Confidence bands

Fig. 12. The relationship between the mean oceanic residence time, T, yr, and the seawater—cmstal rock partition ratio,, of the elements adapted from Ref. 29. , Pretransition metals I, transition metals , B-metals , nonmetals. Open symbols indicate T-values estimated from sedimentation rates. The sohd line indicates the linear regression fit, and the dashed curves show the Working-Hotelling confidence band at the 0.1% significance level. The horizontal broken line indicates the time required for one stirring revolution of the ocean, T. ...

Figure 8. Isoarea map of 80 percent confidence band on 1000 ppm lead concentration (5).

Simulators should be used in a cyclic fashion in order to improve the reliability of the forecast. By simulating small changes in the laboratory organisation and by comparing the forecasts with the real effects, the simulator is refined until the forecasts are within a certain confidence band. 

The confidence band, CB, of the entire calibration straight line as shown in Fig. 6.7 is given by... 

Note how the confidence band changes with changing temperature. As an aside, it ghould be pointed out that at a temperature of about 0°C, the machine torque is below the recommended range but the transducer is still responding to the signal. 

Often it is of interest to obtain a confidence interval for the prediction at a new x value. Even more general, a confidence band for predicted values y as a function of x is given by Massart et al. (1997, p. 195)... 

An example for OLS regression is shown in Figure 4.9 data for x and y are the same as in the next R example. The solid line is the OLS line given by the OLS estimates with intercept b0 and slope b. The points seem to be scattered randomly around the regression line. Additionally, the dashed hyperbolic lines show the 95% confidence band. The true regression line (for the true parameters b0 and b) will fall into this band with probability 95%. 

FIGURE 4.9 Confidence band for predicted values in linear regression. 

Use of Multiple-Curve and Weighted Least-Squares Procedures with Confidence Band Statistics... 

Two procedures for improving precision in calibration curve-based-analysis are described. A multiple curve procedure is used to compensate for poor mathematical models. A weighted least squares procedure is used to compensate for non-constant variance. Confidence band statistics are used to choose between alternative calibration strategies and to measure precision and dynamic range. 

Confidence Band Statistics. The confidence-band statistical approach is described in texts by Natrella (1) and Miller 2) and in three papers from our laboratory (3-5). A computer program, REGRES, (See Appendix) was used to carry out all the computations described in this paper. 

Calibration curve with confidence bands around the curve sample signal, and predicted concentration. (Reproduced with permission from D. G. Mitchell, W. N. Mills, J. S. Garden, and M. Zdeb,... 

These factors are used in the equations given in Table I. The computation requires only that the variance ratios be accurately known. The absolute precision of the method may change from day to day without affecting the validity of either the least-squares curve-of-best fit procedure or the confidence band calculations. (It is not practical to regularly monitor local variances, and errors may develop in variance ratios. Eowever, the error due to incorrect ratios will almost always be much less than the error due to assuming constant variance. Even guessed values of, say, S a concentration are likely to yield more precise data.)... 

Calibration curve quality. Calibration curve quality is usually evaluated by statistical parameters, such as the correlation coefficient and standard error of estimate, and by empirical indexes, such as the length of the linear range. Using confidence band statistics, curve quality can be better described in terms of confidence band widths at several key concentrations. Other semi-quantitative indexes become redundant. Alternatively, the effects of curve quality can be incorporated into statements of sample analysis data quality. 

We suggest using a new parameter, the minimum reportable concentration, defined as the concentration whose confidence band just includes zero (5.). This parameter is obtained by reducing the value of signal Yo, figure 4, until the band around predicted concentration, Xo, just touches zero. For example, for the determination of iron in water by AAS, (data given in Table III) the detection limit, defined as the concentration at which the... 

Confidence bands are direct precision data, and the maximum reportable concentration can be defined as the maximum concentration at which the method yields adequate precision ( ) (excluding measurements near the minimum reportable concentration, where poor precision is unavoidable). Table III shows RCB for the determination of iron in water by AAS. The analyst may consider a RCB of say, 15% to be adequate. The maximum reportable concentration would be 15 pg/ml from a single, weighted least-squares curve, and 20 pg/ml by the multiple-curve method. Samples containing > 20 pg/ml should be diluted to 1-10 pg/ml and analyzed using standards containing 0.05 - 15 pg/mL. (Note that it is always better to include a standard above the maximum desired concentration. The precision of this standard measurement will be poor, but poor data at this level are better than none.)... 

The Linear Catibration Graph and Its Confidence Bands from Regression on Transformed Data... 

KURTZ ET AL. Linear Calibration Graph and Its Confidence Bands... 

The Working-Hotelling confidence band ( ) was then constructed around the estimated regression line. 

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.

The Bonferroni interval estimate of X, given Y, is found in three moves. First, the Working-Hotelling confidence band for the regression line... 

Bonferroni inequality is invoked to combine the two proceeding confidence statements, each made with the confidence (l-a/2), to yield an interval estimate for X with confidence at least (1-a). The confidence band on the regression line and the confidence interval on U are intersected and the Bonferroni interval estimate of X is found by projecting the intersection onto the x-axis. Figure Ic illustrates the procedure. If is in the interval on the Y-axis and if the hyperbolic confidence band contains the line... 

With confidence (l-a/2) the Working-Ho tel ling confidence band contains the true line... 

Move 3 The Working Hotelling confidence band about the regression line... 

A point which may need emphasis, stated clearly in Hunter ( 2 ), is the precise interpretation of the confidence band about the predicted amount. This is important since without a clearly understood meaning, the interval will not be useful for assessing the precision of the predicted amounts or concentrations nor for comparing the results from various laboratories. Another reason the user of these methods must understand the interpretation is because increased precision can be achieved in at least two ways -by additional replication of the standards, which reduces the width of the confidence band about the regression line, and by performing multiple determinations on the unknowns, which reduces the width of the interval about the mean instrument response of the unknown. The interval for U is then given by... 

At any transformation level if the minimum F statistic were less than or equal to the critical F value, our work was done and the confidence band calculations began. Otherwise we either accepted a lack of fit (and would note it in published results), segmented the graph to shorter lengths, or sought a non-linear or higher order model. 

We represent the width of the confidence band numerically by the bandwidth ( 1 ) which is defined ... 

The Bandwidth is essentially a normalized half confidence band. The confidence interval bandwidths for 9 data sets using inverse transformed data are given in Table X. The bandwidths are approximately the vertical widths of response from the line to either band. The best band was found for chlorpyrifos, 1.5%, at the minimum width (located at the mean value of the response) and 4.9% at the minimum or lowest point on the graph. Values for fenvalerate and chlorothalonil were slightly higher, 2.1-2.2% at the mean level. The width at the lowest amount for the former was smaller due to a lower scatter of its points. The same reason explains the difference between fenvalerate and Dataset B. Similarly, the lack of points in Dataset A produced a band that was twice as wide when compared to Dataset B. Dataset C gave a much wider band when compared to Dataset B. 

Residue chemists would be most interested in comparable confidence bands or confidence bandwidths. These values become a performance characteristic for any detection system. The values indicate the precision of not only the prepared standards but also the precision of the overall operating detection system. It is ultimately envisioned that a given system of a separation column in a chromatograph with a certain type of detector should give bands of standardized values. If a chemist finds he has not met... 

The response error bounds and the response error bandwidths for a single unknown determination are given in Tables XI and XII. These values are all much larger than the regression confidence bands because of the much smaller number of data points involved. In comparing the response error bounds (Table XI) to the... 

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