Estimate parameters in the disease model approximating the observed data distribution

Description

approxdist estimates parameters in the disease model given a previously-estimated marginal sensitivity. This estimation is based on approximating the distribution of D* given Z.

Usage

approxdist(Dstar, Z, c_marg, weights = NULL)

Arguments

Details

We are interested in modeling the relationship between binary disease status and covariates Z using a logistic regression model. However, D may be misclassified, and our observed data may not well-represent the population of interest. In this setting, we estimate parameters from the disease model using the following modeling framework.

Notation:

D

Binary disease status of interest.

D*

Observed binary disease status. Potentially a misclassified version of D. We assume D = 0 implies D* = 0.

S

Indicator for whether patient from population of interest is included in the analytical dataset.

Z

Covariates in disease model of interest.

W

Covariates in model for patient inclusion in analytical dataset (selection model).

X

Covariates in model for probability of observing disease given patient has disease (sensitivity model).

Model Structure:

Disease Model

logit(P(D=1|X)) = theta_0 + theta_Z Z

Selection Model

P(S=1|W,D)

Sensitivity Model

logit(P(D* = 1| D = 1, S = 1, X)) = beta_0 + beta_X X

Value

a list with two elements: (1) 'param', a vector with parameter estimates for disease model (logOR of Z), and (2) 'variance', a vector of variance estimates for disease model parameters. Results do not include intercept.

References

Statistical inference for association studies using electronic health records: handling both selection bias and outcome misclassification Lauren J Beesley and Bhramar Mukherjee medRxiv 2019.12.26.19015859

Examples

library(SAMBA)
# These examples are generated from the vignette. See it for more details.

# Generate IPW weights from the true model
expit <- function(x) exp(x) / (1 + exp(x))
prob.WD <- expit(-0.6 + 1 * samba.df$D + 0.5 * samba.df$W)
weights <- nrow(samba.df) * (1  / prob.WD) / (sum(1 / prob.WD))

# Estimate sensitivity by using inverse probability of selection weights
# and P(D=1)
sens <- sensitivity(samba.df$Dstar, samba.df$X, prev = mean(samba.df$D),
                    weights = weights)

approx1 <- approxdist(samba.df$Dstar, samba.df$Z, sens$c_marg,
                     weights = weights)

Estimate parameters in the disease model given sensitivity as a function of covariates.

Description

non-logistic link function for D* given Z and sensitivity. This function assumes that sensitivity as a function of X is known or has been estimated

Usage

nonlogistic(Dstar, Z, c_X, weights = NULL)

Arguments

Details

We are interested in modeling the relationship between binary disease status and covariates Z using a logistic regression model. However, D may be misclassified, and our observed data may not well-represent the population of interest. In this setting, we estimate parameters from the disease model using the following modeling framework.

Notation:

D

Binary disease status of interest.

D*

Observed binary disease status. Potentially a misclassified version of D. We assume D = 0 implies D* = 0.

S

Indicator for whether patient from population of interest is included in the analytical dataset.

Z

Covariates in disease model of interest.

W

Covariates in model for patient inclusion in analytical dataset (selection model).

X

Covariates in model for probability of observing disease given patient has disease (sensitivity model).

Model Structure:

Disease Model

logit(P(D=1|X)) = theta_0 + theta_Z Z

Selection Model

P(S=1|W,D)

Sensitivity Model

logit(P(D* = 1| D = 1, S = 1, X)) = beta_0 + beta_X X

Value

a list with two elements: (1) 'param', a vector with parameter estimates for disease model (logOR of Z), and (2) 'variance', a vector of variance estimates for disease model parameters. Results do not include intercept.

References

Statistical inference for association studies using electronic health records: handling both selection bias and outcome misclassification Lauren J Beesley and Bhramar Mukherjee medRxiv 2019.12.26.19015859

Examples

library(SAMBA)
# These examples are generated from the vignette. See it for more details.

# Generate IPW weights from the true model
expit <- function(x) exp(x) / (1 + exp(x))
prob.WD <- expit(-0.6 + 1 * samba.df$D + 0.5 * samba.df$W)
weights <- nrow(samba.df) * (1  / prob.WD) / (sum(1 / prob.WD))

# Estimate sensitivity by using inverse probability of selection weights
# and P(D=1)
sens <- sensitivity(samba.df$Dstar, samba.df$X, prev = mean(samba.df$D),
                    weights = weights)

nonlog1 <- nonlogistic(samba.df$Dstar, samba.df$Z, c_X = sens$c_X,
                       weights = weights)

Estimate parameters in the disease model using observed data log-likelihood using direct maximization.

Description

obsloglik jointly estimates the disease model and sensitivity model parameters using profile likelihood methods. Estimation involves direct maximization of the observed data log-likelihood.

Usage

obsloglik(Dstar, Z, X, start, beta0_fixed = NULL, weights = NULL,
  expected = TRUE, itnmax = 5000)

Arguments

Details

We are interested in modeling the relationship between binary disease status and covariates Z using a logistic regression model. However, D may be misclassified, and our observed data may not well-represent the population of interest. In this setting, we estimate parameters from the disease model using the following modeling framework. Notation:

D

Binary disease status of interest.

D*

Observed binary disease status. Potentially a misclassified version of D. We assume D = 0 implies D* = 0.

S

Indicator for whether patient from population of interest is included in the analytical dataset.

Z

Covariates in disease model of interest.

W

Covariates in model for patient inclusion in analytical dataset (selection model).

X

Covariates in model for probability of observing disease given patient has disease (sensitivity model).

Model Structure:

Disease Model

logit(P(D=1|X)) = theta_0 + theta_Z Z

Selection Model

P(S=1|W,D)

Sensitivity Model

logit(P(D* = 1| D = 1, S = 1, X)) = beta_0 + beta_X X

Value

A "SAMBA.fit" object with nine elements: 'param', the maximum likelihood estimate of the coeficients, 'variance', the covariance matrix of the final estimate, param.seq', the sequence of estimates at each value of beta0, and 'loglik.seq', the log likelihood at each value. The rest of the elements are Dstar', 'X', 'Z', and 'weights'.

References

Statistical inference for association studies using electronic health records: handling both selection bias and outcome misclassification Lauren J Beesley and Bhramar Mukherjee medRxiv 2019.12.26.19015859

Examples

library(SAMBA)
# These examples are generated from the vignette. See it for more details.

# Generate IPW weights from the true model
expit <- function(x) exp(x) / (1 + exp(x))
prob.WD <- expit(-0.6 + 1 * samba.df$D + 0.5 * samba.df$W)
weights <- nrow(samba.df) * (1  / prob.WD) / (sum(1 / prob.WD))

# Get initial parameter estimates
logit <- function(x) log(x / (1 - x))
fitBeta  <- glm(Dstar ~ X, binomial(), data = samba.df)
fitTheta <- glm(Dstar ~ Z, binomial(), data = samba.df)

sens <- sensitivity(samba.df$Dstar, samba.df$X, mean(samba.df$D), r = 2)
start <- c(coef(fitTheta), logit(sens$c_marg), coef(fitBeta)[2])

# Direct observed data likelihood maximization without fixed intercept

fit1 <- obsloglik(samba.df$Dstar, samba.df$Z, samba.df$X, start = start,
                 weights = weights)
obsloglik1 <- list(param = fit1$param, variance = diag(fit1$variance))

# Direct observed data likelihood maximization with fixed intercept

fit2   <- obsloglik(samba.df$Dstar, samba.df$Z, samba.df$X, start = start,
                 beta0_fixed = logit(sens$c_marg), weights = weights)

# since beta0 is fixed, its variance is NA
obsloglik1 <- list(param = fit2$param, variance = diag(fit2$variance))

Estimate parameters in the disease model using observed data log-likelihood using the expectation-maximization algorithm

Description

obsloglikEM jointly estimates the disease model and sensitivity model parameters using profile likelihood methods. Estimation involves an expectation-maximization algorithm.

Usage

obsloglikEM(Dstar, Z, X, start, beta0_fixed = NULL, weights = NULL,
  expected = TRUE, tol = 1e-06, maxit = 50)

Arguments

Details

We are interested in modeling the relationship between binary disease status and covariates Z using a logistic regression model. However, D may be misclassified, and our observed data may not well-represent the population of interest. In this setting, we estimate parameters from the disease model using the following modeling framework. Notation:

D

Binary disease status of interest.

D*

Observed binary disease status. Potentially a misclassified version of D. We assume D = 0 implies D* = 0.

S

Indicator for whether patient from population of interest is included in the analytical dataset.

Z

Covariates in disease model of interest.

W

Covariates in model for patient inclusion in analytical dataset (selection model).

X

Covariates in model for probability of observing disease given patient has disease (sensitivity model).

Model Structure:

Disease Model

logit(P(D=1|X)) = theta_0 + theta_Z Z

Selection Model

P(S=1|W,D)

Sensitivity Model

logit(P(D* = 1| D = 1, S = 1, X)) = beta_0 + beta_X X

Value

A "SAMBA.fit" object with nine elements: 'param', the final estimate of the coeficients organized as (theta, beta), 'variance', the covariance matrix of the final estimate, param.seq', the sequence of estimates at each step of the EM algorithm, and 'loglik.seq', the log likelihood at each step. The rest of the elements are Dstar', 'X', 'Z', and 'weights'.

References

Statistical inference for association studies using electronic health records: handling both selection bias and outcome misclassification Lauren J Beesley and Bhramar Mukherjee medRxiv 2019.12.26.19015859

Examples

library(SAMBA)
# These examples are generated from the vignette. See it for more details.

# Generate IPW weights from the true model
expit <- function(x) exp(x) / (1 + exp(x))
prob.WD <- expit(-0.6 + 1 * samba.df$D + 0.5 * samba.df$W)
weights <- nrow(samba.df) * (1  / prob.WD) / (sum(1 / prob.WD))

# Get initial parameter estimates
logit <- function(x) log(x / (1 - x))
fitBeta  <- glm(Dstar ~ X, binomial(), data = samba.df)
fitTheta <- glm(Dstar ~ Z, binomial(), data = samba.df)

sens <- sensitivity(samba.df$Dstar, samba.df$X, mean(samba.df$D), r = 2)
start <- c(coef(fitTheta), logit(sens$c_marg), coef(fitBeta)[2])

# Direct observed data likelihood maximization without fixed intercept
fit1 <- obsloglikEM(samba.df$Dstar, samba.df$Z, samba.df$X, start = start,
                 weights = weights)
obsloglik1 <- list(param = fit1$param, variance = diag(fit1$variance))

# Direct observed data likelihood maximization with fixed intercept
fit2   <- obsloglikEM(samba.df$Dstar, samba.df$Z, samba.df$X, start = start,
                 beta0_fixed = logit(sens$c_marg), weights = weights)
# since beta0 is fixed, its variance is NA

list(param = fit2$param, variance = diag(fit2$variance))

Synthetic example data for SAMBA adapted from the vignette

Description

'samba.df' is the sampled data from the entire population

Usage

samba.df

Format

A synthetic data.frame with 4999 observations on 5 variables:

X

Covariate for sensitivity model.

Z

Covariate for disease model.

W

Selection Covariate

D

True disease status.

Dstar

Observed disease status.

Estimate sensitivity

Description

sensitivity estimates (1) marginal sensitivity and (2) sensitivity as a function of covariates X for a misclassified binary outcome.

Usage

sensitivity(Dstar, X, prev, r = NULL, weights = NULL)

Arguments

Details

We are interested in modeling the relationship between binary disease status and covariates Z using a logistic regression model. However, D may be misclassified, and our observed data may not well-represent the population of interest. In this setting, we estimate parameters from the disease model using the following modeling framework.

Notation:

D

Binary disease status of interest.

D*

Observed binary disease status. Potentially a misclassified version of D. We assume D = 0 implies D* = 0.

S

Indicator for whether patient from population of interest is included in the analytical dataset.

Z

Covariates in disease model of interest.

W

Covariates in model for patient inclusion in analytical dataset (selection model).

X

Covariates in model for probability of observing disease given patient has disease (sensitivity model).

Model Structure:

Disease Model

logit(P(D=1|X)) = theta_0 + theta_Z Z

Selection Model

P(S=1|W,D)

Sensitivity Model

logit(P(D* = 1| D = 1, S = 1, X)) = beta_0 + beta_X X

Value

a list with two elements: (1) 'c_marg', marginal sensitivity estimate P(D* = 1|D = 1, S = 1), and (2) 'c_X', sensitivity as a function of X P(D* = 1| D = 1, S = 1, X)

References

Statistical inference for association studies using electronic health records: handling both selection bias and outcome misclassification Lauren J Beesley and Bhramar Mukherjee medRxiv 2019.12.26.19015859

Examples

library(SAMBA)
# These examples are generated from the vignette. See it for more details.

# Generate IPW weights from the true model
expit <- function(x) exp(x) / (1 + exp(x))
prob.WD <- expit(-0.6 + 1 * samba.df$D + 0.5 * samba.df$W)
weights <- nrow(samba.df) * (1  / prob.WD) / (sum(1 / prob.WD))

# Using marginal sampling ratio r ~ 2 and P(D=1)
sens <- sensitivity(samba.df$Dstar, samba.df$X, mean(samba.df$D),
                    r = 2)
# Using inverse probability of selection weights and P(D=1)
sens <- sensitivity(samba.df$Dstar, samba.df$X, prev = mean(samba.df$D),
                    weights = weights)