This page aims to illustrate the Data Generation, Analysis, Simulation Study as well as the Numerical Example for the manuscript “Utilizing a Capture-Recapture Strategy to Accelerate Infectious Disease Surveillance”.
All functions are available in this Github repository and can be loaded as follows.
source("FUN_CRC_project_SeSp.R")Suppose we generate a dataset with total population size
and true disease prevalence
. Data
were generated in such a way that among those with disease, 50% of
individuals exhibited symptoms. In contrast, only 10% of those without
disease showed symptoms. The Stream 1 sample was drawn to reflect
voluntary-based non-representative surveillance data, selecting 80% of
individuals with symptoms for testing as opposed to 10% of those without
symptoms. Stream 2 was generated as the anchor stream independently of
Stream 1, with the sample size
. We
then consider “low” level of misclassification (e.g.,
,
).
This is the same setting as Table 2 in the paper.
Therefore we set the following parameters:
N = 1000
p_case = 0.1
p_case_sym = 0.5
p_not_case_sym = 0.1
N_true = c(rep(1,round(N*p_case)),rep(0,round(N*(1-p_case))))
Npos = sum(N_true)
Nneg = N-Npos
# two-steam surveillance
p_test1_givSym = .8
p_test1_givAsym = .1
Se1 = 0.9
Sp1 = 0.9
n2 = 50
p_stream2 = n2/N
p_test2_givSym = p_stream2
p_test2_givAsym = p_stream2
Se2 = 0.95
Sp2 = 0.95and we can generate the dataset (including testing results in Stream 1
and 2, as well as n1-n9) by the following functions.
set.seed(1111111)
# generate the symptom status
N_sym = rep(0,N)
N_sym[which(N_true == 1)] = rbinom(Npos,1,p_case_sym)
N_sym[which(N_true == 0)] = rbinom(N-Npos,1,p_not_case_sym)
p_sym = sum(N_sym)/N
# generate steam 1
simu1 = simu_sym3(N,N_sym,p_sym,N_true,p_test1_givSym,p_test1_givAsym,Se1,Sp1)
test1 = simu1$test
testpos1 = simu1$testpos
# generate steam 2
simu2 = simu_sym_RS(N,N_true,p_stream2,Se2,Sp2)
test2 = simu2$test
testpos2 = simu2$testpos
obs_summary = two_by_two_table2(test1,testpos1,test2,testpos2)
n1 = obs_summary$n1
n2 = obs_summary$n2
n3 = obs_summary$n3
n4 = obs_summary$n4
n5 = obs_summary$n5
n6 = obs_summary$n6
n7 = obs_summary$n7
n8 = obs_summary$n8
n9 = obs_summary$n9There are two estimators compared in Section 2 of the paper: Random
Sampling (RS) based estimator and Capture-Recapture (CRC) estimator. We
use function RS_mis for RS estimator in eqn. (1)-(3), and
ML_est_SESP_numerical or ML_est_SESP_closedform for CRC estimator in
eqn. (5)-(8).
As an example, we use the following programs to calculate RS estimator:
RS = RS_mis(N, n_stream2=n1+n2+n3+n4+n7+n8, n_pos_stream2=n1+n4+n7, Se2=Se2, Sp2=Sp2)
N_RS = RS$Nhat
N_RS_sd = RS$SEhat
RS = data.frame(est=N_RS, se=N_RS_sd, pct_025=N_RS-1.96*N_RS_sd, pct_975=N_RS+1.96*N_RS_sd,
width=1.96*N_RS_sd*2)
print(RS,row.names=FALSE) ## est se pct_025 pct_975 width
## 188.8889 64.54436 62.38195 315.3958 253.0139
and we calculate the closed form CRC estimator as follows:
MLE = ML_est_SESP_closedform(n1,n2,n3,n4,n5,n6,n7,n8,n9,Se1_Sp1_par=c(Se1,Sp1),
Se2_Sp2_par=c(Se2,Sp2))
N_SESP = MLE$mle
N_SESP_sd = MLE$mle_se
SESP =data.frame(est=N_SESP, se=N_SESP_sd, pct_025=N_SESP-1.96*N_SESP_sd,
pct_975=N_SESP+1.96*N_SESP_sd,width=1.96*N_SESP_sd*2)
print(SESP,row.names=FALSE) ## est se pct_025 pct_975 width
## 125.5507 50.72714 26.12552 224.9759 198.8504
Except for the Wald-type confidence interval for CRC estimator, we also provide a Bayesian credible interval in Section 2.4.
BC_interval = BC_interval_SESP(n1,n2,n3,n4,n5,n6,n7,n8,n9,Se1,Sp1,Se2,Sp2)
BC_interval2 = RS_BC2(N, n_pos_stream2=n1+n4+n7,n_stream2=n1+n2+n3+n4+n7+n8,Se2,Sp2)
if(BC_interval$BC_width>(N * BC_interval2$BC_width)) {
lower = N * BC_interval2$BC_lower
upper = N * BC_interval2$BC_upper
BC_interval = list(BC_lower = lower,BC_upper = upper,BC_width = upper - lower)
}
if (BC_interval$BC_upper >= Npos &&
BC_interval$BC_lower <= Npos) {
BC_interval_coverage = 1
} else{
BC_interval_coverage = 0
}
BC_interval_width = BC_interval$BC_width
SESP_bc =data.frame(est=N_SESP, se=N_SESP_sd, pct_025=BC_interval$BC_lower,
pct_975=BC_interval$BC_upper,width=BC_interval_width)
print(SESP_bc,row.names=FALSE) ## est se pct_025 pct_975 width
## 125.5507 50.72714 53.82643 251.276 197.4496
Meanwhile, we also proposed a MI-based approach when Se and Sp are estimated from the validation data. More details are available in “Numeric Example” below.
To perform simulation study, we use the following loop to evaluate the simulation results and generate Table 2-4.
tmp = NULL
for(p_case in seq(0.1,0.5,0.2)){
for(n2_samples in c(50, 100, 300) ){
p_stream2 = n2_samples/N
res = find_p_p2(p_case,p_stream2, N, Se1,Sp1=Se1,Se2=Se1+0.05,Sp2=Se1+0.05)
tmp = rbind(tmp,t(c(p_case,p_stream2,res$res.true,res$res.RS[1:5],res$res.SESP[1:5],
res$res.BC_width, res$res.BC_pct,
res$res.SESP2, res$res.SESP2_sd, Se1)))
}
}
colnames(tmp) = c('p_case','p_stream2','N_true','RS.mean','RS.sd','RS.avgse','RS.width',
'RS.Clpct','SESP.mean','SESP.sd','SESP.avgse','SESP.width','SESP.Clpct',
'SESP.BCwidth','SESP.BCpct','SESP2.mean','SESP2.sd', 'Se1')
(result = round(tmp[,],2))## p_case p_stream2 N_true RS.mean RS.sd RS.avgse RS.width RS.Clpct
## [1,] 0.1 0.05 100 98.22 54.28 52.74 206.76 91.6
## [2,] 0.1 0.10 100 101.98 37.31 37.39 146.59 94.7
## [3,] 0.1 0.30 100 99.09 20.94 20.07 78.69 93.4
## [4,] 0.3 0.05 300 298.93 72.41 71.70 281.05 93.2
## [5,] 0.3 0.10 300 297.88 50.73 49.66 194.66 94.9
## [6,] 0.3 0.30 300 299.00 25.87 26.17 102.58 95.7
## [7,] 0.5 0.05 500 502.33 76.56 76.99 301.80 94.2
## [8,] 0.5 0.10 500 499.14 52.15 53.29 208.89 96.5
## [9,] 0.5 0.30 500 500.16 27.31 27.92 109.44 95.1
## SESP.mean SESP.sd SESP.avgse SESP.width SESP.Clpct SESP.BCwidth
## [1,] 99.37 41.69 42.92 168.24 93.0 157.59
## [2,] 101.93 30.92 30.98 121.44 93.2 117.60
## [3,] 99.74 17.55 17.66 69.21 94.6 69.16
## [4,] 299.20 58.86 57.66 226.03 92.4 220.54
## [5,] 298.18 40.48 40.32 158.04 94.1 156.34
## [6,] 299.63 21.56 22.38 87.72 95.5 88.31
## [7,] 502.08 64.90 63.36 248.36 92.7 240.00
## [8,] 500.15 42.52 44.15 173.08 95.0 170.49
## [9,] 499.38 23.75 24.18 94.79 94.8 95.36
## SESP.BCpct SESP2.mean SESP2.sd Se1
## [1,] 94.6 99.22 41.77 0.9
## [2,] 94.5 101.91 30.85 0.9
## [3,] 95.4 99.62 17.47 0.9
## [4,] 95.0 299.19 58.85 0.9
## [5,] 95.0 298.19 40.47 0.9
## [6,] 95.7 299.63 21.42 0.9
## [7,] 94.6 502.09 64.89 0.9
## [8,] 95.5 500.11 42.50 0.9
## [9,] 95.2 499.44 23.65 0.9
Now we analyze the numerical example provided in this paper. The data
can be found in (data_numerical_example.R).
library(pander)
set.seed(111111)
source("data_numerical_example.R")The data includes nine cell counts (n1-n9):
data_obs = c(n1,n2,n3,n4,n5,n6,n7,n8,n9)
data_obs## [1] 3 12 0 2 27 130 6 77 743
as well as the 2-by-2 table of testing results from Table 5 of paper: True validation data from Murakami et al. (2023),
n_SE1SP1_validation = c(n11_t1,n10_t1,n01_t1,n00_t1)
matrix(n_SE1SP1_validation,2,2,byrow = T)## [,1] [,2]
## [1,] 65 1
## [2,] 38 552
Se1_initial = n11_t1/(n11_t1+n01_t1)
Sp1_initial = n00_t1/(n10_t1+n00_t1)
c(Se1_initial,Sp1_initial)## [1] 0.6310680 0.9981917
and from Casati et al. (2022),
n_SE2SP2_validation = c(n11_t2,n10_t2,n01_t2,n00_t2)
matrix(n_SE2SP2_validation,2,2,byrow = T)## [,1] [,2]
## [1,] 89 0
## [2,] 6 100
Se2_initial = n11_t2/(n11_t2+n01_t2)
Sp2_initial = n00_t2/(n10_t2+n00_t2)
c(Se2_initial,Sp2_initial)## [1] 0.9368421 1.0000000
In Section 4, we first analyze the data assuming that the misclassification parameters of each data streams are known. Then we can directly calculate the case count estimates and corresponding confidence (or credible) intervals using the approaches introduced in Section 2. That is,
RS = RS_mis(N=sum(data_obs), n_stream2=n1+n2+n3+n4+n7+n8,
n_pos_stream2=n1+n4+n7, Se2=Se2_initial, Sp2=Sp2_initial)
N_RS = RS$Nhat
N_RS_sd = RS$SEhat
RS_m1 = data.frame(est=N_RS, se=N_RS_sd, pct_025=N_RS-1.96*N_RS_sd,
pct_975=N_RS+1.96*N_RS_sd, width=1.96*N_RS_sd*2)
print(RS_m1,row.names=FALSE) ## est se pct_025 pct_975 width
## 117.4157 31.96812 54.75822 180.0732 125.315
and CRC estimate results:
MLE = ML_est_SESP_closedform(n1,n2,n3,n4,n5,n6,n7,n8,n9,Se1_Sp1_par=c(Se1_initial,Sp1_initial),
Se2_Sp2_par=c(Se2_initial,Sp2_initial))
N_SESP = MLE$mle
N_SESP_sd = MLE$mle_se
SESP_m1 =data.frame(est=N_SESP, se=N_SESP_sd, pct_025=N_SESP-1.96*N_SESP_sd,
pct_975=N_SESP+1.96*N_SESP_sd, width=1.96*N_SESP_sd*2)
print(SESP_m1,row.names=FALSE) ## est se pct_025 pct_975 width
## 111.5433 24.67128 63.18762 159.8991 96.71143
as well as CRC estimate with Bayesian credible interval:
tmp2 = BC_interval_SESP(n1,n2,n3,n4,n5,n6,n7,n8,n9,Se1_initial,Sp1_initial,
Se2_initial,Sp2_initial)
SESP_m1_bc = data.frame(est=N_SESP, se=N_SESP_sd, pct_025=tmp2$BC_lower,
pct_975=tmp2$BC_upper,width=tmp2$BC_width)
print(SESP_m1_bc,row.names=FALSE) ## est se pct_025 pct_975 width
## 111.5433 24.67128 75.07522 172.5666 97.49134
With access to validation data for estimating the misclassification
parameters as in Table 5, we recommend the approach introduced in
Section 2.5 that adapts the multiple imputation (MI) paradigm (Rubin,
1987) to account for uncertainty in these parameters. In this
repository, we define a function MI_main to implement MI to calculate
each estimators introduced in the paper. The results are as follows.
tmp = unlist(MI_main(data_obs,n_SE1SP1_validation,n_SE2SP2_validation))
RS_m2 = tmp[1:5]
SESP_m2 = tmp[6:10]
SESP_m2_bc = tmp[c(6,7,11:13)]
RS_m2## RS_MI RS_MI.se RS_MI_lower RS_MI_upper RS_MI_length
## 113.65881 33.30964 48.37191 178.94571 130.57380
SESP_m2## SESP_MI SESP_MI.se SESP_MI_lower SESP_MI_upper SESP_MI_length
## 108.20869 26.04108 57.16818 159.24920 102.08102
SESP_m2_bc## SESP_MI SESP_MI.se SESP_BC_lower.2.5%
## 108.20869 26.04108 68.45380
## SESP_BC_upper.97.5% SESP_BC_length.97.5%
## 172.67935 104.22555
Overall, we can combine all results and compare each methods as follows (same as Table 6 in the paper).
| est | se | 2.5% | 97.5% | width | |
|---|---|---|---|---|---|
| RS_m1 | 117.4 | 32 | 54.8 | 180.1 | 125.3 |
| SESP_m1 | 111.5 | 24.7 | 63.2 | 159.9 | 96.7 |
| SESP_m1_bc | 111.5 | 24.7 | 75.1 | 172.6 | 97.5 |
| RS_m2 | 113.7 | 33.3 | 48.4 | 178.9 | 130.6 |
| SESP_m2 | 108.2 | 26 | 57.2 | 159.2 | 102.1 |
| SESP_m2_bc | 108.2 | 26 | 68.5 | 172.7 | 104.2 |