Package 'binomSamSize' reference manual

Title:	Confidence Intervals and Sample Size Determination for a Binomial Proportion under Simple Random Sampling and Pooled Sampling
Description:	A suite of functions to compute confidence intervals and necessary sample sizes for the parameter p of the Bernoulli B(p) distribution under simple random sampling or under pooled sampling. Such computations are e.g. of interest when investigating the incidence or prevalence in populations. The package contains functions to compute coverage probabilities and coverage coefficients of the provided confidence intervals procedures. Sample size calculations are based on expected length.
Authors:	Michael Hoehle [aut, cre], Wei Liu [ctb]
Maintainer:	Michael Hoehle <[email protected]>
License:	GPL-3
Version:	0.1-6
Built:	2024-10-15 04:00:02 UTC
Source:	https://github.com/mhoehle/binomSamSize

Confidence intervals and sample size determination for a binomial proportion under simple random sampling and pooled sampling

Description

A suite of functions to compute confidence intervals and necessary sample sizes for the parameter $p$ of the Bernoulli $B(p)$ distribution under simple random sampling or under pooled sampling. Such computations are e.g. of interest when investigating the incidence or prevalence in populations.

The package contains functions to compute coverage probabilities and coverage coefficients of the provided confidence intervals procedures. Sample size calculations are based on expected length and coverage probabilities of the resulting confidence intervals.

Details

Package:	binomSamSize
Type:	Package
Version:	0.1-3
Date:	2013-12-10
License:	GPL-3
LazyLoad:	yes

The packages relies heavily on the functions in the package binom by Sundar Dorai-Raj.

Other R packages containing similar functionality for calculating sample sizes for pooled sampling is the binGroup package. Also the epiR has further functionality for computing sample sizes for tests with sensitivity and specificity different from 1.

Functionality for pooled prevalence calculations with a nice web interface can also be found at http://www.ausvet.com.au/pprev/content.php?page=home

Note that the package is still in a development stage. As a consequence, use the results of the package with care because bugs, inconsistencies and errors might exist. In case you find bugs please do not hesitate to report them to the package maintainer.

Author(s)

Michael Höhle with contributions by Wei Liu

Maintainer: Michael Höhle <[email protected]>

References

Piegorsch, W. W. (2004), Sample sizes for improved binomial confidence intervals, Computational Statistics and Data Analysis, 46:309–316.

Calculate fixed width confidence interval for binomial proportion

Description

Calculate a fixed width confidence interval for the the binomial proportion based on one observation from the binomial distribution

Usage

binom.liubailey(x, n, d, lambda=0, conf.level=0.95)
binom.liubailey(x, n, d, lambda=0, conf.level=0.95)

Arguments

`x`	Vector of number of successes in the binomial experiment.
`n`	Vector of number of independent trials in the binomial experiment.
`conf.level`	The level of confidence to be used in the confidence interval
`d`	half width of the confidence interval
`lambda`	Shrinkage factor. `lambda`=0 corresponds to simple $\hat{p} \pm d$ interval.

Details

The confidence interval is as suggested in equation (3.1) of Liu & Bailey (2002).

$(\hat{p}_l,\hat{p}_u) = (C_n(\hat{p}_n)-d,C_n(\hat{p}_n)+d)$

The exact form is as follows: Let $z$ be the appropriate $(1-\code{conf.level})/2$ quantile of the standard normal distribution the interval with shrinkage towards 0.5 is given by:

$(\hat{p}_l,\hat{p}_u) = \hat{p}_n + \frac{\lambda z^2 (0.5-\hat{p}_n)}{n+z^2} \pm d$

The interval is then expanded to a full length of $2d$ using the following transformation:

$\hat{p}_l^* = \max(0,\min( 1-2d, \hat{p}_l))$

$\hat{p}_u^* = \min(1,\max( 2d, \hat{p}_u))$

As a consequence, the computed interval will always have length $2d$ .

If fixed length is a desired property of your CI then this is a way to go. However, the Liu and Bailey (2002) confidence intervals can have a low coverage coefficients when $n$ is very small compared to $d$ . When using the sample size computation procedure in ciss.liubailey one however ensures that $n$ is large enough for the selected $d$ to guarantee the required coverage coefficient. Thus, one should use binom.liubailey in connection with ciss.liubailey.

Value

A data.frame containing the observed proportions and the lower and upper bounds of the confidence interval. The style is similar to the binom.confint function of the binom package

Author(s)

M. Höhle

References

Liu, W. and Bailey, B.J.R. (2002), Sample size determination for constructing a constant width confidence interval for a binomial success probability. Statistics and Probability Letters, 56(1):1-5.

Examples

binom.liubailey(x=0:20,n=20, d=0.1, lambda=0)

#Compute coverage of this interval
cov <- coverage( binom.liubailey, n=20, alpha=0.05, d=0.1, lambda=0,
                 p.grid=seq(0,1,length=1000))

plot(cov,type="l")

binom.liubailey(x=0:20,n=20, d=0.1, lambda=0)

#Compute coverage of this interval
cov <- coverage( binom.liubailey, n=20, alpha=0.05, d=0.1, lambda=0,
                 p.grid=seq(0,1,length=1000))

plot(cov,type="l")

Calculate mid-p confidence interval for binomial proportion

Description

Calculate mid-p confidence interval for the the binomial proportion based on one observation from the binomial distribution

Usage

binom.midp(x, n, conf.level=0.95)
binom.midp(x, n, conf.level=0.95)

Arguments

`x`	Vector of number of successes in the binomial experiment.
`n`	Vector of number of independent trials in the binomial experiment.
`conf.level`	The level of confidence to be used in the confidence interval

Details

The function uses uniroot to determine the upper and lower bounds of the mid-p confidence interval.

The lower bound $p_l$ is found as the solution to the equation

$\frac{1}{2} f(x;n,p_l) + (1-F(x;m,p_l)) = \frac{\alpha}{2}$

The upper bound $p_u$ is found as the solution to the equation

$\frac{1}{2} f(x;n,p_u) + F(x-1;m,p_u) = \frac{\alpha}{2}$

In case x=n then the upper bound is 1.

Value

A data.frame containing the observed proportions and the lower and upper bounds of the confidence interval. The style is similar to the binom.confint function of the binom package

Author(s)

M. Höhle

References

S. E. Vollset (1993), Confidence intervals for a binomial proportion, Statistics in Medicine, 12, 809–824

Fosage, G.T. (2005) Modified exact sample size for a binomial proportion with special emphasis on diagnostic test parameter estimation, Statistics in Medicine 24(18):2857-66.

A. Agresti and A. Gottard (2005), Comment: Randomized Confidence Intervals and the Mid-P Approach, Statistical Science, 20(4):367–371

Examples

binom.midp(x=0:10,n=10)
binom.midp(x=0:5,n=5,conf.level=0.9)
binom.midp(x=0:10,n=10)
binom.midp(x=0:5,n=5,conf.level=0.9)

General purpose sample size calculation based on confidence interval widths

Description

Calculate necessary sample size for estimating a binomial proportion with the confidence interval computed by an arbitrary binom.confint function

Usage

ciss.binom(p0, d, alpha=0.05, ci.fun=binom.confint,
           np02x = function(n, p0) round(n*p0), verbose=FALSE,
           nStart=1,nMax=1e6,...)
ciss.binom(p0, d, alpha=0.05, ci.fun=binom.confint,
           np02x = function(n, p0) round(n*p0), verbose=FALSE,
           nStart=1,nMax=1e6,...)

Arguments

`p0`	hypothesized value of the parameter $p$ in the binomial distribution proportion. This is an upper bound if `p0` is below 1/2, and a lower bound if `p0` is above 1/2.
`d`	half width of the confidence interval. Note: The CI is not necessarily symmetric about the estimate so we just look at its width as determine by $d = 1/2*(CI_upper - CI_lower)$ .
`alpha`	a two-sided $(1-\alpha)\cdot 100\%$ confidence interval is computed
`ci.fun`	Any `binom.confint` like confidence interval computing function. The default is the `binom.confint` function itself. In this case one would have to specify the appropriate method to use using the `method` argument of the `binom.confint` function.
`np02x`	A function specifying how to calculate the value of $x$ which results in an estimator of the proportion being as close as possible to the anticipated value $p_0$ . Typically the value is obtained by rounding the result of $x\cdot p_0$ .
`verbose`	If `TRUE`, additional output of the computations are shown. The default is `FALSE`.
`nStart`	Value where to start the search. The default `n=1` can sometimes lead to wrong answers, e.g. for the Wald-type interval
`nMax`	Max value of the sample size $n$ to try in the iterative search. See details
`...`	Additional arguments sent to `ci.fun` function

Details

Given a pre set $\alpha$ -level and an anticipated value of $p$ , say $p_0$ , the objective is to find the minimum sample size $n$ such that the confidence interval will lead to an interval of length $2\cdot d$ .

Using ciss.binom this is done in a general purpose way by performing an iterative search for the sample size. Starting from $n=nStart$ the appropriate $x$ value, computed as round(x*p0), is found. For this integer $x$ and the current $n$ the corresponding confidence interval is computed using the function ci.fun. This function has to deliver the same type of result as the binom.confint function, i.e. a data frame containing the arguments lower and upper containing the borders of the confidence interval.

The sample size is iteratively increased until the obtained confidence interval has a length smaller than $2\cdot d$ . This might take a while if $n$ is large. It is possible to speed up the search if an appropriate nStart is provided.

A brute force search is used within the function. Note that for many of the confidence intervals explicit expressions exists to calculate the necessary sample size.

Value

the necessary sample size n

Author(s)

M. HÃ¶hle

Examples

#Compute the classical Wald-type interval using brute force search
#Note that nStart=2 needs to be called, because the Wald-intervals
#for x=round(1*0.5)=0 is too short.
ciss.binom(p0=1/2, d=0.1, alpha=0.05, method="asymptotic",nStart=2)
#This could of course be done easier
ciss.wald(p0=1/2, d=0.1, alpha=0.05)

#Same for the Wilson intervals
ciss.binom(p0=1/2, d=0.1, alpha=0.05, method="wilson")
ciss.wilson(p0=1/2, d=0.1, alpha=0.05)

#Now the mid-p intervals 
ciss.binom(p0=1/2, d=0.1, alpha=0.05, ci.fun=binom.midp)
#This search in Fosgate (2005) is a bit different, because interest
#is not directly in the length, but the length is used to derive
#the upper and lower limits and then a search is performed until
#the required alpha level is done. The difference is negliable
ciss.midp(p0=1/2, d=0.1, alpha=0.05)

#Another situation where no closed formula exists
ciss.binom(p0=1/2, d=0.1, alpha=0.05, method="lrt")

#Pooled samples. Now np02x is a func taking three arguments
#The k argument is provided as additional argument
np02x <- function(n,p0,k) round( (1-(1-p0)^k)*n )
ciss.binom( p0=0.1, d=0.05, alpha=0.05, ci.fun=poolbinom.lrt,
            np02x=np02x, k=10,verbose=TRUE)

#Compute the classical Wald-type interval using brute force search
#Note that nStart=2 needs to be called, because the Wald-intervals
#for x=round(1*0.5)=0 is too short.
ciss.binom(p0=1/2, d=0.1, alpha=0.05, method="asymptotic",nStart=2)
#This could of course be done easier
ciss.wald(p0=1/2, d=0.1, alpha=0.05)

#Same for the Wilson intervals
ciss.binom(p0=1/2, d=0.1, alpha=0.05, method="wilson")
ciss.wilson(p0=1/2, d=0.1, alpha=0.05)

#Now the mid-p intervals 
ciss.binom(p0=1/2, d=0.1, alpha=0.05, ci.fun=binom.midp)
#This search in Fosgate (2005) is a bit different, because interest
#is not directly in the length, but the length is used to derive
#the upper and lower limits and then a search is performed until
#the required alpha level is done. The difference is negliable
ciss.midp(p0=1/2, d=0.1, alpha=0.05)

#Another situation where no closed formula exists
ciss.binom(p0=1/2, d=0.1, alpha=0.05, method="lrt")

#Pooled samples. Now np02x is a func taking three arguments
#The k argument is provided as additional argument
np02x <- function(n,p0,k) round( (1-(1-p0)^k)*n )
ciss.binom( p0=0.1, d=0.05, alpha=0.05, ci.fun=poolbinom.lrt,
            np02x=np02x, k=10,verbose=TRUE)

Sample size calculations using the Liu and Bailey (2002) approach

Description

Calculate sample size for a binomial proportion based on the confidence interval width specification in Liu and Bailey (2002).

Usage

ciss.liubailey(alpha, d, lambda.grid = 0:30)
ciss.liubailey(alpha, d, lambda.grid = 0:30)

Arguments

`alpha`	a $(1-\alpha/2)\cdot 100\%$ confidence interval is computed
`d`	half width of the confidence interval
`lambda.grid`	range of lambda values to try

Details

The objective is to find the minimum sample size $n$ so that the minimum coverage probability (aka. as the coverage coefficient) of the confidence interval for the binomial parameter is larger than $1-\alpha$ . In the present approach the confidence interval is of form

$(C_n(\hat{p}_n)-d,C_n(\hat{p}_n)+d)$

as suggested in equation (3.1) of Liu & Bailey (2002):

$(\hat{p}_l,\hat{p}_u) = \hat{p}_n + \frac{\lambda z^2 (0.5-\hat{p}_n)}{n+z^2} \pm d$

where $\hat{p}_n = x/n$ . The interval is then expanded to a full length of $2d$ using the following transformation:

$\hat{p}_l^* = \max(0,\min( 1-2d, \hat{p}_l))$

$\hat{p}_u^* = \min(1,\max( 2d, \hat{p}_u))$

As a consequence, the computed interval will always have length $2d$ .

Given d, fixed lambda and a sample size n, the proportion $p$ in [0,1] where the coverage probability is minimum is computed. The sample size is then gradually increased until this minimum coverage probability becomes larger than $1-\alpha$ . We then change the value of $\lambda$ , and search the minimum sample size that guarantee the $1-\alpha$ confidence level for this lambda value. The smallest minimum sample size over a set of lambda values in lambda.grid is then used as the required sample size; this sample size and the corresponding lambda value are used to calculate the confidence interval given above.

For a general overview of coverage probabilities of confidence intervals for a binomial proportion see Agresti and Coull (1998). Once actual binomial data are obtained the function binom.liubailey can be used to compute the actual confidence interval.

The R function code calls the original Fortran code developed for the Liu and Bailey (2002) article. NAG calls were replaced by R API calls and an R wrapper calling the code as a subroutine was created.

Value

a vector containing the following three elements

`nstar`	sample size at most favorable lambda value in `lambda.grid`
`cp`	coverage probability
`lambda`	value in lambda.grid giving the lowest `nstar` value

Author(s)

M. Höhle and W. Liu

References

Agresti, A. and Coull, B.A. (1998), Approximate is Better than "Exact" for Interval Estimation of Binomial Proportions, The American Statistician, 52(2):119-126.

Liu, W. and Bailey, B.J.R. (2002), Sample size determination for constructing a constant width confidence interval for a binomial success probability. Statistics and Probability Letters, 56(1):1-5.

Examples

ciss.liubailey(alpha=0.1,d=0.05)
ciss.liubailey(alpha=0.1,d=0.05,lambda.grid=5)
ciss.liubailey(alpha=0.1,d=0.05)
ciss.liubailey(alpha=0.1,d=0.05,lambda.grid=5)

Sample size calculations using the Fosgate (2005) approach

Description

Calculate sample size for a binomial proportion based on a mid-p confidence interval width specification.

Usage

ciss.midp(p0, d, alpha, nMax=1e6)
ciss.midp(p0, d, alpha, nMax=1e6)

Arguments

`p0`	hypothesized upper bound (if below 0.5, if above 0.5 then lower bound) on the parameter p in the binomial distribution
`alpha`	an $(1-\alpha/2)\cdot 100\%$ confidence interval is computed
`d`	half width of the confidence interval
`nMax`	Largest n to check. Interrupt iterations when this value is reached

Details

Fosgate (2005) discusses the need for improved sample size calculations in cases where the binomial proportion is close to 0 and 1. To improve on this, calculation on confidence intervals based on the mid-p method are suggested where computation of the upper and lower limit are combined into one formula. Given lower and upper bounds $p_l$ and $p_u$ of the (1-alpha)*100% confidence interval, one finds the sample size $n$ as the solution to

$\frac{1}{2} f(x;n,p_l) + \frac{1}{2} f(x;n,p_u) + (1 - F(x;n,p_l)) + F(x-1;m,p_u) = \alpha$

where $f(x;n,p)$ denotes the probability mass function (pmf) and $F(x;n,p)$ the (cumulative) distribution function of the binomial distribution with size $n$ and proportion $p$ evaluated at $x$ . The function then returns $\lceil n \rceil$ . Note that in this approach $(p_l,p_u) = p_0 \pm d$ , which has to be a subset of $(0,1)$ . Another option would be to choose the lower and upper independent specifically.

In the above, $x$ is found as the integer value, such that $x/n$ is as close as possible to the hypothesized value p0 as possible.

An alternative approach to determine sample sizes based on the mid-p approach is to manually find the sample size $n$ such that the interval obtained by binom.midp has a length less than $2\cdot d$ .

Value

the necessary sample size $n$

Author(s)

M. Höhle

References

Fosage, G.T. (2005) Modified exact sample size for a binomial proportion with special emphasis on diagnostic test parameter estimation, Statistics in Medicine 24(18):2857-66.

Examples

#Fosgate approach
ciss.midp(p0=0.2,alpha=0.05,d=0.05)
#Iterative increase of n using the general purpose function
ciss.binom( p0=0.2, alpha=0.05, ci.fun=binom.midp, d=0.05)

#Fosgate approach
ciss.midp(p0=0.2,alpha=0.05,d=0.05)
#Iterative increase of n using the general purpose function
ciss.binom( p0=0.2, alpha=0.05, ci.fun=binom.midp, d=0.05)

Sample size calculations for fixed pool size and perfect test

Description

Calculate sample size for a binomial proportion based on Wald type confidence interval for pooled samples using a perfect test

Usage

ciss.pool.wald(pi0, alpha, d, k)
ciss.pool.wald(pi0, alpha, d, k)

Arguments

`pi0`	hypothesized upper bound (if below 0.5, if above 0.5 then lower bound) on the parameter $\pi_0$
`alpha`	an $(1-\alpha/2)\cdot 100\%$ confidence interval is computed
`d`	half width of the confidence interval
`k`	The pool size

Details

Sample size calculation based on width of Wald confidence intervals for pooled sample. The equation is

$n = \left\lceil \left( \frac{z_{1-\alpha/2} (1-\pi_0)}{d k} \right)^2 \cdot \left((1-\pi_0)^{-k} - 1\right) \right\rceil$

Value

the necessary sample size $n$

Author(s)

M. Höhle

References

D. D. Worlund and G. Taylor (1983), Estimation of Disease Incidence in Fish Populations, Can. J. Fish. Aquat. Sci., 40:2194-2197.

Examples

k <- 1:50
n <- sapply(k, function(k) ciss.pool.wald(pi0=0.1, alpha=0.05,k=k, d=0.05))
#sizes <- cbind(k=k, n=n, N=n*k)
plot(k, n, type="s",xlab="Pool size",ylab="Number of pools")
plot(k*n,n,type="s",xlab="Total size",ylab="Number of pools")

ciss.pool.wald(pi0=0.1, d=0.01, alpha=0.05, k=10)
#Compare with ciss.binom function
np02x <- function(n,p0,k) round( (1-(1-p0)^k)*n )
(n <- ciss.binom( p0=0.1, d=0.01, alpha=0.05, ci.fun=poolbinom.wald,
            np02x=np02x, k=10))

k <- 1:50
n <- sapply(k, function(k) ciss.pool.wald(pi0=0.1, alpha=0.05,k=k, d=0.05))
#sizes <- cbind(k=k, n=n, N=n*k)
plot(k, n, type="s",xlab="Pool size",ylab="Number of pools")
plot(k*n,n,type="s",xlab="Total size",ylab="Number of pools")

ciss.pool.wald(pi0=0.1, d=0.01, alpha=0.05, k=10)
#Compare with ciss.binom function
np02x <- function(n,p0,k) round( (1-(1-p0)^k)*n )
(n <- ciss.binom( p0=0.1, d=0.01, alpha=0.05, ci.fun=poolbinom.wald,
            np02x=np02x, k=10))

Sample sizes for improved binomial confidence intervals

Description

Calculate sample size for a binomial parameter enhancing the traditional Wald-type interval

Usage

ciss.wald(p0, d, alpha)
ciss.wilson(p0, d, alpha)
ciss.agresticoull(p0, d, alpha)
ciss.wald(p0, d, alpha)
ciss.wilson(p0, d, alpha)
ciss.agresticoull(p0, d, alpha)

Arguments

`p0`	hypothesized upper bound (if below 0.5, if above 0.5 then lower bound) on the parameter p in the binomial distribution
`alpha`	an $(1-\alpha/2)\cdot 100\%$ confidence interval is computed
`d`	half width of the confidence interval

Details

The work in Piegorsch (2004) gives a number of formulas enhancing the traditional Wald-type interval.

Value

the necessary sample size n

Author(s)

M. Höhle

References

Piegorsch, W. W. (2004), Sample sizes for improved binomial confidence intervals, Computational Statistics and Data Analysis, 46:309–316.

Examples


#Simple calculation at one proportion (worst case)
ciss.wald(p0=0.5,alpha=0.1,d=0.05)

#Evaluate for a grid of hypothesized proportion
p.grid <- seq(0,0.5,length=100)
cissfuns <- list(ciss.wald, ciss.wilson, ciss.agresticoull)
ns <- sapply(p.grid, function(p) {
  unlist(lapply(cissfuns, function(f) f(p, d=0.1, alpha=0.05)))
})

matplot(p.grid, t(ns),type="l",xlab=expression(p[0]),ylab="n",lwd=2)
legend(x="topleft", c("Wald", "Wilson","Agresti-Coull"), col=1:3, lty=1:3,lwd=2)

#Simple calculation at one proportion (worst case)
ciss.wald(p0=0.5,alpha=0.1,d=0.05)

#Evaluate for a grid of hypothesized proportion
p.grid <- seq(0,0.5,length=100)
cissfuns <- list(ciss.wald, ciss.wilson, ciss.agresticoull)
ns <- sapply(p.grid, function(p) {
  unlist(lapply(cissfuns, function(f) f(p, d=0.1, alpha=0.05)))
})

matplot(p.grid, t(ns),type="l",xlab=expression(p[0]),ylab="n",lwd=2)
legend(x="topleft", c("Wald", "Wilson","Agresti-Coull"), col=1:3, lty=1:3,lwd=2)

Calculate coverage probability for a binomial proportion confidence interval scheme

Description

For a given true value of the proportion compute the coverage probability of the confidence interval

Usage

coverage(ci.fun, n, alpha=0.05, p.grid=NULL,interval=c(0,1),
         pmfX=function(k,n,p) dbinom(k,size=n,prob=p), ...)
## S3 method for class 'coverage'
plot(x, y=NULL, ...)
coverage(ci.fun, n, alpha=0.05, p.grid=NULL,interval=c(0,1),
         pmfX=function(k,n,p) dbinom(k,size=n,prob=p), ...)
## S3 method for class 'coverage'
plot(x, y=NULL, ...)

Arguments

`ci.fun`	`binom.confint` like function which computes confidence intervals for a binomial proportion.
`n`	Sample size of the binomial distribution.
`alpha`	Level of significance, $1-\alpha$ is the confidence level.
`p.grid`	Vector of proportions where to evaluate the confidence interval function. If `NULL` all those values where the minimum coverage probability can occur is taken. If not `NULL` then the union between `p.grid` and these values is taken.
`interval`	Vector of length two specifying lower and upper border of an interval of interest for the proportion. The intersection of the above grid and this interval is taken.
`pmfX`	A function based on the arguments `k`, `n` and `p`, giving the probability mass function (pmf) $f(x;n,p)=P(X=k;n,p)$ of $X$ . Typically, this will be the pmf of the binomial distribution.
`x`	An object of class `coverage`
`y`	Not used
`...`	Further arguments to be sent to `ci.fun` or the plot function

Details

Compute coverage probabilities for each proportion in p.grid. See actual function code for the exact details, which p.grid is actually chosen.

Value

An object of class coverage containing coverage probabilities, coverage coefficient and more.

Author(s)

M. Höhle

References

Agresti, A. and Coull, B.A. (1998), Approximate is Better than "Exact" for Interval Estimation of Binomial Proportions, The American Statistician, 52(2):119-126.

Examples

#Show coverage of Liu & Bailey interval
cov <- coverage( binom.liubailey, n=100, alpha=0.05,
           p.grid=seq(0,1,length=1000), interval=c(0,1), lambda=0, d=0.1)
plot(cov, type="l")

#Now for some more advanced stuff. Investigate coverage of pooled
#sample size estimators
kk <- 10
nn <- 20
ci.funs <- list(poolbinom.wald, poolbinom.logit, poolbinom.lrt)
covs <- lapply( ci.funs, function(f) {
 coverage( f, n=nn, k=kk, alpha=0.05, p.grid=seq(0,1,length=100),
pmfX=function(k,n,p) dbinom(k,size=n, p=1-(1-p)^kk))
})

par(mfrow=c(3,1))
plot(covs[[1]],type="l",main="Wald",ylim=c(0.8,1))
lines(c(0,1),rep(0.95,2),lty=2,col=2)
plot(covs[[2]],type="l",main="Logit")#,ylim=c(0.8,1))
lines(c(0,1),rep(0.95,2),lty=2,col=2)
plot(covs[[3]],type="l",main="LRT",ylim=c(0.8,1))
lines(c(0,1),rep(0.95,2),lty=2,col=2)

poolbinom.wald(x=1,n=nn,k=kk)
poolbinom.logit(x=1,n=nn,k=kk)
poolbinom.lrt(x=1,n=nn,k=kk)

#Show coverage of Liu & Bailey interval
cov <- coverage( binom.liubailey, n=100, alpha=0.05,
           p.grid=seq(0,1,length=1000), interval=c(0,1), lambda=0, d=0.1)
plot(cov, type="l")

#Now for some more advanced stuff. Investigate coverage of pooled
#sample size estimators
kk <- 10
nn <- 20
ci.funs <- list(poolbinom.wald, poolbinom.logit, poolbinom.lrt)
covs <- lapply( ci.funs, function(f) {
 coverage( f, n=nn, k=kk, alpha=0.05, p.grid=seq(0,1,length=100),
pmfX=function(k,n,p) dbinom(k,size=n, p=1-(1-p)^kk))
})

par(mfrow=c(3,1))
plot(covs[[1]],type="l",main="Wald",ylim=c(0.8,1))
lines(c(0,1),rep(0.95,2),lty=2,col=2)
plot(covs[[2]],type="l",main="Logit")#,ylim=c(0.8,1))
lines(c(0,1),rep(0.95,2),lty=2,col=2)
plot(covs[[3]],type="l",main="LRT",ylim=c(0.8,1))
lines(c(0,1),rep(0.95,2),lty=2,col=2)

poolbinom.wald(x=1,n=nn,k=kk)
poolbinom.logit(x=1,n=nn,k=kk)
poolbinom.lrt(x=1,n=nn,k=kk)

Calculate logit based confidence interval for binomial proportion for pooled samples

Description

Calculate logit based confidence interval for the the Bernoulli proportion of $k\cdot n$ individuals, which are pooled into $n$ pools each of size $k$ . Observed is the number of positive pools $x$ .

Usage

poolbinom.wald(x, k, n, conf.level=0.95)
poolbinom.logit(x, k, n, conf.level=0.95)
poolbinom.wald(x, k, n, conf.level=0.95)
poolbinom.logit(x, k, n, conf.level=0.95)

Arguments

`x`	Number of positive pools (can be a vector).
`k`	Pool size (can be a vector).
`n`	Number of pools (can be a vector).
`conf.level`	The level of confidence to be used in the confidence interval

Details

Assume the individual probability of experiencing the event for each of $k\cdot n$ individuals is $\pi$ , i.e. the response is Bernoulli distributed $X_i \sim B(\pi)$ . For example $\pi$ could be the prevalence of a disease in veterinary epidemiology.

Now, instead of considering each individual the $k\cdot n$ samples are pooled into $n$ pools each of size $k$ . A pool is positive if there is at least one positive in the pool. Let X be the number of positive pools. Then

$X \sim Bin(n, 1-(1-\pi)^k)$

The present function computes an estimator and confidence interval for $\pi$ by computing the MLE and standard error for $\hat{\pi}$ . A Wald confidence interval is formed using $\hat{\pi} \pm z_{1-\alpha/2}\cdot se(\hat{\pi})$ . In case of poolbinom.logit a logit transformation is used, i.e. the standard error for $logit(\hat{\pi})$ is computed and the Wald-CI is derived on the logit-scale which is then backtransformed using the inverse logit function. In case $x=0$ or $x=n$ the logit of $\hat{\pi}$ is not defined and hence the confidence interval is not defined in these two situation. To fix the problem we use the intervals $(0, \hat{\pi}_u(x=0))$ and $(\hat{\pi}_l(x=n),1)$ , respectively, where $\pi_u$ and $\pi_o$ are the respective borders of a corresponding LRT interval.

The poolbinom.wald approach corresponds to method 2 in the Cowling et al. (1999). The logit transformation improves on this procedure, because the method ensures that the interval is in the range (0,1).

Value

A data.frame containing the observed proportions and the lower and upper bounds of the confidence interval. The style is similar to the binom.confint function of the binom package

Author(s)

M. Höhle

References

D. W. Cowling, I. A. Gardner, W. O. Johnson (1999), Comparison of methods for estimation of individual level prevalence based on pooled samples, Preventive Veterinary Medicine, 39:211–225

Examples

poolbinom.wald(x=0, k=10, n=34, conf.level=0.95)
poolbinom.logit(x=0:1, k=10, n=34, conf.level=0.95)
poolbinom.logit(x=1, k=seq(10,100,by=10), n=34, conf.level=0.95)
poolbinom.logit(x=0:34,k=1,n=34)
poolbinom.wald(x=0, k=10, n=34, conf.level=0.95)
poolbinom.logit(x=0:1, k=10, n=34, conf.level=0.95)
poolbinom.logit(x=1, k=seq(10,100,by=10), n=34, conf.level=0.95)
poolbinom.logit(x=0:34,k=1,n=34)

Calculate LRT based confidence interval for binomial proportion for pooled samples

Description

Calculate LRT based confidence interval for the Bernoulli proportion of $k\cdot n$ individuals, which are pooled into n pools each of size $k$ . Observed is the number of positive pools $x$ .

Usage

poolbinom.lrt(x, k, n, conf.level=0.95, bayes=FALSE, conf.adj=FALSE)
poolbinom.lrt(x, k, n, conf.level=0.95, bayes=FALSE, conf.adj=FALSE)

Arguments

`x`	Number of positive pools (can be a vector).
`k`	Pool size (can be a vector).
`n`	Number of pools (can be a vector).
`conf.level`	The level of confidence to be used in the confidence interval
`bayes`	See `binom.bayes`
`conf.adj`	See `binom.bayes`

Details

Compute LRT based intervals for the binomial response $X \sim Bin(n, \theta)$ , where $\theta = 1 - (1-\pi)^k$ . As a consequence,

$\pi = g(\theta) = 1 - (1-\pi)^{1/k}$

One then knows that the borders for $\pi$ are just transformations of the borders of theta using the above $g(\theta)$ function.

For further details about the pooling setup see poolbinom.logit.

Value

A data.frame containing the observed proportions and the lower and upper bounds of the confidence interval. The output is similar to the binom.confint function of the binom package

Author(s)

M. Höhle

Examples

binom.lrt(x=0:34,n=34)
poolbinom.lrt(x=0:34,k=1,n=34)
poolbinom.lrt(x=0:34,k=10,n=34)
binom.lrt(x=0:34,n=34)
poolbinom.lrt(x=0:34,k=1,n=34)
poolbinom.lrt(x=0:34,k=10,n=34)

Package 'binomSamSize'

Help Index

Confidence intervals and sample size determination for a binomial proportion under simple random sampling and pooled sampling

Description

Details

Author(s)

References

See Also

Calculate fixed width confidence interval for binomial proportion

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Calculate mid-p confidence interval for binomial proportion

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

General purpose sample size calculation based on confidence interval widths

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Sample size calculations using the Liu and Bailey (2002) approach

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Sample size calculations using the Fosgate (2005) approach

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Sample size calculations for fixed pool size and perfect test

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Sample sizes for improved binomial confidence intervals

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Calculate coverage probability for a binomial proportion confidence interval scheme

Description

Usage

Arguments