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The “Exact” Binomial Interval

So far we have employed the Normal approximation to the Binomial distribution, and contrasted Wald and Wilson methods. To evaluate formulae against an ideal distribution we need a baseline. We need to calculate P values from first principles. To do this we use the Binomial formula. Recall from Figure 1 that the Binomial distribution is a discrete distribution, i.e. it can be expressed as a finite series of probability values for different values of x = {0, 1, 2, 3, …, n}.

We will consider the lower bound of p, i.e. where P < p (as in Figure 4). There are two interval boundaries on each probability, but the argument is symmetric: we could apply the same calculation substituting q = 1 – p, etc. in what follows.

Consider a coin-tossing experiment where we toss a weighted coin n times and obtain r heads (sometimes called “Bernoulli trials”). The coin has a weight P, i.e. the true value in the population of obtaining a head is P, and the probability of a tail is (1 – P). The coin may be biased, so P need not be 0.5!

The population Binomial distribution of r heads out of n tosses of a coin with weight P is defined in terms of a series of discrete probabilities for r, where the height of each column is defined by the following expression (Sheskin, 1997 Sheskin, D. J. 1997. Handbook of Parametric and Nonparametric Statistical Procedures, Boca Raton, Fl: CRC Press.
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This formula consists of two components: the Binomial combinatorial nCr (i.e. how many ways one can obtain r heads out of n tosses) 5
5There is only 1 way of obtaining all heads (HHHHHH), but 6 different patterns give 1 tail and 5 heads, etc. The expression nCr = n! / {r! (n – r)!}, where “!” refers to the factorial.
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, and the probability of each single pattern of r heads and (n – r) tails appearing, based on the probability of a head being P.

The total area of Binomial columns from x 1 to x 2 inclusive is then the Cumulative Binomial probability:

However, this formula assumes we know P. We want to find an exact upper bound for p = x/n at a given error level . The Clopper-Pearson method employs a computational search procedure to sum the upper tail from x to n to find P where the following holds:

This obtains an exact result for any integer x. The computer modifies the value for P until the formula for the remaining “tail” area under the curve converges on the required value, . We then report P. 6
6This method is Newcombe’s (1998a) method 5 using exact Binomial tail areas. In Figure 6 we estimate the interval for the mean p by summing B(0, r; n, p) < α/2.
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Note how this method is consistent with the idea of a confidence interval on an observation p: to identify a point P, sufficiently distant from p for p to be considered just significantly different from P at the level . As in Section 2.2, we do not know the true population value P but we expect that data would be Binomially distributed around it.

Figure 7 shows the result of computing the lower bound for p = P employing this Binomial formula. We also plot the Wilson formula, with and without an adjustment termed a “continuity correction”, which we will discuss in the next section. As we have noted, the Wilson formula for p is equivalent to a 2 1 goodness of fit based on P. The continuity-corrected formula is similarly equivalent to Yates’ 2 1 .