# Estimate parameters in the disease model using observed data log-likelihood using the expectation-maximization algorithm `obsloglikEM` jointly estimates the disease model and sensitivity model parameters using profile likelihood methods. Estimation involves an expectation-maximization algorithm. ```r obsloglikEM(Dstar, Z, X, start, beta0_fixed = NULL, weights = NULL, expected = TRUE, tol = 1e-06, maxit = 50) ``` ## Arguments - `Dstar`: Numeric vector containing observed disease status. Should be coded as 0/1 - `Z`: Numeric matrix of covariates in disease model. 'Z' should not contain an intercept - `X`: Numeric matrix of covariates in sensitivity model. Set to NULL to fit model with no covariates in sensitivity model. 'X' should not contain an intercept - `start`: Numeric vector of starting values for theta and beta (theta, beta). Theta is the parameter of the disease model, and beta is the parameter of the sensitivity model - `beta0_fixed`: Optional numeric vector of values of sensitivity model intercept to profile over. If a single value, corresponds to fixing intercept at specified value. Default is NULL - `weights`: Optional vector of patient-specific weights used for selection bias adjustment. Default is NULL - `expected`: Whether or not to calculate the covariance matrix via the expected fisher information matrix. Default is TRUE - `tol`: stop estimation when subsequent log-likelihood estimates are within this value - `maxit`: Maximum number of iterations of the estimation algorithm ## Returns 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'. ## 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**: $$ logit(P(D* = 1| D = 1, S = 1, X)) = beta_0 +beta_X X $$ ## Examples ```r 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)) ``` ## 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](https://doi.org/10.1101/2019.12.26.19015859)

obsloglikEM function