Statistics confidence interval

Influent uncertainty Closely related to the preceding point is the fact that the composition of the influent is highly influenced by constraints, which may vary in a random manner depending on human industrial or environmental activities [72]. Again, without suitable on-line sensors to measure these variations, only estimates based on statistical confidence intervals may be used in some cases. 

If a study has negative findings, it should be carefully evaluated with respect to, for example, the power of the study, its concordance or discordance with other related studies, and differences or similarities in study design or end-points with related studies. Results of different studies can be evaluated by comparing statistical confidence intervals. Studies with lower power will tend to yield wider confidence intervals the magnitude of the risks must be considered. Studies with similar risks are important even if statistical significance is not present in all studies. 

An example of a recovery control chart is shown in Figure 4.7. The mean recovery of individual measurements is represented by the centreline. The upper warning limit (UWL) and the lower warning limit (LWL) are calculated as plus/minus two standard deviations (mean recovery + 2s) and correspond to a statistical confidence interval of 95 percent. The upper control limit (UCL) and the lower control limit (LCL) are calculated as plus/minus three standard deviations (mean recovery 3s), and represent a statistical confidence interval of 99 percent. Control limits vary from laboratory to laboratory as they depend on the analytical procedure and the skill of the analysts. 

Inference effects relate to systematic and random errors in modelling inducing problems of drawing extrapolations or logic deductions from small statistical samples, from animal data or experimental data onto humans or from large doses to small doses, etc. All of these are usually expressed through statistical confidence intervals ... 

Veldhuis JD, Evans WS, Johnson ML. Complicating effects of highly correlated model variables on nonlinear least-squares estimates of unique parameter values and their statistical confidence intervals Estimating basal secretion and neurohormone half-life by deconvolution analysis. Methods Neurosci 1995 28 130-8. 

Statistical methods in effluent toxicity evaluations enable the investigator to quantify the observed exposure-response relation, with reference to the desired end point. The resulting statistical confidence interval in the data may then be used to ensure test reproducibility, to compare multiple test results, and for regulatory decision-making. 

The analysis of experimental data shows that the average value 111 g/m2 of all corrosivity data (improved by rejecting outliers) corresponds to the value 140 40 g/m2 indicated in the standard. For the evaluation of the expanded combined uncertainty U with factor k=2 the corrosivity measurement gives the value of 215 g/m2 (at 95% confidence). It means that our data uncertainty is five-times higher than that specified in the standard as the data scattering interval 40 g/m2 and seven times as wide compared the statistic confidence interval in our own experimental data corrosivity (Table 2a and 2b). The main components of the combined uncertainty are mass loss and surface area determination. 

Moving-window PCA (MWPCA) has been proposed to monitor time-varying processes where both the PCA model and the statistical confidence intervals of the monitoring charts are adapted [316]. MWPCA provides recursive adaptation within the moving window to adapt the mean and variance of process variables, the correlation matrix, and the PCA model by recomputing the decomposition. MWPCA is compared to recursive... 

Two points merit emphasis in the above exercise a), The statistical confidence Interval for the outcome s based on S and its SE (using a 2-sided Student s-t) SE but not S is used also for the estimation of Lp. b) The confidence Interval, and Lj, and Lp (and its upper limit) are correct for normally distributed random errors. Faired T, B comparisons and a moderate number of replicates tend to make these assumptions reasonably good this is an important precaution, given the widely varying blank distributions of such difficult measurements. Perhaps the most important consequence of the paired comparison InjJuced, symmetry, is that the expected value for the null signal [B - B ] will be zero -- ie, unbiased. Systematic error bounds, some deeper implications of paired... 

This section discusses the calculation of the uncertainty in the damage quantification stage, by estimating the uncertainty in the value of damage parameter. The classical statistics-based approach calculates statistical confidence intervals on the value of damage parameter, while the Bayesian statistics-based approach directly calculates the probability distribution of the value of the damage parameter. 

Instead, for establishing of the statistical confidence interval, we give up on testing the Hg hypothesis and we write, in general, that... 

The distribution of the /-statistic (x — /ji)s is symmetrical about zero and is a function of the degrees of freedom. Limits assigned to the distance on either side of /x are called confidence limits. The percentage probability that /x lies within this interval is called the confidence level. The level of significance or error probability (100 — confidence level or 100 — a) is the percent probability that /X will lie outside the confidence interval, and represents the chances of being incorrect in stating that /X lies within the confidence interval. Values of t are in Table 2.27 for any desired degrees of freedom and various confidence levels. 

The F statistic, along with the z, t, and statistics, constitute the group that are thought of as fundamental statistics. Collectively they describe all the relationships that can exist between means and standard deviations. To perform an F test, we must first verify the randomness and independence of the errors. If erf = cr, then s ls2 will be distributed properly as the F statistic. If the calculated F is outside the confidence interval chosen for that statistic, then this is evidence that a F 2. 

The probabilistic nature of a confidence interval provides an opportunity to ask and answer questions comparing a sample s mean or variance to either the accepted values for its population or similar values obtained for other samples. For example, confidence intervals can be used to answer questions such as Does a newly developed method for the analysis of cholesterol in blood give results that are significantly different from those obtained when using a standard method or Is there a significant variation in the chemical composition of rainwater collected at different sites downwind from a coalburning utility plant In this section we introduce a general approach to the statistical analysis of data. Specific statistical methods of analysis are covered in Section 4F. 

The equation for the test (experimental) statistic, fexp, is derived from the confidence interval for p,... 

The "feedback loop in the analytical approach is maintained by a quality assurance program (Figure 15.1), whose objective is to control systematic and random sources of error.The underlying assumption of a quality assurance program is that results obtained when an analytical system is in statistical control are free of bias and are characterized by well-defined confidence intervals. When used properly, a quality assurance program identifies the practices necessary to bring a system into statistical control, allows us to determine if the system remains in statistical control, and suggests a course of corrective action when the system has fallen out of statistical control. 

Effects are shown with their 95% confidence intervals. Effects that are similar than their interval are not statistically significant and ate shown with an asterisk. 

The usual practice in these appHcations is to concentrate on model development and computation rather than on statistical aspects. In general, nonlinear regression should be appHed only to problems in which there is a weU-defined, clear association between the independent and dependent variables. The generalization of statistics to the associated confidence intervals for nonlinear coefficients is not well developed. 

First, the parameter estimate may be representative of the mean operation for that time period or it may be representative of an extreme, depending upon the set of measurements upon which it is based. This arises because of the normal fluc tuations in unit measurements. Second, the statistical uncertainty, typically unknown, in the parameter estimate casts a confidence interv around the parameter estimate. Apparently, large differences in mean parameter values for two different periods may be statistically insignificant. 

In the introduction to this section, two differences between "classical" and Bayes statistics were mentioned. One of these was the Bayes treatment of failure rate and demand probttbility as random variables. This subsection provides a simple illustration of a Bayes treatment for calculating the confidence interval for demand probability. The direct approach taken here uses the binomial distribution (equation 2.4-7) for the probability density function (pdf). If p is the probability of failure on demand, then the confidence nr that p is less than p is given by equation 2.6-30. 

Aggregation The statistical combination of several data points to form a single data point and confidence interval. 

Typical-basis The typical property value is an average value. No statistical assurance is associated with this basis. See A-basis B-basis C-basis population confidence interval. 

Up to now (1971) only a limited number of reaction series have been completely worked out in our laboratories along the lines outlined in Sec. IV. In fact, there are rather few examples in the literature with a sufficient number of data, accuracy, and temperature range to be worth a thorough statistical treatment. Hence, the examples collected in Table III are mostly from recent experimental work and the previous ones (1) have been reexamined. When evaluating the results, the main attention should be paid to the question as to whether or not the isokinetic relationship holds i.e., to the comparison of standard deviations of So and Sqo The isokinetic temperature /J is viewed as a mere formal quantity and is given no confidence interval. Comparison with previous treatments is mostly restricted to this value, which has generally and improperly been given too much atention. 

The Production Department was not amused, because lower values had been expected. Quality Control was blamed for using an insensitive, unse-lective, and imprecise test, and thereby unnecessarily frightening top management. This outcome had been anticipated, and a better method, namely polarography, was already being set up. The same samples were run, this time in duplicate, with much the same results. A relative confidence interval of 25% was assumed. Because of increased specificity, there were now less doubts as to the amounts of this particular heavy metal that were actually present. To rule out artifacts, the four samples were sent to outside laboratories to do repeat tests with different methods X-ray fluorescence (XRFi °) and inductively coupled plasma spectrometry (ICP). The confidence limits were determined to be 10% resp. 3%. Figure 4.23 summarizes the results. Because each method has its own specificity pattern, and is subject to intrinsic artifacts, a direct statistical comparison cannot be performed without first correcting the apparent concentrations in order to obtain presumably true... 

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