Estimated amount interval methods

We will describe an accurate statistical method that includes a full assessment of error in the overall calibration process, that is, (I) the confidence interval around the graph, (2) an error band around unknown responses, and finally (3) the estimated amount intervals. To properly use the method, data will be adjusted by using general data transformations to achieve constant variance and linearity. It utilizes a six-step process to calculate amounts or concentration values of unknown samples and their estimated intervals from chromatographic response values using calibration graphs that are constructed by regression. 

One of the opportunities that researchers rarely have is to be in a position of a direct comparison of methods used by several researchers that use the same data. Three of the fenvalerate "unknown" Datasets described and used elsewhere in this volume have been used as primary datasets by 3 research groups in the solution of the calibration problem. Two aspects of the calibration problem, namely, the accuracy of the calibration graph and the description of statistical error as shown by the estimated amount interval are examined here in comparing each of the calibration methods. 

Mitchell s computer program first applied the method to a linear model and then calculated the amount values corresponding to response values of unknowns and the accompanying estimated amount interval calculated as a bandwidth. Bandwidth was defined as the percentage of half the difference of the upper and lower values of the estimated amount interval divided by thex corresponding amount. The standards data was then shortened at the ends, always in such a way to maintain unknowns within the range, and the bandwidth recalculated. Narrower bandwidths were often found in this way. The method also allowed a further recalculation using a second order function model. 

Wegscheider fitted a cubic spline function to the logarithmically transformed sample means of each level. This method obviates any lack of fit, and so it is not possible to calculate a confidence band about the fitted curve. Instead, the variance in response was estimated from the deviations of the calibration standards from their means at an Ot of 0.05. The intersection of this response interval with the fitted calibration line determined the estimated amount interval. 

When I calculated the estimated amount interval from only the response dispersion for the data using Kurtz methods, there was a substantial reduction in the amount bandwidth from the total bandwidth. This calculation was done by intersecting the bounds of the response dispersion with the linear regressed line and projecting these points to the amount axis. This reduction, however, was not nearly enough to account for differences from Wegscheider s calculation to the others. In Table IV the data is... 

Differences in calibration graph results were found in amount and amount interval estimations in the use of three common data sets of the chemical pesticide fenvalerate by the individual methods of three researchers. Differences in the methods included constant variance treatments by weighting or transforming response values. Linear single and multiple curve functions and cubic spline functions were used to fit the data. Amount differences were found between three hand plotted methods and between the hand plotted and three different statistical regression line methods. Significant differences in the calculated amount interval estimates were found with the cubic spline function due to its limited scope of inference. Smaller differences were produced by the use of local versus global variance estimators and a simple Bonferroni adjustment. 

The frequency interpretation of the interval estimates on the unknown amounts is given by the following ( 27 ) With at least 1- a confidence, based on the sampling characteristics of the observations on the standards, at least P proportion of the interval estimates made from a particular calibration will contain the true amounts. The Bonferroni inequality insures the 1-a confidence since the confidence interval about the regression line and the upper bound on cr are each performed using a 1- a/2 confidence coefficient. Hence, the frequency interpretation states that at least (1-a) proportion of the standard calibrations are such that at least P proportion of the intervals produced by the method cover the true unknown amounts. For the remaining a proportion of standard calibrations the proportion of intervals which cover the true unknown values may be less than P. 

First, amount error estimations in Wegscheider s work were the result of only the response uncertainty with no regression (confidence band) uncertainty about the spline. His spline function knots were found from the means of the individual values at each level. Hence the spline exactly followed the points and there was no lack of fit in this method. Confidence intervals around spline functions have not been calculated in the past but are currently being explored ( 5 ). 

Variations in GH isoform detection can seriously affect the interpretation of GH stimulation tests when using different GH assays. For example, one assay may indicate an adequate amount of GH following a stimulation test, whereas another might give a low estimate of die GH concentration that could lead to misclassification of a child as GH deficient. It is important therefore that cutoff values and reference intervals be established separately for each method proposed for routine use to prevent misinterpretation and a misdiagnosis. Moreover, particular care should be taken when comparing serum GH values obtained using different immunoassay systems. 

Using this technique, pool sizes of cholic acid and chenodeoxycholic acid have been estimated to be similar and around 1.0 to 1.5 g each in healthy subjects, with the total bile acid pool amounting to 2 to 4 g (H18, LIO, VIO). Cholic acid turnover is more rapid than for chenodeoxycholic acid, and the rate of hepatic synthesis of cholic acid (300 to 400 mg/day) is therefore approximately double that for chenodeoxycholic acid (150 to 200 mg/day) (H18, VIO). In the steady state, total bile acid synthesis by the liver should equal bile acid loss in the feces, which is around 400 mg/day. Some studies have found that estimates of bile acid synthesis by the isotope dilution technique give values that are higher than those obtained by direct chemical measurement of fecal bile salts (S45), but good agreement has recently been claimed between the two methods (DIO). ITie Lindstedt technique for measuring bile acid turnover and pool size has been modified so that only one bile sample need be collected after intravenous administration of the labeled bile acid. These modified methods measure either pool size alone (D9) or pool size and turnover if both and bile acid are administered at an interval of 24 hours apart (V6). 

Let the effects, sorted in increasing absolute magnitude, and omitting the constant term, be E,. (The constant term is omitted in the case of all methods described in this section.) There are therefore 15 effects. The total probability being unity, the probability interval associated with each effect is 1/15 = 0.067. We associate the first, smallest effect with a cumulative probability of half that amount, 0.033. (Since each datum defines the probability interval, it will be shared between two intervals.) Then for the next smallest, the estimated probability of an effect being less or equal to its value Ej will be 0.033 + 0.067 = 0.1, for the next smallest, E3, the probability will be 0.1+ 0.067, and so on, as shown in table 3.10. 

Klein and Pollauf (31), using a method of micro-extraction and estimation, obtained yields of colchicine from young and old autumn crocus leaves in the ratio of 5/3. The amounts of colchicine in a series of leaf samples picked at intervals throughout the growing season (.32) are shown in Table 3. 

Figure 8.13 Idealized plots according to the Method of Standard Additions. Each point plotted is assumed to be the mean of several replicate determinations. The traditional method (left panel) simply plots the observed analytical signal Y vs the amount of calibration standard added x (the black square corresponds to the nonspiked sample, Y = Yq), and estimates the value of X by extrapolation of a least-squares regression line to Y = 0 (see text) however, this procedure implies that the confidence interval at this point (not shown, compare Figure 8.12) has widened considerably. By using a simple transformation from Y to (Y-Yq) the extrapolation procedure is replaced by one of interpolation, thus improving the precision (more narrow confidence interval). Reproduced from Meier and Ziind, Statistical Methods in Analytical Chemistry, 2nd Edition (2000), with permission of John Wiley Sons Inc.

Condition Evaluation of Concrete Bridges Relative to Reinforcement Corrosion. Vol. 7 Method for Fietd Measurement of Concrete Permeabilitt/. Evaluates a prototype surface air flow (SAP) device for the estimation of concrete surface permeability. A portable field device was constructed that obtains readings at one per minute allowing a large amount of information to be developed at close intervals across a given concrete member. 87 pages. SHRP-S-329, 10... 

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