Package 'binGroup'

Description

This package provides methods for estimation and hypothesis testing of proportions in group testing designs. It involves methods for estimating a proportion in a single population (assuming sensitivity and specificity 1 in designs with equal group sizes), as well as hypothesis tests and functions for experimental design for this situation. For estimating one proportion or the difference of proportions, a number of confidence interval methods are included, which can deal with various different pool sizes. Further, regression methods are implemented for simple pooling and matrix pooling designs.

This package also provides methods for identification of positive items in a number of group testing algorithms. Optimal testing configurations can be found for hierarchical and array-based algorithms. Operating characteristics can be calculated for testing configurations across a wide variety of situations.

Details

1) One-sample case

Methods for calculating confidence intervals for a single population proportion from designs with equal group sizes (as described by Tebbs and Bilder, 2004 and Schaarschmidt, 2007) are implemented in the function bgtCI.

For the problem of choosing an adequate experimental design in the one-sample case with only one group size, the functions estDesign, sDesign, nDesign implement different iterative approaches, as examplified by Swallow (1985) and Schaarschmidt (2007).

If a confidence interval for a single proportion shall be calculated based on a design involving groups of different group sizes, a number of methods described by Hepworth (1999) is available in the function pooledBin. The exact method described by Hepworth (1996) is implemented in the function bgtvs.

2) Two-sample case

The function pooledBinDiff provides a number of confidence interval methods for estimating the difference of proportions from two independent samples, allowing for groups of different group size (Biggerstaff, 2008).

3) Regression models

Two approaches (by Vansteelandt et al., 2000 and Xie, 2001) to estimate parameters of group testing regression models can be applied by calling gtreg. Once fitted, corresponding methods to extract residuals, calculate predictions and summarize the parameter estimates (including hypotheses tests) are available in the S3 methods residuals.gt, predict.gt and summary.gt.

Group testing regression models in settings with matrix pooling (Xie, 2001) can be fit using gtreg.mp.

4) Identification using hierarchical and array-based group testing algorithms

The function OTC implements a number of group testing algorithms, described in Hitt et al. (2018), which calculate the operating characteristics and find the optimal testing configuration over a range of possible initial pool sizes and/or testing configurations (sets of subsequent pool sizes).

Author(s)

Boan Zhang, Christopher Bilder, Brad Biggerstaff, Brianna Hitt, Frank Schaarschmidt

Maintainer: Frank Schaarschmidt <[email protected]>

References

Biggerstaff, B.J. (2008): Confidence interval for the difference of proportions estmimated from pooled samples. Journal of Agricultural Biological and Environmental Statistics, 13(4), 478-496.

Hepworth, G. (1996) Exact confidence intervals for proportions estimated by group testing. Biometrics 52, 1134-1146.

Hepworth, G. (1999): Estimation of proportions by group testing. PhD Dissertation. Melbourne, Australia: The University of Melbourne.

Hitt, B., Bilder, C., Tebbs, J. & McMahan, C. (2018) The Optimal Group Size Controversy for Infectious Disease Testing: Much Ado About Nothing?!. Manuscript submitted for publication. http://www.chrisbilder.com/grouptesting

Schaarschmidt, F. (2007) Experimental design for one-sided confidence intervals or hypothesis tests in binomial group testing. Communications in Biometry and Crop Science 2 (1), 32-40. http://agrobiol.sggw.waw.pl/cbcs/

Swallow, W.H. (1985) Group testing for estimating infection rates and probabilities of disease transmission. Phytopathology 75 (8), 882-889.

Tebbs, J.M. & Bilder, C.R. (2004) Confidence interval procedures for the probability of disease transmission in multiple-vector-transfer designs. Journal of Agricultural, Biological and Environmental Statistics 9 (1), 75-90.

Vansteelandt, S., Goetghebeur, E., and Verstraeten, T. (2000) Regression models for disease prevalence with diagnostic tests on pools of serum samples, Biometrics, 56, 1126-1133.

Xie, M. (2001) Regression analysis of group testing samples, Statistics in Medicine, 20, 1957-1969.

Examples

# 1) One-sample problem

# 1.1) Confidence intervals for designs with equal group size (pool size),
# where 
#  n denotes the number of groups (pools),
#  s denotes the common group size (number of individuals pooled per group),
#  y denotes the number of groups tested positive.

# The following call reproduces the example given 
# by Tebbs and Bilder (2004) for the two-sided 95-percent 
# exact (Clopper-Pearson) interval:

bgtCI(n=24, y=3, s=7, conf.level=0.95,
 alternative="two.sided", method="CP")

# 1.2) Confidence intervals for designs with unequal group size (pool size):
# Keeping notation as above but allowing for (a limited number of) different
# group size s, the examples given in Hepworth (1996), Table 5 can be 
# reproduced by calling:

 bgtvs(n=c(2,3), s=c(5,2), y=c(0,0))
 bgtvs(n=c(2,3), s=c(5,2), y=c(0,1))

# The function pooledBin provides different methods for the same problem,
# where x is the number of positive groups, m is the size of the groups and
# n is the number of groups with the correesponding sizes:

pooledBin(x=c(0,1), m=c(5,2), n=c(2,3), ci.method="score")
pooledBin(x=c(0,1), m=c(5,2), n=c(2,3), ci.method="lrt")
pooledBin(x=c(0,1), m=c(5,2), n=c(2,3), ci.method="bc-skew-score")

# 1.3) For experimental design based on the bias of the point estimate,
# as proposed by Swallow (1985): The values in Table 1 (Swallow, 1985),
# p.885 can be reproduced by calling:

estDesign(n=10, smax=100, p.tr=0.001)
estDesign(n=10, smax=100, p.tr=0.01)

# 2) Two-sample comparison

# Assume a design, where pools 5, 1, 1, 30, and 20 pools of size 10, 4, 1, 25, 50, 
# respectively, are used to estimate the prevalence in two populations.
# In population 1, one out of 5 pools with 10 units is positive,
# while in population 2, two out of five pools with 10 units is positive as well as 
# the one pool with only 1 unit.
# The difference of proportions is to be estimated.

x1 <- c(1,0,0,0,0)
m1 <- c(10,4,1,25,50)
n1 <- c(5,1,1,30,20)

x2 <- c(2,0,1,0,0)
m2 <- c(10,4,1,25,50)
n2 <- c(5,1,1,30,20)

pooledBinDiff(x1=x1, m1=m1,x2=x2, m2=m2, n1=n1, n2=n2, ci.method="lrt")


# 3) Regression models

# 3.1) Fitting a regression model
# A HIV surveillance data (used by Vansteelandt et al. 2000)
# can be analysed for the dependence of HIV prevalence
# on covariates AGE and EDUC., with sensitivity and specificity
# assumed to be 0.9 each.

data(hivsurv)
fit1 <- gtreg(formula = groupres ~ AGE + EDUC., data = hivsurv,
  groupn = gnum, sens = 0.9, spec = 0.9, method = "Xie")
summary(fit1)


# 3.2) Fitting a regression model for matrix pooling data
# The function sim.mp is used to simulate a matrix pooling data set:

set.seed(9128)
sa1a<-sim.mp(par=c(-7,0.1), n.row=c(5,4), n.col=c(6,5),
 sens=0.95, spec=0.95)

str(sa1a)
sa1<-sa1a$dframe


## Not run: 
fit2 <- gtreg.mp(formula = cbind(col.resp, row.resp) ~ x, data = sa1, 
                 coln = coln, rown = rown, arrayn = arrayn, 
                 sens = 0.95, spec = 0.95, n.gibbs = 2000, trace = TRUE)
fit2
summary(fit2)

## End(Not run)

# 4) Identification using hierarchical and array-based group testing algorithms

# 4.1) Finding the optimal testing configuration over a range of initial 
# group sizes, using non-informative three-stage hierarchical testing, where 
#   p denotes the overall prevalence of disease,
#   Se denotes the sensitivity of the diagnostic test, 
#   Sp denotes the specificity of the diagnostic test,
#   group.sz denotes the range of initial pool sizes for consideration, and
#   obj.fn specifies the objective functions for which to find results.

# The following call reproduces results given by Hitt et al. (2018) for 
# informative three-stage hierarchical testing with an overall disease 
# prevalence of E(p_i) = 0.01 and sensitivity and specificity equal to 0.95.

# This example takes approximately 2.5 minutes to run.

## Not run: 
set.seed(1002)
results1 <- OTC(algorithm="ID3", p=0.01, Se=0.95, Sp=0.95, group.sz=3:40, 
obj.fn=c("ET", "MAR"), alpha=2)
results1$opt.ET$OTC
results1$opt.ET$ET/results1$opt.ET$OTC$Stage1
results1$opt.MAR$OTC
results1$opt.MAR$ET/results1$opt.MAR$OTC$Stage1

## End(Not run)

# 4.2) Finding the optimal testing configuration using non-informative
# array testing without master pooling

# The following call reproduces results given by Hitt et al. (2018) for
# non-informative array testing without master pooling with an overall
# disease prevalence of p=0.01 and sensitivity and specificity equal 
# to 0.90.

# This example takes approximately 7 minutes to run.

## Not run: 
results2 <- OTC(algorithm="A2", p=0.01, Se=0.90, Sp=0.90, group.sz=3:40, 
obj.fn=c("ET", "MAR"))
results2$opt.ET$OTC
results2$opt.ET$ET/results2$opt.ET$OTC$Array.sz
results2$opt.MAR$OTC
results2$opt.MAR$ET/results2$opt.MAR$OTC$Array.sz

## End(Not run)

Description

Calculate the accuracy measures for each individual in a pool used with informative Dorfman testing.

Usage

accuracy.dorf(p, se, sp)

Arguments

Details

This function calculates the pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for each individual belonging to a pool of size greater than or equal to one used with informative Dorfman testing. Calculations of these measures are done using the equations presented in McMahan et al. (2012).

Value

a list containing:

Author(s)

This function was originally written by Christopher S. McMahan for McMahan et al. (2012). The function was obtained from http://chrisbilder.com/grouptesting.

References

McMahan, C., Tebbs, J., Bilder, C. (2012). “Informative Dorfman Screening.” Biometrics, 68(1), 287–296. ISSN 0006341X, doi:10.1111/j.1541-0420.2011.01644.x, http://www.ncbi.nlm.nih.gov/pubmed/21762119.

See Also

http://chrisbilder.com/grouptesting

Other Informative Dorfman functions: characteristics.pool, inf.dorf.measures, opt.info.dorf, opt.pool.size, pool.specific.dorf, thresh.val.dorf

Examples

# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(8135)
p.vec <- p.vec.func(p=0.02, alpha=1, grp.sz=10)
accuracy.dorf(p=p.vec[1:3], se=0.90, sp=0.90)

Description

Calculate the expected number of tests and accuracy measures for each individual using array testing without master pooling

Usage

Array.Measures(p, se, sp)

Arguments

Details

This function calculates the operating characteristics for array testing without master pooling. Operating characteristics calculated are expected number of tests, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for each individual.

Value

A list containing:

Note

This function returns the pooling positive and negative predictive values for all individuals in the array even though these measures are diagnostic specific; i.e., PPV (NPV) should only be considered for those individuals who have tested positive (negative).

Author(s)

This function was originally written by Christopher S. McMahan for McMahan et al. (2012). The function was obtained from http://chrisbilder.com/grouptesting.

References

McMahan, C., Tebbs, J., Bilder, C. (2012). “Two-Dimensional Informative Array Testing.” Biometrics, 68(3), 793–804. ISSN 0006341X, doi:10.1111/j.1541-0420.2011.01726.x, http://www.ncbi.nlm.nih.gov/pubmed/22212007.

See Also

MasterPool.Array.Measures for calculating operating characteristics under non-informative array testing with master pooling, hierarchical.desc2 for three-stage hierarchical and non-informative two-stage hierarchical testing, and inf.dorf.measures for informative two-stage hierarchical testing. See p.vec.func for generating a vector of individual risk probabilities for informative group testing and Informative.array.prob for arranging individual risk probabilities in a matrix for informative array testing.

http://chrisbilder.com/grouptesting

Other Operating characteristic functions: MasterPool.Array.Measures, hierarchical.desc2, inf.dorf.measures

Examples

# Calculate the operating characteristics for 
#   non-informative array testing without master
#   pooling, with a 5x5 array and an overall disease 
#   risk of p = 0.02.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
p.mat <- matrix(data=0.02, ncol=5, nrow=5)
Array.Measures(p=p.mat, se=0.95, sp=0.95)

# Calculate the operating characteristics for 
#   informative array testing without master
#   pooling, with a 3x3 array and an overall disease
#   risk of p = 0.03 and alpha = 2.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(8791)
p.vec <- p.vec.func(p=0.03, alpha=2, grp.sz=9)
p.mat <- Informative.array.prob(prob.vec=p.vec, nr=3, 
nc=3, method="gd")
Array.Measures(p=p.mat, se=0.99, sp=0.99)

Description

Get the expected value of order statistics, E(p(i)), from a beta distribution by specifying an average probability and shape parameters for the beta distribution.

Usage

beta.dist(p = 0.05, alpha = 1, beta = NULL, grp.sz = 10,
  simul = FALSE, plot = FALSE, rel.tol = ifelse(a >= 1,
  .Machine$double.eps^0.25, .Machine$double.eps^0.1))

Arguments

Details

If alpha = 0, this function uses an extreme value distribution based on a Bernoulli(p) distribution to find the individual probabilities, p_i. If alpha is infinite, this function uses $p_i=p$ for all i.

If beta is not specified, it is calculated from the average probability p as $b=a*\frac{1}{p-1}$. If beta is specified, this function ignores p unless alpha is infinite.

Depending on the specified probability, alpha level, and overall group size, simulation may be necessary in order to generate the vector of individual probabilities. In this case, the user should specify simul=TRUE and set a seed in order to reproduce results. See Black et al. (2015) for additional details.

Value

A vector of ordered probabilities, p_i.

Author(s)

This function was originally written by Michael S. Black for Black et al. (2015). The function was obtained from http://chrisbilder.com/grouptesting.

References

Black, M., Bilder, C., Tebbs, J. (2015). “Optimal retesting configurations for hierarchical group testing.” Journal of the Royal Statistical Society. Series C: Applied Statistics, 64(4), 693–710. ISSN 14679876, doi:10.1111/rssc.12097.

See Also

http://chrisbilder.com/grouptesting

p.vec.func for generating a vector of individual risk probabilities for informative group testing (after checking whether simulation is needed) and Informative.array.prob for arranging a vector of individual risk probabilities in a matrix for informative array testing without master pooling.

Other Individual risk probability functions: Informative.array.prob, p.vec.func

Examples

# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(8791)
beta.dist(p=0.05, alpha=1, grp.sz=30)

# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(1002)
beta.dist(p=0.02, alpha=2, grp.sz=50, simul=TRUE)

# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(5732)
beta.dist(alpha=2, beta=5, grp.sz=15, plot=TRUE)

Description

Calculates the point estimate, the exact Clopper-Pearson and Blaker CI, the Score test derived Wilson and Agresti-Coull CI, the asymptotic second-order corrected interval fo Cai and the Wald CI for a single binomial proportion estimated from a binomial group testing trial. Assumes equal group sizes, an assay method classifying a group as positive if at least one unit in the group is positive, individuals units randomly assigned to the groups.

Usage

bgtCI(n, s, y, conf.level = 0.95,
 alternative = "two.sided", method = "CP")

Arguments

Details

This function allows the computation of confidence intervals for binomial group testing as described in Tebbs & Bilder (2004) and Schaarschmidt (2007). If an actual confidence level greater or equal to that specified in the conf.level argument shall always be guaranteed, the exact method of Clopper-Pearson (method="CP") can be recommended for one-sided and the improved method of Blaker (2000) (method="Blaker") can be recommended for two-sided hypotheses. If a mean confidence level close to that specified in the argument conf.level is required, but moderate violation of this level is acceptable, the Second-Order corrected (method="SOC"), Wilson Score (method="Score") or Agresti-Coull (method="AC") might be used (Brown, Cai, DasGupta, 2001; Cai 2005).

Value

A list containing:

Author(s)

Frank Schaarschmidt

References

Blaker H (2000). Confidence curves and improved exact confidence intervals for discrete distributions. The Canadian Journal of Statistics 28 (4), 783-798.

Brown LD, Cai TT, DasGupta A (2001). Interval estimation for a binomial proportion. Statistical Science 16 (2), 101-133.

Cai TT (2005). One-sided confidence intervals in discrete distributions. Journal of Statistical Planning and Inference 131, 63-88.

Schaarschmidt F (2007). Experimental design for one-sided confidence intervals or hypothesis tests in binomial group testing. Communications in Biometry and Crop Science 2 (1), 32-40. http://agrobiol.sggw.waw.pl/cbcs/

Tebbs JM & Bilder CR (2004). Confidence interval procedures for the probability of disease transmission in multiple-vector-transfer designs. Journal of Agricultural, Biological and Environmental Statistics, 9 (1), 75-90.

See Also

pooledBin for asymptotic confidence intervals and bgtvs for an exact confidence interval when designs with different group sizes are used

bgtTest: for hypothesis tests in binomial group testing

Examples

# See the example in Tebbs and Bilder (2004)
# the two.sided 95-percent 
# Clopper-Pearson as default method:

bgtCI(n=24,y=3,s=7)
bgtCI(n=24,y=3,s=7,conf.level=0.95,
 alternative="two.sided", method="CP")

# other methods:
# Blaker CI is exact but shorter
# than Clopper-Pearson, only two.sided

bgtCI(n=24,y=3,s=7, alternative="two.sided",
 method="Blaker")

# the asymptotic Wilson CI might even
# be shorter: 

bgtCI(n=24,y=3,s=7, alternative="two.sided",
 method="Score")

# one-sided confidence intervals:

bgtCI(n=24,y=3,s=7, alternative="less", method="CP")

# Wilson Score interval is less conservative 
bgtCI(n=24,y=3,s=7, alternative="less", method="Score")

# the second-order corrected CI is even shorter
# in this situation:
bgtCI(n=24,y=3,s=7, alternative="less", method="SOC")

Description

Closed calculation of the Power to reject a hypothesis in a binomial group testing experiment using confidence intervals for decision. Closed calculation of bias of the point estimator for a given experimental design n, s and the true, unknown proportion.

Usage

bgtPower(n, s, delta, p.hyp, conf.level = 0.95,
 method = "CP", alternative = "two.sided")

Arguments

Details

The power of a confidence interval here is defined as the probability that a confidence interval or limit excludes the threshold parameter (p.hyp) of the null hypothesis, as described in Schaarschmidt(2007). I.e., the null hypothesis H0: p >= p.hyp might be rejected, if an upper confidence limit for p does not contain p.hyp. Due to discreteness, the power does not increase monotone for increasing number of groups n or group size s, but exhibits local maxima and minima, depending on n,s, conf.level, p.hyp. The power can be identical for different methods, depending on the particular combination of n, s, p.hyp, conf.level. Note that coverage probability and power are not necessarily symmetric for upper and lower bounds of binomial CI, especially for Wald, Wilson Score and Agresti-Coull CI.

Additional to the power, bias of the point estimator is calculated according to Swallow (1985).

If vectors are specified for n, s, and (or) delta, a matrix will be constructed and power and bias are calculated for each line in this matrix.

Value

A matrix containing the columns

References

Schaarschmidt F (2007). Experimental design for one-sided confidence intervals or hypothesis tests in binomial group testing. Communications in Biometry and Crop Science 2 (1), 32-40. http://agrobiol.sggw.waw.pl/cbcs/

Swallow WH (1985). Group testing for estimating infection rates and probabilities of disease transmission. Phytopathology Vol.75, N.8, 882-889.

See Also

nDesign: stepwise increasing n (for a fixed group size s) until a certain power is reached within a restriction of bias of the estimator sDesign: stepwise increasing s (for a fixed number of groups) until a certain power is reached within a restriction of bias of the estimator estDesign: selection of an appropriate design to achieve minimal mean square error of the estimator

Examples

# Calculate the power for the design
# in the example given in Tebbs and Bilder(2004):
# n=24 groups each containing 7 insects
# if the true proportion of virus vectors
# in the population would be 0.04 (4 percent),
# the power to reject H0: p>=0.1 using an
# upper Clopper-Pearson ("CP") confidence interval
# can be calculated using the following call:

bgtPower(n=24, s=7, delta=0.06, p.hyp=0.1,
 conf.level=0.95, alternative="less", method="CP")


# c(), seq() or rep() might be used to explore development
# of power and bias for varying n, s, delta. How much can
# we decrease the number of groups (costly assays to be performed)
# by pooling the same number of 320 individuals to groups of
# increasing size without largely decreasing power?

bgtPower(n=c(320,160,80,64,40,32,20,10,5),
 s=c(1,2,4,5,8,10,16,32,64),
 delta=0.01, p.hyp=0.02)


# How does power develop for increasing differences
# delta between the true proportion and the threshold proportion?

bgtPower(n=50, s=10, delta=seq(from=0, to=0.01, by=0.001),
 p.hyp=0.01, method="CP")


# use a more liberal method:

bgtPower(n=50, s=10, delta=seq(from=0, to=0.01, by=0.001),
 p.hyp=0.01, method="SOC")

Description

Calculates p values for hypotheses tests of binomial proportions estimated from binomial group testing experiments against a threshold proportion in the hypotheses. Exact test, Score test and Wald test are available methods. Assumes equal group sizes, 100 percent sensitivity and specificity of the assays to test the groups, and individuals units randomly assigned to the groups with identical true probability of success.

Usage

bgtTest(n, y, s, p.hyp, alternative = "two.sided",
 method = "Exact")

Arguments

Value

A list containing:

References

Swallow WH (1985) Group testing for estimating infection rates and probabilities of disease transmission. Phytopathology 75 (8), 882-889.

Blyth, C and Still, H. (1983) Binomial confidence intervals. Journal of the American Statistical Association 78, 108-116.

Santner TJ and Duffy DE (1989) The statistical analysis of discrete data. Springer New York.

Remund KM, Dixon DA, Wright DL, Holden LR (2001) Statistical considerations on seed purity testing on transgenic traits. Seed Science Research (11), 101-119.

See Also

bgtCI for confidence intervals in binomial group testing

Examples

# Assume the experiment: Assays are performed on
# n=10 groups, each group is a bulk sample
# of s=100 individuals, aim is to show that 
# less than 0.5 percent ('p<0.005') of the units
# of the population show a detrimental trait (positive assay).
# The assay is senstive to show a positive result if only 1
# unit in the bulk sample of 100 units is positive.
# y=1 positive assay and 9 negative assays are observed.

bgtTest(n=10,y=1,s=100,alternative="less",method="Exact",p.hyp=0.005)

# The exact test corresponds to the 
# limits of the Clopper-Pearson confidence interval
# in the example of Tebbs and Bilder(2004):

bgtTest(n=24, y=3, s=7, alternative="two.sided",
 method="Exact", p.hyp=0.0543)

bgtTest(n=24, y=3, s=7, alternative="two.sided",
 method="Exact", p.hyp=0.0038)

# Further methods:

bgtTest(n=24, y=3, s=7, alternative="two.sided",
 method="Score", p.hyp=0.0516)

bgtTest(n=24, y=3, s=7, alternative="two.sided",
 method="Wald", p.hyp=0.0401)

Description

Calculates confidence intervals for a single proportion in binomial group testing if groups of different size are evaluated

Usage

bgtvs(n, s, y, conf.level = 0.95, alternative = "two.sided",
 maxiter = 100)

Arguments

Details

Hepworth (1996) describes methods for constructing confidence intervals in binomial group testing, if groups of different size are used. Currently, only the exact method (Hepworth, 1996, equation5, Table.5) is implemented. Note, that the algorithm becomes computationally very expensive if the number of different groups becomes larger than 3.

Value

A list containing

moreover, some of the input arguments.

Author(s)

Frank Schaarschmidt

References

Hepworth, G (1996): Exact confidence intervals for proportions estimated by group testing. Biometrics 52, 1134-1146.

See Also

pooledBin for asymptotic methods to calculate confidence intervals for one proportion in designs with a number of different pool sizes. Note that pooledBin can handle larger number of different pool sizes than bgtvs

Examples

# Consider a very simple example,
# given in Hepworth (1996), table 5:
# 2 groups each containing 5 units,
# and 3 groups, each containing 2 units

# In the first setting (n=2, s=5) y=1 positive group 
# has been observed, in the second setting (n=3, s=2),
# y=2 positive have been observed.

bgtvs(n=c(2,3), s=c(5,2), y=c(1,2)) 

###############################################

# Recalculate the example given in
# Hepworth (1996), table 5:

 bgtvs(n=c(2,3), s=c(5,2), y=c(0,0))
 bgtvs(n=c(2,3), s=c(5,2), y=c(0,1))
 bgtvs(n=c(2,3), s=c(5,2), y=c(0,2))
 bgtvs(n=c(2,3), s=c(5,2), y=c(0,3))
 bgtvs(n=c(2,3), s=c(5,2), y=c(1,0))
 bgtvs(n=c(2,3), s=c(5,2), y=c(1,1))
 bgtvs(n=c(2,3), s=c(5,2), y=c(1,2))
 bgtvs(n=c(2,3), s=c(5,2), y=c(1,3))
 bgtvs(n=c(2,3), s=c(5,2), y=c(2,0))
 bgtvs(n=c(2,3), s=c(5,2), y=c(2,1))
 bgtvs(n=c(2,3), s=c(5,2), y=c(2,2))
 bgtvs(n=c(2,3), s=c(5,2), y=c(2,3))

Description

Calculation of expected value of the width of confidence intervals for one proportion in binomial group testing, in dependence of the number of groups, group size, confidence level and an assumed true proportion. Available for the confidence interval methods in bgtCI(binGroup).

Usage

bgtWidth(n, s, p, conf.level = 0.95, alternative = "two.sided",
 method = "CP")

Arguments

Details

For calculation of expected interval width in the standard binomial estimation see, e.g., Brown et al. (2001). The calculation in case of binomial group testing is simply done by replacing the binomial probabilities by the probabilities P(Y=y) for group testing (see Tebbs and Bilder, 2004)

Value

A matrix containing the columns

and the calculated

Author(s)

Frank Schaarschmidt

References

Brown LD, Cai TT, DasGupta A (2001) Interval estimation for a binomial proportion. Statistical Science 16 (2), 101-133.

Schaarschmidt F (2007) Experimental design for one-sided confidence intervals or hypothesis tests in binomial group testing. Communications in Biometry and Crop Science 2 (1), 32-40. http://agrobiol.sggw.waw.pl/cbcs/

Tebbs JM & Bilder CR (2004) Confidence interval procedures for the probability of disease transmission in multiple-vector-transfer designs. Journal of Agricultural, Biological and Environmental Statistics 9 (1), 75-90.

Examples

# There is a minimal expected CI length, if 
# group size s is increased (fixed other parameters)
# the corresponding group size might be chosen:

bgtWidth(n=20, s=seq(from=1, to=200, by=10),
 p=0.01, alternative="less", method="CP" )

# and this depends largely on the assumed proportion p: 

bgtWidth(n=20, s=seq(from=1, to=200, by=10),
 p=0.05, alternative="less", method="CP" )

bgtWidth(n=20, s=seq(from=1, to=200, by=10),
 p=0.005, alternative="less", method="CP" )

Description

Calculates the exact Clopper-Pearson and Blaker, the asymptotic second-order corrected, Wilson, Agresti-Coull and Wald confidence interval for a single binomial proportion

Usage

binCI(n, y, conf.level = 0.95, alternative = "two.sided",
 method = "CP")

binCP(n, y, conf.level=0.95, alternative="two.sided")
binBlaker(n,y,conf.level=0.95, tolerance=1e-04, alternative="two.sided")
binAC(n, y, conf.level=0.95, alternative="two.sided")
binSOC(n, y,conf.level=0.95,alternative="two.sided")
binWald(n, y, conf.level=0.95, alternative="two.sided")
binWilson(n, y,conf.level=0.95,alternative="two.sided")

Arguments

Details

This function allows computation of confidence intervals for a binomial proportion from a standard binomial experiment. If an actual confidence level greater or equal to that specified in the conf.level argument shall always be guaranteed, the exact method of Clopper-Pearson (method="CP") can be recommended for one-sided and the improved method of Blaker (method="Blaker") can be recommended for two-sided hypotheses. If a mean confidence level close to that specified in the argument conf.level is required, but moderate violation of this level is acceptable, the Second-Order corrected (method="SOC"), Wilson Score (method="Wilson") or Agresti-Coull (method="AC") might be used, where SOC has the most symmetric coverage and Wilson and Agresti-Coull are in tendency conservative for the upper bound and proportions close to 0 and for the lower bound and proportions close to 1. The Wald CI might be used for large number of observations n>10000 or intermediate proportions.

For discussion of CI for a single binomial proportion see Brown et al. (2001) for two-sided and Cai (2005) for one-sided intervals.

Value

A list containing:

And the method, conf.level and alternative specified in the function call.

Author(s)

Frank Schaarschmidt

References

Blaker H (2000) Confidence curves and improved exact confidence intervals for discrete distributions. The Canadian Journal of Statistics 28 (4), 783-798.

Brown LD, Cai TT, DasGupta A (2001) Interval estimation for a binomial proportion. Statistical Science 16 (2), 101-133.

Cai TT(2005) One-sided confidence intervals in discrete distributions. Journal of Statistical Planning and Inference 131, 63-88.

See Also

binom.test for the exact confidence interval and test, binTest to calculate p.values of the exact, Score and Wald test.

Examples

# Default method is the two-sided 95% Clopper-Pearson CI:

binCI(n=200, y=10)

# other methods might result in 
# shorter intervals (but asymetric coverage):

binCI(n=200,y=10, method="Blaker")
binCI(n=200,y=10, method="Score")

Description

This function increases the sample size until a maximal sample size or a prespecified power is achieved.

Usage

binDesign(nmax, delta, p.hyp, conf.level = 0.95,
 power = 0.8, method = "CP", alternative = "two.sided")

Arguments

Details

The power of a confidence interval here is defined as the probability that a confidence interval or limit excludes the threshold parameter (p.hyp) of the hypothesis.

This function increases the number of trials (number of individuals under observation) until a pre-specified power is reached. Since the power does not increase monotone with increasing n for binomial proportions but oscillates between local maxima and minima, the simple iteration given here will generally result in selecting those n, for which the given CI method shows a local minimum of coverage if the null hypothesis is true. The power can be identical for different methods, depending on the particular combination of n, p.hyp, conf.level.

Especially for large n, the calculation time may become large (particularly for Blaker). Then only the range of sample size which is of interest can be specified in nmax, f.e. as: nmax=c(150,300). Alternatively, the function binPower might be used instead to calculate power and bias only for some particular combinations of n, delta, p.hyp,... .

Note that coverage probability and power are not necessarily symmetric for upper and lower bound of binomial CI.

The results can be visualized by application of the function plot() to the returned object of this function.

Value

a list

and additional input and iteration values needed only for the function plot.binDesign.

Author(s)

Frank Schaarschmidt

References

Schaarschmidt, F. (2007). Experimental design for one-sided confidence intervals or hypothesis tests in binomial group testing. Communications in Biometry and Crop Science 2 (1), 32-40. http://agrobiol.sggw.waw.pl/cbcs/

See Also

binPower for calculation of power, plot.binDesign for plot of the results

Examples

# Find a sample size for which the power to
# reject H0: p >= 0.1 in favor of HA: p < 0.1 is
# at least 0.9 (90 percent) in case that the
# true proportion is 0.04 (i.e. an absolute delta 
# of 0.06 to the threshold proportion p.hyp=0.1)
# The exact one sided Clopper-Pearson CI shall be
# used with default confidence level = 0.95.


sasi<-binDesign( nmax=200, delta=0.06,
 p.hyp=0.1, alternative="less", method="CP", power=0.9)
sasi

#### One can also plot the result:

 plot(sasi)


# For larger sample sizes iteration can be very time consuming.
# Better to use only a smaller range of n then:

sasi<-binDesign( nmax=c(200,300), delta=0.03, p.hyp=0.1,
 alternative="less", method="CP", power=0.9)
sasi

Description

Closed calculation of the power to reject a hypothesis using confidence intervals for a single binomial proportion, for specified sample size n, conf.level and an assumed absolute difference to the threshold parameter under the null hypothesis.

Usage

binPower(n, delta, p.hyp, conf.level = 0.95, 
alternative = "two.sided", method = "CP")

Arguments

Details

The power of a confidence interval here is defined as the probability that a confidence interval or limit excludes the threshold parameter (p.hyp) of the null hypothesis. E.g., the null hypothesis H0: p>= 0.005 can be rejected, if an upper confidence limit for p does not contain p.hyp=0.005. In case that a delta of 0.002 shall be detectable, this function calculates the probability, that an interval of a given method will exclude p.hyp=0.005 if the true proportion = 0.003. Due to discreteness, the power does not increase monotone for increasing sample size (number of trials or indivoiduals under observation) n, but exhibits local maxima and minima, depending on n, conf.level and p.hyp. The power can be identical for different methods, depending on the particular combination of n, p.hyp, conf.level. Note, that coverage probability and power are not necessarily symmetric for upper and lower bound of binomial CI, especially for Wald, Wilson Score and Agresti-Coull CI.

Value

A list containing

Author(s)

Frank Schaarschmidt

See Also

binDesign for iteration of a sample size n for which a specified power is reached

Examples

# What is the probability to reject the null hypothesis
# H0: p >= 0.02 in order to show that the alternative
# hypothesis HA: p < 0.02 is very likely in the first
# example of if 200 seeds are taken from a seed lot and
# are checked for the proportion of defectives.
# Assume a true proportion under the alternative:
# p = 0.01, i.e. a absolute difference delta = 0.01 
# to the threshold proportion p.hyp=0.02.
# The null hypothesis can be rejected if the threshold
# p.hyp=0.02 is excluded by an 95 percent upper bound of the 
# Clopper-Pearson CI. What is the probability  of this event?

binPower(n=200, delta=0.01, p.hyp=0.02,
 alternative="less", method="CP")

# Assuming a lower true proportion (if one is satisfied
# also with the situation that we can only reject H0
# in case that the seed lot has a very high purity, e.g.
# only a proportion of 0.001 defectives )

binPower(n=200, delta=0.019, p.hyp=0.02,
 alternative="less", method="CP")

# Or use a higher sample size:

binPower(n=600, delta=0.01, p.hyp=0.02,
 alternative="less", method="CP")

Description

Calculates p-values for hypothesis tests of a single binomial proportion.

Usage

binTest(n, y, p.hyp, alternative = "two.sided",
 method = "Exact")

Arguments

Value

A list containing:

Author(s)

Frank Schaarschmidt

References

Santner, T.J. and Duffy, D.E. (1989) The statistical analysis of discrete data. Springer Verlag New York Berlin Heidelberg. Chapter 2.1.

See Also

binom.test(stats) for the exact test and corresponding confindence interval

Examples

# 200 seeds are taken from a seed lot.
# 2 are found to be defective.
# H0: p >= 0.02  shall be rejected in favor of  HA: p < 0.02.
# The exact test shall be used for decision:

binTest(n=200, y=2, p.hyp=0.02, alternative="less", method="Exact" )

Description

Calculation of expected value of the width of confidence intervals in a binomial experiment, in dependence of the number of trials (number of individuals under observation), confidence level and an assumed true proportion. Available for the confidence interval methods in binCI(binGroup).

Usage

binWidth(n, p, conf.level = 0.95,
 alternative = "two.sided", method = "CP")

Arguments

Details

For calculation of expected interval width in the standard binomial estimation see Brown et al. (2001).

Value

A list containing:

and the alternative, p and n which are specified in the function call.

Author(s)

Frank Schaarschmidt

See Also

binDesign for experimental design for hypothesis testing

Examples

# methods differ slightly in length when sample sizes are large:

binWidth(n=200,p=0.02,alternative="two.sided",
 method="CP")$expCIWidth

binWidth(n=200,p=0.02,alternative="two.sided",
 method="Blaker")$expCIWidth

binWidth(n=200,p=0.02,alternative="two.sided",
 method="Score")$expCIWidth

# but do more for small sample sizes and intermediate p:

binWidth(n=20,p=0.2,alternative="two.sided",
 method="CP")$expCIWidth

binWidth(n=20,p=0.2,alternative="two.sided",
 method="Blaker")$expCIWidth

binWidth(n=20,p=0.2,alternative="two.sided",
 method="Score")$expCIWidth

Description

Calculate the expectation and variation of the testing expenditure of a pool used with informative Dorfman testing.

Usage

characteristics.pool(p, se, sp)

Arguments

Details

This function calculates the expected value and variance of the testing expenditure of a pool of size greater than or equal to one used with informative Dorfman testing. Calculations of these measures are done using the equations presented in McMahan et al. (2012).

Value

a list containing:

Author(s)

This function was originally written by Christopher S. McMahan for McMahan et al. (2012). The function was obtained from http://chrisbilder.com/grouptesting.

References

McMahan, C., Tebbs, J., Bilder, C. (2012). “Informative Dorfman Screening.” Biometrics, 68(1), 287–296. ISSN 0006341X, doi:10.1111/j.1541-0420.2011.01644.x, http://www.ncbi.nlm.nih.gov/pubmed/21762119.

See Also

http://chrisbilder.com/grouptesting

Other Informative Dorfman functions: accuracy.dorf, inf.dorf.measures, opt.info.dorf, opt.pool.size, pool.specific.dorf, thresh.val.dorf

Examples

# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(8135)
p.vec <- p.vec.func(p=0.02, alpha=1, grp.sz=10)
characteristics.pool(p=p.vec[1:3], se=0.90, sp=0.90)

Description

Find the group size s for a fixed number of assays n and an assumed true proportion p.tr for which the mean squared error (mse) of the point estimator is minimal and bias is within a restriction. For experimental design in binomial group testing as recommended by Swallow (1985), if main objective is estimation.

Usage

estDesign(n, smax, p.tr, biasrest = 0.05)

Arguments

Details

Swallow (1985) recommends to use the upper bound of the expected range of true proportion p.tr for optimization of tzhe design. For further details see the reference. Up to now, specify n<1020.

Value

the group size s, for which the mse of the estimator is minimal for the given n, p.tr or the group size s for which bias restriction biasrest is just not violated, and for this particular group size s: a list containing:

Author(s)

Frank Schaarschmidt

References

Swallow WH (1985) Group testing for estimating infection rates and probabilities of disease transmission. Phytopathology Vol.75, N.8, 882-889.

See Also

nDesign, sDesign for choice of the binomial group testing design according to the power in a hypothesis test

Examples

### Compare table 1 in Swallow(1985),885:

estDesign(n=10, smax=100, p.tr=0.001)

estDesign(n=10, smax=100, p.tr=0.01)

estDesign(n=25, smax=100, p.tr=0.05)

estDesign(n=40, smax=100, p.tr=0.25)

estDesign(n=200, smax=100, p.tr=0.3)

Description

Auxiliary function to control fitting parameters of EM algorithm used internally in gtreg.mp and EM.mp or gtreg, EM, and EM.ret.

Usage

gt.control(tol = 0.0001, n.gibbs = 1000, n.burnin = 20,
 maxit = 500, trace = FALSE, time = TRUE)

Arguments

Value

A list with components named as the arguments

Examples

# The default settings:
gt.control()

Description

gtreg is a function to fit the group testing regression model specified through a symbolic description of the linear predictor and descriptions of the group testing setting.

Usage

gtreg(formula, data, groupn, retest = NULL, sens = 1,
 spec = 1, linkf = c("logit", "probit", "cloglog"),
 method = c("Vansteelandt", "Xie"), sens.ind = NULL,
 spec.ind = NULL, start = NULL, control = gt.control(...), ...)

gtreg.fit(Y, X, groupn, sens, spec, linkf, start=NULL)

EM(Y, X, groupn, sens, spec, linkf, start = NULL,
 control = gt.control())

EM.ret(Y, X, groupn, ret, sens, spec, linkf, sens.ind,
 spec.ind, start = NULL, control = gt.control())

Arguments

Details

A typical predictor has the form groupresp ~ covariates where response is the (numeric) group response vector and covariates is a series of terms which specifies a linear predictor for individual responses. Note that it is actually the unobserved individual responses, not the observed group responses, which are modeled by the covariates here. In groupresp, a 0 denotes a negative response and a 1 denotes a positive response, where the probability of an individual positive response is being modeled directly. A terms specification of the form first + second indicates all the terms in first together with all the terms in second with duplicates removed. The terms in the formula will be re-ordered so that main effects come first, followed by the interactions, all second-order, all third-order and so on; to avoid this pass a terms object as the formula.

A specification of the form first:second indicates the set of terms obtained by taking the interactions of all terms in first with all terms in second. The specification first*second indicates the cross of first and second. This is the same as first + second + first:second.

Three workhorse functions gtreg.fit, EM and EM.ret, where the first corresponds to Vansteelandt's method and the last two corresponds to Xie's method, are called by gtreg to carry out the model fitting. The gtreg.fit function uses the optim function with default method "Nelder-Mead" to maximize the likelihood function of the observed group responses. If this optimization method produces a Hessian matrix of all zero elements, the "SANN" method in optim is employed to find the coefficients and Hessian matrix. For "SANN" method, the number of iterations in optim is set to be 10000.

The EM and EM.ret function apply Xie's EM algorithm to the likelihood function written in terms of the unobserved individual responses; the functions use glm.fit to update the parameter estimates within each M step. The EM function is used when there are no retests and EM.ret is used when individual retests are available. Thus, within retest, individual observations in observed positive groups are 0 (negative) or 1 (positive); the remaining individual observations are NAs meaning that no retest is performed for them. Retests cannot be used with Vansteelandt's method; a warning message will be given in this case, and the individual retests will be ignored in the model fitting. There could be slight differences in the estimates between the Vansteelandt's and Xie's methods (when retests are not available) due to different convergence criteria.

The data used here should be in the form of simple pooling - meaning that each individual appears in exactly one pool. When only the group responses are observed, the null degrees of freedom are the number of groups minus 1 and the residual degrees of freedom are the number of groups minus the number of parameters. When individual retests are observed too, it is an open research question for what the degrees of freedom and the deviance for the null model should be; therefore the degrees of freedom and null.deviance will not be displayed.

For the background on the use of optim, see help(optim).

Value

gtreg returns an object of class "gt". See later in this section. The function summary (i.e., summary.gt) can be used to obtain or print a summary of the results. The group testing functions predict (i.e., predict.gt) and residuals (i.e., residuals.gt) can be used to extract various useful features of the value returned by gtreg. An object of class "gt" is a list containing at least the following components:

Author(s)

Boan Zhang

References

Xie, M. (2001), Regression analysis of group testing samples, Statistics in Medicine, 20, 1957-1969.

Vansteelandt, S., Goetghebeur, E., and Verstraeten, T. (2000), Regression models for disease prevalence with diagnostic tests on pools of serum samples, Biometrics, 56, 1126-1133.

See Also

summary.gt, predict.gt and residuals.gt for gt methods. gtreg.mp for the group testing regression model in the matrix pooling setting.

Examples

data(hivsurv)

fit1 <- gtreg(formula = groupres ~ AGE + EDUC., data = hivsurv,
           groupn = gnum, sens = 0.9, spec = 0.9, method = "Xie")
fit1

## --- Continuing the Example from  '?sim.gt':

set.seed(46)
gt.data <- sim.gt(par = c(-12, 0.2), sample.size = 700, group.size = 5)
fit2 <- gtreg(formula = gres ~ x, data = gt.data, groupn = groupn)
fit2

set.seed(21)
gt.data <- sim.gt(par = c(-12, 0.2), sample.size = 700, group.size = 6,
               sens = 0.95, spec = 0.95, sens.ind = 0.98, spec.ind = 0.98)
fit1 <- gtreg(formula = gres ~ x, data = gt.data, groupn = groupn, 
        retest = retest, method = "X", sens = 0.95, spec = 0.95, sens.ind = 0.98,
        spec.ind = 0.98, trace = TRUE)
summary(fit1)

Description

gtreg.halving is a function to fit the group testing regression model under the halving protocol specified through a symbolic description of the linear predictor and descriptions of the group testing setting.

Usage

gtreg.halving(formula, data, groupn, subg, retest, sens = 1,
 spec = 1, linkf = c("logit", "probit", "cloglog"),
 sens.ind = NULL, spec.ind = NULL, start = NULL, control = gt.control(...), ...)

EM.halving(Y, X, groupn, subg, ret, sens, spec, linkf, sens.ind,
 spec.ind, start = NULL, control = gt.control())

Arguments

Details

A typical predictor has the form groupresp ~ covariates where response is the (numeric) group response vector and covariates is a series of terms which specifies a linear predictor for individual responses. Note that it is actually the unobserved individual responses, not the observed group responses, which are modeled by the covariates here. In groupresp, a 0 denotes a negative response and a 1 denotes a positive response, where the probability of an individual positive response is being modeled directly. A terms specification of the form first + second indicates all the terms in first together with all the terms in second with duplicates removed. The terms in the formula will be re-ordered so that main effects come first, followed by the interactions, all second-order, all third-order and so on; to avoid this pass a terms object as the formula.

A specification of the form first:second indicates the set of terms obtained by taking the interactions of all terms in first with all terms in second. The specification first*second indicates the cross of first and second. This is the same as first + second + first:second.

The EM.halving function apply Xie's EM algorithm to the likelihood function written in terms of the unobserved individual responses; the functions use glm.fit to update the parameter estimates within each M step. In the halving protocol, if the initial group tests positive, it is split into two subgroups. The two subgroups are subsequently tested and if either subgroup tests positive, the third and final step is to test all individuals within the subgroup. Thus, within subg, subgroup responses in observed positive groups are 0 (negative) or 1 (positive); the remaining subgroup responses are NAs meaning that no tests are performed for them. The individual retests are similarly coded.

Value

gtreg.halving returns an object of class "gt". See later in this section. The function summary (i.e., summary.gt) can be used to obtain or print a summary of the results. The group testing functions predict (i.e., predict.gt) and residuals (i.e., residuals.gt) can be used to extract various useful features of the value returned by gtreg.halving. An object of class "gt" is a list containing at least the following components:

Author(s)

Boan Zhang

References

Xie, M. (2001), Regression analysis of group testing samples, Statistics in Medicine, 20, 1957-1969.

See Also

summary.gt, predict.gt and residuals.gt for gt methods.

Examples

## --- Continuing the Example from  '?sim.halving':

set.seed(46)
gt.data <- sim.halving(par = c(-6, .1), gshape = 17, gscale = 1.4,
               sample.size = 5000, group.size = 5,
               sens = 0.95, spec = 0.95)
fit1 <- gtreg.halving(formula = gres ~ x, data = gt.data, groupn = groupn,
        start = c(-6, .1), subg = subgroup, retest = retest,
        sens = .95, spec = .95, trace = TRUE)
summary(fit1)

Description

gtreg.mp is a function to fit the group testing regression model in the matrix pooling setting specified through a symbolic description of the linear predictor and descriptions of the group testing setting.

Usage

gtreg.mp(formula, data, coln, rown, arrayn, retest = NULL,
 sens = 1, spec = 1,  linkf = c("logit", "probit", "cloglog"),
 sens.ind = NULL, spec.ind = NULL,  start = NULL,
 control = gt.control(...), ...)

EM.mp(col.resp, row.resp, X, coln, rown, sqn, ret, sens, spec,
 linkf, sens.ind, spec.ind, start = NULL, control = gt.control())

Arguments

Details

With matrix pooling, individual samples are placed in a matrix-like grid where samples are pooled within each row and within each column. This leads to two kinds of group responses: row and column group responses. Thus, a typical predictor has the form cbind(col.resp, row.resp) ~ covariates where col.resp is the (numeric) column group response vector and row.resp is the (numeric) row group response vector. The covariates term is a series of terms which specifies a linear predictor for individual responses. Note that it is actually the unobserved individual responses, not the observed group responses, which are modeled by the covariates. In col.resp and row.resp, a 0 denotes a negative response and 1 denotes a positive response, where the probability of an individual positive response is being modeled directly. A terms specification of the form first + second indicates all the terms in first together with all the terms in second with duplicates removed. The terms in the formula will be re-ordered so that main effects come first, followed by the interactions, all second-order, all third-order and so on; to avoid this pass a terms object as the formula.

A specification of the form first:second indicates the set of terms obtained by taking the interactions of all terms in first with all terms in second. The specification first*second indicates the cross of first and second. This is the same as first + second + first:second.

EM.mp is the workhorse function. It applies Xie's EM algorithm to the likelihood function written in terms of the unobserved individual responses. In each E step, the Gibbs sampling technique is used to estimate the conditional probabilities. Because of the large number of Gibbs samples needed to achieve convergence, the model fitting process could be quite slow, especially when multiple positive rows and columns are observed. In this case, we can either increase the Gibbs sample size to help achieve convergence or loosen the convergence criteria by increasing tol at the expense of perhaps poorer estimates. If follow-up retests are performed, the retest results going into the model will help achieve convergence faster with the same Gibbs sample size and convergence criteria. In each M step, we use glm.fit to update the parameter estimates

Value

gtreg.mp returns an object of class "gt.mp" which inherits from the class "gt". See later in this section. The function summary (i.e., summary.gt.mp) can be used to obtain or print a summary of the results. The group testing function predict (i.e., predict.gt) can be used to make predictions on "gt.mp" objects. An object of class "gt.mp" is a list containing at least the following components:

Author(s)

Boan Zhang

References

Xie, M. (2001), Regression analysis of group testing samples, Statistics in Medicine, 20, 1957-1969.

See Also

summary.gt.mp and predict.gt for gt.mp methods. gtreg for the group testing regression model in the simple pooling setting.

Examples

## --- Continuing the Example from  '?sim.mp':
# 5*6 and 4*5 matrix
set.seed(9128)
sa1a<-sim.mp(par=c(-7,0.1), n.row=c(5,4), n.col=c(6,5),
 sens=0.95, spec=0.95)
sa1<-sa1a$dframe


## Not run: 
fit1 <- gtreg.mp(formula = cbind(col.resp, row.resp) ~ x, data = sa1, 
                 coln = coln, rown = rown, arrayn = arrayn, 
                 sens = 0.95, spec = 0.95, tol = 0.005, n.gibbs = 2000, trace = TRUE)
fit1
summary(fit1)


## End(Not run)

## Here is an example of how long this fitting process may take. For the 
## following simulated data, it takes a computer with 2.2GHZ processor and 
## 3GB RAM about 6 minutes to achieve convergence.
set.seed(9012)
sa2a<-sim.mp(par=c(-7,0.1), n.row=c(10,10,10,10), n.col=c(10,10,10,10), 
             sens=0.95, spec=0.95)
sa2<-sa2a$dframe

## Not run: 
fit2 <- gtreg.mp(formula = cbind(col.resp, row.resp) ~ x, data = sa2, 
                 coln = coln, rown = rown, arrayn = arrayn, retest = retest,
                 sens = 0.95, spec = 0.95, start = c(-7, 0.1), tol = 0.005)

fit2
summary(fit2)


## End(Not run)

Description

Calculate operating characteristics for hierarchical group testing with up to four stages, given a vector of individual probabilities and a testing configuration.

Usage

hierarchical.desc2(p, I2 = NULL, I3 = NULL, se = 1, sp = 1,
  order.p = TRUE)

Arguments

Details

This function calculates the operating characteristics for hierarchical group testing with up to four stages of testing. Operating characteristics calculated are expected number of tests, and pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value, for each individual and for the configuration overall.

If I2 is NULL, there are two stages of testing. If I3 is NULL but I2 has values, there are three stages of testing. If both I2 and I3 have values, there are four stages of testing.

Vectors I2 and I3 should be entered using notation that keeps track of all individuals through all stages (e.g. for a group of 10 individuals that splits into 5, 4, and 1 individual at stage 2, then into 3, 2, 2, 1, and 1 individual at stage 3 before individual testing at stage 4, then I2=c(5,4,1) and I3=c(3,2,2,1,1,1) so that the specimen that was tested individually at stage 2 is still numbered at stage 3 even though it will not be tested again).

The displayed pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value are weighted averages of the corresponding individual accuracy measures for all individuals within the initial group for a hierarchical algorithm. Expressions for these averages are provided in the Supplementary Material for Hitt et al. (2018). These expressions are based on accuracy definitions given by Altman and Bland (1994a, 1994b).

Value

A list containing:

Note

This function returns the pooling positive and negative predictive values for all individuals even though these measures are diagnostic specific; i.e., PPPV (PNPV) should only be considered for those individuals who have tested positive (negative).

Author(s)

This function was originally written by Michael S. Black for Black et al. (2015). The function was obtained from http://chrisbilder.com/grouptesting. Minor modifications were made to the function for inclusion in the binGroup package.

References

Black, M., Bilder, C., Tebbs, J. (2015). “Optimal retesting configurations for hierarchical group testing.” Journal of the Royal Statistical Society. Series C: Applied Statistics, 64(4), 693–710. ISSN 14679876, doi:10.1111/rssc.12097.

See Also

Array.Measures for calculating operating characteristics under array testing without master pooling, MasterPool.Array.Measures for non-informative array testing with master pooling, and inf.dorf.measures for informative two-stage hierarchical testing. See p.vec.func for generating a vector of individual risk probabilities for informative group testing.

http://chrisbilder.com/grouptesting

Other Operating characteristic functions: Array.Measures, MasterPool.Array.Measures, inf.dorf.measures

Examples

# Calculate the operating characteristics for 
#   non-informative two-stage hierarchical testing
#   with an overall disease prevalence of p = 0.015
#   and an initial group size of 12.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
p.vec <- rep(x=0.015, times=12)
hierarchical.desc2(p=p.vec, I2=NULL, I3=NULL, se=0.95, 
sp=0.95, order.p=FALSE)

# Calculate the operating characteristics for 
#   non-informative three-stage hierarchical testing
#   with an overall disease prevalence of p = 0.04, 
#   where an initial group of 20 individuals is 
#   split into equally sized subgroups of 5 each.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
p.vec <- rep(x=0.04, times=20)
hierarchical.desc2(p=p.vec, I2=rep(x=5, times=4),
I3=NULL, se=0.99, sp=0.99, order.p=FALSE)

# Calculate the operating characteristics for 
#   informative three-stage hierarchical testing
#   where an initial group of 10 individuals is 
#   split into subsequent groups of 5, 4, and 1 
#   individual.
# A vector of individual probabilities is generated using
#   the expected value of order statistics from a beta 
#   distribution with p = 0.02 and a heterogeneity level 
#   of alpha = 0.5. Depending on the specified probability, 
#   alpha level, and overall group size, simulation may 
#   be necessary in order to generate the vector of individual
#   probabilities. This is done using p.vec.func() and 
#   requires the user to set a seed in order to reproduce 
#   results.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(1002)
p.vec <- p.vec.func(p=0.02, alpha=0.5, grp.sz=10)
hierarchical.desc2(p=p.vec, I2=c(5,4,1), I3=NULL,
se=0.90, sp=0.90, order.p=TRUE)

Description

The hivsurv data set comes from an HIV surveillance project discussed in Verstraeten et al. (1998) and Vansteelandt et al. (2000). The purpose of the study was to estimate the HIV prevalence among pregnant Kenyan women in four rural locations of the country, using both individual and group testing responses. Blood tests were administered to each participating woman, and 4 covariates were obtained on each woman. Because the original group responses are unavailable, individuals are artificially put into groups of 5 here to form group responses. Only the 428 complete observations are given here.

Usage

data(hivsurv)

Format

A data frame with 428 observations on the following 8 variables.

DATE

the date when each sample was collected

PAR.

parity (number of children)

AGE

age (in years)

MA.ST.

marital status (1: single; 2: married (polygamous); 3: married (monogamous); 4: divorced; 5: widow)

EDUC.

highest attained education level (1: no schooling; 2: primary school; 3: secondary school; 4: higher)

HIV

individual response of HIV diagnosis (0: negative; 1: positive)

gnum

the group number that designates individuals into groups

groupres

the group response calculated from artificially formed groups

Source

Verstraeten, T., Farah, B., Duchateau, L., Matu, R. (1998), Pooling sera to reduce the cost of HIV surveillance: a feasibility study in a rural Kenyan district, Tropical Medicine & International Health, 3, 747-750.

Vansteelandt, S., Goetghebeur, E., and Verstraeten, T. (2000), Regression models for disease prevalence with diagnostic tests on pools of serum samples, Biometrics, 56, 1126-1133.

Examples

data(hivsurv)

str(hivsurv)

Description

Find the optimal testing configuration (OTC) for informative array testing without master pooling and calculate the associated operating characteristics.

Usage

Inf.Array(p, Se, Sp, group.sz, obj.fn, weights = NULL, alpha = 2)

Arguments

Details

This function finds the OTC and computes the associated operating characteristics for informative array testing without master pooling, implemented using the gradient arrangement described in McMahan et al. (2012). Array testing without master pooling involves amalgamating specimens in rows and columns for the first stage of testing. This function uses only square arrays, which is the way array-based group testing is carried out in most real-world applications. Operating characteristics calculated are expected number of tests, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for the algorithm. See Hitt et al. (2018) or McMahan et al. (2012) at http://chrisbilder.com/grouptesting for additional details on the implementaion of informative array testing without master pooling.

The value(s) specified by group.sz represent the initial group (row/column) size. If a single value is provided for group.sz, operating characteristics will be calculated and no optimization will be performed. If a range of group sizes is specified, the OTC will be found over all group sizes.

The displayed pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value are weighted averages of the corresponding individual accuracy measures for all individuals within the initial group for a hierarchical algorithm, or within the entire array for an array-based algorithm. Expressions for these averages are provided in the Supplementary Material for Hitt et al. (2018). These expressions are based on accuracy definitions given by Altman and Bland (1994a, 1994b).

Value

A list containing:

Author(s)

Brianna D. Hitt

References

Altman, D., Bland, J. (1994). “Diagnostic tests 1: sensitivity and specificity.” BMJ, 308, 1552.

Altman, D., Bland, J. (1994). “Diagnostic tests 2: predictive values.” BMJ, 309, 102.

Graff, L., Roeloffs, R. (1972). “Group testing in the presence of test error; an extension of the Dorfman procedure.” Technometrics, 14(1), 113–122. ISSN 15372723, doi:10.1080/00401706.1972.10488888, https://www.tandfonline.com/doi/abs/10.1080/00401706.1972.10488888.

Hitt, B., Bilder, C., Tebbs, J., McMahan, C. (2018). “The Optimal Group Size Controversy for Infectious Disease Testing: Much Ado About Nothing?!” Manuscript submitted for publication.

Malinovsky, Y., Albert, P., Roy, A. (2016). “Reader reaction: A note on the evaluation of group testing algorithms in the presence of misclassification.” Biometrics, 72(1), 299–302. ISSN 15410420, doi:10.1111/biom.12385.

McMahan, C., Tebbs, J., Bilder, C. (2012). “Two-Dimensional Informative Array Testing.” Biometrics, 68(3), 793–804. ISSN 0006341X, doi:10.1111/j.1541-0420.2011.01726.x, http://www.ncbi.nlm.nih.gov/pubmed/22212007.

See Also

NI.Array for non-informative array testing without master pooling, NI.A2M for non-informative array testing with master pooling, and OTC for finding the optimal testing configuration for a number of standard group testing algorithms.

http://chrisbilder.com/grouptesting

Other OTC functions: Inf.D3, Inf.Dorf, NI.A2M, NI.Array, NI.D3, NI.Dorf, OTC

Examples

# Find the OTC for informative array testing without 
#   master pooling over a range of group (row/column) sizes.
# A vector of individual probabilities is generated using
#   the expected value of order statistics from a beta 
#   distribution with p = 0.03 and a heterogeneity level 
#   of alpha = 2. Depending on the specified probability, 
#   alpha level, and overall group size, simulation may 
#   be necessary in order to generate the vector of individual
#   probabilities. This is done using p.vec.func() and 
#   requires the user to set a seed in order to reproduce 
#   results.
# This examples takes approximately 30 seconds to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
## Not run: 
set.seed(1002)
Inf.Array(p=0.03, Se=0.99, Sp=0.99, group.sz=3:20, 
obj.fn=c("ET", "MAR"), alpha=2)
## End(Not run)

# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(1002)
Inf.Array(p=0.03, Se=0.99, Sp=0.99, group.sz=3:5, 
obj.fn=c("ET", "MAR"), alpha=2)

# Find the OTC for a specified group (row/column) size 
#   for informative array testing without master pooling.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(14849)
Inf.Array(p=p.vec.func(p=0.05, alpha=0.5, grp.sz=100), 
Se=0.95, Sp=0.95, group.sz=10, obj.fn=c("ET", "MAR", "GR"),
weights=matrix(data=c(1,1,10,10), nrow=2, ncol=2, byrow=TRUE),
alpha=NA)

Description

Find the optimal testing configuration (OTC) for informative three-stage hierarchical testing and calculate the associated operating charcteristics.

Usage

Inf.D3(p, Se, Sp, group.sz, obj.fn, weights = NULL, alpha = 2)

Arguments

Details

This function finds the OTC and computes the associated operating characteristics for informative three-stage hierarchical testing. This function finds the optimal testing configuration by considering all possible testing configurations. Operating characteristics calculated are expected number of tests, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for the algorithm. See Hitt et al. (2018) or Black et al. (2015) at http://chrisbilder.com/grouptesting for additional details on the implementation of informative three-stage hierarchical testing.

The value(s) specified by group.sz represent the initial (stage 1) group size. If a single value is provided for group.sz, the OTC will be found over all possible testing configurations for that initial group size. If a range of group sizes is specified, the OTC will be found over all group sizes.

The displayed pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value are weighted averages of the corresponding individual accuracy measures for all individuals within the initial group for a hierarchical algorithm, or within the entire array for an array-based algorithm. Expressions for these averages are provided in the Supplementary Material for Hitt et al. (2018). These expressions are based on accuracy definitions given by Altman and Bland (1994a, 1994b).

Value

A list containing:

Author(s)

Brianna D. Hitt

References

Altman, D., Bland, J. (1994). “Diagnostic tests 1: sensitivity and specificity.” BMJ, 308, 1552.

Altman, D., Bland, J. (1994). “Diagnostic tests 2: predictive values.” BMJ, 309, 102.

Black, M., Bilder, C., Tebbs, J. (2015). “Optimal retesting configurations for hierarchical group testing.” Journal of the Royal Statistical Society. Series C: Applied Statistics, 64(4), 693–710. ISSN 14679876, doi:10.1111/rssc.12097.

Graff, L., Roeloffs, R. (1972). “Group testing in the presence of test error; an extension of the Dorfman procedure.” Technometrics, 14(1), 113–122. ISSN 15372723, doi:10.1080/00401706.1972.10488888, https://www.tandfonline.com/doi/abs/10.1080/00401706.1972.10488888.

Hitt, B., Bilder, C., Tebbs, J., McMahan, C. (2018). “The Optimal Group Size Controversy for Infectious Disease Testing: Much Ado About Nothing?!” Manuscript submitted for publication.

Malinovsky, Y., Albert, P., Roy, A. (2016). “Reader reaction: A note on the evaluation of group testing algorithms in the presence of misclassification.” Biometrics, 72(1), 299–302. ISSN 15410420, doi:10.1111/biom.12385.

See Also

NI.D3 for non-informative three-stage hierarchical testing and OTC for finding the optimal testing configuration for a number of standard group testing algorithms.

http://chrisbilder.com/grouptesting

Other OTC functions: Inf.Array, Inf.Dorf, NI.A2M, NI.Array, NI.D3, NI.Dorf, OTC

Examples

# Find the OTC for informative three-stage hierarchical 
#   testing over a range of group sizes.
# A vector of individual probabilities is generated using
#   the expected value of order statistics from a beta 
#   distribution with p = 0.05 and a heterogeneity level 
#   of alpha = 0.5. Depending on the specified probability, 
#   alpha level, and overall group size, simulation may 
#   be necessary in order to generate the vector of individual
#   probabilities. This is done using p.vec.func() and 
#   requires the user to set a seed in order to reproduce 
#   results.
# This example takes approximately 20 seconds to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
## Not run: 
set.seed(8318)
Inf.D3(p=0.05, Se=0.99, Sp=0.99, group.sz=3:30, 
obj.fn=c("ET", "MAR"), alpha=0.5)
## End(Not run)

# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(8318)
Inf.D3(p=0.05, Se=0.99, Sp=0.99, group.sz=10:15, 
obj.fn=c("ET", "MAR"), alpha=0.5)

# Find the OTC out of all possible testing configurations
#   for a specified group size and vector of individual 
#   probabilities.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(1216)
p.vec <- p.vec.func(p=0.10, alpha=2, grp.sz=12)
Inf.D3(p=p.vec, Se=0.99, Sp=0.99, group.sz=12,
obj.fn=c("ET", "MAR", "GR"), weights=matrix(data=c(1,1), 
nrow=1, ncol=2, byrow=TRUE), alpha=NA)

Description

Find the optimal testing configuration (OTC) for informative two-stage hierarchical (Dorfman) testing and calculate the associated operating characteristics.

Usage

Inf.Dorf(p, Se, Sp, group.sz, obj.fn, weights = NULL, alpha = 2)

Arguments

Details

This function finds the OTC and computes the associated operating characteristics for informative two-stage hierarchical (Dorfman) testing, implemented via the pool-specific optimal Dorfman (PSOD) method described in McMahan et al. (2012). This function finds the optimal testing configuration by considering all possible testing configurations instead of using the greedy algorithm proposed for PSOD testing. Operating characteristics calculated are expected number of tests, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for the algorithm. See Hitt et al. (2018) or McMahan et al. (2012) at http://chrisbilder.com/grouptesting for additional details on the implementation of informative two-stage hierarchical (Dorfman) testing.

The value(s) specified by group.sz represent the overall block size used in the pool-specific optimal Dorfman (PSOD) method, where the overall group size is not tested. Instead, multiple initial pool sizes within this block are found and tested in the first stage of testing. The second stage of testing consists of individual retesting. For more details on informative two-stage hierarchical testing implemented via the PSOD method, see Hitt et al. (2018) and McMahan et al. (2012).

If a single value is provided for group.sz, the OTC will be found over all possible testing configurations. If a range of group sizes is specified, the OTC will be found over all group sizes.

The displayed pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value are weighted averages of the corresponding individual accuracy measures for all individuals within the initial group for a hierarchical algorithm, or within the entire array for an array-based algorithm. Expressions for these averages are provided in the Supplementary Material for Hitt et al. (2018). These expressions are based on accuracy definitions given by Altman and Bland (1994a, 1994b).

Value

A list containing:

Author(s)

Brianna D. Hitt

References

Altman, D., Bland, J. (1994). “Diagnostic tests 1: sensitivity and specificity.” BMJ, 308, 1552.

Altman, D., Bland, J. (1994). “Diagnostic tests 2: predictive values.” BMJ, 309, 102.

Dorfman, R. (1943). “The Detection of Defective Members of Large Populations.” The Annals of Mathematical Statistics, 14(4), 436–440. ISSN 0003-4851, doi:10.1214/aoms/1177731363, https://www.jstor.org/stable/2235930.

Graff, L., Roeloffs, R. (1972). “Group testing in the presence of test error; an extension of the Dorfman procedure.” Technometrics, 14(1), 113–122. ISSN 15372723, doi:10.1080/00401706.1972.10488888, https://www.tandfonline.com/doi/abs/10.1080/00401706.1972.10488888.

Hitt, B., Bilder, C., Tebbs, J., McMahan, C. (2018). “The Optimal Group Size Controversy for Infectious Disease Testing: Much Ado About Nothing?!” Manuscript submitted for publication.

Malinovsky, Y., Albert, P., Roy, A. (2016). “Reader reaction: A note on the evaluation of group testing algorithms in the presence of misclassification.” Biometrics, 72(1), 299–302. ISSN 15410420, doi:10.1111/biom.12385.

McMahan, C., Tebbs, J., Bilder, C. (2012). “Informative Dorfman Screening.” Biometrics, 68(1), 287–296. ISSN 0006341X, doi:10.1111/j.1541-0420.2011.01644.x, http://www.ncbi.nlm.nih.gov/pubmed/21762119.

See Also

NI.Dorf for non-informative two-stage hierarchical (Dorfman) testing and OTC for finding the optimal testing configuration for a number of standard group testing algorithms.

http://chrisbilder.com/grouptesting

Other OTC functions: Inf.Array, Inf.D3, NI.A2M, NI.Array, NI.D3, NI.Dorf, OTC

Examples

# Find the OTC for informative two-stage hierarchical 
#   (Dorfman) testing.
# A vector of individual probabilities is generated using
#   the expected value of order statistics from a beta 
#   distribution with p = 0.01 and a heterogeneity level 
#   of alpha = 2. Depending on the specified probability, 
#   alpha level, and overall group size, simulation may 
#   be necessary in order to generate the vector of individual
#   probabilities. This is done using p.vec.func() and 
#   requires the user to set a seed in order to reproduce 
#   results.
# This example takes approximately 20 seconds to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
## Not run: 
set.seed(9245)
Inf.Dorf(p=0.01, Se=0.95, Sp=0.95, group.sz=3:30, 
obj.fn=c("ET", "MAR"), alpha=2)
## End(Not run)

# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(9245)
Inf.Dorf(p=0.01, Se=0.95, Sp=0.95, group.sz=5:10, 
obj.fn=c("ET", "MAR"), alpha=2)

# Find the OTC for informative two-stage hierarchical 
#   (Dorfman) testing, for a specified block size.
# This example uses rbeta() to generate random probabilities 
#   and requires the user to set a seed in order to reproduce 
#   results.
# This example takes approximately 2.5 minutes to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
## Not run: 
set.seed(8791)
Inf.Dorf(p=p.vec.func(p=0.03, alpha=0.5, grp.sz=50), 
Se=0.90, Sp=0.90, group.sz=50, obj.fn=c("ET", "MAR", "GR"),
weights=matrix(data=c(1,1,10,10), nrow=2, ncol=2, byrow=TRUE),
alpha=NA)
## End(Not run)

Description

Calculate the expected number of tests and accuracy measures for each individual using informative two-stage hierarchical (Dorfman) testing, given a vector of individual probabilities and a testing configuration.

Usage

inf.dorf.measures(prob, se, sp, N, pool.sizes)

Arguments

Details

This function utilizes the equations given by McMahan et al. (2012) for informative two-stage hierarchical (Dorfman) testing. It also repurposes functions written by Christopher S. McMahan (see http://chrisbilder.com/grouptesting) for the implementation of informative Dorfman testing and directly uses functions written for the calculation of the associated operating characteristics. This function calculates the operating characteristics for informative two-stage hierarchical (Dorfman) testing. Operating characteristics calculated are expected number of tests, and pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for each individual.

The specified N represents the block size used in the pool-specific optimal Dorfman (PSOD) method. This is the total number of individuals being tested by the algorithm. This block is not initially tested. Instead, multiple initial pool sizes within this block are found and tested in the first stage of testing. The second stage of testing consists of individual retesting. For more information on block size specification, see McMahan et al. (2012).

Value

A list containing:

Note

This function returns the pooling positive and negative predictive values for all individuals even though these measures are diagnostic specific; i.e., PPPV (PNPV) should only be considered for those individuals who have tested positive (negative).

Author(s)

The majority of this function was originally written by Christopher S. McMahan for McMahan et al. (2012). The function was obtained from http://chrisbilder.com/grouptesting. Minor modifications were made to the function for inclusion in the binGroup package.

References

McMahan, C., Tebbs, J., Bilder, C. (2012). “Informative Dorfman Screening.” Biometrics, 68(1), 287–296. ISSN 0006341X, doi:10.1111/j.1541-0420.2011.01644.x, http://www.ncbi.nlm.nih.gov/pubmed/21762119.

See Also

Array.Measures for calculating operating characteristics under array testing without master pooling, MasterPool.Array.Measures for non-informative array testing with master pooling, and inf.dorf.measures for informative two-stage hierarchical testing. See p.vec.func for generating a vector of individual risk probabilities for informative group testing.

This function repurposed code from opt.info.dorf so that PSOD testing could be implemented using all possible testing configurations instead of a greedy algorithm. This function also used characteristics.pool and accuracy.dorf to calculate operating characteristics for the optimal set of pool sizes.

http://chrisbilder.com/grouptesting

Other Operating characteristic functions: Array.Measures, MasterPool.Array.Measures, hierarchical.desc2

Other Informative Dorfman functions: accuracy.dorf, characteristics.pool, opt.info.dorf, opt.pool.size, pool.specific.dorf, thresh.val.dorf

Examples

# Calculate the operating characteristics for 
#   informative two-stage hierarchical (Dorfman) testing
#   with an overall disease prevalence of E(p(i)) = 0.01,
#   where a block size of 50 is split into initial pools
#   of 18, 13, 11, and 8 individuals.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(8791)
inf.dorf.measures(prob=beta.dist(p=0.01, alpha=2, 
grp.sz=50, simul=TRUE), se=0.95, sp=0.95, N=50, 
pool.sizes=c(18, 13, 11, 8))

Description

Arrange a vector of individual risk probabilities in a matrix for informative array testing without master pooling.

Usage

Informative.array.prob(prob.vec, nr, nc, method = "sd")

Arguments

Value

A matrix of probabilities arranged according to the specified method.

Author(s)

This function was originally written by Christopher S. McMahan for McMahan et al. (2012). The function was obtained from http://chrisbilder.com/grouptesting.

References

McMahan, C., Tebbs, J., Bilder, C. (2012). “Two-Dimensional Informative Array Testing.” Biometrics, 68(3), 793–804. ISSN 0006341X, doi:10.1111/j.1541-0420.2011.01726.x, http://www.ncbi.nlm.nih.gov/pubmed/22212007.

See Also

p.vec.func for generating a vector of individual risk probabilities from an overall probability of disease and Array.Measures for calculating operating characteristics for informative array testing without master pooling.

http://chrisbilder.com/grouptesting

Other Individual risk probability functions: beta.dist, p.vec.func

Examples

# Use the gradient arrangement method to create a matrix
#   of individual risk probabilities for a 10x10 array.
# Depending on the specified probability, alpha level, 
#   and overall group size, simulation may be necessary in 
#   order to generate the vector of individual probabilities. 
#   This is done using the p.vec.func() function and requires 
#   the user to set a seed in order to reproduce results.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(1107)
p.vec1 <- p.vec.func(p=0.05, alpha=2, grp.sz=100)
Informative.array.prob(prob.vec=p.vec1, nr=10, nc=10, method="gd")

# Use the spiral arrangement method to create a matrix
#   of individual risk probabilities for a 5x5 array.
# Depending on the specified probability, alpha level, 
#   and overall group size, simulation may be necessary in 
#   order to generate the vector of individual probabilities. 
#   This is done using the p.vec.func() function and requires 
#   the user to set a seed in order to reproduce results.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
set.seed(8791)
p.vec2 <- p.vec.func(p=0.02, alpha=0.5, grp.sz=25)
Informative.array.prob(prob.vec=p.vec2, nr=5, nc=5, method="sd")

Description

Calculate the expected number of tests and accuracy measures for each individual using array testing with master pooling.

Usage

MasterPool.Array.Measures(results, n, pmat, Se, Sp)

Arguments

Details

This function assumes that the array is square (i.e., the row and column size are equal) and utilizes the equations from Kim et al. (2007) for square array testing with master pooling. This function calculates the operating characteristics for array testing with master pooling. Operating characteristics calculated are expected number of tests, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for each individual.

Value

A list containing:

Note

This function returns the pooling positive and negative predictive values for all individuals in the array even though these measures are diagnostic specific; i.e., PPV (NPV) should only be considered for those individuals who have tested positive (negative).

Author(s)

Brianna D. Hitt

References

Kim, H., Hudgens, M., Dreyfuss, J., Westreich, D., Pilcher, C. (2007). “Comparison of group testing algorithms for case identification in the presence of test error.” Biometrics, 63(4), 1152–1163. ISSN 0006341X, doi:10.1111/j.1541-0420.2007.00817.x, http://www.ncbi.nlm.nih.gov/pubmed/17501946.

See Also

Array.Measures for calculating operating characteristics under array testing without master pooling, hierarchical.desc2 for three-stage hierarchical and non-informative two-stage hierarchical testing, and inf.dorf.measures for informative two-stage hierarchical testing.

Other Operating characteristic functions: Array.Measures, hierarchical.desc2, inf.dorf.measures

Examples

# Calculate the operating characteristics for 
#   non-informative array testing with master
#   pooling, with a 6x6 array and an overall 
#   disease risk of p = 0.10.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
p.mat <- matrix(data=0.10, ncol=6, nrow=6)
results <- Array.Measures(p=p.mat, se=0.90, sp=0.90)
MasterPool.Array.Measures(results=results, n=36, 
pmat=p.mat, Se=0.90, Sp=0.90)

Description

Increasing number of groups (assays, bulk samples) for a fixed group size in a binomial group testing design, until a pre-specified power is achieved. At the same time, bias of the estimator is controlled. A hypothetical threshold proportion p.hyp and the absolute difference delta to be detected have to be specified.

Usage

nDesign(nmax, s, delta, p.hyp, conf.level = 0.95, 
power = 0.8, alternative = "two.sided", method = "CP", biasrest = 0.05)

Arguments

Details

The power of a confidence interval here is defined as the probability that a confidence interval or limit excludes the threshold parameter (p.hyp) of the hypothesis.

This function increases the number of groups (i.e. number of observations or assays in binomial group testing) until a pre-specified power is reached or the maximal number of groups nmax (specified in the function call) is reached. Since the power does not increase monotone with increasing n for binomial proportions but oscillates between local maxima and minima, the simple iteration given here will generally result in selecting those n, for which the given CI method shows a local minimum of coverage if the null hypothesis is true. Bias decreases monotone with increasing the number of groups (if other parameters are fixed) The resulting Problems of chosing a number of groups which results in satisfactory power, are solved in the following manner:

In case that the pre-specified power can be reached within the given range of n, the smallest n will be returned for which at least this power is reached, as well as the actual power for this n.

In case that the pre-specified power is not reached within the given value, that n is returned for which maximal power is achieved, and the corresponding value of power.

In case that biasrestriction is violated even for the largest n within the given range of n, simply that n will be returned for which power was largest in the given range. Due to discreteness of binomial distribution, power can be zero for one-sided hypothesis over a range of n.

The power can be identical for different methods, depending on the particular combination of n, s, p.hyp, conf.level.

Especially for large n, the calculation time may become large (particularly for Blaker). Then only the range of sample size which is of interest can be specified in nmax, f.e. as: nmax=c(150,300). Alternatively, the function bgtPower might be used instead to calculate power and bias only for some particular combinations of n, s, delta, p.hyp,... .

Value

A list containing

and a number of values specified in the function call or produced in the iteration, which are only necessary to apply the function plot() on objects of class 'nDesign'

Author(s)

Frank Schaarschmidt

References

Schaarschmidt F (2007). Experimental design for one-sided confidence intervals or hypothesis tests in binomial group testing. Communications in Biometry and Crop Science 2 (1), 32-40. http://agrobiol.sggw.waw.pl/cbcs/

Swallow WH (1985). Group testing for estimating infection rates and probabilities of disease transmission. Phytopathology Vol.75, N.8, 882-889.

See Also

plot.nDesign to plot the iteration of this function

bgtPower: calculation of power and bias depending on n, s, delta, p.hyp, conf.level, method sDesign: function for stepwise increasing group size s for a given n in order to achieve sufficient power within a biasrestriction estDesign: function to choose group size s according to the minimal mse of the estimator, as given in Swallow (1985)

Examples

## Assume one aims to show that a proportion is smaller
## 0.005 (i.e. 0.5 per cent) with a power 
## of 0.80 (i.e. 80 per cent) if the unknown proportion
## in the population is 0.003 (i.e. 0.3 per cent),
## thus, to detect a delta of 0.002.
## The Clopper Pearson CI shall be used. 
## The maximal group size because of limited
## sensitivity of assay might be s=20 and we
## can only afford to perform maximally 100 assays:

nDesign(nmax=100, s=20, delta=0.002, p.hyp=0.005,
 alternative="less", method="CP", power=0.8)

## A power of 80 per cent can not be reached but
## only 30 percent with n=100 
## One might accept to show significance only for a 
## lower true proportion = 0.001 i.e accepting to be
## able to show significance only if true proportion 
## is delta=0.004 smaller than the threshold

nDesign(nmax=100, s=20, delta=0.004, p.hyp=0.005,
 alternative="less", method="CP", power=0.8)


test<-nDesign(nmax=100, s=30, delta=0.004, p.hyp=0.005,
 alternative="less", method="CP", power=0.8)

plot(test)

Description

Find the optimal testing configuration for non-informative array testing with master pooling and calculate the associated operating characteristics.

Usage

NI.A2M(p, Se, Sp, group.sz, obj.fn, weights = NULL)

Arguments

Details

This function finds the OTC and computes the associated operating characteristics for non-informative array testing with master pooling. Array testing with master pooling involves testing all specimens in the array together in one group before any row or column groups are formed. This function uses only square arrays, which is the way array-based group testing is carried out in most real-world applications. Operating characteristics calculated are expected number of tests, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for the algorithm. See Hitt et al. (2018) at http://chrisbilder.com/grouptesting or Kim et al. (2007) for additional details on the implementation of non-informative array testing with master pooling.

The value(s) specified by group.sz represent the row/column size, which is used for the second stage of testing after the entire array is tested together in one group. If a single value is provided for group.sz, operating characteristics will be calculated and no optimization will be performed. If a range of group sizes is specified, the OTC will be found over all group sizes.

The displayed pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value are weighted averages of the corresponding individual accuracy measures for all individuals within the initial group for a hierarchical algorithm. Expressions for these averages are provided in the Supplementary Material for Hitt et al. (2018). These expressions are based on accuracy definitions given by Altman and Bland (1994a, 1994b).

Value

A list containing:

Note

This function returns the pooling positive and negative predictive values for all individuals in the array even though these measures are diagnostic specific; i.e., PPV (NPV) should only be considered for those individuals who have tested positive (negative).

Author(s)

Brianna D. Hitt

References

Altman, D., Bland, J. (1994). “Diagnostic tests 1: sensitivity and specificity.” BMJ, 308, 1552.

Altman, D., Bland, J. (1994). “Diagnostic tests 2: predictive values.” BMJ, 309, 102.

Graff, L., Roeloffs, R. (1972). “Group testing in the presence of test error; an extension of the Dorfman procedure.” Technometrics, 14(1), 113–122. ISSN 15372723, doi:10.1080/00401706.1972.10488888, https://www.tandfonline.com/doi/abs/10.1080/00401706.1972.10488888.

Hitt, B., Bilder, C., Tebbs, J., McMahan, C. (2018). “The Optimal Group Size Controversy for Infectious Disease Testing: Much Ado About Nothing?!” Manuscript submitted for publication.

Kim, H., Hudgens, M., Dreyfuss, J., Westreich, D., Pilcher, C. (2007). “Comparison of group testing algorithms for case identification in the presence of test error.” Biometrics, 63(4), 1152–1163. ISSN 0006341X, doi:10.1111/j.1541-0420.2007.00817.x, http://www.ncbi.nlm.nih.gov/pubmed/17501946.

Malinovsky, Y., Albert, P., Roy, A. (2016). “Reader reaction: A note on the evaluation of group testing algorithms in the presence of misclassification.” Biometrics, 72(1), 299–302. ISSN 15410420, doi:10.1111/biom.12385.

See Also

NI.Array for non-informative array testing without master pooling, Inf.Array for informative array testing without master pooling, and OTC for finding the optimal testing configuration for a number of standard group testing algorithms.

http://chrisbilder.com/grouptesting

Other OTC functions: Inf.Array, Inf.D3, Inf.Dorf, NI.Array, NI.D3, NI.Dorf, OTC

Examples

# Find the OTC for non-informative array testing with 
#   master pooling over a range of group (row/column) sizes.
# This example takes approximately 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
NI.A2M(p=0.04, Se=0.95, Sp=0.95, group.sz=3:10,
obj.fn=c("ET", "MAR"))

# Calculate the operating characteristics for a specified 
#   group (row/column) size for non-informative array 
#   testing with master pooling.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
NI.A2M(p=rep(0.01, 64), Se=0.90, Sp=0.90, group.sz=8,
obj.fn=c("ET", "MAR", "GR"),
weights=matrix(data=c(1,1,10,10), 
nrow=2, ncol=2, byrow=TRUE))

Description

Find the optimal testing configuration (OTC) for non-informative array testing without master pooling and calculate the associated operating characteristics.

Usage

NI.Array(p, Se, Sp, group.sz, obj.fn, weights = NULL)

Arguments

Details

This function finds the OTC and computes the associated operating characteristics for non-informative array testing without master pooling. Array testing without master pooling involves amalgamating specimens in rows and columns for the first stage of testing. This function uses only square arrays, which is the way array-based group testing is carried out in most real-world applications. Operating characteristics calculated are expected number of tests, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for the algorithm. See Hitt et al. (2018) at http://chrisbilder.com/grouptesting or Kim et al. (2007) for additional details on the implementation of non-informative array testing without master pooling.

The value(s) specified by group.sz represent the initial group (row/column) size. If a single value is provided for group.sz, operating characteristics will be calculated and no optimization will be performed. If a range of group sizes is specified, the OTC will be found over all group sizes.

The displayed pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value are weighted averages of the corresponding individual accuracy measures for all individuals within the initial group for a hierarchical algorithm. Expressions for these averages are provided in the Supplementary Material for Hitt et al. (2018). These expressions are based on accuracy definitions given by Altman and Bland (1994a, 1994b).

Value

A list containing:

Author(s)

Brianna D. Hitt

References

Altman, D., Bland, J. (1994). “Diagnostic tests 1: sensitivity and specificity.” BMJ, 308, 1552.

Altman, D., Bland, J. (1994). “Diagnostic tests 2: predictive values.” BMJ, 309, 102.

Graff, L., Roeloffs, R. (1972). “Group testing in the presence of test error; an extension of the Dorfman procedure.” Technometrics, 14(1), 113–122. ISSN 15372723, doi:10.1080/00401706.1972.10488888, https://www.tandfonline.com/doi/abs/10.1080/00401706.1972.10488888.

Hitt, B., Bilder, C., Tebbs, J., McMahan, C. (2018). “The Optimal Group Size Controversy for Infectious Disease Testing: Much Ado About Nothing?!” Manuscript submitted for publication.

Kim, H., Hudgens, M., Dreyfuss, J., Westreich, D., Pilcher, C. (2007). “Comparison of group testing algorithms for case identification in the presence of test error.” Biometrics, 63(4), 1152–1163. ISSN 0006341X, doi:10.1111/j.1541-0420.2007.00817.x, http://www.ncbi.nlm.nih.gov/pubmed/17501946.

Malinovsky, Y., Albert, P., Roy, A. (2016). “Reader reaction: A note on the evaluation of group testing algorithms in the presence of misclassification.” Biometrics, 72(1), 299–302. ISSN 15410420, doi:10.1111/biom.12385.

See Also

Inf.Array for informative array testing without master pooling, NI.A2M for non-informative array testing with master pooling, and OTC for finding the optimal testing configuration for a number of standard group testing algorithms.

http://chrisbilder.com/grouptesting

Other OTC functions: Inf.Array, Inf.D3, Inf.Dorf, NI.A2M, NI.D3, NI.Dorf, OTC

Examples

# Find the OTC for non-informative array testing 
#   without master pooling over a range of group
#   (row/column) sizes.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
NI.Array(p=0.04, Se=0.95, Sp=0.95, group.sz=3:10,
obj.fn=c("ET", "MAR"))

# Calculate the operating characteristics for a specified 
#   group (row/column) size for non-informative array 
#   testing without master pooling.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
NI.Array(p=rep(0.01, 64), Se=0.90, Sp=0.90, group.sz=8,
obj.fn=c("ET", "MAR", "GR"), 
weights=matrix(data=c(1,1,10,10,100,100), 
nrow=3, ncol=2, byrow=TRUE))

Description

Find the optimal testing configuration (OTC) for non-informative three-stage hierarchical testing and calculate the associated operating characteristics.

Usage

NI.D3(p, Se, Sp, group.sz, obj.fn, weights = NULL)

Arguments

Details

This function finds the OTC and computes the associated operating characteristics for non-informative three-stage hierarchical testing, by considering all possible testing configurations. Operating characteristics calculated are expected number of tests, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for the algorithm. See Hitt et al. (2018) at http://chrisbilder.com/grouptesting or Kim et al. (2007) for additional details on the implementation of non-informative three-stage hierarchical testing.

The value(s) specified by group.sz represent the initial (stage 1) group size. If a single value is provided for group.sz, the OTC will be found over all possible testing configurations for that initial group size. If a range of group sizes is specified, the OTC will be found over all group sizes.

The displayed pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value are weighted averages of the corresponding individual accuracy measures for all individuals within the initial group for a hierarchical algorithm. Expressions for these averages are provided in the Supplementary Material for Hitt et al. (2018). These expressions are based on accuracy definitions given by Altman and Bland (1994a, 1994b).

Value

A list containing:

Author(s)

Brianna D. Hitt

References

Altman, D., Bland, J. (1994). “Diagnostic tests 1: sensitivity and specificity.” BMJ, 308, 1552.

Altman, D., Bland, J. (1994). “Diagnostic tests 2: predictive values.” BMJ, 309, 102.

Graff, L., Roeloffs, R. (1972). “Group testing in the presence of test error; an extension of the Dorfman procedure.” Technometrics, 14(1), 113–122. ISSN 15372723, doi:10.1080/00401706.1972.10488888, https://www.tandfonline.com/doi/abs/10.1080/00401706.1972.10488888.

Hitt, B., Bilder, C., Tebbs, J., McMahan, C. (2018). “The Optimal Group Size Controversy for Infectious Disease Testing: Much Ado About Nothing?!” Manuscript submitted for publication.

Kim, H., Hudgens, M., Dreyfuss, J., Westreich, D., Pilcher, C. (2007). “Comparison of group testing algorithms for case identification in the presence of test error.” Biometrics, 63(4), 1152–1163. ISSN 0006341X, doi:10.1111/j.1541-0420.2007.00817.x, http://www.ncbi.nlm.nih.gov/pubmed/17501946.

Malinovsky, Y., Albert, P., Roy, A. (2016). “Reader reaction: A note on the evaluation of group testing algorithms in the presence of misclassification.” Biometrics, 72(1), 299–302. ISSN 15410420, doi:10.1111/biom.12385.

See Also

Inf.D3 for informative three-stage hierarchical testing, NI.Dorf for non-informative two-stage hierarchical (Dorfman) testing, Inf.Dorf for informative two-stage hierarchical testing, and OTC for finding the optimal testing configuration for a number of standard group testing algorithms.

http://chrisbilder.com/grouptesting

Other OTC functions: Inf.Array, Inf.D3, Inf.Dorf, NI.A2M, NI.Array, NI.Dorf, OTC

Examples

# Find the OTC for non-informative three-stage 
#   hierarchical testing over a range of group sizes.
# This example takes approximately 20 seconds to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
## Not run: 
NI.D3(p=0.02, Se=0.90, Sp=0.90, group.sz=3:30, 
obj.fn=c("ET", "MAR"))
## End(Not run)

# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
NI.D3(p=0.02, Se=0.90, Sp=0.90, group.sz=3:12, 
obj.fn=c("ET", "MAR"))

# Find the OTC out of all possible configurations for 
#   a specified group size for non-informative
#   three-stage hierarchical testing.
# This example takes less than 1 second to run.
# Estimated running time was calculated using a 
#   computer with 16 GB of RAM and one core of an 
#   Intel i7-6500U processor.
NI.D3(p=rep(0.005, 15), Se=0.99, Sp=0.99, group.sz=15,
obj.fn=c("ET", "MAR", "GR"), weights=matrix(data=c(1,1,10,10), 
nrow=2, ncol=2, byrow=TRUE))

Description

Find the optimal testing configuration (OTC) for non-informative two-stage hierarchical (Dorfman) testing and calculate the associated operating characteristics.

Usage

NI.Dorf(p, Se, Sp, group.sz, obj.fn, weights = NULL)

Arguments

Details

This function finds the OTC and computes the associated operating characteristics for non-informative two-stage hierarchical (Dorfman) testing. Operating characteristics calculated are expected number of tests, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for the algorithm. See Hitt et al. (2018) at http://chrisbilder.com/grouptesting, Dorfman (1943), or Kim et al. (2007) for additional details on the implementation of non-informative two-stage hierarchical testing.

The value(s) specified by group.sz represent the initial (stage 1) group size. If a single value is provided for group.sz, operating characteristics will be calculated and no optimization will be performed. If a range of group sizes is specified, the OTC will be found over all group sizes.

The displayed pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value are weighted averages of the corresponding individual accuracy measures for all individuals within the initial group for a hierarchical algorithm. Expressions for these averages are provided in the Supplementary Material for Hitt et al. (2018). These expressions are based on accuracy definitions given by Altman and Bland (1994a, 1994b).

Value

A list containing:

Author(s)

Brianna D. Hitt

References

Altman, D., Bland, J. (1994). “Diagnostic tests 1: sensitivity and specificity.” BMJ, 308, 1552.

Altman, D., Bland, J. (1994). “Diagnostic tests 2: predictive values.” BMJ, 309, 102.

Dorfman, R. (1943). “The Detection of Defective Members of Large Populations.” The Annals of Mathematical Statistics, 14(4), 436–440. ISSN 0003-4851, doi:10.1214/aoms/1177731363, https://www.jstor.org/stable/2235930.

Graff, L., Roeloffs, R. (1972). “Group testing in the presence of test error; an extension of the Dorfman procedure.” Technometrics, 14(1), 113–122. ISSN 15372723, doi:10.1080/00401706.1972.10488888, https://www.tandfonline.com/doi/abs/10.1080/00401706.1972.10488888.