Standard error of the regression

If the measurement of LOD is not critical, an estimate can be made from the calibration parameters taking the intercept as the blank measurement and the standard error of the regression as the standard deviation of the blank. Equation (8.1) becomes... 

Sy is standard error of the regression. If a large extrapolation for the intercept... 

This calculated F value is then compared to the critical value, Fa(1) VlV2, where Vj = regression DF = 1, and v2 = residual DF = n — 2. The residual mean square is often written as sfx, a representation denoting that it is the variance of Y after taking into account the dependence of Y on X. The square root of this quantity — that is, Syj—is called the standard error of estimate (occasionally termed the standard error of the regression). The ANOVA calculations are summarized in Table 2.2. 

The final statistical values that are reported for Equation 7.3 are the standard errors of the regression coefficients. These allow us to assess the significance of the individual terms by computing a statistic, called the t statistic, by dividing the regression coefficient by its standard error ... 

Power Consumption. A previous report (2) showed the power consumption of the pellet mill as a function of throughput. Subsequent work has shown there were errors in these measurements and the result is hereby withdrawn. Later work, using processed office waste with added moisture (75% of the samples contained 10 - 30 wt%, 22% of the samples contained 30 - 50 wt%, and the remainder 10 wt%) showed that the relationship between power consumption P(kW), and feedrate to the densifier F (short tons/h, dry wt basis), could be represented by P = 50.5F + 21.7, for 63 determinations, 0.3 < F < 1.7 Mg/h, with a correlation coefficient r = 0.964 and a standard error of the regression line of 5.7. The intercept should be equal to the idling power of the machine which was separately measured as 24 kW. There was no discernible relationship between power consumption and moisture content. 

For the concentration-versus-time data given in Example 3.26, determine the standard errors of the regression parameters Po and Pi and the R value. Plot the 95% confidence intervals for calculations from the model. 

Leverage The tendency of a single point to drag the calibration line towards it and hence increase the value of the standard error of the regression (sy/x). (Section 5.3.1)... 

Neither. These tell you about the linear relation between y and x, true, but in analytical chemistry you are rarely testing the linear model. The standard error of the regression (Sy/X) is a useful number to quote, or calculate 95% confidence intervals on parameters and estimated concentrations of test solutions. Plot residuals against concentration if you are concerned about curvature or heteroscedacity. (Sections 5.3.2, 5.5)... 

Syjx is a cell containing the standard error of the regression (see previous entry) b is the slope of the calibration curve m is the number of repeats of the test solution (if m > 1, yo is the mean of the m replicates) n is the number of points in the calibration curve ybar is the mean of the calibration y values xbar-range is a range containing the x calibration values minus the mean of the x calibration values (mean centered x values)... 

The next step is to calculate the standard error of the regression syjx. First, we must calculate j, the value of Y estimated from the regression for each xt. Hence for each concentration of the standards, xu fv can be calculated using the regression equation we have now established y = 0.0188 + 0.9197V,. The calculated values of j, for each. v, are shown in spreadsheet 5.1. Now we can determine sy/x using... 

If we need only one of the LI NEST outputs in a calculation, it may be extracted by the function =INDEX(array, row, column), without having to display the entire array. For example, the standard error of the regression is =INDEX(LINEST(y-range, x-range, const, stats),3,2) because sy/x is in the third row, second column of LI NEST. 

Calculate and quote the standard error of the regression. Calculate 95% confidence intervals on any concentration calculated from the calibration relation. If required, quote the regression equation with standard deviations (or 95% confidence intervals) on slope and intercept. 

If it is not possible to record a background or blank signal, the standard deviation of the blank can be estimated by the standard error of the regression of the calibration equation (sy ) and /blank by the intercept of the calibration equation a), when eqn [2] becomes... 

The squared multiple correlation coefficient gives a measure of how well a regression model fits the data and the F statistic gives a measure of the overall significance of the fit. What about the significance of individual terms This can be assessed by calculation of the standard error of the regression coefficients, a measure of how much of the dependent variable... 

The values in brackets are the standard errors of the regression coeffidents and it should be noted that R is quoted not as is more usual for multiple regression equations. The second PC was less well described by a shape parameter based on molecular connectivity ( K) and an indicator variable. 

Mandel s test was summarized as a comparison of the residual standard deviation of the linear model with that of the nonlinear mode . Such a definition results in the well-known conceptual Fischer-Snedecor s F test (Fexperimentai = s ylx,Mn / s y x,nof), whcrc s Stands for standard error of the regression, lin for straight line model and non for non-linear model (here a quadratic one, although this is not mandatory). Note that although the term variance should be used senso stricto instead of standard error, this is not relevant for our discussions here. The definition was then resolved to eqn A 1.1 below (numbered 51 in ref. 8) ... 

On the contrary, other authors " " have proposed the use of the standard error of the regression Syi ) instead of the standard error of the slope in the test and in eqns (A2.2) and (A2.3) [eqn (A2.4) is the same in all cases]. Miller also considered this alternative, although the equation he proposed is incorrect and his more recent textbooks did not consider this issue. Accordingly, two alternative equations must be considered (with i+ 2 4 degrees of freedom) ... 

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