
AGRESTI COULL CONFIDENCE LIMITSName:
Confidence intervals for the binomial proportion can be computed using a method recommended by Agresti and Coull and also by Brown, Cai and DasGupta (the methodology was originally developed by Wilson in 1927). This method solves for the two values of p_{0} (say, p_{upper} and p_{lower})) that result from setting z = α/2 and solving for p_{0} = p_{upper}, and then setting z = z = α/2 and solving for p_{0} = p_{lower} where z_{α/2} denotes the variate value from the standard normal distribution such that the area to the right of the value is α/2. The solution for the two values of p_{0} results in the following confidence intervals:
\( L. L. = \frac{\hat{p} + \frac{z_{\alpha/2}^{2}}{2n}  z_{\alpha/2}\sqrt{\frac{\hat{p}(1\hat{p})}{n} + \frac{z_{\alpha/2}^{2}}{4n^2}}} {1 + z_{\alpha/2}^{2}/n} \) This approach can be substantiated on the grounds that it is the exact algebraic counterpart to the (largesample) hypothesis test and is also supported by the research of Agresti and Coull. One advantage of this procedure is that its worth does not strongly depend upon the value of n and/or p, and indeed was recommended by Agresti and Coull for virtually all combinations of n and p. Another advantage is that the limits are in the (0,1) interval. This is not true for the frequently used normal approximation:
<SUBSET/EXCEPT/FOR qualification> where <p> is constant, parameter, or variable that contains the proportion of successes; <n> is constant, parameter, or variable that contains the number of trials; <alpha> is constant or parameter that contains the significance level; <lowlim> is a variable that contains the computed lower AgrestiCoull confidence limit; <upplim> is a variable that contains the computed upper AgrestiCoull confidence limit; and where the <SUBSET/EXCEPT/FOR qualification> is optional. The <p> and <n> arguements can be either parameters or variables. If they are both variables, then the variables must have the same number of elements. The <alpha> argument is alwasys assumed to be either a constant or a parameter. If <p> and <n> are both parameters, then <lowlim> and <upplim> will be parameters. Otherwise, they will be variables.
LET AL AU = AGRESTI COULL CONFIDENCE LIMITS P N ALPHA ... SUBSET TAG > 2
The Brown, Cai, and DasGupta paper studied the coverage properties of various methods. They specifically recommend the Wilson, the adjusted Wald, and a Bayesion method based on a Jeffreys prior as having the best coverage properties. Specifically, they recommend the Wilson and Jeffreys methods for n ≤ 40. For n > 40, the methods have comparable performance. Although they recommend the adjusted Wald in this case, this is primarily for simplicity in classroom presentation. In any event, the March, 2014 version of Dataplot added the following command:
Whenever an AgrestiCoull interval is invoked in Dataplot, this command specifies which interval will be computed. The adjusted Wald interval is
where
\(\tilde{n} = n + (\Phi^{1}(1  \alpha/2))^{2}\) \(\tilde{p} = \frac{\tilde{X}} {\tilde{n}}\) \(\Phi^{1}\) is the percent point function of the normal distribution Note that the adjusted Wald method is never shorter than the Wilson interval. The Jeffreys interval (the derivation for this interval is given in the Brown, Cai, DasGupta paper) is
UCL = BETPPF(1  α/2,n  X + 0.5) where BETPPF is the percent point function of the beta distribution and X is the number of successes. The default method is the Wilson interval.
LET NTRIAL = SIZE Y LET P = YSUM/NTRIAL LET AL AU = AGRESTI COULL CONFIDENCE LIMITS P NTRIAL ALPHA If you have a groupid variable (X), you would do something like
LET YSUM = CROSS TABULATE SUM Y X LET NTRIAL = CROSS TABULATE SIZE Y X LET P = YSUM/NTRIAL LET AL AU = AGRESTI COULL CONFIDENCE LIMITS P NTRIAL ALPHA In this case, P and NTRIAL are now variables rather than parameters.
LET A = TWO SIDED UPPER AGRESTI COULL Y LET A = ONE SIDED LOWER AGRESTI COULL Y LET A = ONE SIDED UPPER AGRESTI COULL Y This command is a Statistics Let Subcommand rather than a Math LET Subcommand. The distinctions are:
Which form of the command to use is determined by the context of what you are trying to do. For details on the "Statistics" version of the command, enter
Brown, L. D. Cai, T. T. and DasGupta, A. (2001), "Interval estimation for a binomial proportion," Statistical Science, 16(2), 101133. Wilson (1927), "Probable inference, the law of succession, and statistical inference," Journal of the American Statistical Association, Vol. 22, pp. 209212.
2014/3: Support for SET BINOMIAL METHOD command LET N = 25 LET P = 0.8 LET ALPHA = 0.95 LET AL AU = AGRESTICOULL CONFIDENCE LIMITS P N ALPHAThe returned value of AL and AU are 0.6086905 and 0.9113942.  
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Date created: 10/05/2010 