Bootstrap standard error estimates

Description

Function to obtain bootstrap standard error estimates for the parameter estimates of the get_estimates function, under the generalized linear model (GLM) or accelerated failure time (AFT) setting for the analysis of a normally-distributed or censored time-to-event primary outcome.

Usage

bootstrap_se(setting = "GLM", BS_rep = 1000, Y = NULL, X = NULL,
  K = NULL, L = NULL, C = NULL)

Arguments

Details

Under the GLM setting for the analysis of a normally-distributed primary outcome Y, bootstrap standard error estimates are obtained for the estimates of the parameters \alpha_0, \alpha_1, \alpha_2, \alpha_3, \sigma_1^2, \alpha_4, \alpha_{XY}, \sigma_2^2 in the models

Y = \alpha_0 + \alpha_1 \cdot K + \alpha_2 \cdot X + \alpha_3 \cdot L + \epsilon_1, \epsilon_1 \sim N(0,\sigma_1^2)

Y^* = Y - \overline{Y} - \alpha_1 \cdot (K-\overline{K})

Y^* = \alpha_0 + \alpha_{XY} \cdot X + \epsilon_2, \epsilon_2 \sim N(0,\sigma_2^2),

accounting for the additional variability from the 2-stage approach.

Under the AFT setting for the analysis of a censored time-to-event primary outcome, bootstrap standard error estimates are similarly obtained of the parameter estimates of \alpha_0, \alpha_1, \alpha_2, \alpha_3, \sigma_1, \alpha_4, \alpha_{XY}, \sigma_2^2

Value

Returns a vector with the bootstrap standard error estimates of the parameter estimates.

Examples

dat <- generate_data(setting = "GLM", n = 100)

# For illustration use here only 100 bootstrap samples, recommended is using 1000
bootstrap_se(setting = "GLM", BS_rep = 100, Y = dat$Y, X = dat$X,
             K = dat$K, L = dat$L)

CIEE: Causal inference based on estimating equations

Description

Functions to perform CIEE under the GLM or AFT setting: ciee obtains point and standard error estimates of all parameter estimates, and p-values for testing the absence of effects; ciee_loop performs ciee in separate analyses of multiple exposure variables with the same outcome measures and factors ond only returns point estimates, standard error estimates and p-values for the exposure variables. Both functions can also compute estimates and p-values from the two traditional regression methods and from the structural equation modeling method.

Usage

ciee(setting = "GLM", estimates = c("ee", "mult_reg", "res_reg", "sem"),
  ee_se = c("sandwich"), BS_rep = NULL, Y = NULL, X = NULL, K = NULL,
  L = NULL, C = NULL)

ciee_loop(setting = "GLM", estimates = c("ee", "mult_reg", "res_reg",
  "sem"), ee_se = c("sandwich"), BS_rep = NULL, Y = NULL, X = NULL,
  K = NULL, L = NULL, C = NULL)

Arguments

Details

For the computation of CIEE, point estimates of the parameters are obtained using the get_estimates function. Robust sandwich (recommended), bootstrap, or naive standard error estimates of the parameter estimates are obtained using the sandwich_se, bootstrap_se or naive_se function. Large-sample Wald-type tests are performed for testing the absence of effects, using either the robust sandwich or bootstrap standard errors.

Regarding the traditional regression methods, the multiple regression or regression of residual approaches can be computed using the mult_reg and res_reg functions. Finally, the structural equation modeling approachcan be performed using the sem_appl function.

Value

Object of class ciee, for which the summary function summary.ciee is implemented. ciee returns a list containing the point and standard error estimates of all parameters as well as p-values from hypothesis tests of the absence of effects, for each specified approach. ciee_loop returns a list containing the point and standard error estimates only of the exposure variables as well as p-values from hypothesis tests of the absence of effects, for each specified approach.

Examples

# Generate data under the GLM setting with default values
maf <- 0.2
n <- 100
dat <- generate_data(n = n, maf = maf)
datX <- data.frame(X = dat$X)
names(datX)[1] <- "X1"
# Add 9 more exposure variables names X2, ..., X10 to X
for (i in 2:10){
  X <- stats::rbinom(n, size = 2, prob = maf)
  datX$X <- X
  names(datX)[i] <- paste("X", i, sep="")
}

# Perform analysis of one exposure variable using all four methods
ciee(Y = dat$Y, X = datX$X1, K = dat$K, L = dat$L)

# Perform analysis of all exposure variables only for CIEE
ciee_loop(estimates = "ee", Y = dat$Y, X = datX, K = dat$K, L = dat$L)

Estimating functions.

Description

Function to compute logL1 and logL2 under the GLM and AFT setting for the analysis of a normally-distributed and of a censored time-to-event primary outcome. logL1 and logL2 are functions which underlie the estimating functions of CIEE for the derivation of point estimates and standard error estimates. est_funct_expr computes their expression, which is then further used in the functions deriv_obj, ciee and ciee_loop.

Usage

est_funct_expr(setting = "GLM")

Arguments

Details

Under the GLM setting for the analysis of a normally-distributed primary outcome Y, the goal is to obtain estimates for the pararameters \alpha_0, \alpha_1, \alpha_2, \alpha_3, \sigma_1^2, \alpha_4, \alpha_{XY}, \sigma_2^2 under the model

Y = \alpha_0 + \alpha_1 \cdot K + \alpha_2 \cdot X + \alpha_3 \cdot L + \epsilon_1, \epsilon_1 \sim N(0,\sigma_1^2)

Y^* = Y - \overline{Y} - \alpha_1 \cdot (K-\overline{K})

Y^* = \alpha_0 + \alpha_{XY} \cdot X + \epsilon_2, \epsilon_2 \sim N(0,\sigma_2^2)

logL1 underlies the estimating functions for the derivation of the first 5 parameters \alpha_0, \alpha_1, \alpha_2, \alpha_3, \sigma_1^2 and logL2 underlies the estimating functions for the derivation of the last 3 parameters \alpha_4, \alpha_{XY}, \sigma_2^2.

Under the AFT setting for the analysis of a censored time-to-event primary outcome Y, the goal is to obtain estimates of the parameters \alpha_0, \alpha_1, \alpha_2, \alpha_3, \sigma_1, \alpha_4, \alpha_{XY}, \sigma_2^2. Here, logL1 similarly underlies the estimating functions for the derivation of the first 5 parameters and logL2 underlies the estimating functions for the derivation of the last 3 parameters.

logL1, logL2 equal the log-likelihood functions (logL2 given that \alpha_1 is known). For more details and the underlying model, see the vignette.

Value

Returns a list containing the expression of the functions logL1 and logL2.

Examples

est_funct_expr(setting = "GLM")
est_funct_expr(setting = "AFT")

Data generation function

Description

Function to generate data with n observations of a primary outcome Y, secondary outcome K, exposure X, and measured as well as unmeasured confounders L and U, where the primary outcome is a quantitative normally-distributed variable (setting = "GLM") or censored time-to-event outcome under an accelerated failure time (AFT) model (setting = "AFT"). Under the AFT setting, the observed time-to-event variable T=exp(Y) as well as the censoring indicator C are also computed. X is generated as a genetic exposure variable in the form of a single nucleotide variant (SNV) in 0-1-2 additive coding with minor allele frequency maf. X can be generated independently of U (X_orth_U = TRUE) or dependent on U (X_orth_U = FALSE). For more details regarding the underlying model, see the vignette.

Usage

generate_data(setting = "GLM", n = 1000, maf = 0.2, cens = 0.3,
  a = NULL, b = NULL, aXK = 0.2, aXY = 0.1, aXL = 0, aKY = 0.3,
  aLK = 0, aLY = 0, aUY = 0, aUL = 0, mu_X = NULL, sd_X = NULL,
  X_orth_U = TRUE, mu_U = 0, sd_U = 1, mu_K = 0, sd_K = 1, mu_L = 0,
  sd_L = 1, mu_Y = 0, sd_Y = 1)

Arguments

Value

A dataframe containing n observations of the variables Y, K, X, L, U. Under the AFT setting, T=exp(Y) and the censoring indicator C (0 = censored, 1 = uncensored) are also computed.

Examples

# Generate data under the GLM setting with default values
dat_GLM <- generate_data()
head(dat_GLM)

# Generate data under the AFT setting with default values
dat_AFT <- generate_data(setting = "AFT", a = 0.2, b = 4.75)
head(dat_AFT)

CIEE parameter point estimates

Description

Function to perform CIEE to obtain point estimates under the GLM or AFT setting for the analysis of a normally-distributed or censored time-to-event primary outcome.

Usage

get_estimates(setting = "GLM", Y = NULL, X = NULL, K = NULL, L = NULL,
  C = NULL)

Arguments

Details

Under the GLM setting for the analysis of a normally-distributed primary outcome Y, estimates of the parameters \alpha_0, \alpha_1, \alpha_2, \alpha_3, \sigma_1^2, \alpha_4, \alpha_{XY}, \sigma_2^2 are obtained by constructing estimating equations for the models

Y = \alpha_0 + \alpha_1 \cdot K + \alpha_2 \cdot X + \alpha_3 \cdot L + \epsilon_1, \epsilon_1 \sim N(0,\sigma_1^2)

Y^* = Y - \overline{Y} - \alpha_1 \cdot (K-\overline{K})

Y^* = \alpha_0 + \alpha_{XY} \cdot X + \epsilon_2, \epsilon_2 \sim N(0,\sigma_2^2).

Under the AFT setting for the analysis of a censored time-to-event primary outcome, estimates of the parameters \alpha_0, \alpha_1, \alpha_2, \alpha_3, \sigma_1, \alpha_4, \alpha_{XY}, \sigma_2^2 are obtained by constructing similar estimating equations based on a censored regression model and adding an additional computation to estimate the true underlying survival times. In addition to the parameter estimates, the mean of the estimated true survival times is computed and returned in the output. For more details and the underlying model, see the vignette.

For both settings, the point estimates based on estimating equations equal least squares (and maximum likelihood) estimates, and are obtained using the lm and survreg functions for computational purposes.

Value

Returns a list with point estimates of the parameters. Under the AFT setting, the mean of the estimated true survival times is also computed and returned.

Examples

dat_GLM <- generate_data(setting = "GLM")
get_estimates(setting = "GLM", Y = dat_GLM$Y, X = dat_GLM$X, K = dat_GLM$K,
              L = dat_GLM$L)

dat_AFT <- generate_data(setting = "AFT", a = 0.2, b = 4.75)
get_estimates(setting = "AFT", Y = dat_AFT$Y, X = dat_AFT$X, K = dat_AFT$K,
              L = dat_AFT$L, C = dat_AFT$C)

Naive standard error estimates

Description

Function to obtain naive standard error estimates for the parameter estimates of the get_estimates function, under the GLM or AFT setting for the analysis of a normally-distributed or censored time-to-event primary outcome.

Usage

naive_se(setting = "GLM", Y = NULL, X = NULL, K = NULL, L = NULL,
  C = NULL)

Arguments

Details

Under the GLM setting for the analysis of a normally-distributed primary outcome Y, naive standard error estimates are obtained for the estimates of the parameters \alpha_0, \alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_{XY} in the models

Y = \alpha_0 + \alpha_1 \cdot K + \alpha_2 \cdot X + \alpha_3 \cdot L + \epsilon_1, \epsilon_1 \sim N(0,\sigma_1^2)

Y^* = Y - \overline{Y} - \alpha_1 \cdot (K-\overline{K})

Y^* = \alpha_0 + \alpha_{XY} \cdot X + \epsilon_2, \epsilon_2 \sim N(0,\sigma_2^2),

using the lm function, without accounting for the additional variability due to the 2-stage approach.

Under the AFT setting for the analysis of a censored time-to-event primary outcome, bootstrap standard error estimates are similarly obtained of the parameter estimates of \alpha_0, \alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_{XY} from the output of the survreg and lm functions.

Value

Returns a vector with the naive standard error estimates of the parameter estimates.

Examples

dat <- generate_data(setting = "GLM")
naive_se(setting = "GLM", Y = dat$Y, X = dat$X, K = dat$K, L = dat$L)

Sandwich standard error estimates

Description

Function to obtain consistent and robust sandwich standard error estimates based on estimating equations, for the parameter estimates of the get_estimates function, under the GLM or AFT setting for the analysis of a normally-distributed or censored time-to-event primary outcome.

Usage

sandwich_se(setting = "GLM", scores = NULL, hessian = NULL)

Arguments

Details

Under the GLM setting for the analysis of a normally-distributed primary outcome Y, robust sandwich standard error estimates are obtained for the estimates of the parameters \alpha_0, \alpha_1, \alpha_2, \alpha_3, \sigma_1^2, \alpha_4, \alpha_{XY}, \sigma_2^2 in the model

Y = \alpha_0 + \alpha_1 \cdot K + \alpha_2 \cdot X + \alpha_3 \cdot L + \epsilon_1, \epsilon_1 \sim N(0,\sigma_1^2)

Y^* = Y - \overline{Y} - \alpha_1 \cdot (K-\overline{K})

Y^* = \alpha_0 + \alpha_{XY} \cdot X + \epsilon_2, \epsilon_2 \sim N(0,\sigma_2^2)

by using the score and hessian matrices of the parameters.

Under the AFT setting for the analysis of a censored time-to-event primary outcome, robust sandwich standard error estimates are similarly obtained of the parameter estimates of \alpha_0, \alpha_1, \alpha_2, \alpha_3, \sigma_1, \alpha_4, \alpha_{XY}, \sigma_2^2. For more details and the underlying model, see the vignette.

Value

Returns a vector with the CIEE sandwich standard error estimates of the parameter estimates.

Examples

# Generate data including Y, K, L, X under the GLM setting
dat <- generate_data(setting = "GLM")

# Obtain estimating functions expressions
estfunct <- est_funct_expr(setting = "GLM")

# Obtain point estimates of the parameters
estimates <- get_estimates(setting = "GLM", Y = dat$Y, X = dat$X,
                           K = dat$K, L = dat$L)

# Obtain matrices with all first and second derivatives
derivobj <- deriv_obj(setting = "GLM", logL1 = estfunct$logL1,
                      logL2 = estfunct$logL2, Y = dat$Y, X = dat$X,
                      K = dat$K, L = dat$L, estimates = estimates)

# Obtain score and hessian matrices
results_scores <- scores(derivobj)
results_hessian <- hessian(derivobj)

# Obtain sandwich standard error estimates of the parameters
sandwich_se(scores = results_scores, hessian = results_hessian)

Score and hessian matrix based on the estimating functions.

Description

Functions to compute the score and hessian matrices of the parameters based on the estimating functions, under the GLM and AFT setting for the analysis of a normally-distributed or censored time-to-event primary outcome. The score and hessian matrices are further used in the functions sandwich_se, ciee and ciee_loop to obtain robust sandwich error estimates of the parameter estimates of \alpha_0, \alpha_1, \alpha_2, \alpha_3, \sigma_1^2, \alpha_4, \alpha_{XY}, \sigma_2^2 under the GLM setting and \alpha_0, \alpha_1, \alpha_2, \alpha_3, \sigma_1, \alpha_4, \alpha_{XY}, \sigma_2^2 under the AFT setting.

Usage

deriv_obj(setting = "GLM", logL1 = NULL, logL2 = NULL, Y = NULL,
  X = NULL, K = NULL, L = NULL, C = NULL, estimates = NULL)

scores(derivobj = NULL)

hessian(derivobj = NULL)

Arguments

Details

For the computation of the score and hessian matrices, first, the help function deriv_obj is used. In a first step, the expression of all first and second derivatives of the parameters is computed using the expressions of logL1 and logL2 from the est_funct_expr as input. Then, the numerical values of all first and second derivatives are obtained for the observed data Y, X, K, L (and C under the AFT setting) and point estimates (estimates) of the parameters, for all observed individuals.

Second, the functions scores and hessian are used to extract the relevant score and hessian matrices with respect to logL1 and logL2 from the output of deriv_obj and piece them together. For further details, see the vignette.

Value

The deriv_obj function returns a list with objects logL1_deriv, logL2_deriv which contain the score and hessian matrices based on logL1, logL2, respectively.

The scores function returns the (n \times 8) score matrix.

The hessian function returns the (n \times 8 \times 8) hessian matrix.

Examples

# Generate data including Y, K, L, X under the GLM setting
dat <- generate_data(setting = "GLM")

# Obtain estimating functions' expressions
estfunct <- est_funct_expr(setting = "GLM")

# Obtain point estimates of the parameters
estimates <- get_estimates(setting = "GLM", Y = dat$Y, X = dat$X,
                           K = dat$K, L = dat$L)

# Obtain matrices with all first and second derivatives
derivobj <- deriv_obj(setting = "GLM", logL1 = estfunct$logL1,
                      logL2 = estfunct$logL2, Y = dat$Y, X = dat$X,
                      K = dat$K, L = dat$L, estimates = estimates)
names(derivobj)
head(derivobj$logL1_deriv$gradient)

# Obtain score and hessian matrices
scores(derivobj)
hessian(derivobj)

Structural equation modeling approach

Description

Function which uses the sem function in the lavaan package to fit the model

L = \alpha_0 + \alpha_1 \cdot X + \epsilon_1, \epsilon_1 \sim N(0,\sigma_1^2)

K = \alpha_2 + \alpha_3 \cdot X + \alpha_4 \cdot L + \epsilon_2, \epsilon_2 \sim~ N(0,\sigma_2^2)

Y = \alpha_5 + \alpha_6 \cdot K + \alpha_{XY} \cdot X + \epsilon_3, \epsilon_3 \sim N(0,\sigma_3^2)

in order to obtain point and standard error estimates of the parameters \alpha_1, \alpha_3, \alpha_4, \alpha_6, \alpha_{XY} for the GLM setting. See the vignette for more details.

Usage

sem_appl(Y = NULL, X = NULL, K = NULL, L = NULL)

Arguments

Value

Returns a list with point estimates of the parameters (point_estimates), standard error estimates (SE_estimates) and p-values from large-sample Wald-type tests (pvalues).

Examples

dat <- generate_data(setting = "GLM")
sem_appl(Y = dat$Y, X = dat$X, K = dat$K, L = dat$L)

Summary function.

Description

Summary function for the ciee and ciee_loop functions.

Usage

## S3 method for class 'ciee'
summary(object = NULL, ...)

Arguments

Value

Formatted data frames of the results of all computed methods.

Examples

maf <- 0.2
n <- 1000
dat <- generate_data(n = n, maf = maf)
datX <- data.frame(X = dat$X)
names(datX)[1] <- "X1"
for (i in 2:10){
  X <- stats::rbinom(n, size = 2, prob = maf)
  datX$X <- X
  names(datX)[i] <- paste("X", i, sep="")
}

results1 <- ciee(Y = dat$Y, X = datX$X1, K = dat$K, L = dat$L)
summary(results1)

results2 <- ciee_loop(Y = dat$Y, X = datX, K = dat$K, L = dat$L)
summary(results2)

Traditional regression approaches.

Description

Functions to fit traditional regression approaches for a quantitative normally-distributed primary outcome (setting = "GLM") and a censoredtime-to-event primary outcome (setting = "AFT"). mult_reg fits the multiple regression approach and res_reg computes the regression of residuals approach.

Usage

mult_reg(setting = "GLM", Y = NULL, X = NULL, K = NULL, L = NULL,
  C = NULL)

res_reg(Y = NULL, X = NULL, K = NULL, L = NULL)

Arguments

Details

In more detail, for a quantitative normally-distributed primary outcome Y, mult_reg fits the model

Y = \alpha_0 + \alpha_1 \cdot K + \alpha_{XY} \cdot X + \alpha_2 \cdot L + \epsilon

and obtains point and standard error estimates for the parameters \alpha_0, \alpha_1, \alpha_{XY}, \alpha_2. res_reg obtains point and standard error estimates for the parameters \alpha_0, \alpha_1, \alpha_2, \alpha_3, \alpha_{XY} by fitting the models

Y = \alpha_0 + \alpha_1 \cdot K + \alpha_2 \cdot L + \epsilon_1

\widehat{\epsilon}_1 = \alpha_3 + \alpha_{XY} \cdot X + \epsilon_2

Both functions use the lm function and also report the provided p-values from t-tests that each parameter equals 0. For the analysis of a censored time-to-event primary outcome Y, only the multiple regression approach is implemented. Here, mult_reg fits the according censored regression model to obtain coefficient and standard error estimates as well as p-values from large-sample Wald-type tests by using the survreg function. See the vignette for more details.

Value

Returns a list with point estimates of the parameters point_estimates, standard error estimates SE_estimates and p-values pvalues.

Examples

dat_GLM <- generate_data(setting = "GLM")
mult_reg(setting = "GLM", Y = dat_GLM$Y, X = dat_GLM$X, K = dat_GLM$K,
         L = dat_GLM$L)
res_reg(Y = dat_GLM$Y, X = dat_GLM$X, K = dat_GLM$K, L = dat_GLM$L)

dat_AFT <- generate_data(setting = "AFT", a = 0.2, b = 4.75)
mult_reg(setting = "AFT", Y = dat_AFT$Y, X = dat_AFT$X, K = dat_AFT$K,
         L = dat_AFT$L, C = dat_AFT$C)