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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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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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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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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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Confidence intervals for correlations when data are not normal.

Anthony J Bishara1, James B Hittner2

  • 1Department of Psychology, College of Charleston, 66 George Street, Charleston, SC, 29424, USA. BisharaA@cofc.edu.

Behavior Research Methods
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Summary
This summary is machine-generated.

The standard Fisher z

Keywords:
Confidence intervalCorrelationFisher zFisher z'NormalRobustr to z

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Area of Science:

  • Statistics
  • Data Analysis

Background:

  • The accuracy of confidence intervals for correlation coefficients is crucial in statistical analysis.
  • Nonnormal data can significantly compromise the reliability of traditional methods like the Fisher z' transformation.

Purpose of the Study:

  • To evaluate the performance of various confidence interval methods for correlation under nonnormal data conditions.
  • To identify robust alternatives to the Fisher z' method when normality assumptions are violated.

Main Methods:

  • Monte Carlo simulations were employed to compare 11 different confidence interval methods.
  • Methods included Fisher z', Spearman rank-order, Box-Cox transformation, rank-based inverse normal (RIN) transformation, and bootstrap techniques.
  • The study assessed interval accuracy (coverage) and length across varying degrees of nonnormality and sample sizes.

Main Results:

  • The Fisher z' confidence interval demonstrated significant inaccuracy with nonnormal data, showing coverage as low as 68% for a 95% interval.
  • Increasing sample size sometimes exacerbated the inaccuracy of the Fisher z' interval.
  • Spearman rank-order and RIN transformation methods proved universally robust to nonnormality.
  • An observed imposed bootstrap method showed good coverage but often produced overly long intervals.

Conclusions:

  • Nonnormality in sample data strongly justifies the use of alternatives to the Fisher z' confidence interval.
  • Spearman rank-order and RIN transformation methods are recommended for robust correlation inference with nonnormal data.
  • Researchers should consider data distribution when selecting a confidence interval method for correlation.