Amount intervals

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. 

Figure 1. Plots showing the Calibration Process. A. Response transformation to constant variance Examples showing a. too little, b. appropriate, and c. too much transformation power. B. Amount Transformation in conforming to a (linear) model. C. Construction of p. confidence bands about the regressed line, q. response error bounds and intersection of these to determine r. the estimated amount interval.

Bonferroni Interval Estimates Interval estimates for the unknown X, referred to as unlimited simultaneous discrimination intervals ( 21 ), are based on the estimated regression line of y, on X., and the confidence interval (on the Y-axis) about the response y for an unknown The resulting interval estimates have the property that for at least 100(1- )% of the different calibration sets, at least 100P% of the amount intervals estimated from that calibration will contain true unknown amounts... 

Estimated Amount Intervals. Estimated amount intervals, calculated from the example data sets, are shown in Table XIII. 

The bandwidths for this data are found in Table XIV. Typical estimated amount intervals are found in the analysis for fenvalerate. At 4 ng this compound gave a range of 3.5 to 4.5 ng at the confidence level described. This range was similar in the analysis for fenvalerate in Dataset E, chlorothalonil, and chlorpyrifos due to tight control of standards. These ranges amounted to bandwidth percentages of from 10 to 14%. In more... 

Table XIII. Estimated Amount Intervals from Inverse Transformed Data. Overall a 0.05 where 95% of the Unknown Amounts Will Lie within the Estimated Amount Interval of the True Amount.

Table XIV. Estimated Amount Interval Bandwidth from Inverse...

Calibration Data Extrapolation Caveat. Since extrapolations cannot be done in performing proper regression line calculations, it follows that there will be concern about interval estimates from data that do occur at the extreme ends of the range of standards. To avoid extrapolation we suggest that response values be limited to those values corresponding to the range of amounts of the calibration standards. At the extreme ends, however, one end of the estimated amount interval would then extend into an extrapolated region, the lower end at the minimum amount and the upper end at the maximum amount. 

For determining the estimated amount interval at the lower or minimum end of the regression line three cases arise 1. The lower end can be calculated and is positive. 2. The lower end can be calculated and is negative and/or the interval is excessively long. 3. The lower end cannot be calculated because it is negatively infinite. A similar situation exists at the maximum end of the regression line. In these cases the true uncertainty is properly reflected by the calculated amount uncertainty. The effects found in cases 2 and 3 could be studied to determine their possibility with various types of calibration data. This paper, however, will not delve into that aspect. 

The value of this work is to illustrate the importance of including the estimated amount interval with every calculated amount estimate in written reports. These can be calculated at any response level. As an illustration of this process we can use the data of Table XIII. If an analysis of fenvalerate were being performed and the standards were those of Dataset A, an amount estimated to be 1.5 ng would be reported as having a total error range of 1.16 to 1.97 ng or in rounded figures 1.2 to 2.0 ng. 

However, with improper transformation the calculation of confidence bands and amount interval estimates is erroneous because of the non-constant variance ... 

Comparison of Calibration Graph Amount and Estimated Amount Intervals... 

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. 

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. 

The Kurtz and Mitchell approaches both result in an amount interval which comes from the intersection of two confidence intervals the confidence interval about the response and the... 

Estimated Amount Interval and Bandwidth Data. The bandwidths for each of the points compared were calculated from Wegscheider s data. They were markedly smaller than those calculated by either of the other two workers. His bandwidths ranged from 4.9 to 16.0% for all of the three data sets. Refer to Table III. 

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... 

It is not the purpose of this paper at this moment to investigate further for more detailed reasons for discrepancies in confidence bands or estimated amount intervals. That will be investigated fully at a later time. I do wish to point out that the assumptions one makes about the information he has and the statistical approaches he makes profoundly affect the resultant error calculations. Far from being a staid and dormant subject matter, statistical estimations of error are currently very actively being studied in order for scientific workers and citizens alike to be informed about the error in their work. 

Bones must be thought of as more like plastic than like stone. They have a good deal of flexibility and can accommodate and distribute stresses within a certain capacity of tolerance. The amount, interval, and impulse of the force, as well as the vector, direction, and individual structure, age, and nutritional status of the bone help to determine the result. Sudden forces may create bending, as in a green-stick fracture in children, or rapture of the components and related structures. Sometimes the force is transmitted to smrounding or imderlying structures of other types. It is difficult to conceive of any instance when a fracture of bone does not involve also soft tissue injmy of the regional structures. 

---