Population estimation using stepwise model selection — estimate_population

Estimates the total population, including the unobserved population, using an extension of the stepwise model-selection procedure of Chan, Silverman and Vincent (2021). Optional bootstrap and jackknife calculations provide BCa confidence intervals while repeating the stepwise selection procedure for each resampled data set.

Usage

estimate_population_stepwise(
  zdat,
  nboot = 0,
  pthresh = 0.02,
  maxorder = Inf,
  iseed = 1234,
  alpha = c(0.025, 0.1, 0.9, 0.975),
  return_details = FALSE
)

Arguments

zdat: A capture history data matrix with \(t+1\) columns. The first \(t\) columns correspond to the capture lists and contain zeros and ones defining the observed capture histories. The final column contains the number of cases having each capture history. List names A, B, and so on are constructed if they are not supplied. Capture histories not explicitly included in the data are assumed to have zero count.
nboot: Non-negative integer giving the number of bootstrap replications. If nboot = 0, only the point estimate and fitted model are returned and no bootstrap or jackknife calculations are performed. The default is 0.
pthresh: P-value threshold used by the stepwise model-selection procedure. The default is 0.02.
maxorder: Maximum order of interactions considered for selection. An integer of at least 2, or Inf to allow interactions of any order. The default is Inf.
iseed: Integer seed used to initialise the random-number generator when nboot > 0. The default is 1234.
alpha: Numeric vector of cumulative probability levels at which the endpoints of the BCa confidence intervals are to be calculated. This argument is used only when nboot > 0. The default is c(0.025, 0.1, 0.9, 0.975).
return_details: Logical. If TRUE, include the bootstrap estimates, BCa acceleration, and effects represented at minus infinity. The default is FALSE.

Value

A list with components:

input: A list containing the original call and data.
method: The character string "stepwise".
estimate: A named numeric vector containing the estimated dark_figure and total population.
fitted_model: The hierarchy selected from the original data.
uncertainty: A two-row matrix of BCa endpoints for the dark figure and total population when nboot > 0; otherwise an explanatory character string.
details: If return_details = TRUE, a list containing minus_infinity_effects, bootstrap_estimates, and bca_acceleration. Otherwise "not requested".

Details

By default, interactions of any order may be considered. Setting maxorder = 2 restricts the procedure to the pairwise method of Chan, Silverman and Vincent (2021). An interaction is considered only when all its lower-order terms are already present, so every candidate model is hierarchical. A masked sparse interaction is not eligible for selection, including when pthresh = 1.

The procedure starts with a main-effects model. At each stage it considers eligible interactions not already included and adds the interaction having the smallest p-value if that p-value does not exceed pthresh. It stops when no eligible interaction meets the threshold. Candidate models are checked for parameter identifiability and existence of the extended MLE.

The procedure is first applied to the observed data to obtain the point estimate and fitted model.

If nboot > 0, multinomial bootstrap samples are generated from the observed capture history counts. The complete stepwise model-selection procedure is repeated for each bootstrap sample, so the resulting inference allows for variation in the selected model rather than treating the model selected from the original data as fixed.

A delete-one jackknife calculation is also carried out to estimate the acceleration parameter required for the BCa confidence intervals. The jackknife calculation takes account of the number of individuals having each observed capture history.

A small positive value of nboot, such as that used in the example, is useful only for checking that the routine runs. A substantially larger number of bootstrap replications should be used for substantive inference.

References

Chan, L., Silverman, B. W., and Vincent, K. (2021). Multiple Systems Estimation for Sparse Capture Data: Inferential Challenges when there are Non-Overlapping Lists. Journal of the American Statistical Association, 116(535), 1297–1306. Available from https://www.tandfonline.com/doi/full/10.1080/01621459.2019.1708748.

DiCiccio, T. J. and Efron, B. (1996). Bootstrap Confidence Intervals. Statistical Science, 11(3), 189–228.

Examples

data(Korea)

# Point estimate and fitted model without bootstrapping
estimate_population_stepwise(Korea)
#> $input
#> $input$call
#> estimate_population_stepwise(zdat = Korea)
#> 
#> $input$data
#>      b c d Count
#> [1,] 0 0 1    41
#> [2,] 0 1 0     5
#> [3,] 1 0 0     5
#> [4,] 1 1 0    54
#> [5,] 1 0 1     6
#> [6,] 0 1 1     0
#> [7,] 1 1 1    12
#> 
#> 
#> $method
#> [1] "stepwise"
#> 
#> $estimate
#> dark_figure       total 
#>    145.7778    268.7778 
#> 
#> $fitted_model
#> [1] "[12,3]"
#> 
#> $uncertainty
#> [1] "not calculated because nboot = 0"
#> 
#> $details
#> [1] "not requested"
#> 

# Allow interactions of any order
estimate_population_stepwise(Kosovo, maxorder = Inf)
#> $input
#> $input$call
#> estimate_population_stepwise(zdat = Kosovo, maxorder = Inf)
#> 
#> $input$data
#>    EXH ABA OSCE HRW Frequency
#> 1    1   0    0   0      1131
#> 2    0   1    0   0       845
#> 3    0   0    1   0       936
#> 4    0   0    0   1       306
#> 5    1   1    0   0       177
#> 6    1   0    1   0       228
#> 7    1   0    0   1       106
#> 8    0   1    1   0       217
#> 9    0   1    0   1        31
#> 10   0   0    1   1       123
#> 11   1   1    1   0       181
#> 12   1   1    0   1        18
#> 13   1   0    1   1        42
#> 14   0   1    1   1        32
#> 15   1   1    1   1        27
#> 
#> 
#> $method
#> [1] "stepwise"
#> 
#> $estimate
#> dark_figure       total 
#>    13993.31    18393.31 
#> 
#> $fitted_model
#> [1] "[134,12,23]"
#> 
#> $uncertainty
#> [1] "not calculated because nboot = 0"
#> 
#> $details
#> [1] "not requested"
#> 

# A very small number of bootstrap replications is used here only
# to keep the example quick.
estimate_population_stepwise(Korea, nboot = 10)
#> $input
#> $input$call
#> estimate_population_stepwise(zdat = Korea, nboot = 10)
#> 
#> $input$data
#>      b c d Count
#> [1,] 0 0 1    41
#> [2,] 0 1 0     5
#> [3,] 1 0 0     5
#> [4,] 1 1 0    54
#> [5,] 1 0 1     6
#> [6,] 0 1 1     0
#> [7,] 1 1 1    12
#> 
#> 
#> $method
#> [1] "stepwise"
#> 
#> $estimate
#> dark_figure       total 
#>    145.7778    268.7778 
#> 
#> $fitted_model
#> [1] "[12,3]"
#> 
#> $uncertainty
#>                  0.025      0.1      0.9    0.975
#> dark_figure   4.982845 113.3239 202.7143 202.7143
#> total       127.982845 236.3239 325.7143 325.7143
#> 
#> $details
#> [1] "not requested"
#>