Compute MLE based on the full information R1, R2, R3 and R4.

Description

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

Usage

conditional_mle(
  R1 = NULL,
  R2 = NULL,
  R3 = NULL,
  R4 = NULL,
  n = R1 + R2 + R3 + R4,
  pi0,
  gamma = 0.05,
  alpha0 = 0,
  alpha = 0,
  beta = 0,
  V = NULL,
  ...
)

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 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(s)

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, pi0 = X$pi0)

# 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)
conditional_mle(R1 = X$R1, R2 = X$R2, R3 = X$R3, R4 = X$R4, pi0 = X$pi0)
conditional_mle(R1 = X$R1, R2 = X$R2, R3 = X$R3, R4 = X$R4, pi0 = X$pi0,
alpha0 = 0.001, alpha = 0.01, beta = 0.05)

COVID-19 Data from Statistics Austria

Description

Data collected in Austria in 2020 (see e.g. SORA, 2020; Kowarik et al., 2021, for more details), allowing to estimate COVID-19 prevalence.

Usage

covid19_austria

Format

A matrix with 2290 rows and 3 variables:

Y

Binary variable, 1 if participant i is tested positive in the survey sample, 0 otherwise.

Z

Binary variable, 1 if participant i was declared positive with the official procedure, 0 otherwise.

weights

Sampling weights.

Source

Statistics Austria. 2020. "Prävalenz von SARS-CoV-2-Infektionen liegt bei 0.031."

Compute sucess probabilities (tau_j's)

Description

Compute joint probabilities of P(W = j, Y = k) for j, k = 0, 1.

Usage

get_prob(theta, pi0, alpha, beta, alpha0)

Arguments

Value

A vector containing tau1, tau2, tau3 and tau4.

Author(s)

Stephane Guerrier

Examples

prob1 = get_prob(theta = 0.02, pi0 = 0.01, alpha = 0, beta = 0, alpha0 = 0)
prob1
sum(prob1)

prob2 = get_prob(theta = 0.02, pi0 = 0.01, alpha = 0.001, beta = 0, alpha0 = 0.001)
prob2
sum(prob2)

Modified log function (internal function)

Description

Simple function that return log(x) if x > 0 and 0 otherwise.

Usage

log_modified(x)

Arguments

Value

Modified log function

Author(s)

Stephane Guerrier

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

Description

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

Usage

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

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(s)

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)

# 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)
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)

Compute moment-based estimator.

Description

Proportion estimated using the moment-based estimator and confidence intervals based the asymptotic distribution of the estimator as well as the Clopper-Pearson approach.

Usage

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

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.

• ci_cp: Confidence interval (1 - gamma confidence level) based on the Clopper-Pearson approach.

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

• method: Estimation method (in this case moment estimator).

• 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(s)

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)
moment_estimator(R3 = X$R3, n = X$n, pi0 = X$pi0)

# With measurement error
X = sim_Rs(theta = 3/100, pi0 = 1/100, n = 1500, alpha0 = 0.001,
alpha = 0.01, beta = 0.05, seed = 18)
moment_estimator(R3 = X$R3, n = X$n, pi0 = X$pi0)
moment_estimator(R3 = X$R3, n = X$n, pi0 = X$pi0, alpha0 = 0.001,
alpha = 0.01, beta = 0.05)

Negative Log-Likelihood function

Description

Log-Likelihood function based on R1, R2, R3 and R4 multiplied by -1.

Usage

neg_log_lik(theta, Rvect, n, pi0, alpha, beta, alpha0, ...)

Arguments

Value

Negative log-likelihood.

Author(s)

Stephane Guerrier

Negative marginalized log-likelihood function based on R0 and R

Description

Log-Likelihood function based on R1 and R3 multiplied by -1.

Usage

neg_log_lik_integrated(theta, Rvect, n, pi0, alpha0, alpha, beta, ...)

Arguments

Value

Negative marginalized log-likelihood.

Author(s)

Stephane Guerrier

Negative Weighted Log-Likelihood function

Description

Log-Likelihood function based on R1, R2, R3 and R4 with sampling weights multiplied by -1.

Usage

neg_log_wlik(theta, Rvect, n, pi0, alpha, beta, alpha0, ...)

Arguments

Value

Negative log-likelihood.

Author(s)

Stephane Guerrier

Print estimation results

Description

Simple print function for the four estimators (i.e. mle, conditional mle, moment based and survey sample).

Usage

## S3 method for class 'cpreval'
print(x, ...)

Arguments

Value

Prints object

Author(s)

Stephane Guerrier

Examples

X = sim_Rs(theta = 3/100, pi0 = 1/100, n = 1500, seed = 18)
survey_mle(X$R, X$n)

Print (simulated) sample

Description

Simple print function for the cpreval_sim objects

Usage

## S3 method for class 'cpreval_sim'
print(x, ...)

Arguments

Value

Prints object

Author(s)

Stephane Guerrier

Examples

# Samples without measurement error
sim_Rs(theta = 3/100, pi0 = 1/100, n = 1500, seed = 18)

# With measurement error
sim_Rs(theta = 3/100, pi0 = 1/100, n = 1500, alpha0 = 0,
alpha = 0.01, beta = 0.05, seed = 18)

Simulate data (R, R0, R1, R2, R3 and R4)

Description

Simulation function for random variables of interest.

Usage

sim_Rs(theta, pi0, n, alpha0 = 0, alpha = 0, beta = 0, seed = NULL, ...)

Arguments

Value

A cpreval_sim object (list) with the structure:

• R: the number of participants in the survey sample that were tested positive.

• R0: the number of participants in the survey sample that were tested positive with the first testing device (and are, thus, members of the sub-population).

• R1: 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).

• R2: the number of participants in the survey sample that are tested positive only with the first testing device (and are, thus, members of the sub-population).

• R3: the number of participants in the survey sample that are tested positive only with the second testing device.

• R4: the number of participants that are tested negative with the second testing device (and are either members of the sub-population and have tested negative with the first testing device or are not members of the sub-population).

• n: the sample size.

• alpha: the False Negative (FN) rate for the sample R.

• beta: the False Positive (FP) rate for the sample R.

• alpha0: the alpha0 probability (as defined above).

• ...: additional arguments.

Author(s)

Stephane Guerrier

Examples

# Samples without measurement error
sim_Rs(theta = 3/100, pi0 = 1/100, n = 1500, seed = 18)

# With measurement error
sim_Rs(theta = 3/100, pi0 = 1/100, n = 1500, alpha0 = 0,
alpha = 0.01, beta = 0.05, seed = 18)

Compute proportion in the survey sample (standard estimator)

Description

Proportion estimated using the survey sample and confidence intervals based on the Clopper-Pearson and the standard asymptotic approach.

Usage

survey_mle(R, n, pi0 = 0, alpha = 0, beta = 0, gamma = 0.05, V = NULL, ...)

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 sample survey).

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

• 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).

• ...: Additional parameters.

Author(s)

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

Examples

# Samples without measurement error
X = sim_Rs(theta = 30/1000, pi0 = 10/1000, n = 1500, seed = 18)
survey_mle(R = X$R, n = X$n)

# With measurement error
X = sim_Rs(theta = 30/1000, pi0 = 10/1000, n = 1500, alpha = 0.01, beta = 0.05, seed = 18)
survey_mle(R = X$R, n = X$n)
survey_mle(R = X$R, n = X$n, alpha = 0.01, beta = 0.05)

Update prevalence using new case prevalence rates

Description

Updated prevalence and confidence intervals using new case prevalence rates

Usage

update_prevalence(
  pi0_new,
  x,
  gamma = 0.05,
  print = NULL,
  plot = NULL,
  col_line = "#2e5dc1",
  col_ci = "#2E5DC133",
  ...
)

Arguments

Value

A matrix object whose colunms corresponds to pi0, estimate, sd and CI.

Author(s)

Stephane Guerrier

Examples

# Austrian data (November 2020)
pi0 = 93914/7166167
data("covid19_austria")

# Weighted sampling
n = nrow(covid19_austria)
R1w = sum(covid19_austria$weights[covid19_austria$Y == 1 & covid19_austria$Z == 1])
R2w = sum(covid19_austria$weights[covid19_austria$Y == 0 & covid19_austria$Z == 1])
R3w = sum(covid19_austria$weights[covid19_austria$Y == 1 & covid19_austria$Z == 0])
R4w = sum(covid19_austria$weights[covid19_austria$Y == 0 & covid19_austria$Z == 0])

# Assumed measurement errors
alpha0 = 0
alpha = 1/100
beta = 10/100

# MME
mme = moment_estimator(R3 = R3w, n = n, pi0 = pi0, alpha = alpha, beta = beta,
                       alpha0 = alpha0, V = mean(covid19_austria$weights^2))

mme

# Update prevalence using a new pi0, say = 1.5%, instead of 1.31%
update_prevalence(1.5/100, mme)

pi0_new = seq(from = 0.005, to = 0.03, length.out = 100)
update_prevalence(pi0_new, mme)