Scatter plot least-square line

Figure 3 Scatter plot of Assay A data with least-squares line.

On average, the tests shown in Fig. 5.28 represent ten microhardness indentations per specimen. When cracks occasionally appear to emanate from the comers of the indentations (as also observed in SiC, seen in Fig. 5.29b, these values were not averaged in. The line in Fig. 5.28 is linear and, thus, microhardness values of the solid solutions may be represented as linear relations plotted against mol% chromia. The large scatter around the least-square line is assumed to be associated with the microindentation crossing many grains. Hot-pressed materials contain a... 

Fig. 7.3. A scatter-plot of molar enthalpies of melting for organic crystals (data from ref. [5]). The least-squares line is y = 0.0606x.

Once the form of the correlation is selected, the values of the constants in the equation must be determined so that the differences between calculated and observed values are within the range of assumed experimental error for the original data. However, when there is some scatter in a plot of the data, the best line that can be drawn representing the data must be determined. If it is assumed that all experimental errors (s) are in thejy values and the X values are known exacdy, the least-squares technique may be appHed. In this method the constants of the best line are those that minimise the sum of the squares of the residuals, ie, the difference, a, between the observed values,jy, and the calculated values, Y. In general, this sum of the squares of the residuals, R, is represented by... 

Fig. 13.4 Scatter plot of mean tasidotin clearance against body surface area for all patients. The solid line is the line of least squares the dashed lines are the 95% Cl forthe slope ofthe line.

Fig. 13.6 Scatter plot of time to disease progression as a function of tasidotin exposure. The solid line is the line of linear least squares.

Where the number of points is sufficiently large, the limits of error of the position of plotted points can be inferred from their scatter. Thus an upper bound and a lower bound can be drawn, and the lines of lintiting slope drawn so as to lie within these bounds. Since the theory of least squares can be applied not only to yield the equation for the best straight line but also to estimate the uncertainties in the parameters entering into the equation (see Chapter XXI), such graphical methods are justifiable only for rough estimates. In either case, the possibility of systematic error should be kept in ntind. 

The RATIO method table (Table I) includes provision for specifying upper and lower limits of integration for both primary and reference bands with the peak area evaluation procedure. The practical limits of the integration can be determined empirically by evaluating a set of spectra stored on microfloppy disks with varying limits set in the appropriate locations in the method table. Optimum limits can be determined from the calibration plots and related error parameters. The calibration plots shown in Figures 4 and 5 indicate that both evaluation procedures, peak height and peak area provide essentially the same level of precision for the linear least squares fit of the data. The error index and correlation coefficients listed on each table are both indicators of the relative scatter in the data from the least squares fit line. The correlation coefficient is calculated as traditionally defined in statistics. 

Within each of the assays A, B, C, and D, least squares linear regression of observed mass will be regressed on expected mass. The linear regression statistics of intercept, slope, correlation coefficient (r), coefficient of determination (r ), sum of squares error, and root mean square error will be reported. Lack-of-fit analysis will be performed and reported. For each assay, scatter plots of the data and the least squares regression line will be presented. 

An alternative approach to linearity analysis is to analyze the linearity of all the data. The SAS code for the overall analysis is the same as shown before except that the line By Assay is deleted from the code. Table 13 shows the summary statistics for the least squares regression, Figure 7 shows a scatter plot... 

HEDTA (31) Fig. 25 Inclined W plot for log Ki values (31) with Gd point belonging to the third tetrad, predicted log KiPm 15.2. Linear relationship within the first three tetrads for the AHi values (plot not shown) with Gd point belonging to the second tetrad, but a scatter for the points Er to Lu, although a least square straight line could be drawn through the points, predicted AHi Pm 4.7. 

Figure 2.6 Scatter plots and box and whisker plots of 5-fluorouracil (5-FU) clearance as a function of patient demographics. Data presented in Table 2.3. Solid line is the least squares fit to the data. Note that some plots are shown on a log-scale.

Figure 4.5 Scatter plot of simulated data from an Ema model with 20% constant coefficient of variation, where Emax = 75 and EC50 = 25. Legend Solid line is weighted least-squares fit with weights l/Y.

Figure 4.13 Scatter plot of maximal change in albumin concentration (top) in patients dosed with XomaZyme-791 and model predicted fit (solid line). Data were fit to a Emax model using ordinary least-squares. Bottom plot is residual plot of squared residuals divided by MSE against predicted values. Solid line is least-squares regression line and dashed lines are 95% confidence interval.

Figure 1 shows a plot of the gas-phase electron affinities of a number of anions plotted against the gas-phase proton affinities. The circles refer to anions where the donor atom is a second-row element, C,N, O, or F. The straight line is a least-squares fit to these values only. The slope is — 0.87 and the correlation coefficient is —0.923. In spite of the scatter, clearly a strong negative correlation between the electron affinity of a radical and the proton affinity of its anion occurs. A strong base, such as CH3-, loses its electron readily, whereas P03- (metaphosphate ion) is a weak base and is difficult to oxidize. 

Fig. 5.8.2. Spectra of light scattered from the protoplasm of Nitella. The horizontal axis is frequency in Hz and the vertical axis is relative intensity. Spectrum (a) was taken at a scattering angle of 19.5 deg. Spectra (b) and (c) were taken at a scattering angle of 36.1 deg. Spectrum (c) was taken from the same point on the cell as spectrum (b), immediately after addition of parachloromercuribenzoate, a streaming inhibitor. (Each of these spectra was collected in about 30 sec. The points are the output of the spectrum analyzer, and the dark lines have been drawn merely to make the data more perspicuous and to emphasize the reproducible features of the data.) Part (d) is a plot of the magnitude of the Doppler shift from a fixed point on a Nitella cell as a function of the sine of the scattering angle 9. The line is the best least-square fit to the points. The predicted linear dependence is verified, and the deviations from the line provide an estimate of the experimental precision. (From Mustacich and Ware, 1974.)...

Fig. 8.5. Scatter plot of average seiche amplitudes (the average of the five highest crests and five deepest troughs of an event) of coinciding seiche events at ROZ and Europe Harbor. Trend line based on least-squares estimate.

We can choose different coefficients for B in Equation (5.40) provided they sum to 1. An approach which does this is known as the least squares filter. It gets its name from the least squares regression technique used to fit a line to a set of points plotted on an XY (scatter) chart. Its principle, as shown in Figure 5.19 is to fit a straight fine to the lastiVpoints. The end of this line is 7 . 

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