Linear plots analysis

A linear regression analysis should not be accepted without evaluating the validity of the model on which the calculations were based. Perhaps the simplest way to evaluate a regression analysis is to calculate and plot the residual error for each value of x. The residual error for a single calibration standard, r , is given as... 

If a standard method is available, the performance of a new method can be evaluated by comparing results with those obtained with an approved standard method. The comparison should be done at a minimum of three concentrations to evaluate the applicability of the new method for different amounts of analyte. Alternatively, we can plot the results obtained by the new method against those obtained by the approved standard method. A linear regression analysis should give a slope of 1 and ay-intercept of 0 if the results of the two methods are equivalent. 

The second form is statistically advantageous, but the first is amenable to linearized graphical analysis. It represents the equation of a straight line. A plot of (Y, — Too)-1 versus time will be a straight line with a slope from which k may be calculated k = slope x (Yq - Fx)/tA]o. Note that the bracketed quantity is the difference in y s (in spectrophotometry, the difference in molar absorptivities). Thus, k = slope X (yA - yP) = slope X (eA - p). 

The activation energy differences of My as well as of and M, and k /kp and kt/kp. were calculated from Arrhenius and Mayo plots, respectively, by linear regression analysis using a computer. Hie AEjjw values given in kcal/mole can be converted to kJ/mole by multiplying with 4.18. 

Linear regression analysis was performed on the relation of G"(s) versus PICO abrasion index. Figure 16.10 plots the correlation coefficient as a function of strain employed in the measurement of loss modulus. The regression results show poor correlation at low strain with increasing correlations at higher strains. These correlations were performed on 189 data points. 

One-dimensional data are plotted versus an experimental variable a prime example is the Lambert-Beer plot of absorbance vs. concentration, as in a calibration run. The graph is expected to be a straight line over an appreciable range of the experimental variable. This is the classical domain of linear regression analysis. 

Temperature error differences (AT), equal to the experimental temperature minus the linear regression temperatures, were then plotted by another linear regression analysis against the EA values to obtain the reactant state effect slope ... 

Lineweaver-Burk plots [11] over the range 0.1 to 1 mM Paraoxon in 100 mM CHES buffer, pH 9.0. Linear regression analysis for Lineweaver-Bulk plot was performed using SigmaPlot software (Systat Software, USA). 

Fig. 33.1. Canonical variate plot for three classes with different thyroid status. The boundaries are obtained by linear discriminant analysis [2].

Measurements of Pe in fixed-pH solutions but at various different stirring speeds need to be made. The double-reciprocal analysis, HPe versus 1/v , for Caco-2 permeability measurements in the Transwell (Corning Costar) system produced a linear plot for x- 0.8 [514]. The intercept yields the membrane permeability for the particular pH value in the study the slope determines the k constant. From the analysis of testosterone transport, for the stirring speed of 25 rpm (planar rotating shaker), the thickness of each UWL (assuming symmetric geometry) was calculated to be 465 pm at 150 rpm, haq= 110 pm [514], Karlsson and Artursson [512] found x = 1.0 to best represent their stirring-based analysis of the UWL permeability. 

Table 8.3. Classical Porod-law analysis of three kinds of scattering curves. Linearizing plots and the values of intercept and slope...

Using the actual dimensions of commercial steel pipe from Appendix F, plot the pipe wall thickness versus the pipe diameter for both Schedule 40 and Schedule 80 pipe, and fit the plot with a straight line by linear regression analysis. Rearrange your equation for the line in a form consistent with the given equation for the schedule number as a function of wall thickness and diameter ... 

The morphological stability of initially smooth electrodeposits has been analyzed by several authors [48-56]. In a linear stability analysis, the current distribution on a low-amplitude sinusoidal surface is found as an expansion around the distribution on the flat surface. The first order current distribution is used to calculate the rate of amplification of the surface corrugation. A plot of amplification rate versus mode number or wavelength separates the regimes of stable and unstable fluctuation and... 

Two data sets of LC50 values for different time periods of exposure were analyzed using a linear regression analysis of the log-log transformation of a plot of C vs t to derive values of n for monomethylhydrazine. 

Linear regression analysis was used to fit the points for the Scatchard plot. 

A direct linear plot made from seven pairs of (v, [S]) data. The dotted lines mark the lowest and highest points of intersection. Clearly, a graph showing 21 horizontal and 21 vertical dotted lines, equivalent to the number of intersections from seven data pairs (see text), would be cluttered and difficult to interpret, and these lines are not shown. Rather, the hatched lines indicate Km and values obtained from nonlinear regression of the same data not surprisingly, these lie close to the median intersection points that would be obtained from a full direct linear analysis... 

Further drawbacks associated with the direct linear plot include the fact that this analysis does not readily lend itself to standard computerized graphing methods (for example, use of GraphPad Prism), although specialized software is available (Henderson, 1993). Of course, one of the major advantages of the direct linear plot is the ability to obtain kinetic constants by eye, without the need for a computer. However, for presentation purposes, the use of graphing software is still desirable. Furthermore, any behavior more complicated than simple, single substrate kinetics - for example, turnover in the presence of an inhibitor, or multisubstrate kinetics - caimot readily be shown on a direct linear plot. This is in contrast with the flexibility afforded by nonhnear regression approaches. 

Figure 7.2. (A) The greased-gap technique for recording depolarizations of the rat isolated vagus nerve. (B) The effect of ondansetron on depolarizations of the rat isolated vagus nerve induced by 5-HT. Symbols indicate controls (9) or in the presence of ondansetron at I x 10 M (O), 3 X 70" M (M), I X 10 M (O) or 3 x 10 M (A). Results are the mean + S.E.M. of at least four determinations. Experiments were performed as described by Ireland and Tyers [21J. (C) Data from the experiments illustrated in B plotted according to Arunlakshana and Schild [22]. Each point is the result from a separate tissue. The gradient of straight line (95 Vo confidence limits) was determined using linear regression analysis.

A plot of [A]q + y versus reciprocal breakthrough volume supports the determination of Bt and by linear regression analysis. As with all FAC methods for ligand characterization, the cartridge does not require precalibration before a measurement is made, because is a product of the measurement. One disadvantage of this method is that any error in the earlier measurements is carried forward in subsequent ones however the speed of these measurements and the ability to accurately measure Vq usually makes this the method of choice. 

As shown in Table VII there appears to be no significant change of k with respect to temperature. These data were plotted using Equation 3 and from linear regression analysis, the heat of solution was tO.IE Kcal/mole. Since Ah should be negative, this low value is obviously caused by experintental error. Furthermore, the Ah calculated from the standard error of the estimate (t1 standard deviation units) of the linear regression line is +0.17 Kcal/mole. Since Ah is zero or is very close to zero. Equation 3 reduces to... 

Figure 12 shows the classical method of obtaining the Mark-Houwink coefficients, K and a, by plotting the log [n](v) vs. log M(v) for this polymer in THF at 50°C. The data points used for the plot in Figure 12 are indicated by the area between the arrows in Figure 10. Linear regression analysis of the data resulted in K o =1.86x10" and a o =0.662 with a correlation coefficient or t =u.9996 for NBS 70o polystyrene.

---