Least squares regression line

Calculations of the confidence intervals about the least-squares regression line, using Eq. (2-100), reveal that the confidence limits are curved, the interval being smallest at Xj = x. 

Abstracted from the compilation by Jaffe, where original references may be found. Value of log k on the least-squares regression line where a = 0 the time unit is seeonds. 

Linearity is often assessed by examining the correlation coefficient (r) [or the coefficient of determination (r )] of the least-squares regression line of the detector response versus analyte concentration. A value of r = 0.995 (r = 0.99) is generally considered evidence of acceptable fit of the data to the regression line. Although the use of r or is a practical way of evaluating linearity, these parameters, by... 

Construct a calibration curve by plotting the peak area against the standard concentration to obtain a least-squares regression line. 

The peak of 2-TFBA Me-ester usually appears at a retention time around 2.4 min. Construct a calibration curve by plotting the natural logarithm of the peak area counts against the natural logarithm of the standard concentration to obtain a least-squares regression line. 

The quantitative description of the relationship between two or more variables is often addressed using a least squares regression line referred to as linear regression. Linear regression, as and example of Y on X linear regression, between two data sets involves the relationship... 

Release rates (%/min) were calculated from the best fits of released drug vs. time plots. The slopes ( ) of the log(released drug) vs. log(time) plots were calculated from the linear least-squares regression lines. A slope of 0.5 in the log-log plot indicates diffusional, square root of time, dependence and a slope of 1.0 indicates zero-order release kinetics [6]. Times of 50% drug release (t50%) were calculated from the best fits of drug released vs. time plots. 

Figure 5 Least-square regression lines fit for the linear relationship between mean and standard deviation of PVT reaction times (msec). Data are from n = 13 subjects undergoing 88 hr (3.67 days) of total sleep deprivation. This figure illustrates that while all subjects experienced a decline in neurobehavioral performance on the PVT, as illustrated by increased reaction times when responding to the visual stimuli, there is a significant degree of interindividual variability in the magnitude of neurobehavioral impairment, evident by the differing lengths of the lines fit to the data. (From Ref. 44.)...

The independent measurements of surface tension were obtained by the tedious Wilhelmy plate method. Figure 3 illustrates such a calibration curve for one set of orifices and for five types of test fluids (methanol-water, ethanol-water, acetone-water, sodium lauryl sulfate in water saturated with methyl methacrylate, and polymethylmethacrylate latices). This is a "universal" calibration curve independent of the fluid being monitored. For the 63 data points shown in Figure 3, the least squares regression line is given by... 

The online statistical calculations can be performed at http //members.aol.com/ johnp71/javastat.html. To carry out linear regression analysis as an example, select Regression, correlation, least squares curve-fitting, nonparametric correlation, and then select any one of the methods (e.g., Least squares regression line, Least squares straight line). Enter number of data points to be analyzed, then data, x and y . Click the Calculate Now button. The analytical results, a (intercept), b (slope), f (degrees of freedom), and r (correlation coefficient) are returned. 

The satisfactory correlation of the data argues for a linear response of the resonance contributions of the thiophene ring. Substituent constants191 aa+ and ap+ relative to the perturbation produced by the substitution of a S atom for a CH=CH moiety in the benzene ring, can be easily calculated from the slopes of the plots, — 0.79 and — 0.52, respectively. (Slopes refer to least-square regression lines with the theoretical origin.)... 

Figure 10.4 Relationship between monthly averages of total nitrogen concentration ( xg N L-1 and phytoplankton biomass (pg Chi. a L 1), during March to October, from 23 locations in Danish coastal waters. Lines represent least squares regression lines fitted on log-transformed data. (Modified from Nielsen et al., 2002.)...

This would be the osmolality of the drug if the activity coefficient were equal to 1 in the full-strength preparation. The osmolalities of serial dilutions of the drug were plotted against the concentrations of the solution, and a least-squares regression line was drawn. The value for the osmolality of the full-strength solution was then estimated from the line. This is the calculated available milliosmoles. ... 

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. 

Results for the Whatman 541 filter paper (23-28), flat plates (14-22), and Petri dishes (31,32) are shown in figures 7, 8, and 9 respectively. The data for the filter paper and flat plates have been presented as separate points for each measurement reported in the literature. For the Petri dishes, only the mean values and standard errors were available. Least squares regression lines plotted through the points are also shown. Figure 10 presents all three regression lines plotted on the same axes with the predicted sedimentation velocity line. It must be cautioned that a comparison of the regression lines is tenuous due to the spread in the data represented by each line. 

The least squares regression line is the line which minimizes the sum of the square or the error of the data points. It is represented by the linear equation y = ax + b. The variable x is assigned the independent variable, and variable j is assigned the dependent variable. The term b is the y-intercept or regression constant (the value of when x = 0), and the term a is the slope or regression coefficient. 

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.

The high degree of linearity in the temperature dependence of the ring proton shifts is evident from the correlation coefficients of the least squares regression lines (Table IX). The slopes of the lines are all negative and similar in magnitude to that of uranocene. However, the standard deviations of the extrapolated intercepts at T-1=0 indicate that a number of the intercepts are non-zero. Ideally, eq. 3 predicts that all of the intercepts should be zero at T 1=0. 

The standard deviation of the response can be determined based on the standard deviation of either the blank, the residual standard deviation of the least-squares regression line, or the standard deviation of the y intercept of the regression line. The Excel statistical function can be used to obtain the last two. 

It is assumed that the time-based data fit a theoretical dose-response curve as shown in Figure 6. Analysis of such dose-response curves, first described by Bliss (10), consists of finding a convenient method to linearize the data. In such a linearized form, a least-squares regression line and standard estimates of data variance can be obtained. 

Figure 27. Plot of ln(KEg) -F PAVg/RT vs. reciprocal temperature ror the reaction YAG + OH-apatite + 25/4 quartz = 5/4 grossular + 5/4 anorthite + 3 YPO4 monazite + 1/2 H2O. Solid squares = xenotime-bearing assemblages, open squares = xenotime-absent assemblages. Least squares regression line is fit to all data points. Horizontal error bars represent temperature uncertainty of 30°C. Vertical error bars ate la (In Kgq + PAV/RT), derived from propagation of uncertainties in P ( 1000 bars), T ( 30°C), AVrxn (1%), compositional parameters (0.001 mole fraction YAG, 0.01 mole fraction all others), and /(H2O) ( 7.5 1000 trial Monte Carlo simulation). Labels on graph indicate sample numbers. From Pyle et al. (2001).

Product Challenged Growth Studies. To study the inhibitory factors of the acetone-butanol fermentation, the growth rates of Cl. acetobutylicum in the presence of each fermentation product were determined. The end products used in this study included ethanol, butanol, acetone, acetic acid, and butyric acid. From the slopes of the least squares regression lines of optical density vs. time data, the maximum specific growth rates in the presence of varying concentrations of each inhibitor ()j ) were determined. The results for each fermentation product are shown in Figures 1 - 3. There appears to be a threshold concentra-... 

The inhibition constant for each product was calculated from the slope of the least squares regression line of Vi /V4n vs. 

In this approach, a least squares regression line is fit to stability data either from a single batch or from several batches. The expected result for any time point is given by the expression ... 

The concept starts with a definition of a residual for the fth calibration point. The residual g,- is defined to be the square of the difference between the experimental data point y and the calculated data point from the best fit line y- Figure 2.3 illustrates a residual from the author s laboratory where a least squares regression line is fitted from the experimental calibration points for A, A -dimethyl-2-aminoethanol using gas chromatography. Expressed mathematically. 

Equations (2.12) and (2.13) enable the best-fit calibration line to be drawn through the experimental x,y points. Once the slope m and the y intercept b for the least squares regression line are obtained, the calibration line can be drawn by any one of several graphical techniques. A commonly used technique is to use Excel to graphically present the calibration curve. These equations can also be incorporated into computer programs that allow rapid computation. 

The uncertainty in both the slope and intercept of the least squares regression line can be found using the following equations to calculate the standard deviation in the slope, and the standard deviation in the y intercept, sy. 

Figure 2.4 Interpolation of a least squares regression line showing confidence intervals.

The least squares regression line is seen to be shrouded within confidence intervals as shown in Figure 2.4. The derivation of the relationships introduced below was first developed by Hubaux and Vos over 30 years ago (13) The upper and lower confidence limits for y that define the confidence interval for the calibration are obtained for any v (concentration) and are given as follows ... 

The variance in the least squares regression line can further be broken down into the variance in the mean y expressed as Vy and the variance in the slope m expressed in terms of V according to... 

Equation (2.25) is an important relationship if one desires to plot the confidence intervals that surround the least squares regression line. The upper confidence limit for the particular case where x = 0 is obtained from Eq. (2.25) in which a 1 — a probability exists that the Normal distribution of instrument responses falls to within the mean y at x = 0. Mathematically, this instrument response y, often termed a critical response, is defined as... 

To conclude this discussion on IDLs, it is useful to compare Eqs. (2.19) and (2.29). Both equations relate Xp to a ratio of a standard deviation to the slope of the least squares regression line multiplied by a factor. In the simplified case, Eq. (2.19), the standard deviation refers to detector noise found in the baseline of a blank reference standard, whereas the standard deviation in the statistical case, Eq. (2.29), refers to the uncertainty in the least squares regression itself. This is an important conceptual difference in estimating IDLs. This difference should be understood and incorporated into all future methods. 

Lab Y, on the other hand, not only reported detectable levels of the three organics but also gave a confidence interval for each reported concentration. Lab Y demonstrates that reference standards were run to establish a least squares regression line and the goodness of fit expressed by the value of the correlation coefficient. The lab also conducted the required percent recovery studies and demonstrated acceptable precision in the reported % recoveries for each of the three analytes. The lab also reports that it conducted a study of the IDLs and also of the MDLs for the requested method. 

Slope of least squares regression line (linear fit) 4548 MV-s/ppb AR 1242... 

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