8.8: Confidence Intervals
- Page ID
- 50678
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In addition to reporting the p value, report the confidence intervals (CI). Confidence intervals are the range where a set of values will fall. In this case, it is the range of statistical test values. Confidence intervals are important for determining the stability of a result. Because with repeated sampling, results can change. The question is, by how much could the result change, and if it does change, is the result still significant?
Confidence intervals are based on the concept of error. We accept or reject the null hypothesis based on one sample. But if we take another sample, we might get another, different test result.
What is the range of possible test results if we keep sampling? Stated differently, if we collect a different set of observations and re-conduct the statistical test, what will happen? The question is, will we get the same or a different result?
The statistical definition of a confidence interval is – Estimated range of sample test values that likely contains the true population parameter. In English, this statement means that if we keep sampling, our test result will fall into what range of possible values contains the “correct” value. I say “correct” because the population is unknown, and we will never know the correct answer.
The purpose of the CI is to corroborate the p value finding. The confidence interval gives
you a range of values from the lowest to the highest. 95% CI: 95% of the time, we will get a result that will fall within this range of values. The possible CI ranges are 68% CI, 95% CI, and 99% CI. More about these intervals later. For now, note that these intervals are similar to the normal distribution and the standard deviation.
What are you looking for if the CI contains a value of 0 that would lead us to retain the null hypothesis? If the CI range from low to high includes a 0, it means that there is no significant effect. If the CI range from low to high does not include a 0, it means that there is a significant effect, and the CI range corroborates or confirms the p value outcome, which declares that a statistical test is significant.
BTW, if the p value is significant, nearly 100% of the time, the CI does not contain the value 0. If the p value is not significant, nearly 100% of the time, the CI does contain the value 0. I have never encountered a situation where the p value was significant, but the CI contained 0 and indicated no significance. I say if the p value is significant, nearly 100% of the time, the CI will confirm that significant result because I have never seen the CI not confirm a significant p value. Like everything else, anything is possible; that is why I say “nearly” 100% of the time, but in my entire career with statistics, I’ve never seen it. As far as I am concerned, the p value indicates significance; then the CI corroborates that finding 100% of the time.
So, how good is the CI if it almost always confirms the p value result? Sometimes, I do use the CI to evaluate how wide the CI interval is. A wider interval means there is more error and more uncertainty in the estimate of the true value of the population parameter. A shorter interval means there is less error and more precision in estimating the true value of the population parameter. How wide or short does the interval have to be? It depends on how far away the upper and lower bounds are from the test result. There are no hard and fast guidelines to gauge the distance and if it is a wider or shorter interval. You should consult if you are wondering if the CI is too wide or short enough to be considered a good estimate.
In the end, why should you report confidence intervals in addition to the p value? As of 2020, the American Psychological Association (APA) wants you to report it. APA strongly recommends that, in addition to the p values, researchers provide effect sizes and confidence intervals because the all or nothing hypothesis testing does not provide the full range of possible outcomes of statistical analysis (for an explanation, see p. 88, section 3.7, APA manual, 7th edition). If your statistics software program (e.g., SPSS) allows an option for a CI, select it and report it whenever possible.