Intercept variance

The variance of the regression times the diagonal elements of the inverse coefficient mahix gives the variance of the intercept and slope. 

Equations 5.13 for the slope, h, and 5.14 for the y-intercept, ho, assume that indeterminate errors equally affect each value of y. When this assumption is false, as shown in Figure 5.11b, the variance associated with each value of y must be included when estimating [3o and [3i. In this case the predicted slope and intercept are... 

In this study the reader is introduced to the procedures to be followed in entering parameters into the CA program. For this study we will keep Pm = 1.0. We will first carry out 10 runs of 60 iterations each. The exercise described above will be translated into an actual example using the directions in Chapter 10. After the 10-run simulation is completed, determine (x)6o, y)60, and d )6o, along with their respective standard deviations. Do the results of this small sample bear out the expectations presented above Next, plot d ) versus y/n for = 0, 10,20, 30,40, 50, and 60 iterations. What kind of a plot do you get Determine the trendline equation (showing the slope and y-intercept) and the coefficient of determination (the fraction of the variance accounted for by the model) for this study. Repeat this process using 100 runs. Note that the slope of the trendline should correspond approximately to the step size, 5=1, and the y-intercept should be approximately zero. 

By way of illustration, the regression parameters of a straight line with slope = 1 and intercept = 0 are recursively estimated. The results are presented in Table 41.1. For each step of the estimation cycle, we included the values of the innovation, variance-covariance matrix, gain vector and estimated parameters. The variance of the experimental error of all observations y is 25 10 absorbance units, which corresponds to r = 25 10 au for all j. The recursive estimation is started with a high value (10 ) on the diagonal elements of P and a low value (1) on its off-diagonal elements. 

The results of the DFR assessment of different crop zones indicate that low-volume applications result in a more homogeneous distribution over the crop compared to high-volume applications. A recent study on the interception of high-volume applications in the cultivation of chrysanthemums revealed interception ratios from 0.2 to 1 related to the leaf area index (LAI) (Veerman et al., 1994). In our study, it was not easy to assess the LAI because of the structure of the carnation crop. Estimation of the LAI based on the results of estimation of the crop density (leaf volume index) was not reliable enough and resulted in a large variance of the calculated interception ratio (from 0.4 up to 5). 

The denominator n 2 is used here because two parameters are necessary for a fitted straight line, and this makes s2 an unbiased estimator for a2. The estimated residual variance is necessary for constructing confidence intervals and tests. Here the above model assumptions are required, and confidence intervals for intercept, b0, and slope, b, can be derived as follows ... 

Consideration of the effect of experimental design on the elements of the variance-covariance matrix leads naturally to the area of optimal design [Box, Hunter, and Hunter (1978), Evans (1979), and Wolters and Kateman (1990)]. Let us suppose that our purpose in carrying out two experiments is to obtain good estimates of the intercept and slope for the model yj, = Po + Pi i, + r,. We might want to know what levels of the factor x , we should use to obtain the most precise estimates of po and... 

Figure 8.5 shows that can be estimated most precisely when the third experiment is located at x,3 = 0. This is reasonable, for the contribution of the third experiment at x, = 0 to the variance associated with involves no interpolation or extrapolation of a model if the third experiment is carried out at X = 0, then any discrepancy between y,3 and the true intercept must be due to purely experimental uncertainty only. As the third experiment is moved away from x, = 0, does increase, but not drastically the two stationary experiments remain positioned near X, = 0 and provide reasonably good estimates of hg by themselves. 

Prove that the least squares intercept estimator in the classical regression model is the minimum variance linear unbiased estimator. 

For a linear regression one can give the interval estimate of parameter Pi or the slope of the regression line, of its po intercept on the Y-axis, of the true mean Y for any value X ( tY/X=E(Y)) and the true predicted value, Y corresponding to a fixed value of X. The variances of the estimators of these parameters can be shown to be ... 

Lets assume that errors e, and 8, are independent values which have normal distributions with mean values equal to zero, and variances o and o, correspondingly. In this case, if ox is known as the standard uncertainty of the measurement standard (CRM), the slope (P,) and the intercept (P0) of the calibration curve can be estimated by the following equations [6] ... 

If the IS contributes to the signal of the analyte, but the reverse is not true, a linear calibration curve with a positive intercept is obtained. Provided that the variance on the isotope ratios measured is uniform throughout the whole calibration range, linear regression analysis may be applied. Otherwise, weighting factors should be introduced, e.g., the reciprocals of the variances at different concentration levels (Claeys et al., 1977 Schoeller, 1976). 

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