Package 'SurrogateRegression' reference manual

Title:	Surrogate Outcome Regression Analysis
Description:	Performs estimation and inference on a partially missing target outcome (e.g. gene expression in an inaccessible tissue) while borrowing information from a correlated surrogate outcome (e.g. gene expression in an accessible tissue). Rather than regarding the surrogate outcome as a proxy for the target outcome, this package jointly models the target and surrogate outcomes within a bivariate regression framework. Unobserved values of either outcome are treated as missing data. In contrast to imputation-based inference, no assumptions are required regarding the relationship between the target and surrogate outcomes. Estimation in the presence of bilateral outcome missingness is performed via an expectation conditional maximization either algorithm. In the case of unilateral target missingness, estimation is performed using an accelerated least squares procedure. A flexible association test is provided for evaluating hypotheses about the target regression parameters. For additional details, see: McCaw ZR, Gaynor SM, Sun R, Lin X: "Leveraging a surrogate outcome to improve inference on a partially missing target outcome" <doi:10.1111/biom.13629>.
Authors:	Zachary McCaw [aut, cre]
Maintainer:	Zachary McCaw <[email protected]>
License:	GPL-3
Version:	0.6.0.1
Built:	2024-11-10 06:16:10 UTC
Source:	https://github.com/zrmacc/surrogateregression

Bivariate Regression Model

Description

Bivariate Regression Model

Slots

Covariance: Residual covariance matrix.
Covariance.info: Information for covariance parameters.
Covariance.tab: Table of covariance parameters.
Method: Method used for estimation.
Regression.info: Information for regression coefficients.
Regression.tab: Table of regression coefficients.
Residuals: Outcome residuals.

Check Initiation

Description

Check Initiation

Usage

CheckInit(init)
CheckInit(init)

Arguments

init

Optional list of initial parameters for fitting the null model.

Check Test Specification

Description

Check Test Specification

Usage

CheckTestSpec(is_zero, p)
CheckTestSpec(is_zero, p)

Arguments

`is_zero`	Logical vector, with as many entires as columns in the target model matrix, indicating which columns have coefficient zero under the null.
`p`	Number of columns for the target model matrix.

Extract Coefficients from Bivariate Regression Model

Description

Extract Coefficients from Bivariate Regression Model

Usage

## S3 method for class 'bnr'
coef(object, ..., type = NULL)
## S3 method for class 'bnr'
coef(object, ..., type = NULL)

Arguments

`object`	`bnr` object.
`...`	Unused.
`type`	Either Target or Surrogate.

Covariance Information Matrix

Description

Covariance Information Matrix

Usage

CovInfo(data_part, sigma)
CovInfo(data_part, sigma)

Arguments

`data_part`	List of partitioned data. See `PartitionData`.
`sigma`	Target-surrogate covariance matrix.

Value

3x3 Numeric information matrix for the target variance, target-surrogate covariance, and surrogate variance.

Tabulate Covariance Parameters

Description

Tabulate Covariance Parameters

Usage

CovTab(point, info, sig = 0.05)
CovTab(point, info, sig = 0.05)

Arguments

`point`	Point estimates.
`info`	Information matrix.
`sig`	Significance level.

Value

Data.table containing the point estimate, standard error, and confidence interval.

Covariate Update

Description

Covariate Update

Usage

CovUpdate(data_part, b0, a0, b1, a1, sigma0)
CovUpdate(data_part, b0, a0, b1, a1, sigma0)

Arguments

`data_part`	List of partitioned data. See `PartitionData`.
`b0`	Previous target regression coefficient.
`a0`	Previous surrogate regression coefficient.
`b1`	Current target regression coefficient.
`a1`	Current surrogate regression coefficient.
`sigma0`	Initial target-surrogate covariance matrix.

Value

ECM update of the target-surrogate covariance matrix.

Fit Bivariate Normal Regression Model via Expectation Maximization.

Description

Estimation procedure for bivariate normal regression models in which the target and surrogate outcomes are both subject to missingness.

Usage

FitBNEM(
  t,
  s,
  X,
  Z,
  sig = 0.05,
  b0 = NULL,
  a0 = NULL,
  sigma0 = NULL,
  maxit = 100,
  eps = 1e-06,
  report = TRUE
)
FitBNEM(
  t,
  s,
  X,
  Z,
  sig = 0.05,
  b0 = NULL,
  a0 = NULL,
  sigma0 = NULL,
  maxit = 100,
  eps = 1e-06,
  report = TRUE
)

Arguments

`t`	Target outcome vector.
`s`	Surrogate outcome vector.
`X`	Target model matrix.
`Z`	Surrogate model matrix.
`sig`	Type I error level.
`b0`	Initial target regression coefficient.
`a0`	Initial surrogate regression coefficient.
`sigma0`	Initial covariance matrix.
`maxit`	Maximum number of parameter updates.
`eps`	Minimum acceptable improvement in log likelihood.
`report`	Report fitting progress?

Details

The target and surrogate model matrices are expected in numeric format. Include an intercept if required. Expand factors and interactions in advance. Initial values may be specified for any of the target coefficient b0, the surrogate coefficient a0, or the target-surrogate covariance matrix sigma0.

Value

An object of class 'bnr' with slots containing the estimated regression coefficients, the target-surrogate covariance matrix, the information matrices for the regression and covariance parameters, and the residuals.

Fit Bivariate Normal Regression Model via Least Squares

Description

Estimation procedure for bivariate normal regression models in which only the target outcome is subject to missingness.

Usage

FitBNLS(t, s, X, sig = 0.05)
FitBNLS(t, s, X, sig = 0.05)

Arguments

`t`	Target outcome vector.
`s`	Surrogate outcome vector.
`X`	Model matrix.
`sig`	Type I error level.

Details

The model matrix is expected in numeric format. Include an intercept if required. Expand factors and interactions in advance.

Value

Fit Bivariate Normal Regression Model

Description

Estimation procedure for bivariate normal regression models. The EM algorithm is applied if s contains missing values, or if X differs from Z. Otherwise, an accelerated least squares procedure is applied.

Usage

FitBNR(t, s, X, Z = NULL, sig = 0.05, ...)
FitBNR(t, s, X, Z = NULL, sig = 0.05, ...)

Arguments

`t`	Target outcome vector.
`s`	Surrogate outcome vector.
`X`	Target model matrix.
`Z`	Surrogate model matrix. Defaults to `X`.
`sig`	Significance level.
`...`	Additional arguments accepted if fitting via EM. See `FitBNEM`.

Details

The target and surrogate model matrices are expected in numeric format. Include an intercept if required. Expand factors and interactions in advance.

Value

An object of class 'mnr' with slots containing the estimated regression coefficients, the target-surrogate covariance matrix, the information matrices for regression parameters, and the residuals.

Examples


# Case 1: No surrogate missingness.
set.seed(100)
n <- 1e3
X <- stats::rnorm(n)
data <- rBNR(
  X = X,
  Z = X,
  b = 1,
  a = -1,
  t_miss = 0.1,
  s_miss = 0.0
)
t <- data[, 1]
s <- data[, 2]

# Model fit.
fit_bnls <- FitBNR(
  t = t,
  s = s,
  X = X
)

# Case 2: Target and surrogate missingness.
set.seed(100)
n <- 1e3
X <- stats::rnorm(n)
Z <- stats::rnorm(n)
data <- rBNR(
  X = X,
  Z = Z,
  b = 1,
  a = -1,
  t_miss = 0.1,
  s_miss = 0.1
)

# Log likelihood.
fit_bnem <- FitBNR(
  t = data[, 1],
  s = data[, 2],
  X = X,
  Z = Z
)

# Case 1: No surrogate missingness.
set.seed(100)
n <- 1e3
X <- stats::rnorm(n)
data <- rBNR(
  X = X,
  Z = X,
  b = 1,
  a = -1,
  t_miss = 0.1,
  s_miss = 0.0
)
t <- data[, 1]
s <- data[, 2]

# Model fit.
fit_bnls <- FitBNR(
  t = t,
  s = s,
  X = X
)

# Case 2: Target and surrogate missingness.
set.seed(100)
n <- 1e3
X <- stats::rnorm(n)
Z <- stats::rnorm(n)
data <- rBNR(
  X = X,
  Z = Z,
  b = 1,
  a = -1,
  t_miss = 0.1,
  s_miss = 0.1
)

# Log likelihood.
fit_bnem <- FitBNR(
  t = data[, 1],
  s = data[, 2],
  X = X,
  Z = Z
)

Ordinary Least Squares

Description

Fits the standard OLS model.

Usage

fitOLS(y, X)
fitOLS(y, X)

Arguments

`y`	Nx1 Numeric vector.
`X`	NxP Numeric matrix.

Value

List containing the following:

`Beta`	Regression coefficient.
`V`	Outcome variance.
`Ibb`	Information matrix for beta.
`Resid`	Outcome residuals.

Format Output

Description

Format Output

Usage

FormatOutput(data_part, method, b, a, sigma, sig)
FormatOutput(data_part, method, b, a, sigma, sig)

Arguments

`data_part`	List of partitioned data. See `PartitionData`.
`method`	Estimation method.
`b`	Final target regression parameter.
`a`	Final surrogate regression parameter.
`sigma`	Final target-surrogate covariance matrix.
`sig`	Significance level.

Value

Object of class 'bnr'.

Update Iteration

Description

Update Iteration

Usage

IterUpdate(theta0, update, maxit, eps, report)
IterUpdate(theta0, update, maxit, eps, report)

Arguments

`theta0`	List containing the initial parameter values.
`update`	Function to iterate. Should accept and return a list similar parameter values.
`maxit`	Maximum number of parameter updates.
`eps`	Minimum acceptable improvement in log likelihood.
`report`	Report fitting progress?

Matrix Determinant

Description

Calculates the determinant of $A$ .

Usage

matDet(A, logDet = FALSE)
matDet(A, logDet = FALSE)

Arguments

`A`	Numeric matrix.
`logDet`	Return the logarithm of the determinant?

Value

Scalar.

Matrix Inverse

Description

Calcualtes $A^{-1}$ .

Usage

matInv(A)
matInv(A)

Arguments

`A`	Numeric matrix.

Value

Numeric matrix.

Matrix Inner Product

Description

Calculates the product $A'B$ .

Usage

matIP(A, B)
matIP(A, B)

Arguments

`A`	Numeric matrix.
`B`	Numeric matrix.

Value

Numeric matrix.

Matrix Outer Product

Description

Calculates the outer product $AB'$ .

Usage

matOP(A, B)
matOP(A, B)

Arguments

`A`	Numeric matrix.
`B`	Numeric matrix.

Value

Numeric matrix.

Quadratic Form

Description

Calculates the quadratic form $X'AX$ .

Usage

matQF(X, A)
matQF(X, A)

Arguments

`X`	Numeric matrix.
`A`	Numeric matrix.

Value

Numeric matrix.

Matrix Matrix Product

Description

Calculates the product $AB$ .

Usage

MMP(A, B)
MMP(A, B)

Arguments

`A`	Numeric matrix.
`B`	Numeric matrix.

Value

Numeric matrix.

Observed Data Log Likelihood

Description

Observed Data Log Likelihood

Usage

ObsLogLik(data_part, b, a, sigma)
ObsLogLik(data_part, b, a, sigma)

Arguments

`data_part`	List of partitioned data. See `PartitionData`.
`b`	Target regression coefficient.
`a`	Surrogate regression coefficient.
`sigma`	Target-surrogate covariance matrix.

Value

Observed data log likelihood.

Parameter Initialization

Description

Parameter Initialization

Usage

ParamInit(data_part, b0, a0, sigma0)
ParamInit(data_part, b0, a0, sigma0)

Arguments

`data_part`	List of partitioned data. See `PartitionData`.
`b0`	Initial target regression coefficient.
`a0`	Initial surrogate regression coefficient.
`sigma0`	Initial covariance matrix.

Value

List containing initial values of beta, alpha, sigma.

Partition Data by Outcome Missingness Pattern.

Description

Partition Data by Outcome Missingness Pattern.

Usage

PartitionData(t, s, X, Z = NULL)
PartitionData(t, s, X, Z = NULL)

Arguments

`t`	Target outcome vector.
`s`	Surrogate outcome vector.
`X`	Target model matrix.
`Z`	Surrogate model matrix.

Value

List containing these components:

'Orig' original data.
'Dims' dimensions and names.
'Complete', data for complete cases.
'TMiss', data for subjects with target missingness.
'SMiss', data for subjects with surrogate missingness.
'IPs', inner products.

Examples

# Generate data.
n <- 1e3
X <- rnorm(n)
Z <- rnorm(n)
data <- rBNR(X = X, Z = Z, b = 1, a = -1)
data_part <- PartitionData(
  t = data[, 1], 
  s = data[, 2], 
  X = X, 
  Z = Z
)
# Generate data.
n <- 1e3
X <- rnorm(n)
Z <- rnorm(n)
data <- rBNR(X = X, Z = Z, b = 1, a = -1)
data_part <- PartitionData(
  t = data[, 1], 
  s = data[, 2], 
  X = X, 
  Z = Z
)

Print for Bivariate Regression Model

Description

Print for Bivariate Regression Model

Usage

## S3 method for class 'bnr'
print(x, ..., type = "Regression")
## S3 method for class 'bnr'
print(x, ..., type = "Regression")

Arguments

`x`	`bnr` object.
`...`	Unused.
`type`	Either Regression or Covariance.

Simulate Bivariate Normal Data with Missingness

Description

Function to simulate from a bivariate normal regression model with outcomes missing completely at random.

Usage

rBNR(
  X,
  Z,
  b,
  a,
  t_miss = 0,
  s_miss = 0,
  sigma = NULL,
  include_residuals = TRUE
)
rBNR(
  X,
  Z,
  b,
  a,
  t_miss = 0,
  s_miss = 0,
  sigma = NULL,
  include_residuals = TRUE
)

Arguments

`X`	Target design matrix.
`Z`	Surrogate design matrix.
`b`	Target regression coefficient.
`a`	Surrogate regression coefficient.
`t_miss`	Target missingness in [0,1].
`s_miss`	Surrogate missingness in [0,1].
`sigma`	2x2 target-surrogate covariance matrix.
`include_residuals`	Include the residual? Default: TRUE.

Value

Numeric Nx2 matrix. The first column contains the target outcome, the second contains the surrogate outcome.

Examples

set.seed(100)
# Observations.
n <- 1e3
# Target design.
X <- cbind(1, matrix(rnorm(3 * n), nrow = n))
# Surrogate design.
Z <- cbind(1, matrix(rnorm(3 * n), nrow = n))
# Target coefficient.
b <- c(-1, 0.1, -0.1, 0.1)
# Surrogate coefficient.
a <- c(1, -0.1, 0.1, -0.1)
# Covariance structure.
sigma <- matrix(c(1, 0.5, 0.5, 1), nrow = 2)
# Data generation, target and surrogate subject to 10% missingness.
y <- rBNR(X, Z, b, a, t_miss = 0.1, s_miss = 0.1, sigma = sigma)
set.seed(100)
# Observations.
n <- 1e3
# Target design.
X <- cbind(1, matrix(rnorm(3 * n), nrow = n))
# Surrogate design.
Z <- cbind(1, matrix(rnorm(3 * n), nrow = n))
# Target coefficient.
b <- c(-1, 0.1, -0.1, 0.1)
# Surrogate coefficient.
a <- c(1, -0.1, 0.1, -0.1)
# Covariance structure.
sigma <- matrix(c(1, 0.5, 0.5, 1), nrow = 2)
# Data generation, target and surrogate subject to 10% missingness.
y <- rBNR(X, Z, b, a, t_miss = 0.1, s_miss = 0.1, sigma = sigma)

Regression Information

Description

Regression Information

Usage

RegInfo(data_part, sigma, as_matrix = FALSE)
RegInfo(data_part, sigma, as_matrix = FALSE)

Arguments

`data_part`	List of partitioned data. See `PartitionData`.
`sigma`	Target-surrogate covariance matrix.
`as_matrix`	Return as an information matrix? If FALSE, returns a list.

Value

List containing the information matrix for beta (Ibb), the information matrix for alpha (Iaa), and the cross information (Iba).

Tabulate Regression Coefficients

Description

Tabulate Regression Coefficients

Usage

RegTab(point, info, sig = 0.05)
RegTab(point, info, sig = 0.05)

Arguments

`point`	Point estimates.
`info`	Information matrix.
`sig`	Significance level.

Value

Data.table containing the point estimate, standard error, confidence interval, and Wald p-value.

Regression Update

Description

Regression Update

Usage

RegUpdate(data_part, sigma)
RegUpdate(data_part, sigma)

Arguments

`data_part`	List of partitioned data. See `PartitionData`.
`sigma`	Target-surrogate covariance matrix.

Value

List containing the generalized least squares estimates of beta and alpha.

Extract Residuals from Bivariate Regression Model

Description

Extract Residuals from Bivariate Regression Model

Usage

## S3 method for class 'bnr'
residuals(object, ..., type = NULL)
## S3 method for class 'bnr'
residuals(object, ..., type = NULL)

Arguments

`object`	A `bnr` object.
`...`	Unused.
`type`	Either Target or Surrogate.

Schur complement

Description

Calculates the efficient information $I_{bb}-I_{ba}I_{aa}^{-1}I_{ab}$ .

Usage

SchurC(Ibb, Iaa, Iba)
SchurC(Ibb, Iaa, Iba)

Arguments

`Ibb`	Information of target parameter
`Iaa`	Information of nuisance parameter
`Iba`	Cross information between target and nuisance parameters

Value

Numeric matrix.

Score Test via Expectation Maximization.

Description

Performs a Score test of the null hypothesis that a subset of the regression parameters for the target outcome are zero.

Usage

ScoreBNEM(
  t,
  s,
  X,
  Z,
  is_zero,
  init = NULL,
  maxit = 100,
  eps = 1e-08,
  report = FALSE
)
ScoreBNEM(
  t,
  s,
  X,
  Z,
  is_zero,
  init = NULL,
  maxit = 100,
  eps = 1e-08,
  report = FALSE
)

Arguments

`t`	Target outcome vector.
`s`	Surrogate outcome vector.
`X`	Target model matrix.
`Z`	Surrogate model matrix.
`is_zero`	Logical vector, with as many entires as columns in the target model matrix, indicating which columns have coefficient zero under the null.
`init`	Optional list of initial parameters for fitting the null model.
`maxit`	Maximum number of parameter updates.
`eps`	Minimum acceptable improvement in log likelihood.
`report`	Report model fitting progress? Default is FALSE.

Value

A numeric vector containing the score statistic, the degrees of freedom, and a p-value.

Show for Bivariate Regression Model

Description

Show for Bivariate Regression Model

Usage

## S4 method for signature 'bnr'
show(object)
## S4 method for signature 'bnr'
show(object)

Arguments

object

bnr object.

SurrogateRegression: Surrogate Outcome Regression Analysis

Description

This package performs estimation and inference on a partially missing target outcome while borrowing information from a correlated surrogate outcome. Rather than regarding the surrogate outcome as a proxy for the target outcome, this package jointly models the target and surrogate outcomes within a bivariate regression framework. Unobserved values of either outcome are treated as missing data. In contrast to imputation-based inference, no assumptions are required regarding the relationship between the target and surrogate outcomes. However, in order for surrogate inference to improve power, the target and surrogate outcomes must be correlated, and the target outcome must be partially missing. The primary estimation function is FitBNR. In the case of bilateral missingness, i.e. missingness in both the target and surrogate outcomes, estimation is performed via an expectation conditional maximization either (ECME) algorithm. In the case of unilateral target missingness, estimation is performed using an accelerated least squares procedure. Inference on regression parameters for the target outcome is performed using TestBNR.

Author(s)

Zachary R. McCaw

Test Bivariate Normal Regression Model.

Description

Performs a test of the null hypothesis that a subset of the regression parameters for the target outcome are zero in the bivariate normal regression model.

Usage

TestBNR(t, s, X, Z = NULL, is_zero, test = "Wald", ...)
TestBNR(t, s, X, Z = NULL, is_zero, test = "Wald", ...)

Arguments

`t`	Target outcome vector.
`s`	Surrogate outcome vector.
`X`	Target model matrix.
`Z`	Surrogate model matrix.
`is_zero`	Logical vector, with as many entires as columns in the target model matrix, indicating which columns have coefficient zero under the null.
`test`	Either Score or Wald. Only Wald is available for LS.
`...`	Additional arguments accepted if fitting via EM. See `FitBNEM`.

Value

A numeric vector containing the test statistic, the degrees of freedom, and a p-value.

Examples


# Generate data.
set.seed(100)
n <- 1e3
X <- cbind(1, rnorm(n))
Z <- cbind(1, rnorm(n))
data <- rBNR(X = X, Z = Z, b = c(1, 0), a = c(-1, 0), t_miss = 0.1, s_miss = 0.1)

# Test 1st coefficient.
wald_test1 <- TestBNR(
  t = data[, 1], 
  s = data[, 2], 
  X = X, 
  Z = Z,
  is_zero = c(TRUE, FALSE),
  test = "Wald"
)

score_test1 <- TestBNR(
  t = data[, 1], 
  s = data[, 2], 
  X = X, 
  Z = Z,
  is_zero = c(TRUE, FALSE),
  test = "Score"
)

# Test 2nd coefficient.
wald_test2 <- TestBNR(
  t = data[, 1], 
  s = data[, 2], 
  X = X, 
  Z = Z,
  is_zero = c(FALSE, TRUE),
  test = "Wald"
)

score_test2 <- TestBNR(
  t = data[, 1], 
  s = data[, 2], 
  X = X, 
  Z = Z,
  is_zero = c(FALSE, TRUE),
  test = "Score"
)

# Generate data.
set.seed(100)
n <- 1e3
X <- cbind(1, rnorm(n))
Z <- cbind(1, rnorm(n))
data <- rBNR(X = X, Z = Z, b = c(1, 0), a = c(-1, 0), t_miss = 0.1, s_miss = 0.1)

# Test 1st coefficient.
wald_test1 <- TestBNR(
  t = data[, 1], 
  s = data[, 2], 
  X = X, 
  Z = Z,
  is_zero = c(TRUE, FALSE),
  test = "Wald"
)

score_test1 <- TestBNR(
  t = data[, 1], 
  s = data[, 2], 
  X = X, 
  Z = Z,
  is_zero = c(TRUE, FALSE),
  test = "Score"
)

# Test 2nd coefficient.
wald_test2 <- TestBNR(
  t = data[, 1], 
  s = data[, 2], 
  X = X, 
  Z = Z,
  is_zero = c(FALSE, TRUE),
  test = "Wald"
)

score_test2 <- TestBNR(
  t = data[, 1], 
  s = data[, 2], 
  X = X, 
  Z = Z,
  is_zero = c(FALSE, TRUE),
  test = "Score"
)

Matrix Trace

Description

Calculates the trace of a matrix $A$ .

Usage

tr(A)
tr(A)

Arguments

`A`	Numeric matrix.

Value

Scalar.

EM Update

Description

EM Update

Usage

UpdateEM(data_part, b0, a0, sigma0)
UpdateEM(data_part, b0, a0, sigma0)

Arguments

`data_part`	List of partitioned data. See `PartitionData`.
`b0`	Initial target regression coefficient.
`a0`	Initial surrogate regression coefficient.
`sigma0`	Initial covariance matrix.

Value

List containing updated values for beta 'b', alpha 'a', 'sigma', the log likelihood 'loglik', and the change in log likelihood 'delta'.

Extract Covariance Matrix from Bivariate Normal Regression Model

Description

Returns the either the estimated covariance matrix of the outcome, the information matrix for regression coefficients, or the information matrix for covariance parameters.

Usage

## S3 method for class 'bnr'
vcov(object, ..., type = "Regression", inv = FALSE)
## S3 method for class 'bnr'
vcov(object, ..., type = "Regression", inv = FALSE)

Arguments

`object`	`bnr` object.
`...`	Unused.
`type`	Select "Covariance","Outcome",or "Regression". Default is "Regression".
`inv`	Invert the covariance matrix? Default is FALSE.

Wald Test via Expectation Maximization.

Description

Performs a Wald test of the null hypothesis that a subset of the regression parameters for the target outcome are zero.

Usage

WaldBNEM(
  t,
  s,
  X,
  Z,
  is_zero,
  init = NULL,
  maxit = 100,
  eps = 1e-08,
  report = FALSE
)
WaldBNEM(
  t,
  s,
  X,
  Z,
  is_zero,
  init = NULL,
  maxit = 100,
  eps = 1e-08,
  report = FALSE
)

Arguments

`t`	Target outcome vector.
`s`	Surrogate outcome vector.
`X`	Target model matrix.
`Z`	Surrogate model matrix.
`is_zero`	Logical vector, with as many entries as columns in the target model matrix, indicating which columns have coefficient zero under the null.
`init`	Optional list of initial parameters for fitting the null model, with one or more of the components: a0, b0, S0.
`maxit`	Maximum number of parameter updates.
`eps`	Minimum acceptable improvement in log likelihood.
`report`	Report model fitting progress? Default is FALSE.

Value

A numeric vector containing the Wald statistic, the degrees of freedom, and a p-value.

Wald Test via Least Squares.

Description

Performs a Wald test of the null hypothesis that a subset of the regression parameters for the target outcome are zero.

Usage

WaldBNLS(t, s, X, is_zero)
WaldBNLS(t, s, X, is_zero)

Arguments

`t`	Target outcome vector.
`s`	Surrogate outcome vector.
`X`	Model matrix.
`is_zero`	Logical vector, with as many entires as columns in the target model matrix, indicating which columns have coefficient zero under the null.

Value

A numeric vector containing the Wald statistic, the degrees of freedom, and a p-value.

Package 'SurrogateRegression'

Help Index

Bivariate Regression Model

Description

Slots

Check Initiation

Description

Usage

Arguments

Check Test Specification

Description

Usage

Arguments

Extract Coefficients from Bivariate Regression Model

Description

Usage

Arguments

Covariance Information Matrix

Description

Usage

Arguments

Value

Tabulate Covariance Parameters

Description

Usage

Arguments

Value

Covariate Update

Description

Usage

Arguments

Value

Fit Bivariate Normal Regression Model via Expectation Maximization.

Description

Usage

Arguments

Details

Value

Fit Bivariate Normal Regression Model via Least Squares

Description

Usage

Arguments

Details

Value

Fit Bivariate Normal Regression Model

Description

Usage

Arguments

Details

Value

Examples

Ordinary Least Squares

Description

Usage

Arguments

Value

Format Output

Description

Usage

Arguments

Value

Update Iteration

Description

Usage

Arguments

Matrix Determinant

Description

Usage

Arguments

Value

Matrix Inverse

Description

Usage

Arguments

Value

Matrix Inner Product

Description

Usage

Arguments

Value