Residuals, in regression

Residual in regression analysis, the difference between an observed value and the value predicted by the regression equation in analysis of variance, the error remaining after all desired main effects and interactions have been calculated. 

Assuming all noise is removed then the result is the true spectrum. Conversely, from Equation (33), if the smoothed spectrum is subtracted from the original, raw data, then a noise spectrum is obtained. The distribution of this noise as a function of wavelength may provide information regarding the source of the noise in spectrometers. The procedure is analogous to the analysis of residuals in regression analysis and modelling. 

Gray, J.B., and Woodall, W.H. The maximum size of standardized and internally studentized residuals in regression analysis. American Statistician 1994 48 111-113. 

Plot of the residual error in y as a function of X. The distribution of the residuals in (a) indicates that the regression model was appropriate for the data, and the distributions in (b) and (c) indicate that the model does not provide a good fit for the data. 

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. 

B observations in each of several centres) and also with more complex structures which form the basis of ANCOVA and regression. For example, in regression the assumption of normality applies to the vertical differences between each patient s observation y and the value of y on the underlying straight line that describes the relationship between x andy. We therefore look for the normality of the residuals the vertical differences between each observation and the corresponding value on the fitted line. 

Let e, be the rth residual in the ordinary least squares regression of y on X in the classical regression model and let s, be the corresponding true disturbance. Prove that plim(e, - e,) = 0. 

Prove that Newton s method for minimizing the sum of squared residuals in the linear regression model will converge to the minimum in one iteration. 

The retention times of peptides with fewer than 20 residues in reversed-phase chromatography can be predicted with a high degree of accuracy based on their amino acid composition and the characteristics of their N-terminal and C-terminal amino acids. A number of researchers (66 -75) have studied the role of amino acids in peptide retention and have established retention coefficients for the different amino acids. The retention coefficient value of each amino acid is normally calculated by regression analysis of the retention times for peptides of known composition. 

A procedure to determine PIR residues in bovine milk using HPLC with the derivatization step for UV detection has also been published (211). The PIR was extracted from milk after protein precipitation and a two-step LLE procedure. The extract was evaporated to dryness, dissolved in dilute base, and derivatized with 9-fluorenylmethyl chloroformate (FMOC). The de-rivatized extract was analyzed by reversed-phase HPLC. Overall recovery was 89%, with 4% for coefficient of variation. A linear regression analysis of HPLC/UV results was compared with the HPLC/MS assay (209,210). The procedure takes about 2.5 hours to complete six or eight samples. Pirlimycin is stable in milk frozen to —60°C or below for at least 3 months. 

The HDE s to the hands of the thinners versus the total residue found on the leaves is shown in Figure 2. The regression equation is Y = 690X-45 where Y is the HDE in (Jg/h and X is the leaf residue in pg/cm2, and the correlation coefficient was 0.99. Since the regression line does not go through zero, the value of the X intercept corresponds to non-dislodgeable residues which is only 4% when the initial value was 1.71 pg/cm2 and 9% when the initial value was 0.70 pg/cm2. 

In the case of the slope we are now sure that the value used in our regression model is justified, but in the case of the offset we run into the dilemma of statistical significance versus practical relevance should we use the model equation y = a-i -x for statistical reasons or still continue with model Eq. 2-48 because of possibly smaller residuals. In most cases people will prefer the second method. 

A graphic that can be produced to better describe the actual situation is the plot of b against y-y, where symbolizes the Euclidean vector norm [4CM16], As Figure 5.14 discloses, an /.-shaped curve results with the best model occurring at the bend, which reflects a harmonious model with the least amount of compromise in the trade-off between minimization of the residual and regression vector norms. The regression vector norm acts as an indicator of variance for the... 

To find the outlying observations, two strategies can be followed. The first approach is to apply a classical method, followed by the computation of several diagnostics that are based on the resulting residuals. Consider, for example, the Cook s distance in regression. For each observation 1 = 1,. .., n, it is defined as... 

The inverse correlation between the bioavailability of Zn and MnAmAo "was relatively weak (r=0.Ul). The positive residuals of this relationship were largely data collected during the winter. The winter influx of fresh water into San Francisco Bay was accompanied by an increase in the sediments of hydroxylamine-extractable Fe (presumably, freshly precipitated Fe) and humic substances (2T). We have data from too few San Francisco Bay stations to include the humics in regression calculations. However, the increase in hydroxylamine-Fe generally coincided with increases in Zn concentrations in Macoma thus, the combined Fe/Mn ratio in Equation 2 explained 60 percent of the temporal and spatial variance in the Zn concentrations of the bivalve when all the data were considered. 

The significance of the Regression Line is estimated by comparing its mean square with the residual in the usual way. Here it lies between the 20% and 5% levels. 

---