Weighted regression lines

It will be seen that the weights have been scaled so that their sum is equal to the number of points on the graph this simplifies the subsequent calculations. The slope and the intercept of the regression line are then given by  

In equation (5.16) y and x represent the coordinates of the weighted centroid, through which the weighted regression line must pass. These coordinates are given as expected by x = %WiXjn and = %w,yjn. 

Calculate the unweighted and weighted regression lines for the following calibration data. For each line calculate also the concentrations of test samples with absorbances of 0.100 and 0.600. 

Application of equations (5.4) and (5.5) shows that the slope and intercept of the unweighted regression line are respectively 0.0725 and 0.0133. The concentrations corresponding to absorbances of 0.100 and 0.600 are then found to be 1.20 and 8.09 pg mh respectively. 

The weighted regression line is a little harder to calculate in the absence of a suitable computer program it is usual to set up a table as follows. 

---

Determine the slopes and intercepts of the unweighted and weighted regression lines. Calculate, using both regression lines, the confidence limits for the concentrations of solutions with fluorescence intensities of 15 and 90 units. 

The MSWD and probability of fit All EWLS algorithms calculate a statistical parameter from which the observed scatter of the data points about the regression line can be quantitatively compared with the average amount of scatter to be expected from the assigned analytical errors. Arguably the most convenient and intuitively accessible of these is the so-called ATS ITD parameter (Mean Square of Weighted Deviates McIntyre et al. 1966 Wendt and Carl 1991), defined as ... 

The method assumes that the data are independent and normally distributed, and it is sensitive to outliers. The Y-axis (or horizontal) component plays an extremely important part in developing the least square fit. All points have equal weight in determining the height of a regression line, but extreme x-axis values unduly influence the slope of the line. 

Decision diamond Are the classical assumptions for fitting regression lines met N0 Clearly the measurements at the different x-levels differ in their variability. This can be shown by using the F-test. Another method is outlined in another chapter of this text (10). In this case weighted least squares will resolve the problem of heteroscedasticity or unequal variance across the graph. I have chosen weights of 1, 1, 0.1, 0.01 and 0.01 for the resolution of this problem. 

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. 

In Hitchell s work unequal variance of the response data was compensated for by weighting the data by the variance at each level. The regression parameters and the confidence band around the regression line were estimated by least squares ( ) The overall level of uncertainty, OL, was divided between the variation in response values and the variance in the regression estimation. His overall a was 0.05. The prediction interval was estimated around a single response determination. 

It is often assumed in regression calculations that the experimental error only affects the y value and is independent for the concentration, which is typically placed on the x axis. Should this not be the case, the data points used to estimate the best parameters for a straight line do not have the same quality. In such cases, a coefficient Wj is applied to each data point and a weighted regression is used. A variety of formulae have been proposed for this method. 

Regression analysis requires that a new table be constructed listing the individual tablet weight (column 1), corresponding assay (column 2), and percentage of purity of the raw material used to compound the tablet (column 3). From these data, regression lines and confidence intervals can be plotted to complement the usual statistics. 

The silent intervals are weighted with the optimal weighting factor (0.4). The correlations and standard errors that are given are derived from a first (Rl, SI) and second (R2, S2) order regression line. The second order regression line is drawn line. 

Fig. 2.1 Correlation between molecular weight and cumulative recovery (mean SD) of rINF a-2a (MW 19 kDa), cytochrome C (MW 12.3 kDa), inulin (MW 5.2 kDa) and 5-fluoro-2 -deoxyuridine (FUDR) (MW 246 Da) in the efferent lymph from the right popliteal lymph node following subcutaneous administration into the lower part of the right hind leg in sheep (n = 3). The linear regression line has a correlation coefficient r = 0.998 (p <0.01) (from [18]).

Transformations Transformation of the data set can be used to correct for het-eroskedasticity or for linearization of the regression line. A major drawback of transformations is that, when correcting for heteroskedasticity, the assumption of linearity may become violated. Therefore, weighted regression should be preferred for correcting heteroskedasticity. 

Linearity A calibration curve was obtained using the eight calibration standards that were described above. A 1/x2 weighted least squares linear regression using the area ratios of analyte/intemal standard against the nominal concentration was performed. A regression line was obtained from these data, which was used for back calculation of the concentration for unknowns and quality controls. 

In a preliminary experiment, weanling male Wistar rats were depleted of zinc by feeding a low zinc basal diet (0.6 yg/g zinc) for two weeks and then repleted by adding 12 yg/g zinc as zinc sulphate. The analysis of the zinc content of the different tissues at weekly intervals for four weeks revealed that the body weight and the total femur zinc were the parameters of choice because the responses were linear with duration of feeding. Moreover, the relative errors of the slopes of the regression lines were minimal (5). The results of this experiment also showed that since depletion did not reduce the variability in these parameters, it was not essential for the assay. 

Fig. 17.2. Absolute risk of non-fatal, fatal myocardial infarction (Ml), fatal Ml, all Ml and any vascular death plotted against mean follow-up with fitted regression lines obtained through weighted meta-regressions from a systematic review of 39 studies reporting relevant risks in patients with transient ischemic attack and ischemic stroke. Each circle represents a study and its size is inversely proportional to the within-trial variance (Touze et ai. 2005).

---