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A Comparison of Binomial Proportion Interval Estimation Methods John Ulicny, Precision Metrics Inc., Valley Forge PA ABSTRACT Some recent research has confirmed that the standard Wald (normal) confidence interval approximation for the binomial proportion is inadequate in key respects, especially for extreme values of p . Even for moderately large values of n the coverage probabilities can prove to be erratic for various specific values of p . This fact makes general rules for the use of the Wald method problematic. Alternative methods have been proposed, and this paper compares some of those alternatives with new capabilities built in to SAS Version 8.2 to calculate exact intervals. A decision rule is proposed for the selection of an appropriate technique. Also, a few extensions to other discrete distributions will be considered. This topic is relevant to both statisticians and programmers because of the fact that binomial proportion estimation is so commonly needed in a wide variety of situations, and the Wald (normal approximation) technique is taught in even the most basic statistics courses. Non-statisticians therefore often handle the task of interval estimation of this type. INTRODUCTION Binomial random variables arise frequently in data analysis. They are based on a sum of much simpler variables called Bernoulli random variables. When a situation exists where one of only two possible mutually exclusive outcomes can occur, then this forms a particular kind of sample space. It is common to assign a designation of “success” to one of the events in the sample space and “failure” to the other. Furthermore it is helpful to map these designations into a numeric variable. We can create a variable, Y, with a value 1 corresponding to a successful outcome and a 0 for failure. A variable that can take on a value of 1 or 0 in this way is called a Bernoulli random variable. A sum of such random variables is called a binomial random variable when each term in the sum occurs independently of all others, and the probability of a success is constant. The probability of observing exactly “ x ” successes in a sum of “ n ” Bernoulli trials is expressed as (1.1) n P ( X = x) = p x (1 − p ) n − x x For example the probability of observing 2 heads in 10 tosses of a fair coin could be stated as 10 P ( X = 2) = (.5) 2 (.5)10− 2 = .044 2 In practice the value of the population parameter p is usually unknown and must be estimated from a sample of data. This sample estimate is obtained by counting the successes and dividing by the total number of trials. Let’s call p̂ the estimate. Then (1.2) pˆ = ∑ yi / n = x / n But since the estimate of p is a function of the observed values of a random variable, this estimate is itself a random variable and is subject to random fluctuation. By building a confidence interval using our estimate of p and some assumptions we are able to specify two values between which the true p will lie with a stated probability. We can call this probability the “nominal confidence level”. If we call pL and pU lower and upper confidence limits respectively, then we can attempt to make a statement such as (1.3) P ( pL ≤ p ≤ pU ) = .95 where pL and pU are random quantities between which we claim to be 95 percent confident that the true value of p lies. Here I will mention that the value of p is fixed but unknown, and it is the confidence interval around p that varies randomly. For simplicity I will use a confidence level of .95 throughout this paper. The same conclusions will apply to other levels. In addition it is common to state the confidence level as (1 − α ) because it makes it easier to relate confidence coefficients to alpha levels in hypothesis tests. In this paper I will use α =.05. The term “coverage” is used to denote the probability that a confidence interval computed from a sample of data will actually enclose the true value of p . This coverage will in general vary depending on the values of p and n . In addition, the “confidence coefficient” is defined as the smallest coverage probability across all possible values of p. One of our primary goals is to choose a method with coverage as close as possible to the nominal confidence level. Finding values for the confidence limits in equation (1.3) that provide a level of confidence exactly equal to .95 for all values of p is not possible. A confidence interval is defined by the formula that generates pL and pU along with a confidence coefficient. This way of defining the interval implies that an “exact “ interval with a nominal confidence level of .95 needs to meet the requirement that its minimum coverage is .95, as opposed to a requirement that coverage is .95 everywhere. This makes exact methods very conservative. Moreover the term “exact” is a bit counterintuitive. METHODS PROC FREQ in SAS/BASE version 8 calculates exact binomial confidence intervals as well as intervals based on the normal approximation. The other method you can request from PROC FREQ is the normal approximation. It uses the fact n becomes large, p̂ becomes that as approximately normally distributed with mean p and variance p (1 − p ) / n . This method does not guarantee coverage of (1 − α ) and in fact has very erratic coverage properties. The interval limits are: pL = pˆ − zα / 2 pˆ (1 − pˆ ) / n and pU = pˆ + zα / 2 pˆ (1 − pˆ ) / n Consider this simple example: Let’s assume that a machine produces widgets with a defect rate of 10 percent. If we select a random sample of 10 widgets produced by the machine, then there are 11 possible values for the random variable X, the number of defects. Each item produced has an independent probability of being defective. The following tables list the exact and normal approximation confidence interval methods and their coverage properties: Table 1A Binomial Confidence Intervals with p=.1 and n=10 Nominal Confidence Level=.95 To obtain an exact confidence interval we must solve the pair of equations for pL and pU n n k=x x n k =0 ∑ k p k L (1 − pL ) n −k = α / 2 k U (1 − pU ) n −k = α / 2 and ∑ k p The following equivalent formulas are the ones SAS uses. They are derived form the above equations and involve an F-table lookup which is much faster. n − x +1 pL = 1 + xF2 x ,2( n − x +1),1−α / 2 Normal Approximation Method X P(X=x) 0 0.34868 1 0.38742 2 0.19371 3 0.0574 4 0.01116 5 0.00149 6 0.00014 7 0.00001 8 3.60E-07 9 9.00E-09 10 1.00E-10 Total Coverage Lower 0 0 0 0.016 0.0964 0.1901 0.2964 0.416 0.5521 0.7141 1 Upper 0 0.2859 0.4479 0.584 0.7036 0.8099 0.9036 0.984 1 1 1 Coverage Coverage Flag Probability 0 0 1 0.3874 1 0.1937 1 0.0574 1 0.0112 0 0 0 0 0 0 0 0 0 0 0 0 0.6497 −1 n−x pU = 1 + ( x + 1) F2( x +1),2( n − x ),α / 2 −1 Table 1A lists each possible value of x that could have occurred along with the confidence interval that it would have generated based on the normal method. If the confidence interval covered the value of p , which in this case is .1, then the coverage flag column is set to 1. The coverage probability is just the coverage flag times P(X=x). The total coverage for this method with p =.1 and n =10 is defined as the sum of coverage probabilities over all x . As you can see the coverage value of .6479 is quite substantially below the nominal coverage of .95. Table 1B Binomial Confidence Intervals with p=.1 and Exact Method X P(X=x) 0 0.34868 1 0.38742 2 0.19371 3 0.0574 4 0.01116 5 0.00149 6 0.00014 7 0.00001 8 4.00E-07 9 9.00E-09 10 1.00E-10 Lower 0 0.0025 0.0252 0.0667 0.1216 0.1871 0.2624 0.3475 0.4439 0.555 0.6915 Upper 0.3085 0.445 0.5561 0.6525 0.7376 0.8129 0.8784 0.9333 0.9748 0.9975 1 Coverage Coverage Flag Probability 1 0.3487 1 0.3874 1 0.1937 1 0.0574 1 0.0112 0 0 0 0 0 0 0 0 0 0 0 0 Total Coverage 0.9984 n=10 Nominal Confidence Level=.95 It’s easy to see in Table 1B that the exact method’s confidence interval captures the true value of p when x =0. This is not the case with the normal approximation, and so the exact method’s coverage comes much closer to the stated nominal confidence level of .95. In fact it’s actually quite conservative. This is sample code to generate some of the results in the preceding tables for x=3= ∑y data FreqDemo; input y @@; datalines; 1 0 0 0 0 1 0 1 0 0 ; run; proc sort data=FreqDemo; by descending y; run; proc freq data=freqdemo order=data; tables y/binomial; run; And here is a partial listing of results: Binomial Proportion for y = 1 Proportion 0.3000 ASE 95% Lower Conf Limit 95% Upper Conf Limit 0.1449 0.0160 0.5840 Exact Conf Limits 95% Lower Conf Limit 95% Upper Conf Limit 0.0667 0.6525 In the previous example the number of observations was kept small to keep the calculations simple, but the two methods present problems with coverage properties even for larger n and p . Now I would like to discuss and evaluate two other intervals suggested by Brown et al. The first is the Wilson interval and the second is the Jeffreys interval. The Wilson interval is based on inverting the standard test for an asymptotically normally distributed parameter estimate. Instead of using the standard error of the estimate, it uses the standard error of the parameter value under the null hypothesis. That is, instead of using pˆ (1 − pˆ ) / n it uses p (1 − p ) / n . That change results in the following confidence limits: X + z2 / 2 z n ± n + z2 n + z2 ˆ ˆ + z 2 /(4n) pq The Jeffreys interval takes a Baysian approach. A significant difference between this approach and the others is that the population parameter in the Baysian approach is considered to be a random variable. Using Baysian methodology we assume that the parameter, p , has a prior distribution which reflects our beliefs about the likely values of p before the sample was drawn. For technical reasons the choice for this “prior” distribution for p is often a Beta with two parameters a1 and a2 . The distribution of p is then essentially revised by information obtained in the data sample and thus a new “posterior” distribution is obtained. If p has a beta prior then its mean is a1 /( a1 + a2 ) . The estimate of p after the sample is drawn is x a1 + a2 a1 n pˆ J = + a1 + a2 + n n a1 + a2 + n a1 + a2 Close inspection of this equation reveals that it is ˆ (= x / n) and essentially a weighted average of p the mean of the prior distribution, a1 /( a1 + a2 ) . As n gets larger, more weight is given to information from the sample. The beta distribution as used here is called the “natural conjugate prior” for the binomial and has the very desirable property that the posterior distribution for p is also a beta. Brown et al assume a prior of Beta(1/2,1/2). The confidence interval then becomes simply the posterior beta variate evaluated at the appropriate quantiles. Hence: pL = B(.025; x + .5, n − x + .5) , x > 0 pL = 0 , x = 0 and pU = B(.975, x + .5, n − x + .5) , x < n pU = 1 , x = 1 Using the Wilson method to calculate confidence intervals in example 1 would result in a coverage probability of .930 while the Jeffreys method gives coverage of .987. It is clear that for the situation portrayed in example 1 the Wilson method is superior in terms of coverage. Segments of macro code listed below show calculations of the Wilson and Jeffreys intervals. The variable ‘PR’ contains P ( X = x ) and ‘PHAT’ is p̂ . Note the use of the PDF, PROBIT and BETAINV functions. These SAS functions make the calculation of probabilities and quantiles easy. * -------------------------------------; * Section to process the Wilson method ; * -------------------------------------; %WilsonMth: data _zzbintmp; do x=&xlow to &xhigh; pr=pdf('binomial',x,&popp,&num); z=probit(&clevel+(1-&clevel)/2); x1=x+z**2/2; n1=&num+z**2; phat=x1/n1; offset=(z*(&num**.5)/n1)* ((x/&num)*(1x/&num)+z**2/(4*&num))**.5; *-----ll=lower limit, ul=upper limit; ll=max(phat-offset,0); ul=min(phat+offset,1); *----set coverage flag to 1 if interval ; *----covers the population proportion; cflag = (ll le &popp) and (ul ge &popp); output; end; run; %goto ProcessResult; * -----------------------------------; * Section to process Jeffreys method ; * -----------------------------------; %JeffMth: data _zzbintmp; do x=&xlow to &xhigh; pr=pdf('binomial',x,&popp,&num); *-----ll=lower limit, ul=upper limit; ll=max(betainv((1-&clevel)/2, x+.5,&num-x+.5),0); ul=min(betainv(1-(1-&clevel)/2, x+.5,&num-x+.5),1); if x=0 then ll=0; if x=&num then ul=1; *----set coverage flag to 1 if interval ; *----covers the population proportion; cflag = (ll le &popp) and (ul ge &popp); output; end; run; %goto ProcessResult; OTHER CONSIDERATIONS Three other considerations that are important in evaluating a confidence interval are • • • MIL - Mean interval length MC - Mean coverage RMSE - root mean squared error. They are defined below. Define the expected interval length as n n µ n , p = ∑ ( pU − pL ) p x (1 − p) n − x x =0 x Then the mean interval length with respect to a uniformly distributed p is 1 MILn = ∫ µ n , p dp . 0 Mean coverage is defined similarly as CONCLUSIONS 1 MCn = ∫ Cn , p dp . 0 And the root mean squared error is 1/ 2 1 RMSEn = ∫ (Cn , p − (1 − α )) 2 dp 0 . In the above two equations Cn , p is the coverage for a given method at specific values of n and p . A shorter confidence interval is preferred, all other things being equal, and so a smaller MIL is better. The MC and RMSE give a summary of the coverage characteristics over a range (uniformly weighted in this case) of population proportion values. Other distributions for p can be specified, and once again the most commonly used distribution is a beta. Refer to table 2 for a summary of these measures for various values of n . A numerical technique known as Romberg integration was programmed in SAS and used to produce the measures. Romberg integration repeatedly applies the trapezoidal rule to the functions involved until convergence to within a certain tolerance is achieved. All measures are thus approximate. Note that as n increases, the measures improve for all four methods. For instance the exact method has This happens coverage of 100% when n =5. because the exact method needs to be very conservative to ensure that coverage never falls below 95% for a given combination of n , which is equal to 5 in this case, and p . But as n increases, the mean coverage for the exact method falls all the way to 96.67%. Note also that the normal approximation method never really achieves mean coverage as good as any of the other methods. According to SAS version 8 documentation, “Exact statistics can be useful in situations where the asymptotic assumptions are not met, and so the asymptotic p-values are not close approximations for the true p-values.” While this paper has addressed confidence intervals rather than hypothesis tests and hence p-values, the same advantage of the exact method over the normal (asymptotic) method can be seen by referencing the performance measures in table 2. The SAS documentation goes on to state that “Asymptotic results may also be unreliable when the distribution of the data is sparse, skewed or heavily tied.” The distribution of a binomial random variable will be heavily skewed when p is extreme. A major advantage of the exact method is that it provides a guarantee that coverage will never fall below the nominal confidence level. The price to be paid for this guarantee however is the extreme conservativeness of the method. The Wilson and Jeffreys methods both provide a significantly closer match to nominal confidence, especially at small values of n . Moreover, these two methods provide for smaller mean interval widths at all values of n . The addition of exact methods to the statistical capabilities of SAS is welcome. This paper has attempted to compare four methods, two of which are built-in to SAS (the normal approximation method and the exact method). The normal approximation (Wald) method should virtually never be used. The exact method can be used in situations where the researcher must guarantee coverage at least as large as the nominal stated coverage. The Wilson or Jeffreys methods are appropriate when a high degree of accuracy is required, but coverage may be allowed to fall below the nominal level. The Wilson method is simpler to apply. The Jeffreys method relies on Bayesian techniques and will be more appropriate to use in situations amenable to such techniques. Table 2 Binomial Confidence Interval Comparisons* Nominal Confidence Level=.95 Method Mean Coverage Root Mean Squared Error Mean Interval Length n=5 Normal Exact Wilson Jeffreys 64.17% 100.00% 95.43% 96.02% 39.85% 5.00% 3.54% 3.54% 0.4600 0.6779 0.5581 0.5606 n=10 Normal Exact Wilson Jeffreys 76.97% 98.43% 95.44% 95.33% 28.61% 3.39% 3.39% 3.13% 0.4034 0.5085 0.4354 0.4326 n=15 Normal Exact Wilson Jeffreys 82.22% 97.90% 95.26% 95.15% 23.47% 2.62% 2.97% 2.62% 0.3531 0.4210 0.3687 0.3656 n=25 Normal Exact Wilson Jeffreys 86.55% 97.46% 95.11% 95.03% 18.25% 3.37% 2.81% 2.22% 0.2881 0.3277 0.2943 0.2920 n=50 Normal Exact Wilson Jeffreys 90.09% 96.90% 95.71% 95.71% 2.19% 2.50% 2.05% 2.05% 0.2112 0.2306 0.2129 0.2117 n=100 Normal Exact Wilson Jeffreys 92.23% 96.67% 95.08% 95.00% 2.02% 2.30% 2.45% 1.99% 0.1518 0.1614 0.1522 0.1516 Sample Size * All values are approximations based on numerical integration. REFERENCES Agresti, A. and Coull, B. A. (1998). Approximate is Better than “Exact” for Interval Estimation of Binomial Proportions. The American Statistician 52, 119-126. Brown, L. D., Cai T. T. and DasGupta A. (2001). Interval Estimation for a Binomial Proportion. Statistical Science (forthcoming). Collett, D. (1991). Modelling Binary Data, London: Chapman and Hall. Leemis, L. M. and Trivedi, K. S. (1996). A Comparison of Approximate Interval Estimators for the Bernoulli Parameter. The American Statistician 50, 63-68. Burden, R., Faires J. D., and Reynolds A. C. (1981). Numerical Analysis 2nd Edition. Boston: PWS Publishing. SAS Institute Inc. SAS/STAT User’s Guide, Version 8. Cary NC: SAS Institute Inc., 1999. Casella, G. and Berger, R. L. (1990). Inference, Belmont: Duxbury Press. Acknowledgements: Statistical SAS is a Registered Trademark of the SAS Institute, Inc. of Cary, North Carolina.