Description Introduction Proportion methods grouping Proportion function naming convention Reproducibility of reference papers References

Proportion: Let x denote the number of successes in n independent Bernoulli trials with X ~ Binomial (n, p) then phat = x/n denotes the sample proportion.

Objective of this package is to present interval estimation procedures for 'p'
outlined above in a more comprehensive way.Quality assessment procedures such as statistic
based on coverage probability, Expected length, Error, p-confidence and p-bias are also included.
Also, an array of Bayesian computations (Bayes factor, Empirical Bayesian,
Posterior predictive computation, and posterior probability) with conjugate prior are made available.
The proportion package provides three categories of important functions:
**Confidence Intervals**, **metrics on confidence intrvals** (coverage probability, length, p-confidence and
p-bias, error and long term power) and **other methods** (hypothesis
testing and general/simulation methods).

For finding confidence interval for p we have included

Methods based on the asymptotic normality of the sample proportion and estimating standard error

Exact methods based on inverting equal-tailed binomial tests of H0 : p = p0,

Methods based on likelihood ratios

Bayesian approaches with beta priors or other suitable priors.

The general guideline for finding functions are given below:

Short names for concepts: ci - Confidence Interval, covp - Coverage Probability, expl - Expected length (simulation), length - Sum of length, pCOpBI - p-Confidence and p-Bias, err - Error and long term power

Short names for methods: AS - ArcSine, LR - Likelihood Ratio, LT - Logit Wald, SC - Score (also know as Wilson), TW - Wald-T, WD - Wald, BA - Bayesian and EX - Exact in general form that includes Mid-P and Clopper-Pearson.

For adjusted methods "A" is added to the function name while "C" will be added if it is continuity corrected.

For generic functions BAF - Bayesian Factor, SIM - Simulation, GEN - Generic, PRE - Predicted, POS - Posterior

Combining the above you should be able to identify the function. For example, function for coverage probability (covp) using ArcSine (AS) method will be covpAS(). If we need the adjusted coverage probability (covp) using ArcSine (AS) method, then it will be covpAAS().

Wherever possible, results are consolidated for all

`x (0,1...n)`

and specific`x`

(function name succeeds with`x`

). For example, if we run`ciAS(n=5, alp=0.05)`

the output of`x=5`

will be the same as`ciASx(x=5, n=5,alp=0.05)`

. In the first case the output is printed for all the values of`x`

till`x=n`

.All refers to six approximate methods (Wald, Score, Likelihood Ratio, ArcSine, Logit Wald and Wald-T) - AAll (Adjusted All) refers to six methods adjusted with adding factor

`h`

(Wald, Score, Likelihood Ratio, ArcSine, Logit Wald and Wald-T)CAll (Continuity corrected All) refers to five methods (Wald, Score, ArcSine, Logit Wald and Wald-T) with continuity correction

`c`

Grouping functions for plots end with "g" (PlotciAllxg is the same as PlotciAllx, except the results are grouped by x)

For almost all the functions, corrosponding plot function is implemented, which plots the output in an apporiate graph. For example, the function

`covpAll()`

will give the numeric output for the coverage probability of the six approximate methods (see explanation of All above). Prefixing this with Plot makes it`PlotcovpAll()`

and will display the plot for the same six approximate methods.

To help the researcher reporduce results in existing papers we have taken six key papers (see references below) [3], [8], [9], [10], [11], [12] and reproduced the results and suggested further items to try. Details are in the vignette.

[1] 1993 Vollset SE. Confidence intervals for a binomial proportion. Statistics in Medicine: 12; 809 - 824.

[2] 1998 Agresti A and Coull BA. Approximate is better than "Exact" for interval estimation of binomial proportions. The American Statistician: 52; 119 - 126.

[3] 1998 Newcombe RG. Two-sided confidence intervals for the single proportion: Comparison of seven methods. Statistics in Medicine: 17; 857 - 872.

[4] 2001 Brown LD, Cai TT and DasGupta A. Interval estimation for a binomial proportion. Statistical Science: 16; 101 - 133.

[5] 2002 Pan W. Approximate confidence intervals for one proportion and difference of two proportions Computational Statistics and Data Analysis 40, 128, 143-157.

[6] 2008 Pires, A.M., Amado, C. Interval Estimators for a Binomial Proportion: Comparison of Twenty Methods. REVSTAT - Statistical Journal, 6, 165-197.

[7] 2014 Martin Andres, A. and Alvarez Hernandez. M. Two-tailed asymptotic inferences for a proportion. Journal of Applied Statistics, 41, 7, 1516-1529

[8] 2005 Vos PW and Hudson S. Evaluation Criteria for Discrete Confidence Intervals: Beyond Coverage and Length. The American Statistician: 59; 137 - 142.

[9] 2005 Joseph L and Reinhold C. Statistical Inference for Proportions American Journal of Radiologists 184; 1057 - 1064

[10] 2008 Zhou, X. H., Li, C.M. and Yang, Z. Improving interval estimation of binomial proportions. Phil. Trans. R. Soc. A, 366, 2405-2418

[11] 2012 Wei Yu, Xu Guo and Wangli Xua. An improved score interval with a modified midpoint for a binomial proportion, Journal of Statistical Computation and Simulation, 84, 5, 1-17

[12] 2008 Tuyl F, Gerlach R and Mengersen K . A comparison of Bayes-Laplace, Jeffreys, and Other Priors: The case of zero events. The American Statistician: 62; 40 - 44.

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