Intercept standard deviation

How many significant figures should we keep for the least-squares line The standard deviations give us the answer. The slope has a standard deviation of 1.0, and so we write the slope as 53.8 l.o at best. The intercept standard deviation is 0.42, so for the slope we write 0.6 0.4. See also Example 3.22. 

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. 

Application of IP and NCS in conjunction with specification tolerance limits enables to substantiate acceptance criteria for linear regression metrological characteristics (residual standard deviation, correlation coefficient, y-intercept), accuracy and repeatability. Acceptance criteria for impurity influence (in spectrophotometric assay), solution stability and intermediate precision are substantiated as well. 

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. 

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. 

Estimated standard deviation (ESD) of the estimated intercept (blank) a... 

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 ... 

Intercepts and slopes of Fig. 6. The errors are standard deviations. c Number of points. 

Phosphoric Acid. The 2nd-order rate method for analyzing the TGA data was statistically best (Table IV) for the cellulose/H PO samples. This suggests that the conclusions from a prior study which assumed a lst-order reaction (29) may need to be reexamined. While Wilkinson s approximation method gave high r values, the rate constant is determined by the intercept rather than the slope in this method. Thus, the standard deviation of the rates determined by Wilkinson s approximation method is still relatively high when compared to the other methods. In addition, the reaction order as determined by the Wilkinson approximation method was unrealistically high, ranging from 2.6 to 5.8. 

From this the standard deviations of the slope, sm, and intercept, Sb, are... 

Also determine the slope and intercept of the least-squares line for this set of data. Determine the concentration and standard deviation of an analyte that has an absorbance of 0.335. 

Figures 7 and 8 illustrate the behavior of the intercepts and slopes from Figure 6 corresponding to the functional forms of Equation 29. The error bars on Figures 7 and 8 represent one standard deviation as determined from isothermal fits. The intercepts have deviations on the order of 0.5% which is consistent with an apparatus analysis. The slopes, however, have much larger uncertainties ranging up to 15%. Increasing the pressure range would greatly reduce this large and important error.

In practice, the interpretation of test results strongly depends on the accuracy of the estimated intercepts of the theoretical isobole with the axis, which represents the doses of the single compounds that induce the desired effect. In fact, large standard deviations of these intercepts prevent a rehable conclusion as to the deviation from additivity. 

Now we apply the usual linear regression procedure, which delivers slope and intercept of the calibration function. From the residuals, i.e. the verheal distances of the calibration points from the regression line, the residual standard deviation can be ealeulated. This standard deviation is a quality indicator for the calibration functiom... 

Another series of trials, all identical to each other (no changes). This time, the results should be tabulated, and a mean and a standard deviation for the blank and each standard should be calculated and the data graphed (mean response values vs. concentration) to create the standard curve (Figure 3.2). In addition, the slope of the line and the y-intercept are determined, as well as the correlation coefficient. If the results look good, one moves on to Step 5, or makes some change to try to improve the results and repeat the above process. 

First Control Run. A large number (7 to 15) of sets of standards and blanks are run and the results tabulated, as in the trial runs. These data are then plotted (responses vs. concentration for all data points, on one graph) and the means, standard deviations, RSDs, the slope, y-intercept, and correlation coefficient are determined. The smaller the value of the y-intercept, the better (the less chance for a contamination or interference problem). The closer the slope is to 1, the better (the more sensitive). At higher concentrations, the standard deviation should get larger, and the RSDs should get smaller (while approaching some limit). If the RSDs are between 30% and 100%, a close approach to the detection limit is indicated. 

The release tests were performed using the USP dissolution method (apparatus I) and utilized 1000 ml of pH 1.2 simulated gastric fluid (USP XXI) or pH 6.8 simulated intestinal fluid (USP XXI) without enzymes, equilibrated to 37°C and stirred with the basket rotating at 50 or 150 rev/ min. Drug concentrations were assayed by UV spectrophotometry at 255 nm. The experiments were continued until 100% dissolution was achieved. The release data were analysed to zero order, calculating the slope and intercept of the line. Each data point is the average of six individual determinations. In all cases the standard deviation was less than 9%. 

Another approach estimates the DL from the standard deviation of the response and the slope of the calibration curve. The standard deviation can be determined either from the standard deviation of multiple blank samples or from the standard deviation of the intercepts of the regression lines done in the range of the DL. This estimate will need to be subsequently validated by the independent analysis of a suitable number of samples near or at the DL ... 

Standard deviation of slope Sj, Standard deviation of intercept ... 

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