Application of reference materials for quality assessment in neutron activation analysis-use of information theory

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  • Journal of Radioanalytical and Nuclear Chemisoy, Articles, Vol. 169, No. 2 (1993) 347-361

    APPLICATION OF REFERENCE MATERIALS FOR QUALITY ASSESSMENT IN NEUTRON ACTIVATION

    ANALYSIS - USE OF INFORMATION THEORY

    1. OBRUSNIK,* K. ECKSCHLAGER**

    *Nuclear Physics hlstitute, Czechoslovak A cademy of Sciences, 250 68 ~e~ (Czechoslovakia) **Faculty of Natural Sciences, Charles University, 128 40 Prague (Czechoslovakia)

    (Received November 16, 1992)

    It is generally accepted that an analytical procedure can be regarded as an information production system yielding information on the composition of the analyzed sample. Thus, information theory can be useful and the quantities characterizing the information properties of an analytical method may be applied not only as evaluation criteria but also as objective functions in the optimization. The usability of information theory is demonstrated on the example of neutron activation analysis. Both precision and biasof NAA results are taken into account together with the possible use of reference materials for quality assessment. The influence of the above-mentioned parameters on information properties such as informatiori gain and profitability of NAA results is discussed in detail. It has been proved that information theory is especially useful in choosing suitable reference materials for the quality assessment of routine analytical procedures not only with respect to matrix and analyte concentration in the sample but also to concentrations and uncertainties of certified values in the CRM used. In the extreme trace analysis, CRMs with relatively large uncertainties and very low certified concentrations can still yield rather high information gain of results.

    Introduction

    The use of information theory in analytical chemistry can yield rather

    useful and interesting results. Both principles and applications of

    information theory to analytical problems have been described in several

    review articles and monographs. 1-4 As a rule, analytical information serves as

    a basis for decision making in ecology, hygiene, medicine, economy, etc. As

    the importance of such decisions has rapidly increased, the demands on the

    quality of analytical results and their information content have also

    increased.

    The amount of information obtained from the analysis (Information Gain,

    IG) is determined by metrological properties (precision, accuracy, detection

    limit, etc.) of the analytical method used. Information Profitability (IP)

    introduces the time or economic factoi" and the' relevamSe of results into

    decision making. The information properties (gain, profitability) can be used

    for comparison and optimization, o f various analytical metl{ods and procedures.

    Elsevier Sequoia S. A., Lausanne, A kaddmiai Kiad& Budapest

  • I. OBRUSNIK, K. ECKSCHLAGER: APPLICATION OF REFERENCE

    Up to now information theory has been applied to the evaluation and

    optimization of various analytical methods including Neutron Activation

    Analysis (NAA). s Instrumental (lNAA) procedures have been optimized by means

    of information gain and profitability with respect to irradiation, decay and

    counting times, precision, relevance and costs. In another work, e a possible

    bias of ~t-ray spectrometric results has also been included in the optimization

    procedure.

    In this work, we have tried to investigate the use of information theory

    for the optimization and evaluation of selected cases of INAA procedures

    yielding biased results. Moreover, the use of Reference Materials (RMs) for

    Quality Assessment (QA) has also been discussed. As the relations used for

    optimization are derived for quantitative analysis we assume that the value of

    a (method precision) usually does not exceed 10% relative.

    We can suppose for simplicity that a can be estimated by the counting

    (statistical) error. The value of this error can be controlled to some extent

    by altering the decay, irradiation and especially the counting times, by

    lowering of spectral background or by including a radiochemical separation

    into the NAA procedure in extreme cases.

    Theory

    In general, the Information Gain IG is given by the extended divergence

    measure as

    x~-x I 1 ,. ,, (3 z

    E -~2~e "~ 2

    We assume that the value of the elemental concentration x is known before

    the measurement (a priori) to be only within the broad interval (xl, x ) with

    the same probability at any x (rectangular distribution) and that after the

    measurement (a posteriori), the value has been .determined with a standard

    deviation a (normal distribution). The exponent k = (Or~a)" takes into account by means of a the uncertainty of an elemental concentration in an RM used for

    r

    QA, and 6 in the second term on the right hand side of Eq. (I) denotes a mean

    error (bias). 4'6'7 For accurate results (6 ~ 0) this term vanishes.

    In instrumental analytical technique, bias rather arises from improper

    calibration or from the calibration procedure or standards available being

    inadequate for a perfect elimination of errors appearing during the analysis.

    Other sources of bias are inirerferences and contamination or losses in the

    348

  • I, OBRUSNIK, K. ECKSCHLAGER: APPLICATION OF REFERENCE

    elements determined. Eq. (1) can be applied in the cases where the bias 6 is

    either known (or estimated) from the theory or can be established 9 6

    experimentally.

    Analytical information serves as a basis for making decisions on some

    often non-chemical hypothesis. Information enabling a correct decision is

    called relevant, s In analytical practice we should often take into account the

    cost of an analysis. For this purpose another information property -

    Information Profitability IP can be advantageously used and for multicomponent

    analysis obtained as

    ~. IG t . k t I P = (,2)

    "17

    where IG i is the information gain and k i the relevance coefficient for the

    i-th element, r denotes the cost of an analysis. In quantitative

    multicomponent analysis the coefficients k i are often calculated as a function

    of IG i (dynamic model). In some simple cases a static model can also be

    untilized. 3'8

    Resu l ts and d i scuss ion

    Influence of bias 6

    Though INAA can often produce rather accurate results (with low or

    practically no bias), biased results cannot I~e quite avoided. Therefore, for

    application of information theory, the general Eq. (!) for IG should be used.

    It enables a comparison of analytical procedures yielding accurate and

    inaccurate (biased) results directly from the value of the information gain.

    Fig. 1 shows the dependence of IG on the value of a for three different values

    of bias 6 (0%, 3% and 10% relative), calculated from Eq. (1).

    It can be seen from the figure that 1G for unbiased results increases with

    decreasing cr even for very low a values (see the curve for (S = 0%). However,

    for biased results (3% or 10%), the IG curves decrease rather rapidly in the

    region of highly precise (low a) results. The higher the level of bias the

    lower value of IG is obtained. In general, the cases with IG < 0 can be 9

    interpreted as a situation where incorrect results misinform us.

    Fig. J can easily be applied to investigate the influence of the

    cal ibrat ion procedure on IG of INAA results. The calibration by means of

    synthetic standards prepared from pure elements and compounds can produce an

    error (bias of the resu l ts )up to about 3% relative, Then, IG obtainable by

    INAA with this kind of calibration is depicted by the area between IG curves

    for 0% and 3% bias. This information gain is reasonably high. Only for highly

    349

  • I. OBRUSNIIC K. ECKSCHLAGER: APPLICATION OF REFERENCE

    6

    0 5 10 15 20

    sigma (~) Fig. i. Dependence os IG on 6" accord ing to Eq. (1)

    6 = constant (0%, 3% and 10%)

    xe-x I = I000 ppm, x = I00 ppm, k = 1

    precise results (below e = 3%) IG decreases as the 6 value becomes

    statistically signif icant. On the other hand, by using certi f ied reference

    materials (CRMs) for calibration with c, up to 20% relative (it corresponds to r

    an uncertainty of about 10% relative) for some elements, a rather wide range

    (and mostly lower values) of IG can be obtained. Thus, information theory

    shows very clearly the disadvantages of using CRMs for calibration in INAA, as

    pointed out by BECKER 1~ and HEYDORN. n

    hl/luence of the cost and relevance

    It can be derived from the Eqs (1) and (2) that the cost of an analysis

    usually grows more rapidly than the information gain of results. In the

    analyses involving the measurement of activity, the value of ~r is inversely

    proportional to the square root of the counting t ime te, and thus of the cost

    of measurement. In this work, we assume for simplicity that the cost of

    measurement is given only by the price of counting time t . Fig. 2 shows e

    dif ferent shapes of the dependence of IG and costs, respectively, on the a

    value.

    350

  • I. OBRUSNIK, K. ECKSCHLAGER: APPLICATION OF REFERENCE

    Fig. 2.

    O 9

    ~.4

    .c 9

    02 t4---

    .c

    0- 0 4 8 12

    sigma (s~) Dependence os IG and the COst os ana lys ts on i f '

    ( in a rb i t ra ry un i t s ) fo r ana lyses invo lv ing the mea-

    surement os act iv i ty, other para~ters" l tke tn F ig .1

    However, for calculation of information profitability (see Eq. (2)), not

    only IG and the cost r but also the relevance of information 3,s should be

    taken into account. This case is shown in Figs. 3a and 3b. The IP curve in

    Fig. 3a exhibits a maximum while the IG curve does not (for unbiased results).

    We applied the dynamic model for calculation of the relevance

    coefficient, s with the assumption that we need the results having a between 1%

    and 10% relative for our decision making. It can be seen from the figure that

    highly precise results (low a) are rarely obtained as the cost of an analysis

    grows too rapidly.

    Fig. 3b shows the dependence of IP on a computed for several bias levels

    and for the same way of relevance coefficient calculation. The higher 6 value

    the lower level of IP is obtained. Moreover, the maximum :on the IP curve

    shifts in direction to higher cr for highly biased results. In our case, the

    information profitability will reach a maximum for a between 5 and 7% relative

    as a further decrease of ~, by prolongation of t , is too expensive.

    In the case of multielement INAA, where a group of several elements should

    be determined with a good information gain (Mgb relevance), the counting time

    35 t

  • I. OBRUSNIK, K. ECKSCHLAGER: APPLICATION OF REFERENCE

    6

    (13

    C5

    b4

    _I~ 3 LP

    0

    2 m

    (_3

    0 i i i i i " i

    2 4 6 8 10 12

    sigma (%) Fig. 3a. Dependence os IG and IP on ~ ( in a rb i t ra ry un i ts ) DYnamic

    ~de l aS re levance c~s163 ca lcu la t ion s resu l tm With ~r

    bet~en I% and IO~ - see 3,s d = 0%; o ther paramters l i ke in F ig . I

    2.C j

    1.8

    , , ~ ,

    1.5 - - / /~\ ,~ 1.3

    _.Q

    ;4-~ 1.0

    // 9 (3

    ,-~ 0.5

    0.3

    0,0 i J i J i w r ~ J I ~ i i i i F I I I I 1

    0 4 8 2

    sigma (~) Fig. 3b. Dependence 0s IP on ~ ( in a rb i t ra ry un i ts ) re levance coes163 ca lcu la t ion l i ke in F ig .3a 3 leve ls os 6 (0%, 5%

    and 10%) ; o~her parameters 1 i ke in F ig . 1 352

  • I. OBRUSNIK, K. ECKSCHLAGER: APPLICATION OF REFERENCE

    should be chosen with respect to achieving a satisfactory a level for elements

    having the highest detection limits in the group. Otherwise, we will obtain

    zero values for some IG i and, consequently, rather low values of IP (Eq. (2)).

    In general, an optimization of irradiation , decay and counting times should

    be carried out (seeS).

    Moreover, in the cases, where we need highly accurate results (precise and

    practically without bias) to make a decision, we have to use a different model

    for relevance coefficient calculation to obtain maximum information

    profitability IP for such highly precise (and expensive) results.

    Figs 3a and 3b show quite clearly that it is often not effective to

    measure with extremely high precision especially if a relatively high nonzero

    bias of results can be present.

    Eq. (1) should be discussed in more detail. Any value 6 > 0 should be

    substituted regardless of its statistical significance. However, in analytical

    practice, when testing by means of reference materials (quality assessment),

    we can find out that: 7

    a) 5 _< at (m,c0/~n, where t(m,c~) is the critical value of the Student

    distribution with m = n-I degrees of freedom, i.e., the bias is not

    statistically significant on the level c~; then we only know about the true

    information gain of the results that

    x~- -x I 1 x2 - -x t In t2 (m,~) ~ IG , ~ In (3 )

    6" ,~2ne ~ 2 6" ~2n'e ~

    b) 6 > at (m,~)/,] n, i.e., bias is statistically significant and the 6

    value found by means of RM should be substituted in Eq. (1).

    Fig. 4 shows an example where 5% bias has been found and a CRM with a = r 5% (uncertainty about 2.5%) has been applied for QA. One or two samples of CRM

    have been analyzed with each batch of samples. When we apply three measure-

    ments (n=3), the area between the curves for 0 and 3% bias depicting the

    obtainable information gain (see Eq. (3) for o > 2)) is rather wide and IG can

    reach low levels. Below cr = 2% Eq. (1) has to be applied. The information gain

    can be improved by using more, e.g. n=5, measurements. Then Eq. (3) should

    be applied for o > 4, and the area depicting a possible IG is much narrower

    and closer to higher levels of IG. Fig. 4 clearly shows that a simple increase

    of the number of measurements will significantly improve the information gain

    of the results.

    353

  • I. OBRUSNIK, K. ECKSCHLAGER: APPLICATION OF REFERENCE

    Fig. 4.

    3-

    .c

    . c 1

    o

    0

    L

    5 10 15 20

    sigma Dependence os IG on ~" ,i"or b iased resu l ts and

    d is163 number os measurements

    CRH wi th ~r=6% used s QA, 6=6%; o ther parameLers

    l i ke in F ig . l

    ~ /z~ IG For Pesu l ts with s ta t i s t i ca l ly ins ign is

    b ias ( s ~ 2: 2%) s n = 3

    I~ . IG s resu l t s w i th s ta t i s t i ca l l y ins ign is

    b ias ( s ~ ~ 4%) s n = 5

    Influence of the quality of the reference material

    Eq. (1) includes many parameters, among them the parameter k =

    (~/~)2 characterizing the rel iabi l i ty of a quality assessment procedure. The

    general Eq. (1) for 6 = 0 models a- case where the existence of a nonzero bias

    is admitted but it is proved experimental ly that 6 = 0. Therefore, for k < 1

    the IG according to Eq. (1) for ~ = 0 is higher than that according to Eq. (I)

    without the second term on the right hand side (presence of bias is not

    assumed). The dif ference is a contribution to the information gain following

    from the fact that a quality assessment is used. This di f ference depends on

    the qual ity (rel iabil ity) of the reference material used (at). It can be as

    high as 0.5 natural units (see Fig. 5).

    354

  • 5-

    w ithou 2 RM for QA_

    5.

    ' - 1-

    -1

    Fig. 5.

    I. OBRUSNIK, K. ECKSCHLAGER: APPLICATION OF REFERENCE

    i i i i t i r...

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