lik eliho o d function of the prop osed mo del (2.6). W e use as initial p oint the estimates from ˆ β = P n i =1 ( X i − ¯ x )( Y i − ¯ Y ) / P n i =1 ( X i − ¯ X ) 2 and ˆ σ 2 ǫ = P n i =1 ( Y 0 i − ¯ Y 0 ) 2 /n , whic h are the estimators fro m the ho- moscedastic con trolled calibration mo del when σ 2 δ is unkno w [6]. T able 5 presen ts estimates of α , β , X 0 , V ( ˆ X 0 ) and the expanded uncer- tain ty , U ( X 0 ), from the prop osed mo del ( Prop ose d-M) of ch romium, cad- mium a nd lead elemen ts. Also, w e pr esen t the estimates obtained from usual calibration mo del (Usual-M) to observ e the p erformance of bo t h mo dels. In T able 5, for cadmium a nd lead elemen ts, w e observ e that the estimates of α , β and X 0 from the Prop osed-M and Us ual- M are the same . F or the c hromium elemen t, there are small differences. Also, w e o bserve that for the c hromium elemen t there is a small difference b et we en the estimates of X 0 and U ( X 0 ) r espectiv ely obtained from the usual mo del and t he prop o sed mo del. Despite the relev an t estimates o f α, β a nd X 0 from b oth approache s for cadmium and lead elemen t match, the estimates of V ( X 0 ) and U ( X 0 ) dif- fer considerably , the estimates obtained adopting the usual model is greater than the es timates outcome supplied b y the prop osed mo del. 5 Conclud ing remarks The expanded uncertain ty of X 0 from the prop osed mo del arises from the errors app earing in the b oth pro cess, the r eading of equipmen t and the het- eroscedastic error in the prepara tion of standard solutions. W e observ e that, 9 T able 5: Estimates of α, β , X 0 , V ( ˆ X 0 ) and U ( X 0 ) r elat ed to usual and het- eroscedastic mo del, for the c hromium, cadmium and lead elemen t. Chromium element Cadmium element Lead element Pa rameters Us ual-M Prop osed-M Usual-M P roposed-M Usual- M Proposed-M α 134.9469 124.28 01 0.454801 0.454801 -0.3822126 -0.3822126 β 123003.7 1230 27.3 10.54381 10.54381 94.29881 94.29881 X 0 0.08302691 0.08309769 0.08123556 0.08123 556 0.05770535 0.05770535 V ( ˆ X 0 ) 4.357870e-06 4.474395 e-06 7.898643e-05 4.440342e-06 0.0001181 068 7.237226e-08 U ( X 0 ) 0.004091601 0 .004145942 0.01741936 0.004130135 0.02130068 0.000 527281 despite the classical mo de l only considers the error orig inated from equip- men t reading, there are some applications in whic h the expanded uncertaint y is greater t ha n the one obtained through the new approac h. V arious asp ects of the mo del studied a b o ve dese rve atten tion in future researc h, e.g. it is not considerated the error arisen fro m the test sample so- lution preparation, the prop osed mo del can b e studied by conside ring other t yp e of distribution of the errors, suc h as sk ew normal distribution [1 2]. In particular, one of the drawbac ks of t he usual mo del is that it do es not con- sider the error in the independent v ariable, w e b eliev e that despite that this error b eing v ery small, it mus t b e considered as an imp ortant prop ert y of the calibratio n mo del. W e will concen trate o n one of the problems describ ed ab o ve in a f ut ure w ork. Ac knowle dgmen ts The authors are grat eful to Prof. Dr. Heleno Bolfarine for carefully reading the man uscript and Dr. Olga Sato mi from ”Instituto de P esquisas T ecnol´ ogicas” - IPT. Betsab´ e G . B. Ac hic has been supp orted b y a grant from CNP q. 10 A Us ual calibration mo del The first and second stage equations of the usual linear calibration mo del are giv en, resp ectiv ely , by Y i = α + β x i + ǫ i , i = 1 , 2 · · · , n, (A.1) Y 0 i = α + β X 0 + ǫ i , i = n + 1 , n + 2 , · · · , n + k . (A.2) It is considered the follo wing assumptions: • x 1 , x 2 , · · · , x n tak e fixed v alues, whic h are considered as true v alues. • ǫ 1 , ǫ 2 , · · · , ǫ n + k are indep enden t and normally distributed with mean 0 and v ariance σ 2 ǫ . The mo del parameters a r e α, β , X 0 and σ 2 ǫ and the main in terest is to estimate the quan t ity X 0 . The maxim um like liho o d estimators of the usual calibration mo de l are giv en b y ˆ α = ¯ Y − ˆ β ¯ x, ˆ β = S xY S xx , ˆ X 0 = ¯ Y 0 − ˆ α ˆ β , (A.3) σ 2 ǫ = 1 n + k [ n X i =1 ( Y i − ˆ α − ˆ β x i ) 2 + n + k X i = n +1 ( Y 0 i − ¯ Y 0 ) 2 ] , (A.4) where ¯ x = 1 n n X i =1 x i , ¯ Y = 1 n n X i =1 Y i , S xY = 1 n n X i =1 ( x i − ¯ x )( Y i − ¯ Y ) , S xx = 1 n n X i =1 ( x i − ¯ x ) 2 , ¯ Y 0 = 1 n n + k X i = n +1 Y 0 i . The appro ximation of order n − 1 for the v ariance of ˆ X 0 is giv en by V 1 ( ˆ X 0 ) = σ 2 ǫ β 2 " 1 k + 1 n + ( ¯ x − X 0 ) 2 nS xx # . (A.5) In order t o construct a confidence interv al for X 0 , w e consider that ˆ X 0 − X 0 q ˆ V ( ˆ X 0 ) D − → N (0 , 1) , (A.6) hence, the appro ximated confidence in t erv al for X 0 with a confidence lev el (1 − α ), is giv en by  ˆ X 0 − z α 2 q ˆ V ( ˆ X 0 ) , ˆ X 0 + z α 2 q ˆ V ( ˆ X 0 )  , (A.7) where z α 2 is t he quantile of o r der (1 − α 2 ) of t he standard normal distribution. 11 References [1] Sh ukla, G .K. On the problem of calibration. T echnome trics 14 (1972) 547. [2] EURA CHEM/CIT A C Guide(2000) Quantifying unc ertaint y in analyt- ical measuremen t, 2nd edn., F inal D raft April 20 0 0. EURACHE M: h ttp://www.measuremen tuncertaint y .org [3] S. H. Ch ui, Q; Zucc hini R. R; and Lic htig J. Qua lida de de medi¸ c˜ oes em qu ´ ımica anal ´ ıtica. Estudo de caso: determina¸ c˜ ao de c´ admio p o r es- p ectrometria de a bsor¸ c˜ ao atˆ omica com c hama. Qu ´ ımica No v a 24 (2001) 374. [4] Bruggemann, L. and W enrich R. 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