Skip to main content

Main menu

  • Home
  • Content
    • Current issue
    • Past issues
    • Early releases
    • Collections
    • Sections
    • Blog
    • Infographics & illustrations
    • Podcasts
    • COVID-19 Articles
  • Authors
    • Overview for authors
    • Submission guidelines
    • Submit a manuscript
    • Forms
    • Editorial process
    • Editorial policies
    • Peer review process
    • Publication fees
    • Reprint requests
    • Open access
  • CMA Members
    • Overview for members
    • Earn CPD Credits
    • Print copies of CMAJ
  • Subscribers
    • General information
    • View prices
  • Alerts
    • Email alerts
    • RSS
  • JAMC
    • À propos
    • Numéro en cours
    • Archives
    • Sections
    • Abonnement
    • Alertes
    • Trousse média 2022
  • CMAJ JOURNALS
    • CMAJ Open
    • CJS
    • JAMC
    • JPN

User menu

Search

  • Advanced search
CMAJ
  • CMAJ JOURNALS
    • CMAJ Open
    • CJS
    • JAMC
    • JPN
CMAJ

Advanced Search

  • Home
  • Content
    • Current issue
    • Past issues
    • Early releases
    • Collections
    • Sections
    • Blog
    • Infographics & illustrations
    • Podcasts
    • COVID-19 Articles
  • Authors
    • Overview for authors
    • Submission guidelines
    • Submit a manuscript
    • Forms
    • Editorial process
    • Editorial policies
    • Peer review process
    • Publication fees
    • Reprint requests
    • Open access
  • CMA Members
    • Overview for members
    • Earn CPD Credits
    • Print copies of CMAJ
  • Subscribers
    • General information
    • View prices
  • Alerts
    • Email alerts
    • RSS
  • JAMC
    • À propos
    • Numéro en cours
    • Archives
    • Sections
    • Abonnement
    • Alertes
    • Trousse média 2022
  • Visit CMAJ on Facebook
  • Follow CMAJ on Twitter
  • Follow CMAJ on Pinterest
  • Follow CMAJ on Youtube
  • Follow CMAJ on Instagram
Practice

If we're so different, why do we keep overlapping? When 1 plus 1 doesn't make 2

Rory Wolfe and James Hanley
CMAJ January 08, 2002 166 (1) 65-66;
Rory Wolfe
Dr. Wolfe is with the Department of Epidemiology and Preventive Medicine, Monash University, Melbourne, Victoria, Australia. Dr. Hanley is with the Department of Epidemiology and Biostatistics, McGill University, Montreal, Que.
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
James Hanley
Dr. Wolfe is with the Department of Epidemiology and Preventive Medicine, Monash University, Melbourne, Victoria, Australia. Dr. Hanley is with the Department of Epidemiology and Biostatistics, McGill University, Montreal, Que.
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site

In the last decade, guidelines for the presentation of statistical results in medical journals have emphasized confidence intervals (CIs) as an adjunct to, or even a replacement for, statistical tests and p values. Because of the intimate links between the 2 concepts, authors now use statements like “the 95% CI overlaps 0” where they would formerly have stated “the difference is not statistically significant at the 5% level.” Although this interchangeability is technically correct in 1-sample situations, it does not carry over fully to comparisons involving 2 samples. A frequently encountered misconception is that if 2 independent 95% CIs overlap each other, as they do in Fig. 1, then a statistical test of the difference will not be statistically significant at the 5% level.

Figure
  • Download figure
  • Open in new tab
  • Download powerpoint

Fig. 1: Group means with confidence intervals that overlap.

Why is this not necessarily so? Consider the means in 2 independent groups, meanA and meanB, with for simplicity meanA being the smaller of the 2. The 95% CI for the mean in group A is approximately given by meanA plus or minus twice the standard error of the mean for that group, SEA, and correspondingly for group B. A mathematical check for whether these CIs overlap is given by adding the distance 2SEA (from meanA to the upper bound of the CI) to 2SEB and comparing this sum with the distance between the 2 means, that is, meanB minus meanA (Fig. 2). The CIs overlap when

Figure
  • Download figure
  • Open in new tab
  • Download powerpoint

Fig. 2: Confidence intervals and comparison of 2 group means (hypothetical clinical trial data: SEA = SEB = 1.8, means differ by 3 SE; assuming n >> 30 and independent samples, the 2-sided p value for testing the difference in means is approximately 0.036). SE = standard error of the mean.

[Equation 1]

Formula

But overlapping confidence intervals do not demonstrate that group means are not statistically significantly different from each other. In a 2-sample t-test to compare 2 means, significance is attained at the 0.05 level if the t statistic exceeds the critical value of about 2, which occurs when the difference between the means exceeds twice its standard error, namely, if

[Equation 2]

Formula

This standard error reflects the fact that the standard error of a difference involves summing the standard error of each estimate, but doing so by “adding in quadrature,” for example,

[Equation 3]

Formula

Thus, to evaluate the overlap of 2 95% CIs and to determine whether at the same time the difference between the means is significant at the 0.05 level, the following rough rule can be used:

[Equation 4]

Formula

If SEA and SEB are equal, the condition is as follows:

[Equation 5]

Formula

When one SE is 25% larger than the other, the boundaries are 3.2 and 4.5 times the smaller SE. As the lower boundary remains close to 3, Moses1 was prompted to display group means with error bars that were 1.5 SE around the mean in order to have a “by eye” test of significance between the 2 group means while presenting the information in the 2 groups separately.

Footnotes

  • This article has been peer reviewed.

    Contributors: Both authors independently conceived of the material for this article. Both were involved in writing the article, and both have seen and approved the final version.

    Competing interests: None declared.

Reference

  1. 1.
    Moses LE. Graphical methods in statistical analysis. Annu Rev Public Health 1987;8:309-53.

Content

  • Current issue
  • Past issues
  • Collections
  • Sections
  • Blog
  • Podcasts
  • Alerts
  • RSS
  • Early releases

Information for

  • Advertisers
  • Authors
  • Reviewers
  • CMA Members
  • Media
  • Reprint requests
  • Subscribers

About

  • General Information
  • Journal staff
  • Editorial Board
  • Advisory Panels
  • Governance Council
  • Journal Oversight
  • Careers
  • Contact
  • Copyright and Permissions
  • Accessibiity
  • CMA Civility Standards
CMAJ Group

Copyright 2022, CMA Impact Inc. or its licensors. All rights reserved. ISSN 1488-2329 (e) 0820-3946 (p)

All editorial matter in CMAJ represents the opinions of the authors and not necessarily those of the Canadian Medical Association or its subsidiaries.

To receive any of these resources in an accessible format, please contact us at CMAJ Group, 500-1410 Blair Towers Place, Ottawa ON, K1J 9B9; p: 1-888-855-2555; e: cmajgroup@cmaj.ca

Powered by HighWire