Precision standard errors

The practices described by the method provide instructions for sampling coal from beneath the exposed surface of the coal at a depth (approximately 24 in., 61 cm) where drying and oxidation have not occurred. The purpose is to avoid collecting increments that are significantly different from the majority of the lot of coal being sampled due to environmental effects. However, samples of this type do not satisfy the minimum requirements for probability sampling and, as such, cannot be used to draw statistical inferences such as precision, standard error, or bias. Furthermore, this method is intended for use only when sampling by more reliable methods that provide a probability sample is not possible. 

Precision. Standard errors between 2 and 5% are usually obtained. 

PM parameters are most often estimated by assuming asymptotic normality. Often the standard errors or confidence intervals for PM parameters are not accurately estimated because distributions are heavily tailed or skewed or contain influence data. In addition, when sample sizes are small, it is impossible to obtain accurate or precise standard errors of parameters. 

At this point is worthwhile commenting on the computer standard estimation errors of the parameters also shown in Table 16.24. As seen in the last four estimation runs we are at the minimum of the LS objective function. The parameter estimates in the run where we optimized four only parameters (K2, kt, K k3) have the smallest standard error of estimate. This is due to the fact that in the computation of the standard errors, it is assumed that all other parameters are known precisely. In all subsequent runs by introducing additional parameters the overall uncertainty increases and as a result the standard error of all the parameters increases too. 

Precision. The uncertainty of calibration and prediction of unknown concentrations are expressed by the standard error of calibration (SEC), defined as... 

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). 

For fitting such a set of existing data, a much more reasonable approach has been used (P2). For the naphthalene oxidation system, major reactants and products are symbolized in Table III. In this table, letters in bold type represent species for which data were used in estimating the frequency factors and activation energies contained in the body of the table. Note that the rate equations have been reparameterized (Section III,B) to allow a better estimation of the two parameters. For the first entry of the table, then, a model involving only the first-order decomposition of naphthalene to phthalic anhydride and naphthoquinone was assumed. The parameter estimates obtained by a nonlinear-least-squares fit of these data, are seen to be relatively precise when compared to the standard errors of these estimates, s0. The residual mean square, using these best parameter estimates, is contained in the last column of the table. This quantity should estimate the variance of the experimental error if the model adequately fits the data (Section IV). The remainder of Table III, then, presents similar results for increasingly complex models, each of which entails several first-order decompositions. 

Substitution of the typical variances and covariance into Equation (19) suggests that the for tho MC-ICPMS measurements of Mg in solutions is on the order of +0.010%o. This is regarded as an internal precision for an individual solution measurement. We note, however, that the reported measurements represent averages of several replicate analyses of the same solution and so more realistic assessments of the internal precision for A Mg data presented here would be obtained from the imcertainties in the means (standard errors). For example, four analyses of the same solution yields a standard error for A Mg of +0.005%o (this is stiU regarded as an internal precision because the effects of column chemistry and sample dissolution are not included). No attempt has been made here to review all of the raw data sets to calculate standard errors for each datum in Table 1. However, the distribution of data indicates that +0.010%o Icj is an overestimate of the internal precision of A Mg values and that a more realistic imcertainty is closer to a typical standard error, which in most cases will be < +0.005%o (since the number of replicates is usually >4, e.g., Galy et al. 2001). 

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). 

Measurement of precision. Measurement of data quality is valuable for both the analyst and the data user. Least-squares curve-of-best-fit statistical programs usually provide some information on precision (correlation coefficient, standard error of estimate). However, these are not sufficiently quantitative and often overstate the quality parameters of the data. 

This example shows that the standard deviation of the sampling distribution is less than that of the population. In fact, this reduction in the variability is related to the sample size used to calculate the sample means. For example, if we repeat the sampling experiment, but this time based on 15 rather than 10 random samples, the resulting standard deviation of the sampling is 0.159, and on 25 random samples it is 0.081. The precise relationship between the population standard deviation a and the standard error of the mean is ... 

It is not possible at this stage to say precisely what we mean by small and large in this context, we need the concept of the confidence interval to be able to say more in this regard and we will cover this topic in the next chapter. For the moment just look upon the standard error as an informal measure of precision high values mean low precision and vice versa. Further if the standard error is small, it is likely that our estimate x is close to the true mean, p,. If the standard error is large, however, there is no guarantee that we will be close to the true mean. 

So we need to have some measure of precision and reliability and this is provided by the standard error ofxj — X2. Again we have a formula for this ... 

More generally, whatever statistic we are interested in, there is always a formula that allows us to calculate its standard error. The formulas change but their interpretation always remains the same a small standard error is indicative of high precision, high reliability. Conversely a large standard error means that the observed value of the statistic is an unreliable estimate of the true (population) value. It is also always the case that the standard error is an estimate of the standard deviation of the list of repeat values of the statistic that we would get were we to repeat the sampling process, a measure of the inherent sampling variability. 

Note the role played by the standard error in the formula for the confidence interval. We have previously seen that the standard error of the mean provides an indirect measure of the precision with which we have calculated the mean. The confidence interval has now translated the numerical value for the standard error into something useful in terms of being able to make a statement about where jl lies. A large standard error will lead to a wide confidence interval reflecting the imprecision and resulting poor information about the value of jjl. In contrast a... 

Assume that we have decided on the best measure for the treatment effect. If this is expressed as a difference, for example, in the means, then there will be an associated standard error measuring the precision of that difference. If the... 

For each experimental design used for ruggedness results are calculated separately for each level tested. The usual results estimated by these designs are a main effect for each factor, interaction effects for all combinations of factors and a standard error that estimates the precision achieved throughout the study. 

The standard error reflects the statistical relevance of all the main effects (i.e. a main effect with a smaller value than the standard error is not statistically relevant). In this instance a main effect must be larger than -0.7 to be considered a real effect and not just a reflection of the overall precision of the method. The results for each factor are given in Table 5.18. The largest effect is around 3% and is due to the change in the acid type used to control the pH of the mobile phase. All observed effects were unlikely to cause a lack of method ruggedness as no effect caused a critical reduction in the plate count. 

Finally, the precision of the multivariate calibration method can be estimated from the standard error of prediction (SEP) (also called standard error of performance), corrected for bias ... 

XRF analyses of Silver Lake and Sycan Marsh obsidian source samples suggested the possibility that the two sources could be differentiated based on small differences in strontium concentrations. However, when the standard error for strontium was taken into account, both groups overlapped at one standard deviation. Because of higher instrumental detection limits for strontium, NAA could not discriminate between the two sources. LA-ICP-MS analyses were conducted to determine if the sensitivity and precision of this analytical technique was sufficient to confirm the existence of the two compositional... 

The automatic apparatus consists of a viscosimeter and phototran-sistorized sensing devices mounted in a precision thermostat ( 0.005°C) connected to a cooled prethermostat ( 0.1°C). The base apparatus is commercially available (Schott Viscotimer, Jenaer Glaswerk, Schott Gen., Mainz), but the viscosimeter control functions and the time measurements are performed by using an electronic computer-controlled interface. This modification enables one to follow slow reactions and to reduce standard errors on the outflow times to 2 msec. The final results are evaluated numerically by an on-line computer-plotter system. 

Precision measure of the maximum random error or deviation of a single observation. It may be expressed as the standard error or a multiple thereof, depending on the probability level desired. 

Another scale of measurement of precision is standard error of mean (.M) which is the ratio of the standard deviation to the square root of number of measurements ( ). 

The larger the sample size, the more likely it is that the sample mean is a good representation of the population mean If the sample were large enough to contain the entire population, the sample mean would be identical to the population mean. In a clinical trial, the sample size is precisely known. Knowledge of the sample SD and the sample size (N) facilitates precise calculation of the sample standard error of the mean (SEM) ... 

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