Estimated variance

Finally variance estimates of the parameters are calculated by means of the Cj/ elements, as in the following relationships (where var and cov represent variance and covariance, respectively). 

There is a plethora of model adequacy tests that the user can employ to decide whether the assumed mathematical model is indeed adequate. Generally speaking these tests are based on the comparison of the experimental error variance estimated by the model to that obtained experimentally or through other means. 

If the predicted values x / i and Et/t-i are already computed, the minimum variance estimates of the states are obtained as the solution of the minimization problem... 

If the errors are normally distributed, the OLS estimates are the maximum likelihood estimates of 9 and the estimates are unbiased and efficient (minimum variance estimates) in the statistical sense. However, if there are outliers in the data, the underlying distribution is not normal and the OLS will be biased. To solve this problem, a more robust estimation methods is needed. 

Implementation of a covariance structure into this numerical scheme is described in Tarantola and Valette (1982). In essence, an a priori covariance structure is assumed for the whole set of observations and parameters, which should be tightened by iterative refinements since we are still dealing with a minimum variance estimate. 

I.irgest variance estimate smallest variance estimate... 

Differences in calibration graph results were found in amount and amount interval estimations in the use of three common data sets of the chemical pesticide fenvalerate by the individual methods of three researchers. Differences in the methods included constant variance treatments by weighting or transforming response values. Linear single and multiple curve functions and cubic spline functions were used to fit the data. Amount differences were found between three hand plotted methods and between the hand plotted and three different statistical regression line methods. Significant differences in the calculated amount interval estimates were found with the cubic spline function due to its limited scope of inference. Smaller differences were produced by the use of local versus global variance estimators and a simple Bonferroni adjustment. 

A second reason is that the use of local variance versus global variance can result in markedly different bands. The separate calculations of variance at levels throughout the range of standards produces a wider confidence interval at lower values as seen in Kurtz method. If a common variance is used as the variance estimate then a lower confidence interval is calculated at each point as is probably the case in Wegscheider s method. 

In Section 6.2, the standard uncertainty of the parameter estimate was obtained by taking the square root of the product of the purely experimental uncertainty variance estimate, and the matrix (see Equation 6.3). A single number was... 

Considering the relatively less computational effort required to solve problem (7.20), the value of N is typically chosen to be quite larger than N in order to obtain an accurate estimation of (Verweij et al., 2003). Since x is a feasible point to the true problem, we have vn > v. Hence, vn is a statistical upper bound to the true problem with a variance estimated by Equation 7.21 ... 

The univariate response data on all standard biomarker data were analysed, ineluding analysis of variance for unbalaneed design, using Genstat v7.1 statistical software (VSN, 2003). In addition, a-priori pairwise t-tests were performed with the mean reference value, using the pooled variance estimate from the ANOVA. The real value data were not transformed. The average values for the KMBA and WOP biomarkers were not based on different flounder eaptured at the sites, but on replicate measurements of pooled liver tissue. The nominal response data of the immunohistochemical biomarkers (elassification of effects) were analysed by means of a Monte... 

To cany out a Lagrange multiplier test of the hypothesis of equal variances, we require the separate and common variance estimators based on the restricted slope estimator. This, in turn, is the pooled least squares estimator. For the combined sample, we obtain... 

In order to compute a two step GLS estimate, we can use either the original variance estimates based on the separate least squares estimates or those obtained above in doing the LM test. Since both pairs are consistent, both FGLS estimators will have all of the desirable asymptotic properties. For our estimator, we... 

The mean and variance estimates are obtained from expressions ... 

Insignificant, there is no reason to doubt lack of fit of the obtained regression model, and mean sums of squares of experimental error and lack of fit can be used for variance estimate a2. 

Hence, based on calculated effects of individual factors and interactions from relation (2.70), we obtain a variance estimate, which is further analyzed by the analysis of variance method. Here, one should remember once again the difference between two cases in estimating residual variance ... 

Variance estimates based on f=l degrees of freedom are calculated from effects from Eq. (2.70) ... 

---