Binomial proportion confidence interval

A confidence interval of an unknown probability of a confidence interval ( confidence interval ) for the parameter p of the binomial distribution ( after observation of k n hits in a sample of length).

Exact confidence intervals, as the Clopper -Pearson interval is obtained using the binomial distribution. There are also methods of approximation, the (mostly) to the approximation to the binomial distribution, based by the normal distribution.

Introductory Example

To estimate the unknown relative proportion p of a political party A in the electorate, 400 people are asked whether they will choose the Party A in an opinion poll n =. The exact procedure is: We choose 400 times a random person from the electorate and ask them. Here we do not consider whether or not that person has already been interviewed. So it can happen, although it is accordingly unlikely even with a large electorate, that same person is repeatedly questioned. The number X of respondents who indicated that they choose party A is dependent on chance and therefore a random variable. Since the respondents are randomly and independently selected, the random variable X has a binomial distribution with parameters n = 400 and the unknown parameter p. Suppose that in the survey have k = 20 respondents indicated that they choose party A. It calculates an estimate of p as:

This is called a point estimate because only one value is calculated as an estimate of p.

The true value of the relative proportion p can be both smaller and larger than the point estimate. Certainly applies only that p can assume any value between 0 and 1. Desirable would be a confidence interval [ pi, po ] for p. When multiple repeating the procedure, the calculated confidence intervals are included in " most cases " the parameter p. How often this should be the case, is expressed by the confidence probability (or confidence level) γ. The calculated interval [ pi, po ] is called a confidence interval (or confidence interval ). It is often chosen γ equal to 95 %. This means that when repeating the process for 95 % of all samples, the statement p ∈ [ pi, po ] is correct.

Clopper -Pearson interval

C. Clopper Egon and Pearson (1934 ) derives the following precise method to determine the lower limit and the upper limit Pu po. It is, as before, n is the sample size, k is the number of successes and the confidence level is 95%.

The upper limit is determined from P ( X ≤ k, po) = 0.025 and the lower limit of P (X ≥ k; pu) = 0.025, see figure. The lower limit can be used for k = 0 because this formula does not specify.

Explanation: If the probability of achieving a maximum of k successes is less than the limit of 0.025 for a share value of p, it can be excluded at a significance level of more than 2.5% that p is the desired percentage value. Thus po is the largest value of p, in which the given confidence level can be accepted nor that k or fewer successes occur. For larger values ​​of p, this appears to be unlikely.

For the lower limit shall apply accordingly: pi is the smallest value of p, which is still assumed that k or more successes can occur. For smaller values ​​of p, this appears to be unlikely, here too the probability of error is 2.5 %. Thus, it is at least 95 % of all cases with the statement " p ≤ po and p ≥ pu" correctly.

Practical calculation

The two values ​​pi, po can be, for example, calculated using the BETAINV Excel function. The function returns the quantile of the beta distribution given back and is obtained due to the context of the binomial and beta distribution for the resolution of the equation p_p (X ≤ k) = α: p = BETAINV (1- α, k 1, n -k). In the following we assume, as before, the confidence level γ, also had α = 1- γ.

In the example with n = 400, k = 20 and 95% confidence level, one obtains for the lower limit of the confidence interval: P ( X ≥ 20 ) = 0.025 P (X ≤ 19) = 0.975 pu = BETAINV ( 1-.975, 20, 400-19 ) = 0.03081

And the upper limit of the confidence interval: P ( X ≤ 20 ) = 0.025 po = BETAINV ( 1-.025, 21, 400-20 ) = 0.07617

Explanation in words: Even with a vote share of only 3.1 %, the probability that in the sample are at least 20 persons, a further 2.5 %. And accordingly: even with a vote share of 7.6 %, the probability that in the sample are more than 20 persons, 2.5%

The method is summarized in the following table:

Analysis of the coverage probability

If C (k ), the confidence interval for k successes, one requires by definition that for all p, the coverage probability is greater than or equal to γ. For continuous distributions, confidence intervals can be found, so here equality is present (ie γ = instead of > = γ ). This is not possible for the discrete binomial distribution. The following figure shows the coverage probability is shown as a function of p. The Clopper -Pearson interval is the way, why exactly known, because it ensures a coverage probability that is actually for all p greater than or equal to the required confidence level. For the approximation discussed in the following, this is not the case here, there is often overlap probabilities, which are smaller than the required level!

The coverage probability is calculated as a function of p and n are as follows:

Here is the indicator function. She takes the value 1 if p is in the confidence interval, and otherwise 0 For k = 0 we set pi = 0

Simple approximation by the normal distribution

Often one uses the following simple approximate formula:

Which is also called a standard interval.

When this formula is used, it should k ≥ 50 and n - k be ≥ 50. Nevertheless, the use of the standard interval can be problematic. In the figure, for example, the coverage probability for n = 400 and illustrated. It is often below the required level of 0.95.

Wilson interval

This interval was proposed in 1927 by Edwin Bidwell Wilson and is more accurate than the simple approximation by the normal distribution. It applies, with the same c in the previous section,

Apparently converge the interval boundaries for large n ( as n increases and go to zero ) against the limits of the standard interval.

Rearranging yields the in Brown / Cai / DasGupta on page 107 specified formula (4):

In the figure, the coverage probability for n = 1, ..., 40 are illustrated.

The number shown on page 228 in Henze formula which still differs (applied to k of 0.5 or -0.5 ) by a continuity correction.

Agresti - Coull interval

For that interval are employed, and used (as described above), a simple approximation with these parameters:

The center of the interval is identical to that of Wilson interval and the interval is shorter than one Wilson interval.

If γ is = 0.95, then and give you a simple rule:, and.

Discussion of the advantages and disadvantages of the method

The described methods are compared in the seminal article by Brown / Cai / DasGupta. There, the standard interval is also called forest - interval (after Abraham Wald ). Brown / Cai / DasGupta recommend three interval methods: the Wilson interval, the Agresti - Coull interval and the Jeffreys interval, which we have not discussed here. For a graphical comparison of the upper and lower limits for the four methods, see the figure at the beginning of this page.