Residual confidence interval

Figure 2.8. The slopes and residuals are the same as in Figure 2.4 (50,75,100, 125, and 150% of nominal black squares), but the A -values are more densely clustered 90, 95, 100, 105, and 110% of nominal (gray squares), respectively 96, 98, 100, 102, and 104% of nominal (white squares). The following figures of merit are found for the sequence bottom, middle, top the residual standard deviations +0.00363 in all cases the coefficients of determination 0.9996, 0.9909, 0.9455 the relative confidence intervals of b +3.5%, +17.6%, 44.1%. Obviously the extrapolation penalty increases with decreasing Sx.x, and can be readily influenced by the choice of the calibration concentrations. The difference in Sxx (6250, 250 resp. 40) exerts a very large influence on the estimated confidence limits associated with a, b, Y(x), and X( y ).

Conclusions the residual standard deviation is somewhat improved by the weighting scheme note that the coefficient of determination gives no clue as to the improvements discussed in the following. In this specific case, weighting improves the relative confidence interval associated with the slope b. However, because the smallest absolute standard deviations. v(v) are found near the origin, the center of mass Xmean/ymean moves toward the origin and the estimated limits of detection resp. quantitation, LOD resp. 

The target number of commodity samples to be obtained in the OPMBS was 500, as determined using statistical techniques. A sample size of 500 provided at least 95% confidence that the 99th percentile of the population of residues was less than the maximum residue value observed in the survey. In other words, a sample size of 500 was necessary to estimate the upper limit of the 95% confidence interval around the 99th percentile of the population of residues. 

Figure 14.22 Hydrogen isotope ratios of CJ6 oand Cis.-o fatty acids in modern horse milk and adipose fat, both from Kazakhstan (a), displaying seasonal influences [ellipses are 68 % (la) confidence intervals], compared (b) with 8 D values recorded for fatty acids from Eneolithic cooking pot sherds from Botai residues assigned as equine fats based on 813C analysis. Confidence ellipses are la and correspond to the modern reference equine fat values (Outram et al. 2009)...

The numbers following the signs define the 95 % confidence interval on A, based upon residual mean squares. 

If a single reaction order must be selected, an examination of the 95 % confidence intervals (not shown) indicates that the two-thirds order is a reasonable choice. For this order, however, estimates of the forward rate constants deviate somewhat from an Arrhenius relationship. Finally, some trend of the residuals (Section IV) of the transformed dependent variable with time exists for this reaction order. 

Both assumptions are mainly needed for constructing confidence intervals and tests for the regression parameters, as well as for prediction intervals for new observations in x. The assumption of normal distribution additionally helps avoid skewness and outliers, mean 0 guarantees a linear relationship. The constant variance, also called homoscedasticity, is also needed for inference (confidence intervals and tests). This assumption would be violated if the variance of y (which is equal to the residual variance a2, see below) is dependent on the value of x, a situation called heteroscedasticity, see Figure 4.8. 

The denominator n 2 is used here because two parameters are necessary for a fitted straight line, and this makes s2 an unbiased estimator for a2. The estimated residual variance is necessary for constructing confidence intervals and tests. Here the above model assumptions are required, and confidence intervals for intercept, b0, and slope, b, can be derived as follows ... 

The 95% confidence interval amounted to 1.3 log units, which corresponds to a scatter in the ATdoc values by a factor of 20. Burkhard (2000) argues that the uncertainty in Eq. 3.25 originates partly from inter-laboratory variation, and partly from differences in DOC quality. In view of the wide range of DOC sources included in Eq. 3.25, it does not seem likely that the uncertainties in any ATdoc encountered would be more than 1.3 log units. Hence, the worst-case estimate (maximum sorption of dissolved phase residues by DOC) can be described by... 

An approach that is sometimes helpful, particularly for recent pesticide risk assessments, is to use the parameter values that result in best fit (in the sense of LS), comparing the fitted cdf to the cdf of the empirical distribution. In some cases, such as when fitting a log-normal distribution, formulae from linear regression can be used after transformations are applied to linearize the cdf. In other cases, the residual SS is minimized using numerical optimization, i.e., one uses nonlinear regression. This approach seems reasonable for point estimation. However, the statistical assumptions that would often be invoked to justify LS regression will not be met in this application. Therefore the use of any additional regression results (beyond the point estimates) is questionable. If there is a need to provide standard errors or confidence intervals for the estimates, bootstrap procedures are recommended. 

If there is no theory available to determine a suitable transformation, statistical methods can be used to determine a transformation. The Box-Cox transformation [18] is a common approach to determine if a transformation of a response is needed. With the Box-Cox transformation the response, y, is taken to different powers A, (e.g. -2transformed response can be fitted by a predefined (simple) model. Both an optimal value and a confidence interval for A can be estimated. The transformation which results in the lowest value for the residual variance is the optimal value and should give a combination of a homoscedastical error structure and be suitable for the predefined model. When A=0 the trans-... 

The following statistics were reported n = 20, r2(adj) = 0.922, s = 0.329, F = 225. This relationship is shown in Figure 20.1. Investigation of the residuals of predicted toxicity, as well as the standard errors from the above equation, suggests that the toxicity of chemicals may be predicted within a 95% confidence interval of 0.64 log unit. 

The discrimination among rival models has to take into account the fact that, in general, when the number of parameters of a model increases, the quality of fit, evaluated by the sum S(a) of squared deviations, increases, but that, at the same time, the size of confidence regions for parameters also increases. Thus, there is, in most cases, a compromise between the wish to lower both the residuals and the confidence intervals for parameters. The simplest way to achieve the discrimination of models consists of comparing their respective experimental error variances. Other methods and examples have been given in refs. 25, 32 and 195—207. 

The Greek letter alpha a is used to represent this small residual risk of a false positive. Alpha obviously depends upon what confidence interval is inspected. In the case of the standard 95 per cent Cl, the remaining risk is 100 — 95 = 5 per cent. However, if we were particularly anxious to avoid the risk of a false positive, we might calculate a 98 per cent Cl and only declare a positive finding if that wider interval excluded zero. In that case alpha would be only 2 per cent. 

Based on published variability in pharmacokinetic studies of ethinylestradiol in lean subjects, taking confidence intervals of 80-125%, residual variance ranged between 10 and 33%. Based on these residual variance values, calculated samples sizes ranged between 6 and 30 (subjects). For example, based on a residual variance value of 17.5%, a sample size of 14 was calculated. 

The interpretation of the pharmacokinetic variables Cmax, AUCs and MRT of insulin glulisine was based on 95 % confidence intervals, after ln-transformation of the data. These 95 % confidence intervals were calculated for the respective mean ratios of pair-wise treatment comparisons. In addition, the test treatment was compared to the reference treatment with respect to the pharmacokinetic variables using an ANOVA with subject, treatment and period effects, after ln-transformation of the data. The subject sum of squares was partitioned to give a term for sequence (treatment by period interaction) and a term for subject within sequence (a residual term). Due to the explorative nature of the study, no adjustment of the a-level was made for the multiple testing procedure. 

The primary parameter AUCo-oo was subjected to an analysis of variance (ANOVA) including sequence, subject nested within sequence (subject (sequence)), period and treatment (non-fasting/fasting) effects. The sequence effect was tested using the subject (sequence) mean square from the ANOVA as an error term. All other main effects were tested against the residual error (error mean square) from the ANOVA. The ANOVA was performed on ln-transformed data. Lor ratios 90 % confidence intervals were constructed. The point estimates and confidence limits were calculated as antilogs and were expressed as percentages. The... 

Figure 3. System characterization for shape and position of absorbance-derived signals for urea unfolding gradients alone. Upper panel depicts me average of 8 runs measured at each detector and the associated high/low statistical error limits computed at a confidence interval of 99.9%. Inset to upper panel depicts measured urea phase delay between the averaged runs. The lower panel depicts the averaged data phase-corrected. The lower panel inset depicts the residual differences in urea concentrations between the two absorbance detectors.

Fig. 5 Parity plot experimental As contents of the pyrolysis residues (o for labscale, for TG experiments) are compared with the calculated values, using the first order single reaction kinetic scheme. The errorbars represent 95 Vc confidence intervals.

The greatest sensitivity is observed for plots of residual errors. Residual errors normalized by the value of the impedance are presented in Figures 20.5(a) and (b), respectively, for the real and imaginary parts of the impedance. The experimentally measured standard deviation of the stochastic part of the measurement is presented as dashed lines in Figure 20.5. The interval between the dashed lines represents the 95.4 percent confidence interval for the data ( 2cr). Significant trending is observed as a function of frequency for residual errors of both real and imaginary parts of the impedance. 

Examine the imaginary residual errors to determine whether they fall within the error structure. Should a few points lie outside the error structure at intermittent frequency values, do not be concerned. Assess prediction of the real part of the impedance by examining real residual plots with confidence intervals displayed. Real residual data points that are outside the confidence interval are considered to be inconsistent with the Kramers-Kronig relations and should be removed from the data set. 

The following is an example of a mathematical/statistical calculation of a calibration curve to test for true slope, residual standard deviation, confidence interval and correlation coefficient of a curve for a fixed or relative bias. A fixed bias means that all measurements are exhibiting an error of constant value. A relative bias means that the systematic error is proportional to the concentration being measured i.e. a constant proportional increase with increasing concentration. 

A 95% confidence interval of the true slope is between 1.017 and 0.997, therefore the relative bias will lie between 1.7% and 0.7%. In this case the interval is wider than the line through the centroid even with smaller t-value and residual standard deviations, and the fact that Cx2 was used instead of ]C(x — x)2, and because of this it is expected that the line through the origin would give a narrower confidence interval. There are many reasons for this and they are beyond the scope of this book. 

Acceptable population models resulted in successful minimization, with at least three significant digits for any parameter, a successful estimation of the covariance, and the absolute value of last iteration gradients greater than 0.001 but smaller than 100. Confidence intervals of structural parameters should not include value zero correlation between any two structural parameters should never be greater than 0.95. Acceptable models should not lead to trends in the distribution of weighted residuals versus model predictions and versus independent variable. They should not be oversensitive to initial estimates nor lead to differences between the population parameters and the corresponding medians of individual POSTHOC parameters. The predictions versus observations data should be evenly distributed around the unit line. If constraints were applied on parameters, no final estimate should be equal to one of the boundaries. 

The final model (coding 1) was compliant with the above model-acceptance criteria since the run finished successfully with more than three significant digits and the covariance, the 95% confidence intervals of all the parameters did not include zero, none of the correlation between the structural parameters was above 0.95, and the weighted residuals versus model predictions and versus independent variable data were evenly distributed around the zero line. However, slight trends toward overprediction were seen in the plot of predictions versus observations possible explanations are given in Section 44.2.3. 

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