Confidence Intervals

Confidence Intervals

by Jean Digitale -

On the slides, there is the following definition of CIs:

"Over many repetitions of the study (with different samples) it is expected that 95% of the 95% CI will include the true (population) value"


Then in the book there is the following:

"You can think of this 95% CI as answering the question, “If the results observed in this study were the true values, and I repeated the study many times, what range would include 95% of the observed results?”"


I understand the definition on the slides and am so used to thinking about CIs this way that I am confused by the sentence in the book. I am also a bit unclear as to how the two definitions fit together. 


Is the key that in the book you say, "if the results observed in this study were the true values"?

Hence, the sentence is the book is really also saying "what range would include 95% of the TRUE VALUES"? 

Is it true that the book sentence is generally NOT correct about observed results, it is only correct if we are assuming they are equal to the true values?


Thanks,

Jean

Re: Confidence Intervals

by Michael Kohn -

Thanks for the question Jean.

When you use the normal approximation to calculate the 95% confidence interval for an observed proportion like 12/20 = 0.6, you assume that the observed proportion is actually the true value of the p (the probability of a success) and can then calculate the boundaries for an interval that has 95% probability.  In this example, if you assume that p = 0.6, then the interval from 0.385 to 0.815 has 95% probability, but that is based on assuming that p = 0.6, which is simply the observed result.  the true value of p could be very different.

The definition for the CI given in the previous paragraph, which gives the exact 95% interval, is as follows:

the 95% confidence interval gives the range of hypotheses about the probability of heads that would not be rejected at the 0.025 significance level on either side.  More generally, the (1 − α) confidence interval is the range of hypotheses that would not have been rejected at significance level α/2.  This is a hard definition to follow, but it is the one I have come to prefer.

--MAK


Re: Confidence Intervals

by Laura Koth -

I too am quite confused by the chapter and additional reading on CI's.

I find it more helpful to focus on what the CI is, and not try to memorize what the CI's are not which seems to be the approach with the readings.

that being said, I do not understand the concept of what a CI as described in the chapter.

can someone us lay terms/language to summarize it in a way i could tell a medical resident or med student and make the concept more likely to be remembered?

Dr. Kohn posted this definition but is it possible to use lay terms (to encourage long term memory)?

the 95% confidence interval gives the range of hypotheses about the probability of heads that would not be rejected at the 0.025 significance level on either side.  More generally, the (1 − α) confidence interval is the range of hypotheses that would not have been rejected at significance level α/2.  This is a hard definition to follow, but it is the one I have come to prefer.


Re: Confidence Intervals

by Michael Kohn -

Chapter 11, Page 12:

Of course, if you don't want to spend a couple of paragraphs explaining their exact meaning, a nonquantitative definition works almost as well: the confidence interval indicates a range of values consistent with what was observed in the study. The higher the “level of confidence” (e.g., 99% vs. 95%), the wider the interval will be, corresponding to a looser definition of “consistent.” 

Stick with this definition of the confidence interval.  It is the one I use with most people.

It is very difficult to give a correct quantitative definition of a confidence interval.  You end up with my preferred definition, which admittedly requires a lot of time and effort to understand.  It certainly took me a long time and much effort.  

I know it's frustrating, but we think it's important to emphasize that a 95% confidence interval is NOT what we all want it to be: the range of values with a 95% probability of containing the true value.


Bottom line: stick with the non-quantitative definition: the range of values for the parameter of interest that is consistent with the results of the study.