Calculating Bone-Lead Measurement Variance.The technique of [sup.109]Cd-based X-ray fluorescence X-ray fluorescence (XRF) is the emission of characteristic "secondary" (or fluorescent) X-rays from a material that has been excited by bombarding with high-energy X-rays or gamma rays. (XRF XRF X-Ray Fluorescence XRF X-Ray Flash XRF Cross Reference XRF Extended Recovery Facility (IBM) XRF Extended Reliability Feature XRF Cross Reference File XRF External Reference ) measurements of lead in bone is well established. A paper by some XRF researchers [Gordon CL, et al. The Reproducibility of [sup.109]Cd-based X-ray Fluorescence Measurements of Bone Lead. Environ Health Perspect 102:690-694 (1994)] presented the currently practiced method for calculating the variance of an in vivo in vivo /in vi·vo/ (ve´vo) [L.] within the living body. in vi·vo adj. Within a living organism. in vivo adv. measurement once a calibration calibration /cal·i·bra·tion/ (kal?i-bra´shun) determination of the accuracy of an instrument, usually by measurement of its variation from a standard, to ascertain necessary correction factors. line has been established. This paper corrects typographical errors typographical error - (typo) An error while inputting text via keyboard, made despite the fact that the user knows exactly what to type in. This usually results from the operator's inexperience at keyboarding, rushing, not paying attention, or carelessness. Compare: mouso, thinko. in the method published by those authors; presents a crude estimate of the measurement error that can be acquired without computational peak fitting programs; and draws attention to the measurement error attributable to covariance Covariance A measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means returns vary inversely. , an important feature in the construct of the currently accepted method that is flawed flaw 1 n. 1. An imperfection, often concealed, that impairs soundness: a flaw in the crystal that caused it to shatter. See Synonyms at blemish. 2. under certain circumstances. Key words: bone, lead, measurement error, X-ray fluorescence. Environ Health Perspect 108:383-386 (2000). [Online 15 March 1999] http://ehpnet1.niehs.nih.gov/docs/2000/108p383-386todd/abstract.html The in vivo measurement of lead in human bone using [sup.109]Cd-based fluorescence fluorescence (fl  rĕs`əns), luminescence in which light of a visible color is emitted from a substance under stimulation or excitation by light or other forms of electromagnetic of the K-shell X-rays of lead (KXRF) is a
well-established technique that has been widely applied to studies of
the human health effects of lead and has been reviewed, most recently,
by Todd and Chettle (1) in a technical manner and by Hu et al. (2) in a
conceptual manner. This paper addresses the method for calculating the
measurement uncertainty in a bone-lead measurement given in a 1994 paper
by Gordon et al. (3).In [sup.109]Cd-based KXRF, the 88.034 keV [Gamma]-rays from [sup.109]Cd are used to fluoresce fluo·resce intr.v. fluo·resced, fluo·resc·ing, fluo·resc·es To undergo, produce, or show fluorescence. [Back-formation from fluorescence. the K-shell X-rays of lead (in increasing energy, those with Siegbahn notation The Siegbahn notation is used in x-ray spectroscopy to name the spectral lines that are characteristic to elements. It was created by Manne Siegbahn. The characteristic lines in x-ray emission spectra correspond to electronic transitions where an electron jumps down to an : K[[Alpha].sub.2], K[[Alpha].sub.1], K[[Beta].sub.1], K[[Beta].sub.3], and K[[Beta].sub.2]). The [sup.109]Cd [Gamma]-rays can also elastically scatter scat·ter v. 1. To cause to separate and go in different directions. 2. To separate and go in different directions; disperse. 3. To deflect radiation or particles. n. off of the calcium and phosphorus phosphorus (fŏs`fərəs) [Gr.,=light-bearing], nonmetallic chemical element; symbol P; at. no. 15; at. wt. 30.97376; m.p. 44.1°C;; b.p. about 280°C;; sp. gr. 1.82 at 20°C;; valence −3, +3, or +5. (and, to a lesser extent, oxygen) atoms in bone and inelastically scatter off all of the elements in the sample undergoing measurement (principally the bone, soft tissue, and skin). The photons are recorded by a spectroscopy spectroscopy Branch of analysis devoted to identifying elements and compounds and elucidating atomic and molecular structure by measuring the radiant energy absorbed or emitted by a substance at characteristic wavelengths of the electromagnetic spectrum (including gamma ray, system that yields an energy distribution of the recorded photons that is then fitted using a nonlinear A system in which the output is not a uniform relationship to the input. nonlinear - (Scientific computation) A property of a system whose output is not proportional to its input. least-squares technique with a mathematical function A rule for creating a set of new values from an existing set; for example, the function f(x) = 2x creates a set of even numbers (if x is a whole number). to extract the amplitudes of the X-ray and elastic scatter peaks. The ratio of the X-ray-to-elastic peaks is the response of the system and is regressed, for each X-ray peak under analysis, against the lead concentration of the calibration standards to produce a calibration line. The in vivo signal from a subject is measured for each lead X-ray to be analyzed and is compared to the established calibration line to obtain one or more estimates of the subject's bone-lead level. The individual X-ray estimates are then combined, usually in an inverse-variance weighted manner, to produce the result. The remainder of this paper addresses methods for the mathematical treatment of the measurement uncertainty; corrects typographical errors in the published method of Gordon et al. (3); presents a crude estimate of the measurement error that can be acquired without computational peak-fitting programs; and addresses the measurement error attributable to covariance. Materials and Methods Gordon et al. (3) published a study of the reproducibility of [sup.109]Cd-based X-ray fluorescence (XRF) measurements of bone lead. Their paper contained an "Appendix" wherein where·in adv. In what way; how: Wherein have we sinned? conj. 1. In which location; where: the country wherein those people live. 2. they gave a near-complete description of the mathematical method by which they calculated the variance of an in vivo bone-lead measurement. In brief, they made multiple measurements of a series of plaster-of-paris phantoms doped dope n. 1. Informal a. A narcotic, especially an addictive narcotic. b. Narcotics considered as a group. c. An illicit drug, especially marijuana. 2. with a range of lead concentrations. The spectrum of scattered Scattered Used for listed equity securities. Unconcentrated buy or sell interest. radiation showed characteristic peaks from the emission of lead K X-rays that varied in size depending, in part, on the lead concentration of the phantom. Gordon et al. used four of the lead K X-rays for analysis: those with Siegbahn (International Union of Pure and Applied Chemistry International Union of Pure and Applied Chemistry (IUPAC), an international organization est. 1919 to advance the chemical sciences and contribute to the application of chemistry to the service of humanity. notation in parentheses See parenthesis. parentheses - See left parenthesis, right parenthesis. ) K[[Alpha].sub.1] (K-[L.sub.3]), K[[Alpha].sub.2] (K-[L.sub.2]), K[[Beta].sub.1] (K-[M.sub.3]), and K[[Beta].sub.3] (K-[M.sub.2]). For clarity and ease of comparison, I will use the notation of Gordon et al.: [x.sub.i] denotes the amplitude amplitude (ăm`plĭt  d'), in physics, maximum displacement from a zero value or rest position. of each X-ray peak, coh denotes the coherent peak amplitude, and
[R.sub.i] denotes the ratio of the two peak amplitudes. Peak amplitudes
and SDs are extracted from the spectra by applying a nonlinear
least-squares technique. A calibration line is constructed for the ratio
of the X-ray-to-coherent peak amplitudes against lead concentration. The
ratio is used because it is independent, to a good approximation approximation /ap·prox·i·ma·tion/ (ah-prok?si-ma´shun)1. the act or process of bringing into proximity or apposition. 2. a numerical value of limited accuracy. , of two important factors that affect in vivo and phantom measurements; namely, source-to-skin distance and overlying overlying suffocation of piglets by the sow. The piglets may be weak from illness or malnutrition, the sow may be clumsy or ill, the pen may be inadequate in size or poorly designed so that piglets cannot escape. tissue thickness. Each calibration line is calculated using least-squares regression. I perform weighted least-squares regression and I suspect that Gordon et al. did also, although they did not state what method they used. However, the method of least-squares regression is irrelevant to the arguments of this paper. Regression gives estimates of the calibration line slope ([m.sub.i]), the slope's variance [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ], the intercept ([C.sub.i]), the intercept's variance [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the covariance between the slope and intercept ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). The X-ray-to-coherent ratios from an in vivo measurement can be converted, using the calibration lines, into estimates of the in vivo lead concentration ([Pb.sub.i]). A matrix correction term accounts for the difference between phantom (plaster-of-paris) and human (bone) matrices, and the estimates of the in vivo bone-lead concentration are combined into an inverse-variance weighted mean to give a single estimate ([Pb.sub.[Mu]]). The inverse-variance weighted estimate has a variance that is denoted [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For each of the lead X-rays in use [Gordon 1] [Pb.sub.i] = 1.46 [R.sub.i] - [C.sub.i]/[m.sub.i], where 1.46 is "the ratio of coherent scattering cross-sections The scattering cross-section, σscat, relates the scattering of light or other radiation to the number of particles present. Its SI unit is the square metre, m², although smaller units are usually used in practice. of bone mineral to hydrated hy·drat·ed adj. Chemically combined with water, especially existing in the form of a hydrate. Adj. 1. hydrated - containing combined water (especially water of crystallization as in a hydrate) hydrous plaster of paris at 88 keV and 160 [degrees]" (3) and [Gordon 2] [R.sub.i] = [x.sub.i]/coh. The variance of the ratio, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is given by [Gordon 3] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] An expression for "a crude underestimate of the measurement variance [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]" (3), which ignores the calibration line uncertainties, is given by [Gordon 4] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Gordon et al. then gave two expressions for the inverse-variance weighted mean ([Pb.sub.[Mu]]) and the variance thereof ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) derived flow the individual lead X-ray peak estimates: [Gordon 5] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which has a variance of [Gordon 6] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] An expression that combines the variance of the XRF response and the variance of the calibration line is then given: [Gordon 7] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] This formula for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is incorrect in the fourth term inside the square bracket square bracket n. One of a pair of marks, [ ], used to enclose written or printed material or to indicate a mathematical expression considered in some sense a single quantity. ; this term should be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] giving a corrected equation of [Gordon 7 (Corrected)] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The typographical error in Equation 7 of Gordon et al. is propagated through their Equations 8 and 9. Gordon Equation 8 gave the variance of the lead concentration obtained from an inverse-variance weighted mean of the estimates from each of the lead X-rays considered. The final term inside the bracket of Gordon Equation 8, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] should be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] giving a corrected version of Gordon Equation 8 of [Gordon 8 (corrected)] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Similarly, the final term of Gordon Equation 9, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] should be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] resulting in the corrected version of Gordon Equation 9: [Gordon 9 (corrected)] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Gordon et al. (3) then pointed out that "each of the terms ([x.sub.i]/coh) has a mutual dependence on coh, the coherent scatter amplitude," and that this dependence can be accounted for by adding a further term to Equation Gordon 9 (corrected): [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Discussion Crude estimates of measurement error. The crude estimate of measurement error given by Gordon et al. (Equation Gordon 4) can be simplified into a form that can be obtained at the time of measurement ("online") and with slightly less computational effort than the estimate of Gordon et al. A cruder estimate of measurement error can be obtained by using the fact that the variance of the X-ray peak amplitude (or area) dominates the variance of the ratio of the X-ray to coherent peak amplitudes (or areas). The fractional fractional size expressed as a relative part of a unit. fractional catabolic rate the percentage of an available pool of body component, e.g. protein, iron, which is replaced, transferred or lost per unit of time. error in the ratio is therefore approximately equal to the fractional error in the X-ray peak: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] whereupon where·up·on conj. 1. On which. 2. In close consequence of which: The instructor entered the room, whereupon we got to our feet. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Several spectroscopy package regions of interest give [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] allowing an online estimate of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to be obtained (assuming the calibration line slope is already known). An online estimate of the measurement error has been useful when physicians require rapid assessment of a patient, and it may prove useful if a target measurement error is needed for all subjects. The expression for variance that accounts for only the X-ray amplitude is indistinguishable from the expression that accounts for the variances in both the coherent and the X-ray amplitudes. Table 1 illustrates this using the data of Gordon et al. Table 1. The proportional contribution of the variance in the X-ray peak to the variance in the X-ray-to-coherent peak ratio for two human subjects measured by Gordon et al. Subject, peak Coherent [Alpha]1 [Alpha]2 Subject B (male) Amplitude(a) 2,523 421.5 313.9 Amplitude [+ or -](a) 15.16 24.81 38.52 Error in peak amplitude (%) 0.601 5.886 12.271 Error in peak/coherent (%) - 5.917 12.286 (Error in peak amplitude)/ - 99.483 99.880 (Error in peak/coherent) (%) Subject C (female) Amplitude(a) 3,436 31.74 85.46 Amplitude [+ or -](a) 17.69 29.96 50.07 Error in peak amplitude (%) 0.515 94.392 58.589 Error in peak/coherent (%) - 94.393 58.591 (Error in peak amplitude)/ - 99.999 99.996 (Error in peak/coherent) (%) Subject, peak [Beta]1 [Beta]3 Subject B (male) Amplitude(a) 81.29 34.61 Amplitude [+ or -](a) 7.399 7.532 Error in peak amplitude (%) 9.102 21.762 Error in peak/coherent (%) 9.122 21.771 (Error in peak amplitude)/ 99.783 99.962 (Error in peak/coherent) (%) Subject C (female) Amplitude(a) 9.106 6.438 Amplitude [+ or -](a) 8.102 8.442 Error in peak amplitude (%) 88.974 131.128 Error in peak/coherent (%) 88.976 131.129 (Error in peak amplitude)/ 99.998 99.999 (Error in peak/coherent) (%) (a) Data from Gordon et al. (3), Table A2. The error arising from covariance. Equation Gordon 7 warrants further examination because it contains two assumptions that are not explicitly stated by Gordon et al. Equation Gordon 7 is derived from a generalized treatment of error propagation The transmission (spreading) of signals from one place to another. that can be represented in matrix form (4): [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [V.sub.y] is the variance in y = f([x.sub.1], [x.sub.2], [x.sub.3]), cov is the covariance, and var is the variance. Upon multiplication multiplication, fundamental operation in arithmetic and algebra. Multiplication by a whole number can be interpreted as successive addition. For example, a number N multiplied by 3 is N + N + N. : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] When I revert re·vert v. 1. To return to a former condition, practice, subject, or belief. 2. To undergo genetic reversion. from generalized notation to the notation of Gordon et al. and write out the summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) terms, the result is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [R.sub.i] = [x.sub.i]/coh, we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] This equation has all of the terms in Gordon's (corrected) Equation 7 but also contains two additional terms that involve the covariance between the response [R.sub.i] and the calibration line slope ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) and intercept ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). The unstated assumptions Unstated assumption is a type of propaganda message which foregoes explicitly communicating the propaganda's purpose and instead states ideas derived from it. This technique is used when a propaganda's main idea lacks credibility, and thus when mentioned directly will result in the of Gordon et al. are that both of these covariances are zero: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] These assumptions are valid and are stated here only for completeness. Gordon et al. did not derive the expression that accounts for the covariance introduced by the mutual dependence of the ratios of X-ray-to-coherent amplitudes ([R.sub.i]) on the same coherent peak amplitude. It may, however, be derived from the product of crude estimates of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from two X-rays i and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] whereupon [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] If we then use [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we obtain [Todd 1] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] It may be that Gordon et al. then ignored all of the terms except the last one in the bracket (the only one that contains the product of the X-ray peak amplitudes) to obtain: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The square root of the expression gives the term given by Gordon et al. if the weighting factors ([w.sub.i] [w.sub.j] are added and the product is evaluated for all pairs of X-rays [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] In the notation of Gordon et al., [w.sub.i] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [w.sub.j] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], making Gordon's actual expression [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] If the assumption about how this term was derived is correct, there is a potential problem because the final term of Equation Todd 1 is not the largest term. Using the data of Table A2 in Gordon et al. (3), Table 2 of this paper shows that the first term, ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), contributes [is greater than] 95% of the value of the whole, whereas the final term used by Gordon et al. contributes very little to the value of the whole. Table 2. The contribution to an expression for the covariance from two terms, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], expressed as a percentage of Equation Todd 1 and derived from the data of Gordon et al. (3).
Human subject B(a)
Covariance [Alpha]2 [Beta]1
[MATHEMATICAL
EXPRESSION NOT
REPRODUCIBLE IN
ASCII]
[Alpha]1 98.7 98.5
[Alpha]2 - 99.3
[Beta]1 - -
[MATHEMATICAL
EXPRESSION NOT
REPRODUCIBLE IN
ASCII]
[Alpha]1 3.715 x [10.sub.-5] 3.708 x [10.sub.-5]
[Alpha]2 - 8.598 x [10.sub.-6]
[Beta]1 - -
Human subject B(a) Human subject C(a)
Covariance [Beta]3 [Alpha]2
[MATHEMATICAL
EXPRESSION NOT
REPRODUCIBLE IN
ASCII]
[Alpha]1 98.9 98.4
[Alpha]2 99.7 -
[Beta]1 99.5 -
[MATHEMATICAL
EXPRESSION NOT
REPRODUCIBLE IN
ASCII]
[Alpha]1 3.721 x [10.sub.-5] 3.703 x [10.sub.-5]
[Alpha]2 8.629 x [10.sub.-6] -
[Beta]1 1.565 x [10.sub.-5] -
Human subject C(a)
Covariance [Beta]1 [Beta]3
[MATHEMATICAL
EXPRESSION NOT
REPRODUCIBLE IN
ASCII]
[Alpha]1 98.1 98.5
[Alpha]2 99.2 99.6
[Beta]1 99.3 -
[MATHEMATICAL
EXPRESSION NOT
REPRODUCIBLE IN
ASCII]
[Alpha]1 3.692 x [10.sub.-5] 3.707 x [10.sub.-5]
[Alpha]2 8.586 x [10.sub.-6] 8.621 x [10.sub.-6]
[Beta]1 1.563 x [10.sub.-5] -
(a) Percentage contribution to the total covariance expression. Irrespective of irrespective of prep. Without consideration of; regardless of. irrespective of preposition despite the assumption about how Gordon et al. derived the covariance term, the quantitative values for the increase in error arising from the covariance term reported by Gordon et al. cannot be reproduced, probably because of rounding errors Noun 1. rounding error - (mathematics) a miscalculation that results from rounding off numbers to a convenient number of decimals; "the error in the calculation was attributable to rounding"; "taxes are rounded off to the nearest dollar but the rounding error is . Table A2 of Gordon et al. gives the measurement uncertainty resulting from accounting for the covariance as 0.009 (3.417 - 3.408) [micro]g Pb/g bone mineral for human subject B. For Gordon et al. human subject C, the measurement uncertainty due to the covariance term is 0.00006 (100.0-99.998% of 2.972). My calculation of the contribution of the covariance term directly from the peak amplitudes given by Gordon et al., but presumably pre·sum·a·ble adj. That can be presumed or taken for granted; reasonable as a supposition: presumable causes of the disaster. rounded, is 0.021 and 0.002 lag Pb/g bone mineral for subjects B and C, respectively. For Gordon et al.'s high-lead subject (B), my calculation of the error increase resulting from adding the covariance term is greater than that calculated by Gordon et al. by a factor of 2.3. For the low-lead subject (C), my calculations yield an increase in error 35 times of that of Gordon et al. For both subjects, the covariance correction is still small. If the differences are indeed a result of rounding errors, I question the use of making a correction to a bone-lead measurement error that can vary so much solely as a result of rounding. Conclusions I have corrected typographical errors in the published method of Gordon et al. (3) and I provided a crude estimate of measurement error that may be of some use. I propose that the correction for the mutual dependence of the X-rays on the coherent peak not be used because of the small size of the covariance correction and the variability due to rounding. REFERENCES AND NOTES (1.) Todd AC, Chettle DR. In vivo X-ray fluorescence of lead in bone: review and current issues. Environ Health Perspect 102:172-177 (1994). (2.) Hu H, Rabinowitz M, Smith D. Bone lead as a biological marker in epidemiologic studies epidemiologic study A study that compares 2 groups of people who are alike except for one factor, such as exposure to a chemical or the presence of a health effect; the investigators try to determine if any factor is associated with the health effect of chronic toxicity chronic toxicity Toxicology A condition caused by repeated or long-term exposure to low doses of a toxic substance : conceptual paradigms. Environ Health Perspect 106:1-8 (1998). (3.) Gordon CL, Webber CE, Chettle DR. The reproducibility of [sup.109]Cd-based X-ray fluorescence measurements of bone lead. Environ Health Perspect 102:690-694 (1994). (4.) Morrison DF. Multivariate The use of multiple variables in a forecasting model. Statistical Methods. 2nd ed. New York New York, state, United States New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of :McGraw-Hill, 1976. Address-correspondence to A.C. Todd, Mount Sinai School of Medicine
Mount Sinai School of Medicine is a medical school found in the borough of Manhattan in New York City. , Department of Community and Preventive Medicine preventive medicine, branch of medicine dealing with the prevention of disease and the maintenance of good health practices. Until recently preventive medicine was largely the domain of the U.S. , 1 Gustave L. Levy Place, New York, NY 10029 USA. Telephone: (212) 241-1668. Fax: (212) 996-0407. E-mail: todd@mssm.edu I thank S. Carroll for manuscript preparation, J.H. Godbold for statistical guidance, and D.R. Chettle for helpful discussions on the Gordon et al. "Appendix." This project was supported by grants ES05697 and ES06616 from the National Institute of Environmental Health Sciences The National Institute of Environmental Health Sciences (NIEHS) is one of 27 Institutes and Centers of the National Institutes of Health (NIH),which is a component of the Department of Health and Human Services (DHHS). The Director of the NIEHS is Dr. David A. Schwartz. . Received 26 July 1999; accepted 21 October 1999. Andrew Christian Todd Mount Sinai School of Medicine, Department of Community and Preventive Medicine, New York, New York, USA |
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