Estimation And Confidence Intervals Pdf
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- 8.1: Basics of Confidence Intervals
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- Confidence intervals for robust estimates of measurement uncertainty
Uncertainties arising at different stages of a measurement process can be estimated using analysis of variance ANOVA on duplicated measurements.
8.1: Basics of Confidence Intervals
Uncertainties arising at different stages of a measurement process can be estimated using analysis of variance ANOVA on duplicated measurements. In some cases, it is also desirable to calculate confidence intervals for these uncertainties. This can be achieved using probability models that assume the measurement data are normally distributed.
In this case, robust ANOVA methods are able to provide variance estimates that are much closer to the parameters of the underlying normal distributions. A method using bootstrapping to calculate confidence intervals from robust estimates of variances is proposed and evaluated and is shown to work well when the number of outlying values is small. The method has been implemented in a visual basic program. The importance of the estimation of measurement uncertainty, including the uncertainty from sampling, by either modelling or empirical methods, is now well established [ 1 ].
One empirical method, recommended because of its cost-effectiveness and simple application, is the duplicate method. In this method, the processes of sampling and analysis are duplicated following pre-determined protocols applied to a number of different sampling targets. This allows the variances at each of these two stages to be separated and estimated by analysis of variance ANOVA.
In some cases, it is useful for the analyst or researcher to quantify the reliability of variance estimates, expressing this reliability as confidence intervals for the variance estimates from the ANOVA. For example, the researcher might estimate uncertainties of measurements on the same targets using more than one analytical method.
Confidence limits i. A further potential application is the comparison of analyte heterogeneity in materials, which can also be estimated using the duplicate method [ 2 ]. An example of this approach is provided by Lyn et al. This study demonstrated two important characteristics of uncertainties estimated by the duplicate method:. As would be expected, the width of the confidence intervals around the uncertainty estimates decreases as the number of sampling targets n increases.
However, many data sets obtained by experiment contain a small proportion of outlying values that may have a disproportionately large effect on these estimates. Robust ANOVA has been widely used for the estimation of measurement uncertainty, especially that arising from the primary sampling process [ 5 , 6 ]. Previously, there has not been a method available to calculate the confidence interval on a value of uncertainty estimated using robust ANOVA.
The overall purpose of this paper is to propose such a method, using a bootstrapping approach, applied with a new computer program CI-RANOVA, and to evaluate its performance for both normal and contaminated data. This is fine for routine use but too slow to permit the extensive simulations needed to accurately estimate coverage probabilities for the confidence intervals.
To enable this to be done, the method was also implemented in Matlab. Having two independent, in the sense that they were coded on different platforms by different programmers, implementations also enabled a validation of the CI-RANOVA coding, and to this end simulations were run using both versions. Validate the confidence limits produced by a bootstrapping method against those calculated by analytical mathematical formulas from the results of a classical ANOVA, using multiple simulations of normally distributed data.
The simplest form of the full three-tiered balanced experimental design is illustrated in Fig. Two samples are acquired from each of these targets by independent duplication of the sampling protocol.
These two samples are then treated individually and are both subjected to the same preparation procedures. Two test portions are then drawn from each of these test samples and analysed individually. The method is described in more detail in the Eurachem guide [ 1 ].
If the variability at each of these three levels can be assumed to be normally distributed, then a confidence interval at the bottom analysis level can be derived from a Chi-squared distribution, and Williams [ 7 ] provides a method of calculating approximate confidence limits at the top 2 target and sample levels. Details can be found in Graybill [ 8 ]. A circumflex denotes an estimate, e. These equations give confidence intervals for variances. To obtain intervals for the standard deviations, as reported below, one simply takes the square root of each of the limits of the intervals for the variances.
At the target and sample levels, the ANOVA estimates of variance are obtained by subtraction, and it is possible for the lower limits or even the variance estimates themselves to be negative. The standard practice of replacing negative estimates or confidence limits by zero, given that the true value of the variance cannot possibly be in the negative part of the interval, is followed here.
It can be seen that this single analytical outlier has had a large effect on both the variances calculated by classical ANOVA and the associated confidence limits. These are no longer representative of the bulk of the data and differ significantly from the input parameters used in the original simulation. The high value outlier in this example is the original value Ideally, values resulting from this type of mistake would not be included in uncertainty estimation [ 10 ], but practically they are not always identified and corrected or removed.
Its presence and the method used to treat it should be reported. The presence of such an outlying value may affect the inferences drawn from the experiment.
Less extreme outlying values can occur for a number of reasons, e. They might also be due to genuine variations in the property being measured and should not be ignored. One potential use of robust statistics in this application is to draw attention to the presence of outliers in experimental data. However, the calculation of confidence intervals in the context of a robust ANOVA is far from straightforward. It cannot be achieved via formulas based on assumed probability distributions, because it is precisely when these assumptions break down that we need to use the robust approach.
An alternative is to use bootstrapping methods. In this computer-based approach, a large number B of independent bootstrap samples are generated. A bootstrap sample is a data set, of the same size and structure as the observed one, generated by random sampling with replacement from the observed data set.
The statistic of interest e. Confidence intervals can then be derived from the empirical distribution of these results [ 11 ]. This is much more efficient than Excel when processing multiple data arrays, allowing a greater number of simulations to be created and analysed within a practical time frame, and in particular allowing the accurate estimation of coverage probabilities, requiring 50 simulations.
Versions of the two programs were produced independently by different researchers based in different institutions. Both programs calculate confidence limits on variances produced by classical ANOVA using a mathematical method based on Eqs. Confidence limits on the variances produced by robust ANOVA are estimated using a bootstrapping method. During development, a problem was encountered if a large number of bootstrap samples are generated on data containing outlying values.
The bootstrapping method generates samples by selecting means and differences at every level at random with replacement. Some of these bootstrap samples therefore contain fewer outliers than the original data set, while some contain more. When a large number e. It was found in practice that a small proportion of the bootstrap samples includes too many large outlying differences for the robust ANOVA to cope with.
For this reason, a winsorization process Fig. This is applied to the input data matrices after the initial robust analysis but prior to generating the bootstrap samples. The winsorization process brings in large outliers in a very similar way to how they are dealt with in the robust analysis, but using wider limits. The idea, due to [ 12 ], is that this will change the results of the robust analysis very little, because the outliers are still outside the limits used in that analysis, but it will limit the damage that they can cause when they occur in large numbers and in particular will avoid the breakdown of the robust analysis that can occur in this situation.
The limit used here, as suggested by Singh [ 12 ], is 1. This results in a spuriously high robust estimate of the target standard deviation, due to breakdown of the robust algorithm. Note that all other data values and standard deviations remain unchanged. The bootstrap samples at the lower levels were based on the differences, so no such correction is needed for the other two variances. The variances for each level are sorted into numerical order.
The 2. However, this simple approach is known not to work particularly well for skewed distributions, especially when the data being bootstrapped are small, and Efron and Tibshirani [ 11 ] recommend what they call the Bca method in this case. This is described in Fig. In the case of contaminated data, outliers were introduced at one of the three levels in each simulation.
These were created by randomly selecting a target or targets from the normally distributed data sets and adding a constant, in this case , to either all four measurements for that target, or both measurements for one of its samples, or one single analytical measurement similar to the method described by Ramsey et al. Tests were also performed to investigate the effects of different severities of the outlying value by varying the magnitudes of the added constant between 0 and The underlying calculations, including the averaging of the results of different simulations, were performed using variances, because variances from classical ANOVA are unbiased estimators of the population variances, whereas their square roots are not unbiased estimators of the corresponding standard deviations.
However, the tables in this section present the results as standard deviations because these have the same units as the original data and are more readily interpretable.
Validation of the bootstrapping method of estimating confidence limits at three different levels was performed by comparison with those produced by the mathematical method.
For each n, variances and confidence intervals were estimating using the Matlab program, with B set to This allowed 50 simulated data matrices to be analysed in a practical time frame to give coverage probabilities accurate to approximately 0. Coverage percentages were estimated by counting the number of times the CI contained the true value of the input parameter. This is a consequence of the approximations involved in the bias correction of the robust estimates in the case of the simultaneous estimation of both mean and variance.
This is consistent with previous findings where bootstrapping methods have been applied to estimate confidence intervals on variances See p. Further simulations were created with outlying values added to the data. Each outlying value was applied by randomly selecting a target without replacement , and adding to the original simulated value s to create one or more extreme outlier s. This table also shows coverage percentages calculated from 50 simulations using the Matlab program.
Note that in the simulations with contaminated data, the coverage probabilities are still calculated as the percentage of intervals that include the parameters used to generate the data before contamination.
This is consistent with earlier work [ 4 , 5 , 13 ]. These biases are most evident at the levels where outliers have been added, and also at any levels above.
For example, adding two sample outliers result in a positive bias at the sample and target levels. The same pattern is seen for all of the outlier scenarios.
This is not because the intervals are too narrow, but because the robust estimates have a slight upwards bias compared to the true values for the uncontaminated data. Overall, these results suggest that the bootstrapping method produces reasonable estimates of the confidence intervals of the robust standard deviations when there are a small number of outlying values i. When a larger number of outliers are present e. This value of n would be fairly unusual in practice, due to the financial cost of obtaining that quantity of duplicated measurements.
It is therefore appropriate to repeat the experiments for lower n values. The widths of the robust confidence intervals decrease as n increases Fig.
Skip to Main Content. A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity. Use of this web site signifies your agreement to the terms and conditions. Confidence Intervals Estimation for Reliability Data of Power Distribution Equipments Using Bootstrap Abstract: The uncertainty assessment in reliability indices is usually performed by the propagation of uncertainties in input data reliability failure rates and repair times for the estimated reliability indices using mathematical models, such as fuzzy sets. This analysis cannot be directly applied in distribution networks when there are significant errors between historical and predicted indices.
In Lesson 4. In real life, we don't typically have access to the whole population. In these cases we can use the sample data that we do have to construct a confidence interval to estimate the population parameter with a stated level of confidence. This is one type of statistical inference. The statistics professors at a university want to estimate the average statistics anxiety score for all of their undergraduate students. It would be too time consuming and costly to give every undergraduate student at the university their statistics anxiety survey. Instead, they take a random sample of 50 undergraduate students at the university and administer their survey.
During an election year, we see articles in the newspaper that state confidence intervals in terms of proportions or percentages. Investors in the stock market are interested in the true proportion of stocks that go up and down each week. Businesses that sell personal computers are interested in the proportion of households in the United States that own personal computers. Confidence intervals can be calculated for the true proportion of stocks that go up or down each week and for the true proportion of households in the United States that own personal computers. The procedure to find the confidence interval for a population proportion is similar to that for the population mean, but the formulas are a bit different although conceptually identical. While the formulas are different, they are based upon the same mathematical foundation given to us by the Central Limit Theorem.
Confidence intervals for robust estimates of measurement uncertainty
Items in EconStor are protected by copyright, with all rights reserved, unless otherwise indicated. Confidence interval estimation tasks and the economics of overconfidence. Experiments in psychology, where subjects estimate confidence intervals to a series of factual questions, have shown that individuals report far too narrow intervals. This has been interpreted as evidence of overconfidence in the preciseness of knowledge, a potentially serious violation of the rationality assumption in economics. Following these results a growing literature in economics has incorporated overconfidence in models of, for instance, financial markets.
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