Precision of estimation

In the work described earlier, the applicability of the Weibull model was further tested by assessing the precision of estimation [expressed by the CV defined as the standard error of estimates divided by the estimated value] and the relative accuracy of estimation of the model parameters (based on the difference of the estimates from the actual value, divided by the actual value). Regarding the precision of estimates, for data with SD = 2 the maximum CV value for Wo, b, and c was 13%, 52%, and 16%, respectively, whereas the corresponding numbers for data with SD = 4 were 33%, 151%, and 34%, respectively. As expected, the precision of the estimates decreases as the SD of the data increases, with the poorest precision for the b estimates and the best for the Wo estimates. Additionally, the maximum CV values were associated with low c values (c = 0.5). 

Equation 7.1 is one of the most important relationships in the area of experimental design. As we have seen in this chapter, the precision of estimated parameter values is contained in the variance-covariance matrix V the smaller the elements of V, the more precise will be the parameter estimates. As we shall see in Chapter 11, the precision of estimating the response surface is also directly related to V the smaller the elements of V, the less fuzzy will be our view of the estimated surface. 

Recalculate Example 8.2 for four experiments, the fourth experiment located at x, = 0. Assuming s is the same as for the three experiment case, what effect does this fourth experiment have on the precision of estimating Po and P, ... 

A powerful argument in favor of Group Additivity is that a start has already been made in the application of Group Additivity to condensed phases. In 1969, groups were derived (Ref 11) for the heat capacities of liquids at 298°K that improved the precision of estimation from + 4 to better than +1.5 cal/(mole-K). 

Blocking can also improve the precision of estimation if within-block variation is small compared to between-block variation. Sample-to-sample variability and variation in the amounts deposited on different arrays can be blocked effectively by hybridizing each replicate sample to a different array and considering the factor array as a blocking factor in modeling the data. 

Another important recent contribution is the provision of a good measurement of the precision of estimated reactivity ratios. The calculation of independent standard deviations for each reactivity ratio obtained by linear least squares fitting to linear forms of the differential copolymer equations is invalid, because the two reactivity ratios are not statistically independent. Information about the precision of reactivity ratios that are determined jointly is properly conveyed by specification of joint confidence limits within which the true values can be assumed to coexist. This is represented as a closed curve in a plot of r and r2- Standard statistical techniques for such computations are impossible or too cumbersome for application to binary copolymerization data in the usual absence of estimates of reliability of the values of monomer feed and copolymer composition data. Both the nonlinear least squares and the EVM calculations provide computer-assisted estimates of such joint confidence loops [15]. 

Bayes theorem (Bayes 1763 Box and Tiao 1973, 1992) permits estimation of the error covariance matrix S from a multiresponse data set, along with the parameter vector 0 of a predictive model. It is also possible, under further assumptions, to shorten the calculations by estimating 6 and I separately, as we do in the computer package GREGPLUS provided in Athena. We can then analyze the goodness of fit, the precision of estimation of parameters and functions of them, the relative probabilities of alternative models, and the choice of additional experiments to improve a chosen information measure. This chapter summarizes these procedures and their implementation in GREGPLUS details and examples are given in Appendix C. 

Replicate results show how variable the response is within treatments. They allow you to compare the differences among treatments in the context of the variability within treatments - you can do this via statistical tests such as analysis of variance (Chapter 41). Larger sample sizes tend to increase the precision of estimates of statistical parameters and increase the chances of showing a significant difference between treatments if one exists. For statistical reasons (weighting, ease of calculation, fitting data to certain tests), it is best to keep the number of replicates even. 

D-optimal criterion being the most usual, based on optimization of the overall precision of estimation of the coefficients of the modelJ This method and type of design is extremely flexible because ... 

We used Monte Carlo simulation to gain insight of the performances of the designs and the analysis methods. As initial explorations, the simulations focused on the accuracy and precision of estimating relative bioavailabihty. For potential regulatory applications, it is important to study the effect of error model (mis)specification and the influence of structural model misspecification. Of particular interest are the following issues ... 

The final conclusion is that the solutions that have been presented herein should encourage investigators to consider carrying out experiments in soil columns with sinusoidal inlet boundary conditions. With the ability to obtain repeated sets of data at the column outlet, one set of data for each period of the sine wave, should be an attractive and economical method for improving the precision of estimates of dispersion coefficients. 

The strength of various natural and anthropogenic aerosol sources is tabulated in Table 17 (SMIC, 1971). The precision of estimates is represented by the intervals given. For further details, the interested reader is referred to the original work. 

Altham, P.M.E. Improving the precision of estimation by fitting a model. Journal of the Royal Statistical Society, Series B 1984 46 118-119. 

Up to now, we have the results of only 5 experiments carried out under distinct conditions. At least one other experiment is needed, but there is no satisfactory way of adding only one experiment to the design, in terms of precision of estimate and orthogonality. 

It has been questioned whether the assessment of micronutrient intake over 4 days is representative of habitual diets however, increasing the recording period from 4 to 7 days is associated with only a marginal increase in the precision of estimated habitual nutrient intakes (Gay,... 

Alternatively, some practitioners find it easier to specify the relative precision of estimate desired in a study. Instead of setting p e, relative precision range ofp Rp. The proportion R is termed the relative precision in contrast to the precision e, which, for clarity, can be ... 

The MADC values are slightly different due to the changes in crude protein content. The precision of estimations of total and partial digestibility in the small intestine vs. large intestine respectively has been improved due to the new data, obtained mainly in France (INRA), the Netherlands (PR) and the USA. The tables are also more complete. 

The precision of estimates of degradation rate from accelerated degradation tests is much improved by ... 

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