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A confidence interval is a better estimate of the population than a point estimate, as it uses a range of values from a sample instead of a single value.
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The confidence interval is the range of values around the mean that contains the true mean. It is expressed as a probability percentage. The interpretation of a 95% confidence interval, for instance, is that the statistician is 95% confident that the true mean falls within the interval. The upper and lower limits of this range are known as confidence limits. The confidence limits for the true mean are estimated from the sample's mean, the standard deviation, and the statistical factor...
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The accurate values of population parameters such as population proportion, population mean, and population standard deviation (or variance) are usually unknown. These are fixed values that can only be estimated from the data collected from the samples. The estimates of each of these parameters are sample proportion, the sample mean, and sample standard deviation (or variance). To obtain the values of these sample statistics, data are required that have particular distribution and central...
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The confidence coefficient is also known as the confidence level or degree of confidence. It is the percent expression for the probability, 1-α, that the confidence interval contains the true population parameter assuming that the confidence interval is obtained after sufficient unbiased sampling; for example, if the CL = 90%, then in 90 out of 100 samples the interval estimate will enclose the true population parameter. Here α is the area under the curve, distributed equally under...
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A point estimate of the population mean is obtained from a single sample. Such a point estimate does not represent a population well because it needs to account for variability in the population. Single point estimate can also be biased despite the sample being selected randomly. Thus, a point estimate is often unreliable. A confidence interval is needed to reduce this unreliability.
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An unbiased point estimate is often insufficient to predict a population estimate, such as population mean or population proportion. In this scenario, a confidence interval is used. A confidence interval is an estimate similar to a  sample proportion. However, unlike the point estimate which is a single value, the confidence interval  contains a range of values. These values have lower and upper limits, known as confidence limits, and can be designated as L1 and L2, respectively.
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Capturing the underlying distribution in meta-analysis: Credibility and tolerance intervals.

Michael T Brannick1, Kimberly A French2, Hannah R Rothstein3

  • 1Psychology Department, University of South Florida, Tampa, Florida, USA.

Research Synthesis Methods
|February 5, 2021
PubMed
Summary

Tolerance intervals in meta-analysis aim to capture population effect sizes. While no single method consistently achieved desired coverage, prediction and bootstrap intervals showed promise under specific conditions for better bracket estimation.

Keywords:
Monte Carlo simulationcredibility intervalprediction intervalstandardized mean differencetolerance interval

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Area of Science:

  • Statistics
  • Biostatistics
  • Meta-analysis

Background:

  • Tolerance intervals are crucial for estimating population distribution coverage from sample data.
  • In random-effects meta-analysis, these intervals should encompass specified proportions of population effect sizes.
  • Accurate tolerance intervals are essential for robust meta-analytic conclusions.

Purpose of the Study:

  • To investigate the coverage accuracy of five different tolerance interval estimators in random-effects meta-analysis.
  • To evaluate the performance of Schmidt-Hunter credibility intervals, prediction intervals, content tolerance intervals, and bootstrap tolerance intervals.
  • To provide practical recommendations for selecting appropriate tolerance intervals in meta-analysis.

Main Methods:

  • Monte Carlo simulation was employed to assess the coverage properties of five tolerance interval estimators.
  • The study examined interval performance across various conditions, including the number of primary studies and sample sizes.
  • Real-world meta-analysis data were used to demonstrate the application and implications of the findings.

Main Results:

  • No single tolerance interval estimator achieved nominal coverage rates under all simulated conditions.
  • Prediction intervals performed well except with a small number of primary studies (<30).
  • Bootstrap tolerance intervals demonstrated near-nominal coverage with sufficient primary studies (30+) and large sample sizes (N ≈ 70), though slightly exceeding coverage with numerous large-sample studies.

Conclusions:

  • Tolerance intervals effectively incorporate estimation error for bracketing population effect sizes.
  • Prediction and bootstrap tolerance intervals show potential for improved coverage in meta-analysis under certain conditions.
  • The choice of tolerance interval estimator should consider the number of studies and sample sizes within the meta-analysis.