Compute (marginalized) MLE based on the partial information R1 and R3.

Source: R/estimation.R

Proportion estimated using the MLE and confidence intervals based the asymptotic distribution of the estimator.

marginal_mle(
  R1,
  R3,
  n,
  pi0,
  gamma = 0.05,
  alpha = 0,
  beta = 0,
  alpha0 = 0,
  V = NULL,
  ...
)

Arguments

R1

A numeric that provides the number of participants in the survey sample that were tested positive with both (medical) testing devices (and are, thus, members of the sub-population).

R3

A numeric that provides the number of participants in the survey sample that are tested positive only with the second testing device.

n

A numeric that provides the sample size.

pi0

A numeric that provides the prevalence or proportion of people (in the whole population) who are positive, as measured through a non-random, but systematic sampling (e.g. based on medical selection).

gamma

A numeric that is used to compute a (1 - gamma) confidence region for the proportion. Default value is 0.05.

alpha

A numeric that provides the False Negative (FN) rate for the sample R. Default value is 0.

beta

A numeric that provides the False Positive (FP) rate for the sample R. Default value is 0.

alpha0

A numeric that corresponds to the probability that a random participant has been incorrectly declared positive through the nontransparent procedure. In most applications, this probability is likely very close to zero. Default value is 0.

V

A numeric that corresponds to the average of squared sampling weights. Default value is NULL and for the moment this method is currently only implemented for random sampling.

...

Additional arguments.

Value

A cpreval object with the structure:

  • estimate: Estimated proportion.

  • sd: Estimated standard error of the estimator.

  • ci_asym: Asymptotic confidence interval at the 1 - gamma confidence level.

  • gamma: Confidence level (i.e. 1 - gamma) for confidence intervals.

  • method: Estimation method (in this case marginal mle).

  • measurement: A vector with (alpha0, alpha, beta).

  • beta0: Estimated false negative rate of the official procedure.

  • ci_beta0: Asymptotic confidence interval (1 - gamma confidence level) for beta0.

  • boundary: A boolean variable indicating if the estimates falls at the boundary of the parameter space.

  • pi0: Value of pi0 (input value).

  • sampling: Type of sampling considered ("random" or "weighted").

  • V: Average sum of squared sampling weights if weighted/stratified is used (otherwise NULL).

  • n: Sample size.

  • avar_beta0: Estimated asymptotic variance of beta0

  • ...: Additional parameters

Author

Stephane Guerrier, Maria-Pia Victoria-Feser, Christoph Kuzmics

Examples

# Samples without measurement error
X = sim_Rs(theta = 3/100, pi0 = 1/100, n = 1500, seed = 18)
conditional_mle(R1 = X$R1, R2 = X$R2, R3 = X$R3, R4 = X$R4, n = X$n, pi0 = X$pi0)
#> Method: Conditional MLE
#> 
#> Estimated proportion: 3.2061%
#> Standard error      : 0.3792%
#> 
#> Confidence interval at the 95% level:
#> Asymptotic Approach: 2.4629% - 3.9494%
#> 
#> Assumed measurement error: alpha  = 0%, beta = 0%,
#>                            alpha0 = 0% 
#> 
#> Estimated false negative rate of the
#> official procedure: beta0 = 68.81%
#> CI at the 95% level: 61.58% - 76.04%
#> 
#> Estimated ascertainment rate: 
#> pi0/pi = 31.19%
#> CI at the 95% level: 23.96% - 38.42%
#> 
#> Sampling: Random

# With measurement error
X = sim_Rs(theta = 30/1000, pi0 = 10/1000, n = 1500, alpha0 = 0.001,
alpha = 0.01, beta0 = 0.05, beta = 0.05, seed = 18)
marginal_mle(R1 = X$R1, R3 = X$R3, n = X$n, pi0 = X$pi0)
#> Method: Marginal MLE
#> 
#> Estimated proportion: 3.6683%
#> Standard error      : 0.4160%
#> 
#> Confidence interval at the 95% level:
#> Asymptotic Approach: 2.8528% - 4.4837%
#> 
#> Assumed measurement error: alpha  = 0%, beta = 0%,
#>                            alpha0 = 0% 
#> 
#> Estimated false negative rate of the
#> official procedure: beta0 = 72.74%
#> CI at the 95% level: 66.68% - 78.80%
#> 
#> Estimated ascertainment rate: 
#> pi0/pi = 27.26%
#> CI at the 95% level: 21.20% - 33.32%
#> 
#> Sampling: Random
marginal_mle(R1 = X$R1, R3 = X$R3, n = X$n, pi0 = X$pi0,
alpha0 = 0.001, alpha = 0.01, beta0 = 0.05, beta = 0.05)
#> Method: Marginal MLE
#> 
#> Estimated proportion: 2.6910%
#> Standard error      : 0.4433%
#> 
#> Confidence interval at the 95% level:
#> Asymptotic Approach: 1.8223% - 3.5598%
#> 
#> Assumed measurement error: alpha  = 1%, beta = 5%,
#>                            alpha0 = 0.1% 
#> 
#> Estimated false negative rate of the
#> official procedure: beta0 = 66.46%
#> CI at the 95% level: 55.66% - 77.25%
#> 
#> Estimated ascertainment rate: 
#> pi0/pi = 33.54%
#> CI at the 95% level: 22.75% - 44.34%
#> 
#> Sampling: Random