A note on Wilson coverage intervals for proportions estimated from complex samples
Section 4. Some concluding remarks

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The asymptotic equivalence of a coverage interval based on a logistic transformation to the theoretically grounded Wilson interval is the main contribution of this paper. Although in the asymptotic framework, $P\left(1-P\right)$ is fixed and positive as $n*$ grows large, in practice it is the size of $p\left(1-p\right)n*$ that matters when comparing the Wilson-type and logistic-transformation intervals. This requires that $P\left(1-P\right)$ not be too small.

Brown et al. (2001) show empirically that under simple random sampling (with $n=50),$ coverage intervals derived from the logistic transformation tend to be larger than corresponding Wilson intervals for small values of $P\left(1-P\right).$ Kott and Liu (2009) make the same observation for one-sided intervals based on complex samples, supporting the notion that it is a better choice.

The asymptotic equivalence of the logistic-transformation interval with the Wilson interval explains the former’s empirical superiority in the literature (e.g., in Brown et al., 2001) to an analogous interval constructed using an arcsine transformation. Because arcsin$\left(p\right)$ has a constant large-sample variance under simple random sampling no matter the true value of $P$ (so long as $P\left(1-P\right)>0),$ it has been hoped that the arcsine transformation would be ideal for interval construction.

Better than a Wilson interval, but not yet incorporated into any software package I know of, is the one-sided coverage intervals for $P$ derived using an Edgeworth expansion on $p-P$ in Kott and Liu (2009). That method produces this two-sided interval:

$$\begin{array}{l}p+\frac{1-2p}{\tilde{n}}\left(\frac{1}{6}+\frac{z_{1-\alpha /2}^2}{3}\right)-z_{1-\alpha /2}{\left(\mathrm{var}\left(p\right)+{\left[\frac{1-2p}{\tilde{n}}\left(\frac{1}{6}+\frac{z_{1-\alpha /2}^2}{3}\right)\right]}^2\right)}^{1/2}\text{}\le P\hfill \\ \le p+\frac{1-2p}{\tilde{n}}\left(\frac{1}{6}+\frac{z_{1-\alpha /2}^2}{3}\right)+z_{1-\alpha /2}{\left(\mathrm{var}\left(p\right)+{\left[\frac{1-2p}{\tilde{n}}\left(\frac{1}{6}+\frac{z_{1-\alpha /2}^2}{3}\right)\right]}^2\right)}^{1/2}\text{},\hfill \end{array}$$

where $\tilde{n}=\left[\left(1-2p\right)\mathrm{var}\left(p\right)\right]/\mathrm{cov}\left[\mathrm{var}\left(p\right),p\right],$ and $\mathrm{cov}\left[\mathrm{var}\left(p\right),p\right],$ a consistent estimator for $\text{Cov}\left[\mathrm{var}\left(p\right),p\right],$ exists and equals a consistent estimator for the third moment of $p.$ Note that $\mathrm{cov}\left[\mathrm{var}\left(p\right),p\right]$ doesn’t exist for designs with only two primary sampling units per stratum. Moreover, it is not a consistent estimator for the third moment of $p$ when finite population correction matters.

Observe that $\tilde{n}$ again replaces $n*.$ In addition, $1/6+z_{1-\alpha /2}/3$ replaces $z_{1-\alpha /2}/2,$ which means that the center will often be closer to the $p$ using this interval rather than the Wilson. The good coverage properties of this interval, like the Wilson, breaks down when the skewness coefficient of $p\left({\text{E}\left[{\left(p-P\right)}^3\right]/\left[\text{Var}\left(p\right)\right]}^{3/2}\right)$ gets too large in absolute value, how large has yet to be determined.

Finally, SAS/STAT (SAS Institute Inc., 2010) offers a Wilson coverage interval for estimated proportions in its SURVEYFREQ procedure. The procedure’s method of adjusting the effective sample size, which can $–$ and should $–$ be turned off, is not related to the $\tilde{n}$ discussed here. Instead, it is based on an ad-hoc $t-$ adjustment that sadly is not related to the variance of the denominator variance of the Wilson pivotal.

Acknowledgements

The author thanks Per Gösta Andersson for introducing me to this area of research and an anonymous referee for correcting errors in a previous version of the manuscript. Remaining errors are my own.

References

Brown, L.D., Cai, T. and Dasgupta, A. (2001). Interval estimation for a binomial proportion. Statistical Science, 16, 101-133.

Kott, P.S., and Carr, D.A. (1997). Developing an estimation strategy for a pesticide data program. Journal of Official Statistics, 13, 367-383.

Kott, P.S., and Liu, Y.K. (2009). One-sided coverage intervals for a proportion estimated from a stratified simple random sample. International Statistical Review/Revue Internationale de Statistique, 77, 251-265.

Kott, P.S., Andersson, P.G. and Nerman, O. (2001). Two-sided coverage intervals for small proportion based on survey data. Presented at Federal Committee on Statistical Methodology Research Conference, Washington, DC. http://fcsm.sites.usa.gov/files/2014/05/2001FCSM_Kott.pdf.

SAS Institute Inc. (2010). SAS/STAT^® 9.22 User’s Guide. Cary, NC: SAS Institute Inc. http://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#statug_surveyfreq_a0000000252.htm.

WesVar (2007). WesVar^® 4.3 Users’ Guide, B28-B29.

Wilson, E.B. (1927). Probable inference, the law of succession, and statistical inference. Journal of the American Statistical Association, 22, 209-212.

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