Regression weighed

This problem corresponds to the simple linear regression model (w= 1, n= 1, p=2). Taking as Q,=l (all data points are weighed equally) Equations 3.19a and 3.19b become... 

Concentration-Response Relationship. The simplest model that describes the concentration-response relationship adequately should be used. The selection of weighing and use of a complex regression equation should be justified. 

Method validation is important to ensure that the analytical method is in statistical control. A method may be validated by the so-called method evaluation function (MEF) (Christensen et al., 1993), which is obtained by linear regression analysis of the measured concentrations versus the true concentrations. A true concentration in a solution can be obtained by use of a high purity standard obtained from another manufacturer or batch than the one used for calibration. Both the high purity standard and the solvent are weighed using a traceable calibrated balance. If certified reference material is available this is preferred. The method evaluation includes the most important characteristics of the method as the following elements (see Figure 2.7) ... 

Fig. 3. Time sequence photographs of two mice with solid Sarcoma-180 tumors. The mouse at the top was an untreated negative control. She died on day 21 when the tumor weighed about 3 g. The bottom mouse was in the group treated on day 8 with an intraperitoneal injection of cA-dichlo-rodiammineplatinum(II). Her tumor was completely regressed six days after treatment, and she died of age-related causes almost 3 years later. 

Weighted Regression Heteroskedasticity can be solved by introducing weighing factors (Wj) in the regression model. The weighing factors are inversely related to the variance Wt = 1 /S t. 

Ignoring heteroskedasticity in an ordinary calibration experiment will not lead to major errors when measuring in the middle of the calibration line since the estimated regression parameters are unbiased (it should be recalled that the regression estimators are inefficient, but unbiased). However, the uncertainty of the results is much larger in the lower part of the calibration line when incorrect or no weighing factors have been applied to correct for heteroskedasticity. It should be stressed that extreme errors are introduced by using the calibration line for the estimation of the detection capability, if heteroskedasticity is not corrected. 

Indirect methods, based on allometric inference, from measurements of the diameter at breast height (DBH) and the height of the trees to obtain wood volumes, is the main method adopted to estimate biomass in the Amazon. Forest biomass estimates are made through regression analysis, where several fitting curves are tested to obtain an ideal model that can be applied to the trees. These models are calibrated by direct weighing of the biomass from a subsample of trees (see for example, Jordan and Uhl 1978, Higuchi et al. 1994, Brown et al. 1995), and could also include other compartments besides trees, such as vines or understory. Indirect methods are broadly adopted in the forestry industry to evaluate the volume of commercial wood. 

---