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Related Concept Videos

Interpretation of Confidence Intervals01:19

Interpretation of Confidence Intervals

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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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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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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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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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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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A goodness-of-fit test is conducted to determine whether the observed frequency values are statistically similar to the frequencies expected for the dataset. Suppose the expected frequencies for a dataset are equal such as when predicting the frequency of any number appearing when casting a die. In that case, the expected frequency is the ratio of the total number of observations (n)  to the number of categories (k).
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Likelihood Confidence Intervals for Misspecified Cox Models.

Yongwu Shao1, Xu Guo1

  • 1Gilead Sciences, Foster City, California, USA.

Statistics in Medicine
|March 1, 2026
PubMed
Summary
This summary is machine-generated.

The robust Wald confidence interval (CI) for Cox models can be unreliable with rare events. This study introduces a novel robust likelihood confidence interval for improved accuracy in such scenarios.

Keywords:
asymptotic theorylikelihood ratio testmodel misspecificationrobustnesssurvival data

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Area of Science:

  • Biostatistics
  • Survival Analysis
  • Statistical Modeling

Background:

  • The robust Wald confidence interval (CI) is widely used for Cox models, especially with model misspecification or applied weights.
  • However, the Wald CI performs poorly with few events, common in rare event studies or highly effective treatments, leading to counter-intuitive results.
  • Existing likelihood CIs for Cox models lack a robust version, and standard software may incorrectly provide regular versions even when robustness is requested.

Purpose of the Study:

  • To develop and evaluate a robust likelihood confidence interval (CI) for the Cox model.
  • To address the limitations of the robust Wald CI and the absence of a robust likelihood CI in standard statistical software.
  • To provide a more accurate and reliable CI for survival data, particularly when dealing with rare events or small sample sizes.

Main Methods:

  • Demonstrated that the likelihood ratio test statistic for the Cox model converges to a weighted chi-square distribution under misspecification.
  • Derived the robust likelihood CI by inverting this robust likelihood ratio test.
  • Evaluated the performance of the proposed CIs using simulation studies and real-world data.

Main Results:

  • The proposed robust likelihood confidence intervals demonstrate improved performance compared to the Wald CI, especially in scenarios with few events.
  • Simulation studies confirmed the superior accuracy of the new CIs in matching nominal coverage probabilities.
  • The method was successfully applied to real data from an HIV prevention trial, showing its practical utility.

Conclusions:

  • The developed robust likelihood CI offers a more reliable alternative to the robust Wald CI for Cox models, particularly in challenging data situations.
  • This work fills a critical gap in statistical methodology by providing a robust likelihood CI for the Cox model.
  • A companion R package, "CoxLikelihood," is available, facilitating the application of these new methods by researchers.