Nabil, try the following short article on the topic: http://www.psychologicalscience.org/index.php/publications/observer/2010/april-10/understanding-confidence-intervals-cis-and-effect-size-estimation.html. It also includes a couple of helpful references, especially Cumming & Finch's "Inference by Eye" article.

Here's one relevant paragraph from the short article -

"The overlap misconception. The overlap misconception is a belief that, when comparing means for two independent groups, the means are statistically significantly different at p < .05 when the 95 percent CIs around the means are just touching. In actuality, CIs for independent groups are at p ≈ .05 when they overlap by about ¼ (Figure 2; Cumming & Finch, 2005). When the 95 percent CIs are just touching (no overlap), p ≈ .01. This, however, does not apply to repeated or paired design statistics. For repeated or paired designs, you should create a CI around the difference between the groups."

I believe that overlapping confidence intervals "suggest" insignificant differences or homogeneous subsets. It is quite similar to use boxplots as a "graphical" quick and dirty statistical test.

The use of the word suggest is required as it can be more formal in the setting of simultaneous confidence intervals. The problem with assess two confidence intervals is that one does not know what the overall confidence level is, similar to multiple comparisons in ANOVA. However, tukey's method allows constructing confidence intervals based on the studentized range distribution which turns the first statement into a formal one. Hence, confidence intervals which overlap where a simultaneous confidence level is achieved imply homogenous subsets.

I broadly agree with Kurt's answer, particularly with regard to "notched" boxplots, see e.g. http://en.wikipedia.org/wiki/Box_plot#Variations and cited reference there, and example code below.

However, it's important to note that with dependent (e.g. paired) data, the CIs/notches could overlap substantially even in the presence of a significant difference, if the within-subject variability is much smaller than the between-subject.

Here's a simple example using MATLAB (for those not familiar with it, randn is standard normal, while rand is uniform on [0,1] and hence has non-zero mean). Note that the notches overlap for the two simulated times, but the notches for the paired time difference do not overlap zero, so they are significantly different.

>> randn('state', 0); rand('state', 0); N = 30;

>> subjects = randn(N, 1);

>> time1 = subjects + randn(N,1)/10;

>> time2 = subjects + randn(N,1)/10 + rand(N,1);

>> boxplot([time1 time2 time2-time1], 'notch', true)

Nabil, try the following short article on the topic: http://www.psychologicalscience.org/index.php/publications/observer/2010/april-10/understanding-confidence-intervals-cis-and-effect-size-estimation.html. It also includes a couple of helpful references, especially Cumming & Finch's "Inference by Eye" article.

Here's one relevant paragraph from the short article -

"The overlap misconception. The overlap misconception is a belief that, when comparing means for two independent groups, the means are statistically significantly different at p < .05 when the 95 percent CIs around the means are just touching. In actuality, CIs for independent groups are at p ≈ .05 when they overlap by about ¼ (Figure 2; Cumming & Finch, 2005). When the 95 percent CIs are just touching (no overlap), p ≈ .01. This, however, does not apply to repeated or paired design statistics. For repeated or paired designs, you should create a CI around the difference between the groups."

Thank you all for your comments.

1-

I would like to know if two intervals are not overlapping, can we say that there is a significant different between the two parameters they are related to?

2-

Here is an other question (no relation with CIs):

Let A and B be two dependent events which have the same probability value "p". The Bonferroni's inequality ensures that: P(A and B) >= -1 + 2p.

I am looking for a lower bound that is greater than "2p - 1". Any suggestion?

For the second question-

Start with (p-1)^2=p^2-2p+1>=0, or 2p-1=2p-1, so if I show that ,

2p-1

Dear Hiqmet,

Thank you for your comments.

You suggested an upper bound of P(A and B) while i am looking for a lower bound.

In other words, i am looking for F(p) that satisfies:

1-2p

Hi Nabil,

Oh yes, it is the lowest upper bound.

@Nabil,

Actually if you noticed:

p^3 = p * p^2

Many thanks.

Please, how can one know if confidence intervals overlap? What's the simplest possible way to detect overlapping confidence intervals? Thanks.

This link provides useful information to understand  the discrepency between overlapping confidence intervals and statistical significance

https://www.cscu.cornell.edu/news/statnews/Stnews73insert.pdf

Below is a link some helpful, interactive and free free software (ESCI) by Geoff Cumming that works in excel. It provivdes a great way to explore the overlap in confidence intervals and how this can help with making statistical inferences. https://thenewstatistics.com/itns/esci/

Hope this helps.