Variance of slope

Conventionally, the deviation of a surface from its mean plane is assumed to be a random process for which statistical parameters such as the variances of the height, the slope and curvature are used for characterisation (Nayak, 1971). However, it has been found that the variances of slope and curvature depend strongly on the resolution of the roughnessmeasuring instrument or any other form of filter and are hence not unique (Thomas, 1982 Bhushan et al., 1988 Majumdar and Tien, 1990). It is also well known that surface topography... 

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

Figure 7 shows that for the maximum likelihood estimator the variance in the slope estimate decreases as the telescope aperture size increases. For the centroid estimator the variance of the slope estimate also decreases with increasing aperture size when the telescope aperture is less than the Fried parameter, ro (Fried, 1966), but saturates when the aperture size is greater than this value. 

One possibility is that although averages for polystyrene standards require correction, those for PMMA would not According to symmetrical axial dispersion theory (5) the correction depends upon both the slope of the calibration curve (different for each polymer type) and the variance of the chromatogram of a truly monodisperse sample. Furthermore, the calibration curve to be utilized can be obtained from a broad standard as well as from monodisperse samples. The broad standard method may itself incorporate some axial dispersion correction depending upon how the standard was characterized. 

Mjj and My or [q] for the broad MWD standard are taken as known quantities. Fy(v) is the normalized chromatogram for the broad MWD standard obtained with a mass detector. D2 is the slope of the molecular weight calibration curve at the peak position of the chromatogram (the equation of the tangent is given by M(v) = Dj exp(-D2v). is the variance of the single-species chromatogram... 

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. 

An alternative to the slope approach to determining the appropriate value of n for use in model calculations is based on a determination of the variance of the response of the actual... 

Both TOA and supernatant BOD5, concentrations are expressed as g/1. The mean variance of the TOA concentration from the regression line is 0.208 and the standard error of the slope 0.020. 

Besides the continuous effect models, drugs are often fitted to all-or-none models where the responses such as disappearance of arrhythmia in response to a drug dose cannot be graded the same results hold when we classify the response ending in cure or not, or presence or absence of a given side effect or effects. In such instances, EC50 is the median concentration for which half of the subject population is above the threshold and the slope of the curve becomes the variance of the threshold in that population. 

This situation shows two problems The application of ordinary least squares estimation, which requires constant variance, is not appropriate with untreated data. Then, the large variance of the largest numbers in such data excessively controls the direction or slope of the graph. 

A more complete picture is shown graphically in Figure 7.1 where the value of the element for the estimated variance of fe, is shown as a function of the location of the second experiment. The stationary experiment is shown by a dot at j , = +1. As the two experiments are located farther and farther apart, the uncertainty in the slope of the straight line decreases. Note that the curve in Figure 7.1 is symmetrical about the fixed experiment at j , = +1. 

A more complete picture of the effect of the location of the third experiment on the variance of the slope estimate (fej) is shown in Figure 8.4. From this figure, it is evident that the uncertainty associated with fe, can be decreased by placing the third experiment far away from the other two. Placing the experiment at x,3 = 0 seems to give the worst estimate of b. For the conditions giving rise to Figure 8.4, this is true, but it must not be concluded that the additional third experiment at x,3 = 0 is detrimental the variance of bi (sl = 2) for the three experiments at = -1, x 2 = +1, and Xi3 = 0 is the same as the variance of fc, for only two experiments at = -1 and Xi2 = +1 (see Equations 7.1 and 7.13). 

Suppose that the cla ssica l regression model applies, but the true value of the constant is zero. Compare the variance of the least squares slope estimator computed without a constant term to that of the estimator computed with an unnecessary constant term. 

Consider the multiple regression of y on K variables, X and an additional variable, z. Prove that under the assumptions A1 through A6 of the classical regression model, the true variance of the least squares estimator of the slopes on X is larger when z is included in the regression than when it is not. Does the same hold for the sample estimate of this covariance matrix Why or why not Assume that X and z are nonstochastic and that the coefficient on z is nonzero. 

Suppose that a linear probability model is to be fit to a set of observations on a dependent variable, y, which takes values zero and one, and a single regressor, x, which varies continuously across observations. Obtain the exact expressions for the least squares slope in the regression in terns of the mean(s) and variance of x and interpret the result. 

Comment. The macroscopic equation (8.6) is a differential equation for y, which determines y(t) uniquely when the initial value /(0) is given. In the next approximation (8.12) the evolution of / depends on the variance of the fluctuations as well. The reason is that y fluctuates around / and thereby feels the value of not merely at / but also in the neighborhood. This effect is proportional to the curvature of al the slope of at is ineffective as the fluctuations are symmetric (in this approximation). The magnitude of the fluctuations, however, is determined by the second equation (8.9), which does contain the slope of ax. 

As for sample selection, I will submit two different methods for variable selection one that is relatively simple and one that is more computationally intensive. The simpler method68 involves a series of linear regressions of each X-variable to the property of interest. The relevance of an X-variable is then expressed by the ratio of the linear regression slope (b) to the variance of the elements in the linear regression model residual... 

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 ... 

Returning once again to an analysis of the least-squares line, it is now possible to make confidence statements concerning the slope, the intercept, and any value of y predicted by the equation. The statistical quantities required to make these statements are given in the next four equations and are the residual variances S2(Y), the variance of the slope s2(b), the variance of the mean s2(y),... 

The confidence interval on the slope can be calculated from the variance of the slope in the same manner as was used to determine the confidence range for the mean. Thus for a 95 percent confidence interval on the slope... 

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