Standard errors of the coefficient

The standard error of the coefficient is given for each independent variable the number of cases, overall F for the equation and its level of significance and are also given for each equation. 

Parametric population methods also obtain estimates of the standard error of the coefficients, providing consistent significance tests for all proposed models. A hierarchy of successive joint runs, improving an objective criterion, leads to a final covariate model for the pharmacokinetic parameters. The latter step reduces the unexplained interindividual randomness in the parameters, achieving an extension of the deterministic component of the pharmacokinetic model at the expense of the random effects. Recently used individual empirical Bayes estimations exhibit more success in targeting a specific individual concentration after the same dose. 

In all the tables of estimated coefficients in this section the values in brackets represent the standard errors of the coefficient estimates and indicates the coefficient estimates were at least significant at the 95% level using a student t test. 

FIG. 1. Means and standard errors of the coefficient of phenotypic variance (CVP), the coefficient of additive genetic variation (CVA) and the coefficient of residual variation (CKR). Comparison is made between paired data for sexual and non-sexual characters from the same species (Data from Pomiankowski Mpller 1995, n = 16). 

The standard error of the coefficients in the canonical model will be approximately the same as for the linear and quadratic coefficients in the original model when a rotatable design has been used. 

Column 3 of table 4.7 lists the standard deviations of the coefficients for the solubility example. We will call these standard deviations the standard errors of the coefficients. 

As to the significance of the individual terms in the equation, this can be assessed using the t statistic, which is obtained by dividing the relevant regression coefficient by the standard error of the coefficient. If is the regression coefficient associated with a variable x, then the t statistic is obtained as follows ... 

Exercise 5.11. The result of Exercise 5.10 should show that there is no evidence of lack of fit in the model of Exercise 5.5. Use the residual mean square as an estimate of the variance of the observations and determine the standard errors of the coefficient estimates. Are they statistically significant at the 95% confidence level ... 

The large value of the 6J3 coefficient immediately suggests a strong sjmergistic interaction between components 1 and 3. However, good statistical practice requires that before trying to interpret these results we obtain estimates of their errors. Since the runs were replicated, we can use the variances observed for the response values of each distinct run (last colimui of Table 7.1) to calculate a pooled estimate of the variance of an individual response. From there, by means of Eq. (5.30), we arrive at estimates of the standard errors of the coefficients. With these values we can finally write the complete equation for the fitted model ... 

The standard errors of the coefficients are usually based on the residual error. If the design contains a reasonable number of replicates this estimate can also be based on the standard error of the replicates. The residual based standard error of the Tth regression coefficient Sj can be easily transformed into replicate error based ones by the formula - MS / MSjj . In... 

The value of t for a given coefficient is equal to the value of the coefficient divided by the estimate of the standard error of that coefficient obtained from the regression data. It arises this way because the values of the dependent variable each contain a contribution from the random error in the reference laboratory, if another set of samples were to be used for the calibration, a different set of data would be used in the calibration, even if the data were taken from different aliquots of the same specimens. Then, when the calculations were performed, slightly different answers would result. It is possible to estimate from the data itself how much variation in the coefficients would result from this source. The estimate of the variation is called the standard error of the coefficient and Student s t is a coefficient divided by its standard error. 

Using a constant error for the measurement of the osmotic coefficient, estimate Pitzer s parameters as well as the standard error of the parameter estimates by minimizing the objective function given by Equation 15.1 and compare the results with the reported parameters. 

The ratios of the experimental to the calculated values for deposition efficiency for each generation are shown for each of the experiments in Figures 2-4. The error bars represent the standard error derived from the mean measured deposition fraction for the generation. A coefficient of variation was calculated for the measured deposition in each generation which was used to obtain an estimate of the standard error of the ratio. It was assumed that no additional variability in the ratio was introduced by the calculated deposition fraction. The mean ratio for all six sets of cast measurements is shown in Figure 5. The error bars in Figure 5 represent the standard error of the mean of the six experiments. 

A number of studies reported that several kinetic models can describe rate data well, when based on correlation coefficients and standard errors of the estimates [25,118,131,132]. Despite this, there often is no consistent relation between the equation which gives the best fit and the physicochemical and min-eralogical properties of the adsorbent(s) being studied. Another problem with some of the kinetic equations is that they are empirical and no meaningful rate parameters can be obtained. 

Whereas precision (Section 6.5) measures the reproducibility of data from replicate analyses, the accuracy (Section 6.4) of a test estimates how accurate the data are, that is, how close the data would represent probable true values or how accurate the analytical procedure is to giving results that may be close to true values. Precision and accuracy are both measured on one or more samples selected at random for analysis from a given batch of samples. The precision of analysis is usually determined by running duplicate or replicate tests on one of the samples in a given batch of samples. It is expressed statistically as standard deviation, relative standard deviation (RSD), coefficient of variance (CV), standard error of the mean (M), and relative percent difference (RPD). 

Data from Dzvinchuk and Lozinskii (88ZOR2167). Determined by H-NMR at 25°C (in CHCI3 at 20°C). The values (Xt)o were directly measured, rather than obtained from regression analysis. The standard errors of the slope are in the range 0.01-0.03. n = 8 (X = H, 3-NO2, 4-NO2, 4-Br, 4-Ph, 4-Me, 4-MeCONH, 4-MeO). Molar ratio. This is the coefficient of the Yukawa-Tsuno equation, not the correlation coefficient. The correlation coefficients lie in the range 0.997-0.999. 

Numbers in parentheses are the standard errors of the corresponding regression coefficients. 

Concentrations in yg/m all data dispersion normalized. Values of F and r are given for the overall equation and the standard error of each coefficient is given. 

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