Confidence Intervals Classical Approach

The confidence intervals for 6, and the variance 7 can be determined by assuming that the errors in measuring y are normally and independently distributed so that the distribution of the variable Vj which is defined as 

if the errors are normally and independently distributed, the sampling 

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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. 

The classical, frequentist approach in statistics requires the concept of the sampling distribution of an estimator. In classical statistics, a data set is commonly treated as a random sample from a population. Of course, in some situations the data actually have been collected according to a probability-sampling scheme. Whether that is the case or not, processes generating the data will be snbject to stochastic-ity and variation, which is a sonrce of uncertainty in nse of the data. Therefore, sampling concepts may be invoked in order to provide a model that accounts for the random processes, and that will lead to confidence intervals or standard errors. The population may or may not be conceived as a finite set of individnals. In some situations, such as when forecasting a fnture value, a continuous probability distribution plays the role of the popnlation. 

The classical or frequentist approach to probability is the one most taught in university conrses. That may change, however, becanse the Bayesian approach is the more easily nnderstood statistical philosophy, both conceptually as well as numerically. Many scientists have difficnlty in articnlating correctly the meaning of a confidence interval within the classical frequentist framework. The common misinterpretation the probability that a parameter lies between certain limits is exactly the correct one from the Bayesian standpoint. 

Confidence intervals nsing freqnentist and Bayesian approaches have been compared for the normal distribntion with mean p and standard deviation o (Aldenberg and Jaworska 2000). In particnlar, data on species sensitivity to a toxicant was fitted to a normal distribntion to form the species sensitivity distribution (SSD). Fraction affected (FA) and the hazardons concentration (HC), i.e., percentiles and their confidence intervals, were analyzed. Lower and npper confidence limits were developed from t statistics to form 90% 2-sided classical confidence intervals. Bayesian treatment of the uncertainty of p and a of a presupposed normal distribution followed the approach of Box and Tiao (1973, chapter 2, section 2.4). Noninformative prior distributions for the parameters p and o specify the initial state of knowledge. These were constant c and l/o, respectively. Bayes theorem transforms the prior into the posterior distribution by the multiplication of the classic likelihood fnnction of the data and the joint prior distribution of the parameters, in this case p and o (Fignre 5.4). 

In the rest of this chapter, it will be assumed that some general approach via classical confidence intervals, one-side for the case of noninferiority and two-sided for genuine equivalence) will be used. In fact, for true equivalence there are various technical controversies surrounding the use of two-sided confidence intervals (Mehring, 1993 Senn, 2001b). These will be taken up in Chapter 22. 

A perhaps more intuitive approach was suggested by Ortiz et al. It employs the predicted vs. reference regression line and the confidence intervals associated with the predictions. The mathematical details are outside scope of this section but interested readers are encouraged to consult reference and other papers referred to therein. The main conclusion is that the equations derived for classical univariate regression hold for multivariate regression and that all we need is to regress the concentrations of the analyte predicted by the PLS model for the calibrators against their reference concentrations. The equations are presented here only schematically. 

The Best Estimate Plus Uncertainties (BEPU) analysis is recommended by IAEA in addition or alternatively to the deterministic approach in the safety analysis of nuclear components and systems (IAEA 2009). The aim of the BEPU analysis is to determine a quantile of an output measure of interest (noted R in the following) with a certain level of confidence (usually the 95% quantile obtained with a 95% confidence, denoted Rg gf and to verify that this quantile is below an acceptable limit (acceptance criteria). In order to obtain this quantile, the uncertainty space of input parameters is sampled at random according to their combined probability distribution and a code calculation is performed for each sampled set of parameters. The number of code calculations is determined by the requirement to estimate a tolerance and confidence interval for the quantity of interest. Wilks formula (Wilks 1941) (or Wald formula (Wald 1943) when several criteria must be respected simultaneously) is used to determine the number of calculations to obtain the uncertainty bands and the associated quantile with a given confidence level. In classical BEPU analysis, there is no separation between the aleatory variables and the epistemic variables the epistemic variables, which are often model uncertainties, are generally modeled by uniform probability distributions within intervals provided by expert opinion and propagated in the same way that the aleatory variables by Monte Carlo simulation. 

Often we have data from several populations that we believe follow the same parametric distribution (such as the normal distribution), but may have different values of the parameter (such as the mean). The classical frequentist approach would be to analyze each population separately. The maximum likelihood estimate of the parameter for each population would be estimated from the sample from that population. Simultaneous confidence intervals such as Bonferroni, Tiikey, or Scheff6 intervals would be used for the difference between different population parameter values. These wider intervals would control the overall confidence level, and the overall significance level for testing the hypothesis that the differences between all the population parameters are zero. However, these intervals don t do anything about the parameter estimates themselves. 

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. 

It acknowledges that the true value of the damage parameter , is unknown but deterministic there is a (1 — a)% probability that this interval will contain the true value of the damage parameter. This confidence interval can be calculated at multiple levels of a, but this is not equivalent to estimating the probability distribution of 0,. In fact, in the classical statistics approach, 0, is an unknown but deterministic quantity and hence cannot be assigned a probability distribution. However, with the availability of more data, the confidence interval can be reconstructed, and its... 

Damage quantification using the classical statistics procedure yields a 90 % confidence interval of (24,509 NIm, 24,612 NIni). The Bayesian approach directly calculates the probability distribution of ki as explained in section Uncertainty in Damage Estimation. The overall uncertainty in diagnosis can be calculated as in section Overall Uncertainty in Diagnosis, and the corresponding probability density function is shown in Fig. 4. From Fig. 4, it can be seen that... 

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