OTC1 {binGroup2}

Find the optimal testing configuration for group testing algorithms that use a single-disease assay

Description

Find the optimal testing configuration (OTC) using non-informative and informative hierarchical and array-based group testing algorithms. Single-disease assays are used at each stage of the algorithms.

Usage

OTC1(
  algorithm,
  p = NULL,
  probabilities = NULL,
  Se = 0.99,
  Sp = 0.99,
  group.sz,
  obj.fn = "ET",
  weights = NULL,
  alpha = 2,
  trace = TRUE,
  print.time = TRUE,
  ...
)

Arguments

algorithm

character string defining the group testing algorithm to be used. Non-informative testing options include two-stage hierarchical ("D2"), three-stage hierarchical ("D3"), square array testing without master pooling ("A2"), and square array testing with master pooling ("A2M"). Informative testing options include two-stage hierarchical ("ID2"), three-stage hierarchical ("ID3"), and square array testing without master pooling ("IA2").

p

overall probability of disease that will be used to generate a vector/matrix of individual probabilities. For non-informative algorithms, a homogeneous set of probabilities will be used. For informative algorithms, the expectOrderBeta function will be used to generate a heterogeneous set of probabilities. Further details are given under 'Details'. Either p or probabilities should be specified, but not both.

probabilities

a vector of individual probabilities, which is homogeneous for non-informative testing algorithms and heterogeneous for informative testing algorithms. Either p or probabilities should be specified, but not both.

Se

a vector of sensitivity values, where one value is given for each stage of testing (in order). If a single value is provided, sensitivity values are assumed to be equal to this value for all stages of testing. Further details are given under 'Details'.

Sp

a vector of specificity values, where one value is given for each stage of testing (in order). If a single value is provided, specificity values are assumed to be equal to this value for all stages of testing. Further details are given under 'Details'.

group.sz

a single group size or range of group sizes for which to calculate operating characteristics and/or find the OTC. The details of group size specification are given under 'Details'.

obj.fn

a list of objective functions which are minimized to find the OTC. The expected number of tests per individual, "ET", will always be calculated. Additional options include "MAR" (the expected number of tests divided by the expected number of correct classifications, described in Malinovsky et al. (2016)), and "GR" (a linear combination of the expected number of tests, the number of misclassified negatives, and the number of misclassified positives, described in Graff & Roeloffs (1972)). See Hitt et al. (2019) for additional details. The first objective function specified in this list will be used to determine the results for the top configurations. Further details are given under 'Details'.

weights

a matrix of up to six sets of weights for the GR function. Each set of weights is specified by a row of the matrix.

alpha

a shape parameter for the betadistribution that specifies the degree of heterogeneity for the generated probability vector (for informative testing only).

trace

a logical value indicating whether the progress of calculations should be printed for each initial group size provided by the user. The default is TRUE.

print.time

a logical value indicating whether the length of time for calculations should be printed. The default is TRUE.

...

arguments to be passed to the expectOrderBeta function, which generates a vector of probabilities for informative testing algorithms. Further details are given under 'Details'.

Details

This function finds the OTC for group testing algorithms with an assay that tests for one disease and computes the associated operating characteristics, as described in Hitt et al. (2019).

Available algorithms include two- and three-stage hierarchical testing and array testing with and without master pooling. Both non-informative and informative group testing settings are allowed for each algorithm, except informative array testing with master pooling is unavailable because this method has not appeared in the group testing literature. Operating characteristics calculated are expected number of tests, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for each individual.

For informative algorithms where the p argument is specified, the expected value of order statistics from a beta distribution are found. These values are used to represent disease risk probabilities for each individual to be tested. The beta distribution has two parameters: a mean parameter p (overall disease prevalence) and a shape parameter alpha (heterogeneity level). Depending on the specified p, alpha, and overall group size, simulation may be necessary to generate the vector of individual probabilities. This is done using expectOrderBeta and requires the user to set a seed to reproduce results.

Informative two-stage hierarchical (Dorfman) testing is implemented via the pool-specific optimal Dorfman (PSOD) method described in McMahan et al. (2012a), where the greedy algorithm proposed for PSOD is replaced by considering all possible testing configurations. Informative array testing is implemented via the gradient method (the most efficient array design), where higher-risk individuals are grouped in the left-most columns of the array. For additional details on the gradient arrangement method for informative array testing, see McMahan et al. (2012b).

The sensitivity/specificity values are allowed to vary across stages of testing. For hierarchical testing, a different sensitivity/specificity value may be used for each stage of testing. For array testing, a different sensitivity/specificity value may be used for master pool testing (if included), row/column testing, and individual testing. The values must be specified in order of the testing performed. For example, values are specified as (stage 1, stage 2, stage 3) for three-stage hierarchical testing or (master pool testing, row/column testing, individual testing) for array testing with master pooling. A single sensitivity/specificity value may be specified instead. In this situation, sensitivity/specificity values for all stages are assumed to be equal.

The value(s) specified by group.sz represent the initial (stage 1) group size for hierarchical testing and the row/column size for array testing. For informative two-stage hierarchical testing, the group.sz specified represents the block size used in the pool-specific optimal Dorfman (PSOD) method, where the initial group (block) is not tested. For more details on informative two-stage hierarchical testing implemented via the PSOD method, see Hitt et al. (2019) and McMahan et al. (2012a).

If a single value is provided for group.sz with array testing or non-informative two-stage hierarchical testing, operating characteristics will be calculated and no optimization will be performed. If a single value is provided for group.sz with three-stage hierarchical or informative two-stage hierarchical, the OTC will be found over all possible configurations. If a range of group sizes is specified, the OTC will be found over all group sizes.

In addition to the OTC, operating characteristics for some of the other configurations corresponding to each initial group size provided by the user will be displayed. These additional configurations are only determined for whichever objective function ("ET", "MAR", or "GR") is specified first in the function call. If "GR" is the objective function listed first, the first set of corresponding weights will be used. For algorithms where there is only one configuration for each initial group size (non-informative two-stage hierarchical and all array testing algorithms), results for each initial group size are provided. For algorithms where there is more than one possible configuration for each initial group size (informative two-stage hierarchical and all three-stage hierarchical algorithms), two sets of configurations are provided: 1) the best configuration for each initial group size, and 2) the top 10 configurations for each initial group size provided by the user. If a single value is provided for group.sz with array testing or non-informative two-stage hierarchical testing, operating characteristics will not be provided for configurations other than that specified by the user. Results are sorted by the value of the objective function per individual, value.

The displayed overall 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 (or block) 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. (2019). These expressions are based on accuracy definitions given by Altman and Bland (1994a, 1994b). Individual accuracy measures can be calculated using the operatingCharacteristics1 (opChar1) function.

The OTC1 function accepts additional arguments, namely num.sim, to be passed to the expectOrderBeta function, which generates a vector of probabilities for informative group testing algorithms. The num.sim argument specifies the number of simulations from the beta distribution when simulation is used. By default, 10,000 simulations are used.

Value

A list containing:

algorithm

the group testing algorithm used for calculations.

prob

the probability of disease or the vector of individual probabilities, as specified by the user.

alpha

level of heterogeneity for the generated probability vector (for informative testing only).

Se

the vector of sensitivity values for each stage of testing.

Sp

the vector of specificity values for each stage of testing.

opt.ET, opt.MAR, opt.GR

a list of results for each objective function specified by the user, containing:

OTC

a list specifying elements of the optimal testing configuration, which may include:

Stage1

group size for the first stage of hierarchical testing, if applicable.

Stage2

group sizes for the second stage of hierarchical testing, if applicable.

Block.sz

the block size/initial group size for informative Dorfman testing, which is not tested.

pool.szs

group sizes for the first stage of testing for informative Dorfman testing.

Array.dim

the row/column size for array testing.

Array.sz

the overall array size for array testing (the square of the row/column size).

p.vec

the sorted vector of individual probabilities, if applicable.

p.mat

the sorted matrix of individual probabilities in gradient arrangement, if applicable. Further details are given under 'Details'.

ET

the expected testing expenditure to decode all individuals in the algorithm; this includes all individuals in all groups for hierarchical algorithms or in the entire array for array testing.

value

the value of the objective function per individual.

Accuracy

a matrix of overall accuracy measures for the algorithm. The columns correspond to the pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value for the overall algorithm. Further details are given under 'Details'.

Configs

a data frame containing results for the best configuration for each initial group size provided by the user. The columns correspond to the initial group size, configuration (if applicable), overall array size (if applicable), expected number of tests, value of the objective function per individual, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value. No results are displayed if a single group.sz is provided. Further details are given under 'Details'.

Top.Configs

a data frame containing results for the top overall configurations across all initial group sizes provided by the user. The columns correspond to the initial group size, configuration, expected number of tests, value of the objective function per individual, pooling sensitivity, pooling specificity, pooling positive predictive value, and pooling negative predictive value. No results are displayed for non-informative two-stage hierarchical testing or for array testing algorithms. Further details are given under 'Details'.

group.sz

Initial group (or block) sizes examined to find the OTC.

Note

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

Additionally, only stage dependent sensitivity and specificity values are allowed within the program (no group within stage dependent values are allowed). See Bilder et al. (2019) for additional information.

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.

Bilder, C., Tebbs, J., McMahan, C. (2019). “Informative group testing for multiplex assays.” Biometrics, 75, 278–288.

Graff, L., Roeloffs, R. (1972). “Group testing in the presence of test error; an extension of the Dorfman procedure.” Technometrics, 14, 113–122.

Hitt, B., Bilder, C., Tebbs, J., McMahan, C. (2019). “The objective function controversy for group testing: Much ado about nothing?” Statistics in Medicine, 38, 4912–4923.

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, 299–302.

McMahan, C., Tebbs, J., Bilder, C. (2012a). “Informative Dorfman Screening.” Biometrics, 68, 287–296.

McMahan, C., Tebbs, J., Bilder, C. (2012b). “Two-Dimensional Informative Array Testing.” Biometrics, 68, 793–804.

See Also

Other OTC functions: OTC2()

Examples

# Find the OTC for non-informative
#   two-stage hierarchical (Dorfman) testing.
OTC1(algorithm = "D2", p = 0.05, Se = 0.99, Sp = 0.99,
     group.sz = 2:100, obj.fn = "ET",
     trace = TRUE, print.time = TRUE)

# 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 = 0.5.
set.seed(52613)
OTC1(algorithm = "ID2", p = 0.01, Se = 0.95, Sp = 0.95,
     group.sz = 50, obj.fn = c("ET", "MAR", "GR"),
     weights = matrix(data = c(1, 1, 10, 10, 0.5, 0.5),
     nrow = 3, ncol = 2, byrow = TRUE), alpha = 0.5,
     trace = FALSE, print.time = TRUE, num.sim = 10000)

# Find the OTC over all possible testing configurations
#   for non-informative three-stage hierarchical testing
#   with a specified group size.
OTC1(algorithm = "D3", p = 0.001, Se = 0.95, Sp = 0.95,
     group.sz = 18, obj.fn = "ET",
     trace = FALSE, print.time = FALSE)

# Find the OTC for non-informative three-stage
#   hierarchical testing.
OTC1(algorithm = "D3", p = 0.06, Se = 0.90, Sp = 0.90,
     group.sz = 3:30, obj.fn = c("ET", "MAR", "GR"),
     weights = matrix(data = c(1, 1, 10, 10, 100, 100),
     nrow = 3, ncol = 2, byrow = TRUE))

# Find the OTC over all possible configurations
#   for informative three-stage hierarchical testing
#   with a specified group size and a heterogeneous
#   vector of probabilities.
set.seed(1234)
OTC1(algorithm = "ID3",
     probabilities = c(0.012, 0.014, 0.011,
                       0.012, 0.010, 0.015),
     Se = 0.99, Sp = 0.99, group.sz = 6,
     obj.fn = "ET",
     alpha = 0.5, num.sim = 5000, trace = FALSE)

# Calculate the operating characteristics for
#   non-informative array testing without master pooling
#   with a specified array size.
OTC1(algorithm = "A2", p = 0.005, Se = 0.95, Sp = 0.95,
     group.sz = 8, obj.fn = "ET", trace = FALSE)

# Find the OTC for informative array testing without
#   master pooling.
# 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. The probabilities are then arranged in
#   a matrix using the gradient method.
set.seed(1002)
OTC1(algorithm = "IA2", p = 0.03, Se = 0.95, Sp = 0.95,
     group.sz = 2:20, obj.fn = c("ET", "MAR", "GR"),
     weights = matrix(data = c(1, 1, 10, 10, 100, 100),
                      nrow = 3, ncol = 2, byrow = TRUE),
     alpha = 2)

# Find the OTC for non-informative array testing
#   with master pooling. The calculations may not
#   be completed instantaneously.
OTC1(algorithm = "A2M", p = 0.04, Se = 0.90, Sp = 0.90,
     group.sz = 2:20, obj.fn = "ET")