Slope standard deviation

Depending on method type or intent, the LOD or LOQ may need to be determined. ICH guidelines describe several approaches and allow alternative approaches, if scientifically justifiable. Suggested approaches include calculation based on signal-to-noise ratio (typically set at 3 1 for LOD and 10 1 for LOQ) standard deviation of the response and slope standard deviation of the response of a blank, or a calibration curve. 

The detection limit (signahnoise ratio of 3 1) is evaluated on a plasma sample with low concentration of each of the BAs studied. The detection limit and linear regression (mean slope, standard deviation and correlation coefficient) data are shown in Table 5.4.10. 

The reader is asked to find the standard deviations of the slopes matrix in Problem 9 below. 

Eind the standard deviations of the slopes in matrix (3-78) for row 2, which refers to absorbances measured at 525 nm. 

The scatter of the points around the calibration line or random errors are of importance since the best-fit line will be used to estimate the concentration of test samples by interpolation. The method used to calculate the random errors in the values for the slope and intercept is now considered. We must first calculate the standard deviation Sy/x, which is given by ... 

Now we can calculate the standard deviations for the slope and the intercept. These are given by ... 

There is an obvious similarity between equation 5.15 and the standard deviation introduced in Chapter 4, except that the sum of squares term for Sr is determined relative toy instead of y, and the denominator is - 2 instead of - 1 - 2 indicates that the linear regression analysis has only - 2 degrees of freedom since two parameters, the slope and the intercept, are used to calculate the values ofy . 

A more useful representation of uncertainty is to consider the effect of indeterminate errors on the predicted slope and intercept. The standard deviation of the slope and intercept are given as... 

These standard deviations can be used to establish confidence intervals for the true slope and the true y-intercept... 

The standard deviation about the regression, Sr, suggests that the measured signals are precise to only the first decimal place. For this reason, we report the slope and intercept to only a single decimal place. 

A plot depicting isokinetic relationships, (a) The thermal rearrangement of triarylmethyl azides, reaction (7-35) is shown with different substituents and solvent mixtures. The slope of the line gives an isokinetic temperature of 489 K. Data are from Ref. 8. (b) The complexation of Nr by the pentaammineoxalatocobalt(III) ion in water-methanol solvent mixtures follows an isokinetic relationship with an isokinetic temperature of 331 K. The results for forward (upper) and reverse reactions are shown with the reported standard deviations. Data are from Ref. 9. 

The slopes bj are connected with activation energies of individual reactions, computed with the constraint of a common point of intersection. We called them the isokinetic activation energies (163) (see Sec. VI). The residual sum of squares So has (m - 1)X— 2 degrees of freedom and can thus serve to estimate the standard deviation a. Furthermore, So can be compared to the sum of squares Sqo computed from the free regression lines without the constraint of a common point of intersection... 

The natural and correct form of the isokinetic relationship is eq. (13) or (13a). The plot, AH versus AG , has slope Pf(P - T), from which j3 is easily obtained. If a statistical treatment is needed, the common regression analysis can usually be recommended, with AG (or logK) as the independent and AH as the dependent variable, since errors in the former can be neglected. Then the overall fit is estimated by means of the correlation coefficient, and the standard deviation from the regression line reveals whether the correlation is fulfilled within the experimental errors. 

The test for the significance of a slope b is formally the same as a t-test (Section 1.5.2) if the confidence interval CI( ) includes zero, b cannot significantly differ from zero, thus ( = 0. If a horizontal line can be fitted between the plotted CL, the same interpretation applies, cf. Figures 2.6a-c. Note that si, corresponds to fx ean). that is, the standard deviation of a mean. In the above example the confidence interval clearly does not include zero this remains so even if a higher confidence level with t(f = 3, p = 0.001) = 12.92 is used. 

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 ).

Example 33 Assume that a simple measurement costs 20 currency units n measurements are performed for calibration and m for replicates of each of five unknown samples. Furthermore, the calibration series of n measurements must be paid for by the unknowns to be analyzed. The slope of the calibration line is > = 1.00 and the residual standard deviation is Sres = 3, cf. Refs. 75, 95. The n calibration concentrations will be evenly spaced between 50 and 150% of nominal, that is for n = 4 x, 50, 83, 117, 150. For an unknown corresponding to 130% of nominal, should be below 3.3 units, respectively < 3.3 = 10.89. What combination of n and m will provide the most economical solution Use Eq. (2.4) for S x and Eq. (2.18) for Vx-Solution since Sxx is a function of the x-values, and thus a function of n (e.g. n = 4 Sxx = 5578), solve the three equations in the given order for various combinations of n and m and tabulate the costs per result, c/5 then select the... 

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. 

Program FACTORS in Section 5.2.3 lists the exact procedure, which is only. sketched here. Table 3.2, for each factor and interaction, lists the observed effect, and the specific effect, the latter being a slope or slopes that make(s) the connection between any factor(s) and the corresponding effect. A r-test is conducted on the specific effects, i.e., t = Ej/(Ax se), where Ej is the observed effect y, - yi, se its (estimated) standard deviation, and Ax the change in the factor(s) that produced the effect. If t is larger than the... 

Results The uncertainties associated with the slopes are very different and n = H2, so that the pooled variance is roughly estimated as (V + V2)/2, see case c in Table 1.10 this gives a pooled standard deviation of 0.020 a simple r-test is performed to determine whether the slopes can be distinguished. (0.831 - 0.673)/0.020 = 7.9 is definitely larger than the critical /-value for p - 0.05 and / = 3 (3.182). Only a test for H[ t > tc makes sense, so a one-sided test must be used to estimate the probability of error, most likely of the order p = 0.001 or smaller. 

Legend No number of measurement. Cone concentration in fig, CN"/100 ml Absorb absorbance [AU] slope slope of regression line t CV intercept see slope res. std. dev. residual standard deviation Srts -n number of points in regression LOD limit of detection LOQ limit of quantitation measurements using a 2-fold higher sample amount and 5-cm cuvettes—i.e., measured absorption 0. .. 0.501 was divided by 10. 

The slope and the intercept with the appropriate relative 95% CLs, the residual standard deviation, and r. ... 

Figure 4.31. Key statistical indicators for validation experiments. The individual data files are marked in the first panels with the numbers 1, 2, and 3, and are in the same sequence for all groups. The lin/lin respectively log/log evaluation formats are indicated by the letters a and b. Limits of detection/quantitation cannot be calculated for the log/log format. The slopes, in percent of the average, are very similar for all three laboratories. The precision of the slopes is given as 100 t CW b)/b in [%]. The residual standard deviation follows a similar pattern as does the precision of the slope b. The LOD conforms nicely with the evaluation as required by the FDA. The calibration-design sensitive LOQ puts an upper bound on the estimates. The XI5% analysis can be high, particularly if the intercept should be negative.

Display key results number of points N, intercept a, slope b, both with 95 % confidence limits, coefficient of determination r, residual standard deviation. 

In this study the reader is introduced to the procedures to be followed in entering parameters into the CA program. For this study we will keep Pm = 1.0. We will first carry out 10 runs of 60 iterations each. The exercise described above will be translated into an actual example using the directions in Chapter 10. After the 10-run simulation is completed, determine (x)6o, y)60, and d )6o, along with their respective standard deviations. Do the results of this small sample bear out the expectations presented above Next, plot d ) versus y/n for = 0, 10,20, 30,40, 50, and 60 iterations. What kind of a plot do you get Determine the trendline equation (showing the slope and y-intercept) and the coefficient of determination (the fraction of the variance accounted for by the model) for this study. Repeat this process using 100 runs. Note that the slope of the trendline should correspond approximately to the step size, 5=1, and the y-intercept should be approximately zero. 

This theory appears not to involve adjustable parameters (other than the nuclear radius parameters that were taken from the literature). In particular, it was criticized that the calibration approach involved a slope that is too high by about a factor of two. However, in actual calculations with the linear response approach, it was found that the slope of the correlation line between theory and experiment (dependent on the quantum chemical method) is close to 0.5. Thus, it also requires a scaling factor of about 2 in order to reach quantitative agreement with experiment. The standard deviations between the calibration and linear response approaches are comparable thus indicating that the major error in both approaches still stems from errors in the description of the bonding that is responsible for the actual valence shell electron distribution. 

Data from several laboratories within the Interregional Research Project No. 4 (IR-4) in the USA have been evaluated for determining the values of MDL and MQL. These data have been presented in Table 1. The two-step procedure described in the EPA guideline was used to calculate the values of MDL and MQL. For the first step, the slope, intercept and RMSE values for the first three calibration curves of each study were separately calculated, then the IDL and IQL values calculated and the value of LQQ estimated for the method. These values were compared with the actual values of LLMV. The standard deviation of the spike recoveries at the LLMV (xllmv) was used to calculate the MDL and MQL. The values of LLMV were separately determined by the laboratory not using any of the methods described in this article. 

Famoxadone, IN-JS940, and IN-KZ007 residues are measured in soil (p-g kg ), sediment (p-gkg ), and water (pgL ). Quantification is based on analyte response in calibration standards and sample extract analyses determined as pg mL Calibration standard runs are analyzed before and after every 1 samples in each analytical set. Analyte quantification is based on (1) linear regression analysis of (y-axis) analyte concentration (lagmL Q and (x-axis) analyte peak area response or (2) the average response factor determined from the appropriate calibration standards. The SLOPE and INTERCEPT functions of Microsoft Excel are used to determine slope and intercept. The AVERAGE and STDEV functions of Microsoft Excel are used to determine average response factors and standard deviations. 

One can apply a similar approach to samples drawn from a process over time to determine whether a process is in control (stable) or out of control (unstable). For both kinds of control chart, it may be desirable to obtain estimates of the mean and standard deviation over a range of concentrations. The precision of an HPLC method is frequently lower at concentrations much higher or lower than the midrange of measurement. The act of drawing the control chart often helps to identify variability in the method and, given that variability in the method is less than that of the process, the control chart can help to identify variability in the process. Trends can be observed as sequences of points above or below the mean, as a non-zero slope of the least squares fit of the mean vs. batch number, or by means of autocorrelation.106... 

To construct the Hill plot (Figure 5.10E), it was assumed that fimax was 0.654 fmol/mg dry wt., the Scatchard value. The slope of the plot is 1.138 with a standard deviation of 0.12, so it would not be unreasonable to suppose % was indeed 1 and so consistent with a simple bimolecular interaction. Figure 5.10B shows a nonlinear least-squares fit of Eq. (5.3) to the specific binding data (giving all points equal weight). The least-squares estimates are 0.676 fmol/mg dry wt. for fimax and... 

Figure 17 Permeability of uranine ( ), dextran 4.4K (O), and dextran 150K (A) through cross-linked poly(/V-isopropyl acrylamide-co-butyl methacrylate, 95 5 mol%) membrane. Error bars represent standard deviation in the slope of the curve of the receiver concentration of solute as a function of time at steady state. (From Ref. 37.)...

The DL and QL for chromatographic analytical methods can be defined in terms of the signal-to-noise ratio, with values of 2 1-3 1 defining the DL and a value of 10 1 defining the QL. Alternatively, in terms of the ratio of the standard deviation of the blank response, the residual standard deviation of the calibration line, or the standard deviation of intercept (s) and slope (5) can be used [40, 42], where ... 

Figure 54-1, however, still shows a number of characteristics that reveal the behavior of derivatives. First of all, we note that the first derivative crosses the X-axis at the wavelength where the absorbance peak has a maximum, and has maximum values (both positive and negative) at the point of maximum slope of the absorbance bands. These characteristics, of course, reflect the definition of the derivative as a measure of the slope of the underlying curve. For Gaussian bands, the maxima of the first derivatives also correspond to the standard deviation of the underlying spectral curve. 

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