prop.gt: EM Algorithm to Estimate the Prevalence of a Disease from...

In groupTesting: Simulating and Modeling Group (Pooled) Testing Data

EM Algorithm to Estimate the Prevalence of a Disease from Group Testing Data

Description

This function implements an expectation-maximization (EM) algorithm to find the maximum likelihood estimate (MLE) of a disease prevalence, p, based on group testing data. The EM algorithm, which is outlined in Warasi (2021), can model pooling data observed from any group testing protocol used in practice, including hierarchical and array testing (Kim et al., 2007).

Usage

prop.gt(
  p0,
  gtData,
  covariance = FALSE,
  nburn = 2000,
  ngit = 5000,
  maxit = 200,
  tol = 0.001,
  tracing = TRUE,
  conf.level = 0.95
)

Arguments

p0	An initial value of the prevalence.
gtData	A matrix or data.frame consisting of the pooled test outcomes and other information from a group testing application. Needs to be specified as shown in the example below.
covariance	When TRUE, the variance is calculated at the MLE.
nburn	The number of initial Gibbs iterates to be discarded.
ngit	The number of Gibbs iterates to be used in the E-step after discarding the initial iterates as a burn-in period.
maxit	The maximum number of EM steps (iterations) allowed in the EM algorithm.
tol	Convergence tolerance used in the EM algorithm.
tracing	When TRUE, progress in the EM algorithm is displayed.
conf.level	Confidence level to be used for the Wald confidence interval.

Details

gtData must be specified as follows. Columns 1-5 consist of the pooled test outcomes (0 for negative and 1 for positive), pool sizes, pool-specific sensitivities, pool-specific specificities, and assay ID numbers, respectively. From column 6 onward, the pool member ID numbers need to be specified. Note that the ID numbers must start with 1 and increase consecutively up to N, the total number of individuals tested. For smaller pools, incomplete ID numbers must be filled out by -9 or any non-positive numbers as shown in the example below.

	Z	psz	Se	Sp	Assay	Mem1	Mem2	Mem3	Mem4	Mem5	Mem6
Pool:1	1	6	0.90	0.92	1	1	2	3	4	5	6
Pool:2	0	6	0.90	0.92	1	7	8	9	10	11	12
Pool:3	1	2	0.95	0.96	2	1	2	-9	-9	-9	-9
Pool:4	0	2	0.95	0.96	2	3	4	-9	-9	-9	-9
Pool:5	1	2	0.95	0.96	2	5	6	-9	-9	-9	-9
Pool:6	0	1	0.92	0.90	3	1	-9	-9	-9	-9	-9
Pool:7	1	1	0.92	0.90	3	2	-9	-9	-9	-9	-9
Pool:8	0	1	0.92	0.90	3	5	-9	-9	-9	-9	-9
Pool:9	0	1	0.92	0.90	3	6	-9	-9	-9	-9	-9

This is an example of gtData, where 12 individuals are assigned to 2 non-overlapping initial pools and then tested based on the 3-stage hierarchical protocol. The test outcomes, Z, from 9 pools are in column 1. In three stages, different pool sizes (6, 2, and 1), sensitivities, specificities, and assays are used. The ID numbers of the pool members are shown in columns 6-11. The row names and column names are not required. Note that the EM algorithm can accommodate any group testing data including those described in Kim et al. (2007). For individual testing data, the pool size in column 2 is 1 for all pools.

The EM algorithm implements a Gibbs sampler to approximate quantities required to complete the E-step. Under each EM iteration, ngit Gibbs samples are retained for these purposes after discarding the initial nburn samples.

The variance of the MLE is calculated by an appeal to the missing data principle and the method outlined in Louis (1982).

Value

A list with components:

param	The MLE of the disease prevalence.
covariance	Estimated variance for the disease prevalence.
iterUsed	The number of EM iterations used for convergence.
convergence	0 if the EM algorithm converges successfully and 1 if the iteration limit maxit has been reached.
summary	Estimation summary with Wald confidence interval.

References

Kim HY, Hudgens M, Dreyfuss J, Westreich D, and Pilcher C. (2007). Comparison of Group Testing Algorithms for Case Identification in the Presence of Testing Error. Biometrics, 63:1152-1163.

Litvak E, Tu X, and Pagano M. (1994). Screening for the Presence of a Disease by Pooling Sera Samples. Journal of the American Statistical Association, 89:424-434.

Liu A, Liu C, Zhang Z, and Albert P. (2012). Optimality of Group Testing in the Presence of Misclassification. Biometrika, 99:245-251.

Louis T. (1982). Finding the Observed Information Matrix when Using the EM algorithm. Journal of the Royal Statistical Society: Series B, 44:226-233.

Warasi M. (2021). groupTesting: An R Package for Group Testing Estimation. Communications in Statistics-Simulation and Computation. Published online on Dec 9, 2021. Available at https://www.tandfonline.com/doi/full/10.1080/03610918.2021.2009867

See Also

hier.gt.simulation and array.gt.simulation for group testing data simulation, and glm.gt for group testing regression models.

Examples

library(groupTesting)

## To illustrate 'prop.gt', we use data simulated by 
## the R functions 'hier.gt.simulation' and 'array.gt.simulation'.

## The simulated data-structures are consistent  
## with the data-structure required for 'gtData'.

## Example 1: MLE from 3-stage hierarchical group testing data.
## The data used is simulated by 'hier.gt.simulation'.

N <- 90               # Sample size
S <- 3                # 3-stage hierarchical testing
psz <- c(6,2,1)       # Pool sizes used in stages 1-3
Se <- c(.95,.95,.98)  # Sensitivities in stages 1-3
Sp <- c(.95,.98,.96)  # Specificities in stages 1-3
assayID <- c(1,2,3)   # Assays used in stages 1-3
p.t <- 0.05           # The TRUE parameter to be estimated

# Simulating data:
set.seed(123)
gtOut <- hier.gt.simulation(N,p.t,S,psz,Se,Sp,assayID)$gtData

# Running the EM algorithm:
pStart <- p.t + 0.2   # Initial value
res <- prop.gt(p0=pStart,gtData=gtOut,covariance=TRUE,
               nburn=2000,ngit=5000,maxit=200,tol=1e-03,
               tracing=TRUE,conf.level=0.95)

# Estimation results:
# > res

# $param
# [1] 0.05158

# $covariance
#              [,1]
# [1,] 0.0006374296

# $iterUsed
# [1] 4

# $convergence
# [1] 0

# $summary
#      Estimate StdErr 95%lower 95%upper
# prop    0.052  0.025    0.002    0.101

## Example 2: MLE from two-dimensional array testing data.
## The data used is simulated by 'array.gt.simulation'.

N <- 100             # Sample size
protocol <- "A2"     # 2-stage array without testing the initial master pool
n <- 5               # Row/column size
Se <- c(0.95, 0.95)  # Sensitivities
Sp <- c(0.98, 0.98)  # Specificities
assayID <- c(1, 1)   # The same assay in both stages
p.true <- 0.05       # The TRUE parameter to be estimated

# Simulating data:
set.seed(123)
gtOut <- array.gt.simulation(N,p.true,protocol,n,Se,Sp,assayID)$gtData

# Fitting the model:
pStart <- p.true + 0.2  # Initial value
res <- prop.gt(p0=pStart,gtData=gtOut,covariance=TRUE)
print(res)


## Example 3: MLE from non-overlapping initial pooled responses.
## The data used is simulated by 'hier.gt.simulation'. 

## Note: With initial pooled responses, our MLE is equivalent  
## to the MLE in Litvak et al. (1994) and Liu et al. (2012).

N <- 1000             # Sample size
psz <- 5              # Pool size
S <- 1                # 1-stage testing
Se <- 0.95            # Sensitivity
Sp <- 0.99            # Specificity
assayID <- 1          # Assay used for all pools
p.true <- 0.05        # True parameter

set.seed(123)
gtOut <- hier.gt.simulation(N,p.true,S,psz,Se,Sp,assayID)$gtData

pStart <- p.true + 0.2   # Initial value
res <- prop.gt(p0=pStart,gtData=gtOut,
               covariance=TRUE,nburn=2000,ngit=5000,
               maxit=200,tol=1e-03,tracing=TRUE)
print(res)

## Example 4: MLE from individual (one-by-one) testing data.
## The data used is simulated by 'hier.gt.simulation'.

N <- 1000             # Sample size
psz <- 1              # Pool size 1 (i.e., individual testing)
S <- 1                # 1-stage testing
Se <- 0.95            # Sensitivity
Sp <- 0.99            # Specificity
assayID <- 1          # Assay used for all pools
p.true <- 0.05        # True parameter

set.seed(123)
gtOut <- hier.gt.simulation(N,p.true,S,psz,Se,Sp,assayID)$gtData

pStart <- p.true + 0.2   # Initial value
res <- prop.gt(p0=pStart,gtData=gtOut,
               covariance=TRUE,nburn=2000,
               ngit=5000,maxit=200,
               tol=1e-03,tracing=TRUE)
print(res)

## Example 5: Using pooled testing data.

# Pooled test outcomes:
Z <- c(1, 0, 1, 0, 1, 0, 1, 0, 0)     

# Pool sizes used:
psz <- c(6, 6, 2, 2, 2, 1, 1, 1, 1)

# Pool-specific Se & Sp:
Se <- c(.90, .90, .95, .95, .95, .92, .92, .92, .92)
Sp <- c(.92, .92, .96, .96, .96, .90, .90, .90, .90)

# Assays used:
Assay <- c(1, 1, 2, 2, 2, 3, 3, 3, 3)

# Pool members:
Memb <- rbind( 
   c(1, 2,  3,  4,  5,  6),
   c(7, 8,  9, 10, 11, 12),
   c(1, 2, -9, -9, -9, -9),
   c(3, 4, -9, -9, -9, -9),
   c(5, 6, -9, -9, -9, -9),
   c(1,-9, -9, -9, -9, -9),
   c(2,-9, -9, -9, -9, -9),
   c(5,-9, -9, -9, -9, -9),
   c(6,-9, -9, -9, -9, -9)
)
# The data-structure suited for 'gtData':
gtOut <- cbind(Z, psz, Se, Sp, Assay, Memb)

# Fitting the model:
pStart <- 0.10
res <- prop.gt(p0=pStart,gtData=gtOut,
               covariance=TRUE,nburn=2000,
               ngit=5000,maxit=200,
               tol=1e-03,tracing=TRUE)
print(res)

groupTesting documentation built on Nov. 6, 2023, 9:06 a.m.