Confidence Interval

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A Confidence Interval is a 2-Tuple (Event Interval, Confidence Level) that estimates the likelihood of an event occurring within the interval.

References

2021

  • (Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Confidence_interval Retrieved:2021-5-28.
    • In statistics, a confidence interval (CI) is a type of estimate computed from the statistics of the observed data. This gives a range of values for an unknown parameter (for example, a population mean). The interval has an associated confidence level that gives the probability with which the estimated interval will contain the true value of the parameter. The confidence level is chosen by the investigator. For a given estimation in a given sample, using a higher confidence level generates a wider (i.e., less precise) confidence interval. In general terms, a confidence interval for an unknown parameter is based on sampling the distribution of a corresponding estimator.

      This means that the confidence level represents the theoretical long-run frequency (i.e., the proportion) of confidence intervals that contain the true value of the unknown population parameter. In other words, 90% of confidence intervals computed at the 90% confidence level contain the parameter, 95% of confidence intervals computed at the 95% confidence level contain the parameter, 99% of confidence intervals computed at the 99% confidence level contain the parameter, etc.

      The confidence level is designated before examining the data. Most commonly, a 95% confidence level is used. However, other confidence levels, such as 90% or 99%, are sometimes used.

      Factors affecting the width of the confidence interval include the size of the sample, the confidence level, and the variability in the sample. A larger sample will tend to produce a better estimate of the population parameter, when all other factors are equal. A higher confidence level will tend to produce a broader confidence interval.

      Many confidence intervals are of the form [math]\displaystyle{ (t-c\sigma_T, t+c\sigma_T) }[/math] , where [math]\displaystyle{ t }[/math] is the realization of the dataset, c is a constant and [math]\displaystyle{ \sigma_T }[/math] is the standard deviation of the dataset.

      Another way to express the form of confidence interval is a set of two parameters: (point estimate – error bound, point estimate + error bound), or symbolically expressed as (–EBM, (EBM), where (point estimate) serves as an estimate for m (the population mean) and EBM is the error bound for a population mean.

      The margin of error (EBM) depends on the confidence level.

      A rigorous general definition:

      Suppose a dataset [math]\displaystyle{ x_1, \ldots, x_n }[/math] is given, modeled as realization of random variables [math]\displaystyle{ X_1,\ldots,X_n }[/math] . Let [math]\displaystyle{ \theta }[/math] be the parameter of interest, and [math]\displaystyle{ \gamma }[/math] a number between 0 and 1. If there exist sample statistics [math]\displaystyle{ L_n=g(X_1,\ldots,X_n) }[/math] and [math]\displaystyle{ U_n = h(X_1,\ldots,X_n) }[/math] such that:

      [math]\displaystyle{ P(L_n\lt \theta\lt U_n)=\gamma }[/math] for every value of [math]\displaystyle{ \theta }[/math] then [math]\displaystyle{ (l_n,u_n) }[/math] , where [math]\displaystyle{ l_n=g(x_1,\ldots,x_n) }[/math] and [math]\displaystyle{ u_n=h(x_1,\ldots,x_n) }[/math] , is called a [math]\displaystyle{ \gamma \times 100\% }[/math] confidence interval for [math]\displaystyle{ \theta }[/math] . The number [math]\displaystyle{ \gamma }[/math] is called the confidence level.


2017

    • ABSTRACT: Biomedical research is seldom done with entire populations but rather with samples drawn from a population. Although we work with samples, our goal is to describe and draw inferences regarding the underlying population. It is possible to use a sample statistic and estimates of error in the sample to get a fair idea of the population parameter, not as a single value, but as a range of values. This range is the confidence interval (CI) which is estimated on the basis of a desired confidence level. Calculation of the CI of a sample statistic takes the general form: CI = Point estimate ± Margin of error, where the margin of error is given by the product of a critical value (z) derived from the standard normal curve and the standard error of point estimate. Calculation of the standard error varies depending on whether the sample statistic of interest is a mean, proportion, odds ratio (OR), and so on. The factors affecting the width of the CI include the desired confidence level, the sample size and the variability in the sample. Although the 95% CI is most often used in biomedical research, a CI can be calculated for any level of confidence. A 99% CI will be wider than 95% CI for the same sample. Conflict between clinical importance and statistical significance is an important issue in biomedical research. Clinical importance is best inferred by looking at the effect size, that is how much is the actual change or difference. However, statistical significance in terms of P only suggests whether there is any difference in probability terms. Use of the CI supplements the P value by providing an estimate of actual clinical effect. Of late, clinical trials are being designed specifically as superiority, non-inferiority or equivalence studies. The conclusions from these alternative trial designs are based on CI values rather than the P value from intergroup comparison.

2015

  • (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/confidence_interval Retrieved:2015-2-28.
    • In statistics, a confidence interval (CI) is a type of interval estimate of a population parameter. It is an observed interval (i.e. it is calculated from the observations), in principle different from sample to sample, that frequently includes the parameter of interest if the experiment is repeated. How frequently the observed interval contains the parameter is determined by the confidence level or confidence coefficient. More specifically, the meaning of the term "confidence level" is that, if confidence intervals are constructed across many separate data analyses of repeated (and possibly different) experiments, the proportion of such intervals that contain the true value of the parameter will match the confidence level; this is guaranteed by the reasoning underlying the construction of confidence intervals. Whereas two-sided confidence limits form a confidence interval, their one-sided counterparts are referred to as lower or upper confidence bounds. ...


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2023

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