,
Edward H. Kennedy [aut] | Title: | Efficient and Doubly Robust Population Size Estimation |
|---|---|
| Description: | Estimation of the total population size from capture-recapture data efficiently and with low bias implementing the methods from Das M, Kennedy EH, and Jewell NP (2021) <arXiv:2104.14091>. The estimator is doubly robust against errors in the estimation of the intermediate nuisance parameters. Users can choose from the flexible estimation models provided in the package, or use any other preferred model. |
| Authors: | Manjari Das [aut, cre] ,
Edward H. Kennedy [aut] |
| Maintainer: | Manjari Das <[email protected]> |
| License: | GPL-3 |
| Version: | 0.0.3 |
| Built: | 2025-03-06 06:35:25 UTC |
| Source: | https://github.com/mqnjqrid/drpop |
A function to check whether a given data table/matrix/data frame is in the appropriate for drpop.
informat(data, K = 2)informat(data, K = 2)
data |
The data table/matrix/data frame which is to be checked. |
K |
The number of lists (optional). |
A boolean for whether data is in the appropriate format.
data = matrix(sample(c(0,1), 2000, replace = TRUE), ncol = 2) x = matrix(rnorm(nrow(data)*3, 2,1), nrow = nrow(data)) informat(data = data) #this returns TRUE data = cbind(data, x) informat(data = data) #this returns TRUE informat(data = data, K = 3) #this returns FALSEdata = matrix(sample(c(0,1), 2000, replace = TRUE), ncol = 2) x = matrix(rnorm(nrow(data)*3, 2,1), nrow = nrow(data)) informat(data = data) #this returns TRUE data = cbind(data, x) informat(data = data) #this returns TRUE informat(data = data, K = 3) #this returns FALSE
popsize or popsize_cond.Plot estimated confidence interval of total population size from object of class popsize or popsize_cond.
plotci(object, tsize = 12, ...)plotci(object, tsize = 12, ...)
object |
An object of class |
tsize |
The text size for the plots. |
... |
Any extra arguments passed into the function. |
A ggplot object fig with population size estimates and the 95% confidence intervals.
H. Wickham. ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York, 2016.
data = simuldata(n = 10000, l = 1)$data_xstar p = popsize(data = data, funcname = c("logit", "gam")) plotci(p) data = simuldata(n = 10000, l = 1, categorical = TRUE)$data_xstar p = popsize_cond(data = data, condvar = 'catcov') plotci(p)data = simuldata(n = 10000, l = 1)$data_xstar p = popsize(data = data, funcname = c("logit", "gam")) plotci(p) data = simuldata(n = 10000, l = 1, categorical = TRUE)$data_xstar p = popsize_cond(data = data, condvar = 'catcov') plotci(p)
Estimate total population size and capture probability using user provided set of models or user provided nuisance estimates.
popsize( data, K = 2, j, k, margin = 0.005, filterrows = FALSE, nfolds = 5, funcname = c("rangerlogit"), sl.lib = c("SL.gam", "SL.glm", "SL.glm.interaction", "SL.ranger", "SL.glmnet"), getnuis, q1mat, q2mat, q12mat, idfold, TMLE = TRUE, PLUGIN = TRUE, Nmin = 100, ... )popsize( data, K = 2, j, k, margin = 0.005, filterrows = FALSE, nfolds = 5, funcname = c("rangerlogit"), sl.lib = c("SL.gam", "SL.glm", "SL.glm.interaction", "SL.ranger", "SL.glmnet"), getnuis, q1mat, q2mat, q12mat, idfold, TMLE = TRUE, PLUGIN = TRUE, Nmin = 100, ... )
data |
The data frame in capture-recapture format with |
K |
The number of lists that are present in the data. |
j |
The first list to be used for estimation. |
k |
The secod list to be used in the estimation. |
margin |
The minimum value the estimates can attain to bound them away from zero. |
filterrows |
A logical value denoting whether to remove all rows with only zeroes. |
nfolds |
The number of folds to be used for cross fitting. |
funcname |
The vector of estimation function names to obtain the population size. |
sl.lib |
Algorithm library for |
getnuis |
A list object with the nuisance function estimates and the fold assignment of the rows for cross-fitting or a data.frame with the nuisance estimates. |
q1mat |
A dataframe with capture probabilities for the first list. |
q2mat |
A dataframe with capture probabilities for the second list. |
q12mat |
A dataframe with capture probabilities for both the lists simultaneously. |
idfold |
The fold assignment of each row during estimation. |
TMLE |
The logical value to indicate whether TMLE has to be computed. |
PLUGIN |
The logical value to indicate whether the plug-in estimates are returned. |
Nmin |
The cutoff for minimum sample size to perform doubly robust estimation. Otherwise, Petersen estimator is returned. |
... |
Any extra arguments passed into the function. See |
A list of estimates containing the following components for each list-pair, model and method (PI = plug-in, DR = doubly-robust, TMLE = targeted maximum likelihood estimate):
result |
A dataframe of the below estimated quantities.
|
N |
The number of data points used in the estimation after removing rows with missing data. |
ifvals |
The estimated influence function values for the observed data. |
nuis |
The estimated nuisance functions (q12, q1, q2) for each element in funcname. |
nuistmle |
The estimated nuisance functions (q12, q1, q2) from tmle for each element in funcname. |
idfold |
The division of the rows into sets (folds) for cross-fitting. |
Bickel, P. J., Klaassen, C. A., Bickel, P. J., Ritov, Y., Klaassen, J., Wellner, J. A., and Ritov, Y. (1993). Efficient and adaptive estimation for semiparametric models, volume 4. Johns Hopkins University Press Baltimore
van der Vaart, A. (2002a). Part iii: Semiparameric statistics. Lectures on Probability Theory and Statistics, pages 331-457
van der Laan, M. J. and Robins, J. M. (2003). Unified methods for censored longitudinal data and causality. Springer Science & Business Media
Tsiatis, A. (2006). Semiparametric theory and missing data springer. New York
Kennedy, E. H. (2016). Semiparametric theory and empirical processes in causal inference. Statistical causal inferences and their applications in public health research, pages 141-167. Springer
Das, M., Kennedy, E. H., & Jewell, N.P. (2021). Doubly robust capture-recapture methods for estimating population size. arXiv preprint arXiv:2104.14091.
data = simuldata(1000, l = 3)$data qhat = popsize(data = data, funcname = c("logit", "gam"), nfolds = 2, margin = 0.005) psin_estimate = popsize(data = data, getnuis = qhat$nuis, idfold = qhat$idfold) data = simuldata(n = 6000, l = 3)$data psin_estimate = popsize(data = data[,1:2]) #this returns the basic plug-in estimate since covariates are absent. psin_estimate = popsize(data = data, funcname = c("gam", "rangerlogit"))data = simuldata(1000, l = 3)$data qhat = popsize(data = data, funcname = c("logit", "gam"), nfolds = 2, margin = 0.005) psin_estimate = popsize(data = data, getnuis = qhat$nuis, idfold = qhat$idfold) data = simuldata(n = 6000, l = 3)$data psin_estimate = popsize(data = data[,1:2]) #this returns the basic plug-in estimate since covariates are absent. psin_estimate = popsize(data = data, funcname = c("gam", "rangerlogit"))
Estimate total population size and capture probability using user provided set of models conditioned on an attribute.
popsize_cond( data, K = 2, filterrows = FALSE, funcname = c("rangerlogit"), condvar, nfolds = 2, margin = 0.005, sl.lib = c("SL.gam", "SL.glm", "SL.glm.interaction", "SL.ranger", "SL.glmnet"), TMLE = TRUE, PLUGIN = TRUE, Nmin = 100, ... )popsize_cond( data, K = 2, filterrows = FALSE, funcname = c("rangerlogit"), condvar, nfolds = 2, margin = 0.005, sl.lib = c("SL.gam", "SL.glm", "SL.glm.interaction", "SL.ranger", "SL.glmnet"), TMLE = TRUE, PLUGIN = TRUE, Nmin = 100, ... )
data |
The data frame in capture-recapture format for which total population is to be estimated. The first K columns are the capture history indicators for the K lists. The remaining columns are covariates in numeric format. |
K |
The number of lists in the data. typically the first |
filterrows |
A logical value denoting whether to remove all rows with only zeroes. |
funcname |
The vector of estimation function names to obtain the population size. |
condvar |
The covariate for which conditional estimates are required. |
nfolds |
The number of folds to be used for cross fitting. |
margin |
The minimum value the estimates can attain to bound them away from zero. |
sl.lib |
Algorithm library for |
TMLE |
The logical value to indicate whether TMLE has to be computed. |
PLUGIN |
The logical value to indicate whether the plug-in estimates are returned. |
Nmin |
The cutoff for minimum sample size to perform doubly robust estimation. Otherwise, Petersen estimator is returned. |
... |
Any extra arguments passed into the function. See |
A list of estimates containing the following components for each list-pair, model and method (PI = plug-in, DR = doubly-robust, TMLE = targeted maximum likelihood estimate):
result |
A dataframe of the below estimated quantities.
|
N |
The number of data points used in the estimation after removing rows with missing data. |
ifvals |
The estimated influence function values for the observed data. |
nuis |
The estimated nuisance functions (q12, q1, q2) for each element in funcname. |
nuistmle |
The estimated nuisance functions (q12, q1, q2) from tmle for each element in funcname. |
idfold |
The division of the rows into sets (folds) for cross-fitting. |
Das, M., Kennedy, E. H., & Jewell, N.P. (2021). Doubly robust capture-recapture methods for estimating population size. arXiv preprint arXiv:2104.14091.
data = simuldata(n = 10000, l = 2, categorical = TRUE)$data psin_estimate = popsize_cond(data = data, funcname = c("logit", "gam"), condvar = 'catcov', PLUGIN = TRUE, TMLE = TRUE) #this returns the plug-in, the bias-corrected and the tmle estimate for the #two models conditioned on column catcovdata = simuldata(n = 10000, l = 2, categorical = TRUE)$data psin_estimate = popsize_cond(data = data, funcname = c("logit", "gam"), condvar = 'catcov', PLUGIN = TRUE, TMLE = TRUE) #this returns the plug-in, the bias-corrected and the tmle estimate for the #two models conditioned on column catcov
Estimate the total population size and capture probabilities using perturbed true nuisance functions.
popsize_simul( data, n, K = 2, nfolds = 5, pi1, pi2, omega, alpha, margin = 0.005, iter = 100, twolist = TRUE )popsize_simul( data, n, K = 2, nfolds = 5, pi1, pi2, omega, alpha, margin = 0.005, iter = 100, twolist = TRUE )
data |
The data frame in capture-recapture format for which total population is to be estimated. The first K columns are the capture history indicators for the K lists. The remaining columns are covariates in numeric format. |
n |
The true population size. Required to calculate the added error. |
K |
The number of lists in the data. typically the first |
nfolds |
The number of folds to be used for cross fitting. |
pi1 |
The function to calculate the conditional capture probabilities of list 1 using covariates. |
pi2 |
The function to calculate the conditional capture probabilities of list 2 using covariates. |
omega |
The standard deviation from zero of the added error. |
alpha |
The rate of convergence. Takes values in (0, 1]. |
margin |
The minimum value the estimates can attain to bound them away from zero. |
iter |
An integer denoting the maximum number of iterations allowed for targeted maximum likelihood method. |
twolist |
The logical value of whether targeted maximum likelihood algorithm fits only two modes when K = 2. |
A list of estimates containing the following components:
psi |
A matrix of the estimated capture probability for each list pair, model and method combination. In the absence of covariates, the column represents the standard plug-in estimate. The rows represent the list pair which is assumed to be independent conditioned on the covariates. The columns represent the model and method combinations (PI = plug-in, DR = bias-corrected, TMLE = targeted maximum likelihood estimate)indicated in the columns. |
sigma2 |
A matrix of the efficiency bound |
n |
A matrix of the estimated population size n in the same format as |
varn |
A matrix of the variance for population size estimate in the same format as |
N |
The number of data points used in the estimation after removing rows with missing data. |
Das, M., Kennedy, E. H., & Jewell, N.P. (2021). Doubly robust capture-recapture methods for estimating population size. arXiv preprint arXiv:2104.14091
simulresult = simuldata(n = 2000, l = 2) data = simulresult$data psin_estimate = popsize_simul(data = data, pi1 = simulresult$pi1, pi2 = simulresult$pi2, alpha = 0.25, omega = 1)simulresult = simuldata(n = 2000, l = 2) data = simulresult$data psin_estimate = popsize_simul(data = data, pi1 = simulresult$pi1, pi2 = simulresult$pi2, alpha = 0.25, omega = 1)
Estimate marginal and joint distribution of lists j and k using generalized additive models.
qhat_gam(List.train, List.test, K = 2, j = 1, k = 2, margin = 0.005, ...)qhat_gam(List.train, List.test, K = 2, j = 1, k = 2, margin = 0.005, ...)
List.train |
The training data matrix used to estimate the distibution functions. |
List.test |
The data matrix on which the estimator function is applied. |
K |
The number of lists in the data. |
j |
The first list that is conditionally independent. |
k |
The second list that is conditionally independent. |
margin |
The minimum value the estimates can attain to bound them away from zero. |
... |
Any extra arguments passed into the function. |
A list of the marginal and joint distribution probabilities q1, q2 and q12.
Trevor Hastie (2020). gam: Generalized Additive Models. R package version 1.20. https://CRAN.R-project.org/package=gam
## Not run: qhat = qhat_gam(List.train = List.train, List.test = List.test, margin = 0.005) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 ## End(Not run)## Not run: qhat = qhat_gam(List.train = List.train, List.test = List.test, margin = 0.005) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 ## End(Not run)
Estimate marginal and joint distribution of lists j and k using logistic regression.
qhat_logit(List.train, List.test, K = 2, j = 1, k = 2, margin = 0.005, ...)qhat_logit(List.train, List.test, K = 2, j = 1, k = 2, margin = 0.005, ...)
List.train |
The training data matrix used to estimate the distibution functions. |
List.test |
The data matrix on which the estimator function is applied. |
K |
The number of lists in the data. |
j |
The first list that is conditionally independent. |
k |
The second list that is conditionally independent. |
margin |
The minimum value the estimates can attain to bound them away from zero. |
... |
Any extra arguments passed into the function. |
A list of the marginal and joint distribution probabilities q1, q2 and q12.
## Not run: qhat = qhat_logit(List.train = List.train, List.test = List.test, margin = 0.005) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 ## End(Not run)## Not run: qhat = qhat_logit(List.train = List.train, List.test = List.test, margin = 0.005) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 ## End(Not run)
Estimate marginal and joint distribution of lists j and k using multinomial logistic model.
qhat_mlogit(List.train, List.test, K = 2, j = 1, k = 2, margin = 0.005, ...)qhat_mlogit(List.train, List.test, K = 2, j = 1, k = 2, margin = 0.005, ...)
List.train |
The training data matrix used to estimate the distibution functions. |
List.test |
The data matrix on which the estimator function is applied. |
K |
The number of lists in the data. |
j |
The first list that is conditionally independent. |
k |
The second list that is conditionally independent. |
margin |
The minimum value the estimates can attain to bound them away from zero. |
... |
Any extra arguments passed into the function. |
A list of the marginal and joint distribution probabilities q1, q2 and q12.
Croissant Y (2020). Estimation of Random Utility Models in R: The mlogit Package. Journal of Statistical Software, 95(11), 1-41. doi: 10.18637/jss.v095.i11 (URL: https://doi.org/10.18637/jss.v095.i11).
Venables, W. N. & Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth Edition. Springer, New York. ISBN 0-387-95457-0
## Not run: qhat = qhat_mlogit(List.train = List.train, List.test = List.test, margin = 0.005) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 ## End(Not run)## Not run: qhat = qhat_mlogit(List.train = List.train, List.test = List.test, margin = 0.005) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 ## End(Not run)
Estimate marginal and joint distribution of lists j and k using ranger.
qhat_ranger(List.train, List.test, K = 2, j = 1, k = 2, margin = 0.005, ...)qhat_ranger(List.train, List.test, K = 2, j = 1, k = 2, margin = 0.005, ...)
List.train |
The training data matrix used to estimate the distibution functions. |
List.test |
The data matrix on which the estimator function is applied. |
K |
The number of lists in the data. |
j |
The first list that is conditionally independent. |
k |
The second list that is conditionally independent. |
margin |
The minimum value the estimates can attain to bound them away from zero. |
... |
Any extra arguments passed into the function. |
A list of the marginal and joint distribution probabilities q1, q2 and q12.
Marvin N. Wright, Andreas Ziegler (2017). ranger: A Fast Implementation of Random Forests for High Dimensional Data in C++ and R. Journal of Statistical Software, 77(1), 1-17. doi:10.18637/jss.v077.i01
## Not run: qhat = qhat_ranger(List.train = List.train, List.test = List.test, margin = 0.005) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 ## End(Not run)## Not run: qhat = qhat_ranger(List.train = List.train, List.test = List.test, margin = 0.005) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 ## End(Not run)
Estimate marginal and joint distribution of lists j and k using ensemble of ranger and logit.
qhat_rangerlogit( List.train, List.test, K = 2, j = 1, k = 2, margin = 0.005, ... )qhat_rangerlogit( List.train, List.test, K = 2, j = 1, k = 2, margin = 0.005, ... )
List.train |
The training data matrix used to estimate the distibution functions. |
List.test |
The data matrix on which the estimator function is applied. |
K |
The number of lists in the data. |
j |
The first list that is conditionally independent. |
k |
The second list that is conditionally independent. |
margin |
The minimum value the estimates can attain to bound them away from zero. |
... |
Any extra arguments passed into the function. |
A list of the marginal and joint distribution probabilities q1, q2 and q12.
Marvin N. Wright, Andreas Ziegler (2017). ranger: A Fast Implementation of Random Forests for High Dimensional Data in C++ and R. Journal of Statistical Software, 77(1), 1-17. doi:10.18637/jss.v077.i01
Polley, Eric C. and van der Laan, Mark J., (May 2010) Super Learner In Prediction. U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 266. https://biostats.bepress.com/ucbbiostat/paper266
## Not run: qhat = qhat_ranger(List.train = List.train, List.test = List.test, margin = 0.005) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 ## End(Not run)## Not run: qhat = qhat_ranger(List.train = List.train, List.test = List.test, margin = 0.005) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 ## End(Not run)
Estimate marginal and joint distribution of lists j and k using super learner.
qhat_sl( List.train, List.test, K = 2, j = 1, k = 2, margin = 0.005, sl.lib = c("SL.glm", "SL.gam", "SL.glm.interaction", "SL.ranger", "SL.glmnet"), num_cores = NA, ... )qhat_sl( List.train, List.test, K = 2, j = 1, k = 2, margin = 0.005, sl.lib = c("SL.glm", "SL.gam", "SL.glm.interaction", "SL.ranger", "SL.glmnet"), num_cores = NA, ... )
List.train |
The training data matrix used to estimate the distibution functions. |
List.test |
The data matrix on which the estimator function is applied. |
K |
The number of lists in the data. |
j |
The first list that is conditionally independent. |
k |
The second list that is conditionally independent. |
margin |
The minimum value the estimates can attain to bound them away from zero. |
sl.lib |
The functions from the SuperLearner library to be used for model fitting. See |
num_cores |
The number of cores to be used for paralellization in Super Learner. |
... |
Any extra arguments passed into the function. |
A list of the marginal and joint distribution probabilities q1, q2 and q12.
Eric Polley, Erin LeDell, Chris Kennedy and Mark van der Laan (2021). SuperLearner: Super Learner Prediction. R package version 2.0-28. https://CRAN.R-project.org/package=SuperLearner
van der Laan, M. J., Polley, E. C. and Hubbard, A. E. (2008) Super Learner, Statistical Applications of Genetics and Molecular Biology, 6, article 25.
## Not run: qhat = qhat_sl(List.train = List.train, List.test = List.test, margin = 0.005, num_cores = 1) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 # One can specify the number of cores to be used for parallel computing qhat = qhat_sl(List.train = List.train, List.test = List.test, margin = 0.005, num_cores = 2) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 ## End(Not run)## Not run: qhat = qhat_sl(List.train = List.train, List.test = List.test, margin = 0.005, num_cores = 1) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 # One can specify the number of cores to be used for parallel computing qhat = qhat_sl(List.train = List.train, List.test = List.test, margin = 0.005, num_cores = 2) q1 = qhat$q1 q2 = qhat$q2 q12 = qhat$q12 ## End(Not run)
A function to reorder the columns of a data table/matrix/data frame and to change factor variables to numeric.
reformat(data, capturelists)reformat(data, capturelists)
data |
The data table/matrix/data frame which is to be checked. |
capturelists |
The vector of column names or locations for the capture history list columns. |
data With reordered columns so that the capture history columns are followed by the rest.
data = matrix(sample(c(0,1), 2000, replace = TRUE), ncol = 2) x = matrix(rnorm(nrow(data)*3, 2, 1), nrow = nrow(data)) data = cbind(x, data) result<- reformat(data = data, capturelists = c(4,5))data = matrix(sample(c(0,1), 2000, replace = TRUE), ncol = 2) x = matrix(rnorm(nrow(data)*3, 2, 1), nrow = nrow(data)) data = cbind(x, data) result<- reformat(data = data, capturelists = c(4,5))
A function to reorder the columns of a data table/matrix/data frame and to change factor variables to numeric.
simuldata(n, l, categorical = FALSE, ep = 0, K = 2)simuldata(n, l, categorical = FALSE, ep = 0, K = 2)
n |
The size of the population. |
l |
The number of continuous covariates. |
categorical |
A logical value of whether to include a categorical column. |
ep |
A numeric value to change the list probabilities. |
K |
The number of lists. Default value is 2. Maximum value is 3. |
A list of estimates containing the following components:
data |
A dataframe in with |
data_xstar |
A dataframe in with two list capture histories and transformed covariates from a population if true size |
psi0 |
The empirical capture probability for the set-up used. |
pi1 |
The conditional capture probabilities for list 1. |
pi2 |
The conditional capture probabilities for list 2. |
pi3 |
The conditional capture probabilities for list 3 when |
Tilling, K., & Sterne, J. A. (1999). Capture-recapture models including covariate effects. American journal of epidemiology, 149(4), 392-400.
Kennedy, E. H. (2019). Nonparametric causal effects based on incremental propensity score interventions. Journal of the American Statistical Association, 114(526), 645-656.
data = simuldata(n = 1000, l = 2)$data psi0 = simuldata(n = 10000, l = 2)$psi0data = simuldata(n = 1000, l = 2)$data psi0 = simuldata(n = 10000, l = 2)$psi0
Returns the targeted maximum likelihood estimates for the nuisance functions
tmle( datmat, iter = 250, margin = 0.005, stop_margin = 0.005, twolist = FALSE, K = 2, ... )tmle( datmat, iter = 250, margin = 0.005, stop_margin = 0.005, twolist = FALSE, K = 2, ... )
datmat |
The data frame containing columns |
iter |
An integer denoting the maximum number of iterations allowed for targeted maximum likelihood method. Default value is 100. |
margin |
The minimum value the estimates can attain to bound them away from zero. |
stop_margin |
The minimum value the estimates can attain to bound them away from zero. |
twolist |
The logical value of whether targeted maximum likelihood algorithm fits only two modes when K = 2. |
K |
The number of lists in the original data. |
... |
Any extra arguments passed into the function. |
A list of estimates containing the following components:
error |
An indicator of whether the algorithm ran and converged. Returns FALSE, if it ran correctly and FALSE otherwise. |
datmat |
A data frame returning |
van der Laan, M. J. and Rubin, D. (2006). Targeted maximum likelihood learning. The International Journal of Biostatistics, 2(1)
Das, M., Kennedy, E. H., & Jewell, N.P. (2021). Doubly robust capture-recapture methods for estimating population size. arXiv preprint arXiv:2104.14091.
data = matrix(sample(c(0,1), 2000, replace = TRUE), ncol = 2) xmat = matrix(runif(nrow(data)*3, 0, 1), nrow = nrow(data)) datmat = cbind(data, data[,1]*data[,2], xmat) colnames(datmat) = c("yj", "yk", "yjk", "q10", "q02", "q12") datmat = as.data.frame(datmat) result = tmle(datmat, margin = 0.005, stop_margin = 0.00001, twolist = TRUE)data = matrix(sample(c(0,1), 2000, replace = TRUE), ncol = 2) xmat = matrix(runif(nrow(data)*3, 0, 1), nrow = nrow(data)) datmat = cbind(data, data[,1]*data[,2], xmat) colnames(datmat) = c("yj", "yk", "yjk", "q10", "q02", "q12") datmat = as.data.frame(datmat) result = tmle(datmat, margin = 0.005, stop_margin = 0.00001, twolist = TRUE)