Confidence interval

In statistics, a confidence interval (CI) is a tool for estimating a parameter, such as the mean of a population.^[1] To make a CI, an analyst first selects a confidence level, such as 95%. The analyst then follows a procedure that outputs an interval. By following this procedure many times across many experiments, the fraction of intervals that contain the parameter will approach the confidence level. It is a common misconception that the confidence level is the probability that a particular interval contains the parameter. Although these ideas are related, they are subtly different.

Factors affecting the width of the CI include the sample size, the variability in the sample, and the confidence level.^[2] All else being the same, a larger sample produces a narrower confidence interval, greater variability in the sample produces a wider confidence interval, and a higher confidence level produces a wider confidence interval.^[3]

History

Methods for calculating confidence intervals for the binomial proportion appeared from the 1920s.^[4]^[5] The main ideas of confidence intervals in general were developed in the early 1930s,^[6]^[7]^[8] and the first thorough and general account was given by Jerzy Neyman in 1937.^[9]

Neyman described the development of the ideas as follows (reference numbers have been changed):^[8]

[My work on confidence intervals] originated about 1930 from a simple question of Waclaw Pytkowski, then my student in Warsaw, engaged in an empirical study in farm economics. The question was: how to characterize non-dogmatically the precision of an estimated regression coefficient? ...
Pytkowski's monograph ... appeared in print in 1932.^[10] It so happened that, somewhat earlier, Fisher published his first paper^[11] concerned with fiducial distributions and fiducial argument. Quite unexpectedly, while the conceptual framework of fiducial argument is entirely different from that of confidence intervals, the specific solutions of several particular problems coincided. Thus, in the first paper in which I presented the theory of confidence intervals, published in 1934,^[6] I recognized Fisher's priority for the idea that interval estimation is possible without any reference to Bayes' theorem and with the solution being independent from probabilities a priori. At the same time I mildly suggested that Fisher's approach to the problem involved a minor misunderstanding.

In medical journals, confidence intervals were promoted in the 1970s but only became widely used in the 1980s.^[12] By 1988, medical journals were requiring the reporting of confidence intervals.^[13]

Definition

Let $X$ be a random sample from a probability distribution with statistical parameter $(\theta ,\varphi )$ , where $\theta$ is a quantity to be estimated and $\varphi$ represents further quantities which together with $\theta$ determine the probability distribution of the sample, if present, but which are not of immediate interest. A confidence interval for the parameter $\theta$ , with confidence level or coefficient $\gamma$ , is an interval $(u(X),v(X))$ determined by random variables $u(X)$ and $v(X)$ with the property:

P(u(X)<\theta <v(X))=\gamma \quad {\text{ for every }}(\theta ,\varphi ).

The number $\gamma$ , whose typical value is close to but not greater than 1, is sometimes given in the form $1-\alpha$ (or as a percentage $100\%\cdot (1-\alpha )$ ), where $\alpha$ is a small positive number, often 0.05.

In many applications, confidence intervals that have exactly the required confidence level are hard to construct, but approximate intervals can be computed. The rule for constructing the interval may be accepted as providing a confidence interval at level $\gamma$ if

P(u(X)<\theta <v(X))\approx \ \gamma \quad {\text{ for every }}(\theta ,\varphi )

to an acceptable level of approximation. Alternatively, some authors^[14] simply require that

P(u(X)<\theta <v(X))\geq \ \gamma \quad {\text{ for every }}(\theta ,\varphi ).

When it is known that the coverage probability can be strictly larger than $\gamma$ for some parameter values, the confidence interval is called conservative, i.e., it errs on the safe side; which also means that the interval can be wider than need be.

Methods of derivation

There are many ways of calculating confidence intervals, and the best method depends on the situation. Two widely applicable methods are bootstrapping and the central limit theorem.^[15] The latter method works only if the sample is large, since it entails calculating the sample mean ${\bar {X}}_{n}$ and sample standard deviation $S_{n}$ and assuming that the quantity

{\frac {{\bar {X}}_{n}-\mu }{S_{n}/{\sqrt {n}}}}

is normally distributed.

Example

In this bar chart, the top ends of the brown bars indicate observed means and the red line segments ("error bars") represent the confidence intervals around them. Although the error bars are shown as symmetric around the means, that is not always the case. In most graphs, the error bars do not represent confidence intervals (e.g., they often represent standard errors or standard deviations).

Suppose $X_{1},\ldots ,X_{n}$ is an independent sample from a normally distributed population with unknown parameters mean $\mu$ and variance $\sigma ^{2}.$ Define the sample mean ${\bar {X}}$ and unbiased sample variance $S^{2}$ as

{\bar {X}}={\frac {X_{1}+\cdots +X_{n}}{n}},

S^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}.

Then the value

T={\frac {{\bar {X}}-\mu }{S/{\sqrt {n}}}}

has a Student's t distribution with $n-1$ degrees of freedom.^[16] This value is useful because its distribution does not depend on the values of the unobservable parameters $\mu$ and $\sigma ^{2}$ ; i.e., it is a pivotal quantity. Suppose we wanted to calculate a 95% confidence interval for $\mu .$ First, let $c$ the 97.5th percentile of the distribution of $T$ . Then there is a 2.5% chance that $T$ will be less than $-c$ and a 2.5% chance that it will be larger than $+c.$ In other words,

P_{T}(-c\leq T\leq c)=0.95.

Consequently,

P_{X}{\left({\bar {X}}-{\frac {cS}{\sqrt {n}}}\leq \mu \leq {\bar {X}}+{\frac {cS}{\sqrt {n}}}\right)}=0.95.

Here $P_{X}$ is the probability measure for the sample $X_{1},\ldots ,X_{n}$ .

After observing the sample, we find values ${\bar {x}}$ for ${\bar {X}}$ and $s$ for $S,$ from which we compute the confidence interval

\left[{\bar {x}}-{\frac {cs}{\sqrt {n}}},{\bar {x}}+{\frac {cs}{\sqrt {n}}}\right].

Interpretation

Various interpretations of a confidence interval can be given (taking the 95% confidence interval as an example in the following).

The confidence interval can be expressed in terms of a long-run frequency in repeated samples (or in resampling): "Were this procedure to be repeated on numerous samples, the proportion of calculated 95% confidence intervals that encompassed the true value of the population parameter would tend toward 95%."^[17]
The confidence interval can be expressed in terms of probability with respect to a single theoretical (yet to be realized) sample: "There is a 95% probability that the 95% confidence interval calculated from a given future sample will cover the true value of the population parameter."^[9] This essentially reframes the "repeated samples" interpretation as a probability rather than a frequency.
The confidence interval can be expressed in terms of statistical significance, e.g.: "The 95% confidence interval represents values that are not statistically significantly different from the point estimate at the .05 level."^[18]

Common misunderstandings

Confidence intervals and levels are frequently misunderstood, and published studies have shown that even professional scientists often misinterpret them.^[19]

A 95% confidence level does not mean that for a given realized interval there is a 95% probability that the population parameter lies within the interval (i.e., a 95% probability that the interval covers the population parameter).^[20] This distinction can be understood by analogy: If the weather forecast is accurate 95% of the time, it does not follow that it is accurate today with 95% probability. For instance, maybe forecasts of sun are 99% accurate and forecasts of rain are 80% accurate. Then, having seen today's forecast, one would be either 99% or 80% confident, but never 95% confident.

A 95% confidence level does not mean that 95% of the sample data lie within the confidence interval.
A 95% confidence level does not mean that there is a 95% probability of the parameter estimate from a repeat of the experiment falling within the confidence interval computed from a given experiment.^[21]

Examples of how naïve interpretation of confidence intervals can be problematic

Confidence procedure for uniform location

Welch^[22] presented an example which clearly shows the difference between the theory of confidence intervals and other theories of interval estimation (including Fisher's fiducial intervals and objective Bayesian intervals). Robinson^[23] called this example "[p]ossibly the best known counterexample for Neyman's version of confidence interval theory." To Welch, it showed the superiority of confidence interval theory; to critics of the theory, it shows a deficiency. Here we present a simplified version.

Suppose that $X_{1},X_{2}$ are independent observations from a uniform $(\theta -1/2,\theta +1/2)$ distribution. Then the optimal 50% confidence procedure for $\theta$ is^[24]

{\bar {X}}\pm {\begin{cases}{\dfrac {|X_{1}-X_{2}|}{2}}&{\text{if }}|X_{1}-X_{2}|<1/2\\[8pt]{\dfrac {1-|X_{1}-X_{2}|}{2}}&{\text{if }}|X_{1}-X_{2}|\geq 1/2.\end{cases}}

A fiducial or objective Bayesian argument can be used to derive the interval estimate

{\bar {X}}\pm {\frac {1-|X_{1}-X_{2}|}{4}},

which is also a 50% confidence procedure. Welch showed that the first confidence procedure dominates the second, according to desiderata from confidence interval theory; for every $\theta _{1}\neq \theta$ , the probability that the first procedure contains $\theta _{1}$ is less than or equal to the probability that the second procedure contains $\theta _{1}$ . The average width of the intervals from the first procedure is less than that of the second. Hence, the first procedure is preferred under classical confidence interval theory.

However, when $|X_{1}-X_{2}|\geq 1/2$ , intervals from the first procedure are guaranteed to contain the true value $\theta$ : Therefore, the nominal 50% confidence coefficient is unrelated to the uncertainty we should have that a specific interval contains the true value. The second procedure does not have this property.

Moreover, when the first procedure generates a very short interval, this indicates that $X_{1},X_{2}$ are very close together and hence only offer the information in a single data point. Yet the first interval will exclude almost all reasonable values of the parameter due to its short width. The second procedure does not have this property.

The two counter-intuitive properties of the first procedure – 100% coverage when $X_{1},X_{2}$ are far apart and almost 0% coverage when $X_{1},X_{2}$ are close together – balance out to yield 50% coverage on average. However, despite the first procedure being optimal, its intervals offer neither an assessment of the precision of the estimate nor an assessment of the uncertainty one should have that the interval contains the true value.

This example is used to argue against naïve interpretations of confidence intervals. If a confidence procedure is asserted to have properties beyond that of the nominal coverage (such as relation to precision, or a relationship with Bayesian inference), those properties must be proved; they do not follow from the fact that a procedure is a confidence procedure.

Confidence procedure for ω²

Steiger^[25] suggested a number of confidence procedures for common effect size measures in ANOVA. Morey et al.^[20] point out that several of these confidence procedures, including the one for ω², have the property that as the F statistic becomes increasingly small—indicating misfit with all possible values of ω²—the confidence interval shrinks and can even contain only the single value ω² = 0; that is, the CI is infinitesimally narrow (this occurs when $p\geq 1-\alpha /2$ for a $100(1-\alpha )\%$ CI).

This behavior is consistent with the relationship between the confidence procedure and significance testing: as F becomes so small that the group means are much closer together than we would expect by chance, a significance test might indicate rejection for most or all values of ω². Hence the interval will be very narrow or even empty (or, by a convention suggested by Steiger, containing only 0). However, this does not indicate that the estimate of ω² is very precise. In a sense, it indicates the opposite: that the trustworthiness of the results themselves may be in doubt. This is contrary to the common interpretation of confidence intervals that they reveal the precision of the estimate.