R/profile_out.R
In Epic-Doughnut/Combined-Reg: Semiparametric Likelihood Estimation with Errors in Variables
Defines functions
profile_out
#' Profiles out nuisance parameters from the observed-data log-likelihood for a given value of theta
#'
#' For a given vector `theta` to parameterize P(Y|X,Z), this function repeats the EM algorithm to find
#' the values of `gamma` and `p` at convergence. The resulting parameters are used to find the profile
#' log-likelihood for `theta` by plugging them into the observed-data log-likelihood.
#' This function is used by `pl_theta()`.
#
#' @param theta Parameters for the analysis model (a column vector)
#' @param N Phase I sample size
#' @param n Phase II sample size
#' @param Y_unval Column with the unvalidated outcome (can be name or numeric index)
#' @param Y Column with the validated outcome (can be name or numeric index)
#' @param X_unval Column(s) with the unvalidated predictors (can be name or numeric index)
#' @param X Column(s) with the validated predictors (can be name or numeric index)
#' @param Z (Optional) Column(s) with additional error-free covariates (can be name or numeric index)
#' @param Bspline Vector of columns containing the B-spline basis functions (can be name or numeric index)
#' @param comp_dat_all Augmented dataset containing rows for each combination of unvalidated subjects' data with values from Phase II (a matrix)
#' @param theta_pred Vector of columns in \code{data} that pertain to the predictors in the analysis model.
#' @param gamma_pred Vector of columns in \code{data} that pertain to the predictors in the outcome error model.
#' @param gamma0 Starting values for `gamma`, the parameters for the outcome error model (a column vector)
#' @param p0 Starting values for `p`, the B-spline coefficients for the approximated covariate error model (a matrix)
#' @param p_val_num Contributions of validated subjects to the numerator for `p`, which are fixed (a matrix)
#' @param TOL Tolerance between iterations in the EM algorithm used to define convergence.
#' @param MAX_ITER Maximum number of iterations allowed in the EM algorithm.
#'
#' @return Profile likelihood for `theta`: the value of the observed-data log-likelihood after profiling out other parameters.
#'
#' @importFrom stats as.formula
#' @importFrom stats glm
#'
#' @noRd
profile_out <- function(theta, n, N, Y_unval = NULL, Y = NULL, X_unval = NULL, X = NULL, Z = NULL, Bspline = NULL,
comp_dat_all, theta_pred, gamma_pred, gamma0, p0, p_val_num, TOL, MAX_ITER) {
# Determine error setting -----------------------------------------
## If unvalidated variable was left blank, assume error-free ------
errorsY <- errorsX <- TRUE
if (is.null(Y_unval)) {errorsY <- FALSE}
if (is.null(X_unval) & is.null(X)) {errorsX <- FALSE}
## ------ If unvalidated variable was left blank, assume error-free
# ----------------------------------------- Determine error setting
sn <- ncol(p0)
m <- nrow(p0)
prev_gamma <- gamma0
prev_p <- p0
theta_design_mat <- cbind(int = 1, comp_dat_all[-c(1:n), theta_pred])
# For the E-step, save static P(Y|X) for unvalidated --------------
pY_X <- .pYstarCalc(theta_design_mat, n, 0, theta, comp_dat_all, match(Y, colnames(comp_dat_all))-1, vector("numeric",nrow(theta_design_mat)), vector("numeric",nrow(theta_design_mat)))
if (errorsY)
{
gamma_formula <- as.formula(paste0(Y_unval, "~", paste(gamma_pred, collapse = "+")))
gamma_design_mat <- cbind(int = 1, comp_dat_all[, gamma_pred])
}
CONVERGED <- FALSE
CONVERGED_MSG <- "Unknown"
it <- 1
# pre-allocate memory for our loop variables
# improves performance
if (errorsY)
{
pYstar <- vector(mode="numeric", length = nrow(gamma_design_mat) - n)
mu_gamma <- vector(mode="numeric", length = length(pYstar))
mus_gamma <- vector("numeric", nrow(gamma_design_mat) * ncol(prev_gamma))
}
psi_num <- matrix(,nrow = nrow(comp_dat_all)-n, ncol=length(Bspline))
psi_t <- matrix(,nrow = nrow(psi_num), ncol = ncol(psi_num))
w_t <- vector("numeric", length = nrow(psi_t))
u_t <- matrix(,nrow = m * (N-n), ncol = ncol(psi_t))
pX <- matrix(,nrow = m * (N-n) * ifelse(errorsY, 2, 1), ncol = length(Bspline))
# Estimate gamma/p using EM -----------------------------------------
while(it <= MAX_ITER & !CONVERGED)
{
# E Step ----------------------------------------------------------
# P(Y*|X*,Y,X) ---------------------------------------------------
if (errorsY)
{
pYstar <- .pYstarCalc(gamma_design_mat, n, n, prev_gamma, comp_dat_all, match(Y_unval, colnames(comp_dat_all))-1, pYstar, mu_gamma)
}
### -------------------------------------------------- P(Y*|X*,Y,X)
###################################################################
### P(X|X*) -------------------------------------------------------
# these are the slowest lines in this function (13280 ms), but I can't seem to get any faster with C++
# pX <- pXCalc(n, comp_dat_all[-c(1:n), Bspline], errorsX, errorsY, pX, prev_p[rep(seq(1, m), each = (N - n)), ])
if (errorsX & errorsY) {
### p_kj ------------------------------------------------------
### need to reorder pX so that it's x1, ..., x1, ...., xm, ..., xm-
### multiply by the B-spline terms
pX <- prev_p[rep(rep(seq(1, m), each = (N - n)), times = 2), ] * comp_dat_all[-c(1:n), Bspline]
### ---------------------------------------------------------- p_kj
} else if (errorsX) {
### p_kj ----------------------------------------------------------
### need to reorder pX so that it's x1, ..., x1, ...., xm, ..., xm-
### multiply by the B-spline terms
pX <- prev_p[rep(seq(1, m), each = (N - n)), ] * comp_dat_all[-c(1:n), Bspline]
### ---------------------------------------------------------- p_kj
}
### ------------------------------------------------------- P(X|X*)
###################################################################
### Estimate conditional expectations -----------------------------
# this function modifies w_t, u_t, psi_num, and psi_t internally
# condExp <- conditionalExpectations(errorsX, errorsY, pX, pY_X, pYstar, N - n, m, w_t, u_t, psi_num, psi_t)
# w_t <- condExp[["w_t"]]
# u_t <- condExp[["u_t"]]
# psi_t <- condExp[["psi_t"]]
if (errorsY & errorsX) {
### P(Y|X,Z)P(Y*|X*,Y,X,Z)p_kjB(X*) -----------------------------
psi_num <- c(pY_X * pYstar) * pX
### Update denominator ------------------------------------------
#### Sum up all rows per id (e.g. sum over xk/y) ----------------
psi_denom <- rowsum(psi_num, group = rep(seq(1, (N - n)), times = 2 * m))
#### Then sum over the sn splines -------------------------------
psi_denom <- rowSums(psi_denom)
#### Avoid NaN resulting from dividing by 0 ---------------------
psi_denom[psi_denom == 0] <- 1
### And divide them! --------------------------------------------
psi_t <- psi_num / psi_denom
### Update the w_kyi for unvalidated subjects -------------------
### by summing across the splines/ columns of psi_t -------------
w_t <- rowSums(psi_t)
### Update the u_kji for unvalidated subjects ------------------
### by summing over Y = 0/1 w/i each i, k ----------------------
### add top half of psi_t (y = 0) to bottom half (y = 1) -------
u_t <- psi_t[c(1:(m * (N - n))), ] + psi_t[- c(1:(m * (N - n))), ]
} else if (!errorsY && errorsX) {
### P(Y|X,Z)p_kjB(X*) -------------------------------------------
psi_num <- c(pY_X) * pX
### Update denominator ------------------------------------------
#### Sum up all rows per id (e.g. sum over xk) ------------------
psi_denom <- rowsum(psi_num, group = rep(seq(1, (N - n)), times = m))
#### Then sum over the sn splines -------------------------------
psi_denom <- rowSums(psi_denom)
#### Avoid NaN resulting from dividing by 0 ---------------------
psi_denom[psi_denom == 0] <- 1
### And divide them! --------------------------------------------
psi_t <- psi_num / psi_denom
### Update the w_kyi for unvalidated subjects -------------------
### by summing across the splines/ columns of psi_t -------------
w_t <- rowSums(psi_t)
} else if (!errorsX && errorsY) {
### P(Y|X,Z)P(Y*|Y,X,Z) -----------------------------------------
psi_num <- matrix(c(pY_X * pYstar), ncol = 1)
#### Sum up all rows per id (e.g. sum over y) -------------------
psi_denom <- rowsum(psi_num, group = rep(seq(1, (N - n)), times = 2))
#### Avoid NaN resulting from dividing by 0 ---------------------
psi_denom[psi_denom == 0] <- 1
### And divide them! --------------------------------------------
psi_t <- psi_num / psi_denom
### Update the w_kyi for unvalidated subjects -------------------
w_t <- psi_t
}
### ----------------------------- Estimate conditional expectations
# ---------------------------------------------------------- E Step
###################################################################
###################################################################
# M Step ----------------------------------------------------------
###################################################################
if (errorsY)
{
## Update gamma using weighted logistic regression ----------------
w_t <- .lengthenWT(w_t, n)
muVector <- .calculateMu(gamma_design_mat, prev_gamma)
gradient_gamma <- .calculateGradient(w_t, n, gamma_design_mat, comp_dat_all[,c(Y_unval)], muVector)
hessian_gamma <- .calculateHessian(gamma_design_mat, w_t, muVector, n, mus_gamma)
new_gamma <- tryCatch(expr = prev_gamma - (solve(hessian_gamma) %*% gradient_gamma),
error = function(err) {
matrix(NA, nrow = nrow(prev_gamma))
})
if (any(is.na(new_gamma)))
{
suppressWarnings(new_gamma <- matrix(glm(formula = gamma_formula, family = "binomial", data = data.frame(comp_dat_all), weights = w_t)$coefficients, ncol = 1))
# browser()
}
# Check for convergence -----------------------------------------
gamma_conv <- abs(new_gamma - prev_gamma) < TOL
## ---------------- Update gamma using weighted logistic regression
} else { gamma_conv <- TRUE }
###################################################################
## Update {p_kj} --------------------------------------------------
if (errorsX & errorsY)
{
### Update numerators by summing u_t over i = 1, ..., N ---------
new_p_num <- p_val_num +
rowsum(u_t, group = rep(seq(1, m), each = (N - n)), reorder = TRUE)
new_p <- t(t(new_p_num) / colSums(new_p_num))
### Check for convergence ---------------------------------------
p_conv <- abs(new_p - prev_p) < TOL
}
else if (errorsX)
{
### Update numerators by summing u_t over i = 1, ..., N ---------
new_p_num <- p_val_num +
rowsum(psi_t, group = rep(seq(1, m), each = (N - n)), reorder = TRUE)
new_p <- t(t(new_p_num) / colSums(new_p_num))
### Check for convergence ---------------------------------------
p_conv <- abs(new_p - prev_p) < TOL
}
else
{ p_conv <- TRUE }
## -------------------------------------------------- Update {p_kj}
###################################################################
# M Step ----------------------------------------------------------
###################################################################
all_conv <- c(gamma_conv, p_conv)
if (mean(all_conv) == 1) { CONVERGED <- TRUE }
# Update values for next iteration -------------------------------
it <- it + 1
if (errorsY) { prev_gamma <- new_gamma }
if (errorsX) { prev_p <- new_p }
# ------------------------------- Update values for next iteration
}
if(it == MAX_ITER & !CONVERGED) {
CONVERGED_MSG <- "MAX_ITER reached"
if (errorsY) { new_gamma <- matrix(NA, nrow = nrow(gamma0), ncol = 1) } else { new_gamma <- NA }
if (errorsX) { new_p <- matrix(NA, nrow = nrow(p0), ncol = ncol(p0)) } else { new_p <- NA }
}
if(CONVERGED) CONVERGED_MSG <- "converged"
if (!errorsY) { new_gamma <- NA }
if (!errorsX) { new_p <- NA }
# ---------------------------------------------- Estimate theta using EM
return(list("psi_at_conv" = psi_t,
"gamma_at_conv" = new_gamma,
"p_at_conv" = new_p,
"converged" = CONVERGED,
"converged_msg" = CONVERGED_MSG))
}
Epic-Doughnut/Combined-Reg documentation built on Dec. 17, 2021, 6:32 p.m.
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Defines functions profile_out
#' Profiles out nuisance parameters from the observed-data log-likelihood for a given value of theta
#'
#' For a given vector `theta` to parameterize P(Y|X,Z), this function repeats the EM algorithm to find
#' the values of `gamma` and `p` at convergence. The resulting parameters are used to find the profile
#' log-likelihood for `theta` by plugging them into the observed-data log-likelihood.
#' This function is used by `pl_theta()`.
#
#' @param theta Parameters for the analysis model (a column vector)
#' @param N Phase I sample size
#' @param n Phase II sample size
#' @param Y_unval Column with the unvalidated outcome (can be name or numeric index)
#' @param Y Column with the validated outcome (can be name or numeric index)
#' @param X_unval Column(s) with the unvalidated predictors (can be name or numeric index)
#' @param X Column(s) with the validated predictors (can be name or numeric index)
#' @param Z (Optional) Column(s) with additional error-free covariates (can be name or numeric index)
#' @param Bspline Vector of columns containing the B-spline basis functions (can be name or numeric index)
#' @param comp_dat_all Augmented dataset containing rows for each combination of unvalidated subjects' data with values from Phase II (a matrix)
#' @param theta_pred Vector of columns in \code{data} that pertain to the predictors in the analysis model.
#' @param gamma_pred Vector of columns in \code{data} that pertain to the predictors in the outcome error model.
#' @param gamma0 Starting values for `gamma`, the parameters for the outcome error model (a column vector)
#' @param p0 Starting values for `p`, the B-spline coefficients for the approximated covariate error model (a matrix)
#' @param p_val_num Contributions of validated subjects to the numerator for `p`, which are fixed (a matrix)
#' @param TOL Tolerance between iterations in the EM algorithm used to define convergence.
#' @param MAX_ITER Maximum number of iterations allowed in the EM algorithm.
#'
#' @return Profile likelihood for `theta`: the value of the observed-data log-likelihood after profiling out other parameters.
#'
#' @importFrom stats as.formula
#' @importFrom stats glm
#'
#' @noRd
profile_out <- function(theta, n, N, Y_unval = NULL, Y = NULL, X_unval = NULL, X = NULL, Z = NULL, Bspline = NULL,
comp_dat_all, theta_pred, gamma_pred, gamma0, p0, p_val_num, TOL, MAX_ITER) {
# Determine error setting -----------------------------------------
## If unvalidated variable was left blank, assume error-free ------
errorsY <- errorsX <- TRUE
if (is.null(Y_unval)) {errorsY <- FALSE}
if (is.null(X_unval) & is.null(X)) {errorsX <- FALSE}
## ------ If unvalidated variable was left blank, assume error-free
# ----------------------------------------- Determine error setting
sn <- ncol(p0)
m <- nrow(p0)
prev_gamma <- gamma0
prev_p <- p0
theta_design_mat <- cbind(int = 1, comp_dat_all[-c(1:n), theta_pred])
# For the E-step, save static P(Y|X) for unvalidated --------------
pY_X <- .pYstarCalc(theta_design_mat, n, 0, theta, comp_dat_all, match(Y, colnames(comp_dat_all))-1, vector("numeric",nrow(theta_design_mat)), vector("numeric",nrow(theta_design_mat)))
if (errorsY)
{
gamma_formula <- as.formula(paste0(Y_unval, "~", paste(gamma_pred, collapse = "+")))
gamma_design_mat <- cbind(int = 1, comp_dat_all[, gamma_pred])
}
CONVERGED <- FALSE
CONVERGED_MSG <- "Unknown"
it <- 1
# pre-allocate memory for our loop variables
# improves performance
if (errorsY)
{
pYstar <- vector(mode="numeric", length = nrow(gamma_design_mat) - n)
mu_gamma <- vector(mode="numeric", length = length(pYstar))
mus_gamma <- vector("numeric", nrow(gamma_design_mat) * ncol(prev_gamma))
}
psi_num <- matrix(,nrow = nrow(comp_dat_all)-n, ncol=length(Bspline))
psi_t <- matrix(,nrow = nrow(psi_num), ncol = ncol(psi_num))
w_t <- vector("numeric", length = nrow(psi_t))
u_t <- matrix(,nrow = m * (N-n), ncol = ncol(psi_t))
pX <- matrix(,nrow = m * (N-n) * ifelse(errorsY, 2, 1), ncol = length(Bspline))
# Estimate gamma/p using EM -----------------------------------------
while(it <= MAX_ITER & !CONVERGED)
{
# E Step ----------------------------------------------------------
# P(Y*|X*,Y,X) ---------------------------------------------------
if (errorsY)
{
pYstar <- .pYstarCalc(gamma_design_mat, n, n, prev_gamma, comp_dat_all, match(Y_unval, colnames(comp_dat_all))-1, pYstar, mu_gamma)
}
### -------------------------------------------------- P(Y*|X*,Y,X)
###################################################################
### P(X|X*) -------------------------------------------------------
# these are the slowest lines in this function (13280 ms), but I can't seem to get any faster with C++
# pX <- pXCalc(n, comp_dat_all[-c(1:n), Bspline], errorsX, errorsY, pX, prev_p[rep(seq(1, m), each = (N - n)), ])
if (errorsX & errorsY) {
### p_kj ------------------------------------------------------
### need to reorder pX so that it's x1, ..., x1, ...., xm, ..., xm-
### multiply by the B-spline terms
pX <- prev_p[rep(rep(seq(1, m), each = (N - n)), times = 2), ] * comp_dat_all[-c(1:n), Bspline]
### ---------------------------------------------------------- p_kj
} else if (errorsX) {
### p_kj ----------------------------------------------------------
### need to reorder pX so that it's x1, ..., x1, ...., xm, ..., xm-
### multiply by the B-spline terms
pX <- prev_p[rep(seq(1, m), each = (N - n)), ] * comp_dat_all[-c(1:n), Bspline]
### ---------------------------------------------------------- p_kj
}
### ------------------------------------------------------- P(X|X*)
###################################################################
### Estimate conditional expectations -----------------------------
# this function modifies w_t, u_t, psi_num, and psi_t internally
# condExp <- conditionalExpectations(errorsX, errorsY, pX, pY_X, pYstar, N - n, m, w_t, u_t, psi_num, psi_t)
# w_t <- condExp[["w_t"]]
# u_t <- condExp[["u_t"]]
# psi_t <- condExp[["psi_t"]]
if (errorsY & errorsX) {
### P(Y|X,Z)P(Y*|X*,Y,X,Z)p_kjB(X*) -----------------------------
psi_num <- c(pY_X * pYstar) * pX
### Update denominator ------------------------------------------
#### Sum up all rows per id (e.g. sum over xk/y) ----------------
psi_denom <- rowsum(psi_num, group = rep(seq(1, (N - n)), times = 2 * m))
#### Then sum over the sn splines -------------------------------
psi_denom <- rowSums(psi_denom)
#### Avoid NaN resulting from dividing by 0 ---------------------
psi_denom[psi_denom == 0] <- 1
### And divide them! --------------------------------------------
psi_t <- psi_num / psi_denom
### Update the w_kyi for unvalidated subjects -------------------
### by summing across the splines/ columns of psi_t -------------
w_t <- rowSums(psi_t)
### Update the u_kji for unvalidated subjects ------------------
### by summing over Y = 0/1 w/i each i, k ----------------------
### add top half of psi_t (y = 0) to bottom half (y = 1) -------
u_t <- psi_t[c(1:(m * (N - n))), ] + psi_t[- c(1:(m * (N - n))), ]
} else if (!errorsY && errorsX) {
### P(Y|X,Z)p_kjB(X*) -------------------------------------------
psi_num <- c(pY_X) * pX
### Update denominator ------------------------------------------
#### Sum up all rows per id (e.g. sum over xk) ------------------
psi_denom <- rowsum(psi_num, group = rep(seq(1, (N - n)), times = m))
#### Then sum over the sn splines -------------------------------
psi_denom <- rowSums(psi_denom)
#### Avoid NaN resulting from dividing by 0 ---------------------
psi_denom[psi_denom == 0] <- 1
### And divide them! --------------------------------------------
psi_t <- psi_num / psi_denom
### Update the w_kyi for unvalidated subjects -------------------
### by summing across the splines/ columns of psi_t -------------
w_t <- rowSums(psi_t)
} else if (!errorsX && errorsY) {
### P(Y|X,Z)P(Y*|Y,X,Z) -----------------------------------------
psi_num <- matrix(c(pY_X * pYstar), ncol = 1)
#### Sum up all rows per id (e.g. sum over y) -------------------
psi_denom <- rowsum(psi_num, group = rep(seq(1, (N - n)), times = 2))
#### Avoid NaN resulting from dividing by 0 ---------------------
psi_denom[psi_denom == 0] <- 1
### And divide them! --------------------------------------------
psi_t <- psi_num / psi_denom
### Update the w_kyi for unvalidated subjects -------------------
w_t <- psi_t
}
### ----------------------------- Estimate conditional expectations
# ---------------------------------------------------------- E Step
###################################################################
###################################################################
# M Step ----------------------------------------------------------
###################################################################
if (errorsY)
{
## Update gamma using weighted logistic regression ----------------
w_t <- .lengthenWT(w_t, n)
muVector <- .calculateMu(gamma_design_mat, prev_gamma)
gradient_gamma <- .calculateGradient(w_t, n, gamma_design_mat, comp_dat_all[,c(Y_unval)], muVector)
hessian_gamma <- .calculateHessian(gamma_design_mat, w_t, muVector, n, mus_gamma)
new_gamma <- tryCatch(expr = prev_gamma - (solve(hessian_gamma) %*% gradient_gamma),
error = function(err) {
matrix(NA, nrow = nrow(prev_gamma))
})
if (any(is.na(new_gamma)))
{
suppressWarnings(new_gamma <- matrix(glm(formula = gamma_formula, family = "binomial", data = data.frame(comp_dat_all), weights = w_t)$coefficients, ncol = 1))
# browser()
}
# Check for convergence -----------------------------------------
gamma_conv <- abs(new_gamma - prev_gamma) < TOL
## ---------------- Update gamma using weighted logistic regression
} else { gamma_conv <- TRUE }
###################################################################
## Update {p_kj} --------------------------------------------------
if (errorsX & errorsY)
{
### Update numerators by summing u_t over i = 1, ..., N ---------
new_p_num <- p_val_num +
rowsum(u_t, group = rep(seq(1, m), each = (N - n)), reorder = TRUE)
new_p <- t(t(new_p_num) / colSums(new_p_num))
### Check for convergence ---------------------------------------
p_conv <- abs(new_p - prev_p) < TOL
}
else if (errorsX)
{
### Update numerators by summing u_t over i = 1, ..., N ---------
new_p_num <- p_val_num +
rowsum(psi_t, group = rep(seq(1, m), each = (N - n)), reorder = TRUE)
new_p <- t(t(new_p_num) / colSums(new_p_num))
### Check for convergence ---------------------------------------
p_conv <- abs(new_p - prev_p) < TOL
}
else
{ p_conv <- TRUE }
## -------------------------------------------------- Update {p_kj}
###################################################################
# M Step ----------------------------------------------------------
###################################################################
all_conv <- c(gamma_conv, p_conv)
if (mean(all_conv) == 1) { CONVERGED <- TRUE }
# Update values for next iteration -------------------------------
it <- it + 1
if (errorsY) { prev_gamma <- new_gamma }
if (errorsX) { prev_p <- new_p }
# ------------------------------- Update values for next iteration
}
if(it == MAX_ITER & !CONVERGED) {
CONVERGED_MSG <- "MAX_ITER reached"
if (errorsY) { new_gamma <- matrix(NA, nrow = nrow(gamma0), ncol = 1) } else { new_gamma <- NA }
if (errorsX) { new_p <- matrix(NA, nrow = nrow(p0), ncol = ncol(p0)) } else { new_p <- NA }
}
if(CONVERGED) CONVERGED_MSG <- "converged"
if (!errorsY) { new_gamma <- NA }
if (!errorsX) { new_p <- NA }
# ---------------------------------------------- Estimate theta using EM
return(list("psi_at_conv" = psi_t,
"gamma_at_conv" = new_gamma,
"p_at_conv" = new_p,
"converged" = CONVERGED,
"converged_msg" = CONVERGED_MSG))
}
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