Package 'stdReg2'

stdReg2: Regression Standardization for Causal Inference

Description

Contains more modern tools for causal inference using regression standardization. Four general classes of models are implemented; generalized linear models, conditional generalized estimating equation models, Cox proportional hazards models, and shared frailty gamma-Weibull models. Methodological details are described in Sjölander, A. (2016) doi:10.1007/s10654-016-0157-3. Also includes functionality for doubly robust estimation for generalized linear models in some special cases, and the ability to implement custom models.

Author(s)

Maintainer: Michael C Sachs [email protected]

Authors:

• Arvid Sjölander

• Erin E Gabriel

• Johan Sebastian Ohlendorff

• Adam Brand

See Also

Useful links:

• https://sachsmc.github.io/stdReg2/

• Report bugs at https://github.com/sachsmc/stdReg2/issues/

---

Fits shared frailty gamma-Weibull models

Description

parfrailty fits shared frailty gamma-Weibull models. It is specifically designed to work with the function standardize_parfrailty, which performs regression standardization in shared frailty gamma-Weibull models.

Usage

parfrailty(formula, data, clusterid, init)

Arguments

formula	an object of class "formula", in the same format as accepted by the coxph function.
data	a data frame containing the variables in the model.
clusterid	a string containing the name of a cluster identification variable.
init	an optional vector of initial values for the model parameters.

Details

parfrailty fits the shared frailty gamma-Weibull model

$\lambda(t_{ij}|C_{ij})=\lambda(t_{ij};\alpha,\eta)U_i\exp\{h(C_{ij};\beta)\},$

where $t_{ij}$ and $C_{ij}$ are the survival time and covariate vector for subject $j$ in cluster $i$, respectively. $\lambda(t;\alpha,\eta)$ is the Weibull baseline hazard function

$\eta t^{\eta-1}\alpha^{-\eta},$

where $\eta$ is the shape parameter and $\alpha$ is the scale parameter. $U_i$ is the unobserved frailty term for cluster $i$, which is assumed to have a gamma distribution with scale = 1/shape = $\phi$. $h(X;\beta)$ is the regression function as specified by the formula argument, parameterized by a vector $\beta$. The ML estimates $\{\log(\hat{\alpha}),\log(\hat{\eta}),\log(\hat{\phi}),\hat{\beta}\}$ are obtained by maximizing the marginal (over $U$) likelihood.

Value

An object of class "parfrailty" which is a list containing:

est	the Maximum Likelihood (ML) estimates $\{\log(\hat{\alpha}),\log(\hat{\eta}), \log(\hat{\phi}),\hat{\beta}\}$.
vcov	the variance-covariance vector of the ML estimates.
score	a matrix containing the cluster-specific contributions to the ML score equations.

Note

If left truncation is present, it is assumed that it is strong left truncation. This means that even if the truncation time may be subject-specific, the whole cluster is unobserved if at least one subject in the cluster dies before his/her truncation time. If all subjects in the cluster survive beyond their subject-specific truncation times, then the whole cluster is observed (Van den Berg and Drepper, 2016).

Author(s)

Arvid Sjölander and Elisabeth Dahlqwist.

References

Dahlqwist E., Pawitan Y., Sjölander A. (2019). Regression standardization and attributable fraction estimation with between-within frailty models for clustered survival data. Statistical Methods in Medical Research 28(2), 462-485.

Van den Berg G.J., Drepper B. (2016). Inference for shared frailty survival models with left-truncated data. Econometric Reviews, 35(6), 1075-1098.

Examples

require(survival)

# simulate data
set.seed(5)
n <- 200
m <- 3
alpha <- 1.5
eta <- 1
phi <- 0.5
beta <- 1
id <- rep(1:n, each = m)
U <- rep(rgamma(n, shape = 1 / phi, scale = phi), each = m)
X <- rnorm(n * m)
# reparameterize scale as in rweibull function
weibull.scale <- alpha / (U * exp(beta * X))^(1 / eta)
T <- rweibull(n * m, shape = eta, scale = weibull.scale)

# right censoring
C <- runif(n * m, 0, 10)
D <- as.numeric(T < C)
T <- pmin(T, C)

# strong left-truncation
L <- runif(n * m, 0, 2)
incl <- T > L
incl <- ave(x = incl, id, FUN = sum) == m
dd <- data.frame(L, T, D, X, id)
dd <- dd[incl, ]

fit <- parfrailty(formula = Surv(L, T, D) ~ X, data = dd, clusterid = "id")
print(fit)

---

Plots regression standardization fit

Description

This is a plot method for class "std_glm".

Usage

## S3 method for class 'std_glm'
plot(
  x,
  plot_ci = TRUE,
  ci_type = "plain",
  ci_level = 0.95,
  transform = NULL,
  contrast = NULL,
  reference = NULL,
  summary_fun = "summary_std_glm",
  ...
)

Arguments

x	An object of class "std_glm".
plot_ci	if TRUE, add the confidence intervals to the plot.
ci_type	A string, indicating the type of confidence intervals. Either "plain", which gives untransformed intervals, or "log", which gives log-transformed intervals.
ci_level	Coverage probability of confidence intervals.
transform	If set to "log", "logit", or "odds", the standardized mean $\theta(x)$ is transformed into $\psi(x)=\log\{\theta(x)\}$, $\psi(x)=\log[\theta(x)/\{1-\theta(x)\}]$, or $\psi(x)=\theta(x)/\{1-\theta(x)\}$, respectively. If left unspecified, $\psi(x)=\theta(x)$.
contrast	If set to "difference" or "ratio", then $\psi(x)-\psi(x_0)$ or $\psi(x) / \psi(x_0)$ are constructed, where $x_0$ is a reference level specified by the reference argument. If not NULL, a doubly robust estimator of the standardized estimator is used.
reference	If contrast is specified, the desired reference level.
summary_fun	For internal use only. Do not change.
...	Unused.

Value

None. Creates a plot as a side effect

Examples

# see standardize_glm

---

Plots regression standardization fit

Description

This is a plot method for class "std_surv".

Usage

## S3 method for class 'std_surv'
plot(
  x,
  plot_ci = TRUE,
  ci_type = "plain",
  ci_level = 0.95,
  transform = NULL,
  contrast = NULL,
  reference = NULL,
  legendpos = "bottomleft",
  summary_fun = "summary_std_coxph",
  ...
)

Arguments

x	An object of class "std_surv".
plot_ci	if TRUE, add the confidence intervals to the plot.
ci_type	A string, indicating the type of confidence intervals. Either "plain", which gives untransformed intervals, or "log", which gives log-transformed intervals.
ci_level	Coverage probability of confidence intervals.
transform	If set to "log", "logit", or "odds", the standardized mean $\theta(x)$ is transformed into $\psi(x)=\log\{\theta(x)\}$, $\psi(x)=\log[\theta(x)/\{1-\theta(x)\}]$, or $\psi(x)=\theta(x)/\{1-\theta(x)\}$, respectively. If left unspecified, $\psi(x)=\theta(x)$.
contrast	If set to "difference" or "ratio", then $\psi(x)-\psi(x_0)$ or $\psi(x) / \psi(x_0)$ are constructed, where $x_0$ is a reference level specified by the reference argument. If not NULL, a doubly robust estimator of the standardized estimator is used.
reference	If contrast is specified, the desired reference level.
legendpos	position of the legend; see legend.
summary_fun	For internal use only. Do not change.
...	Unused.

Value

None. Creates a plot as a side effect

---

Prints summary of regression standardization fit

Description

Prints summary of regression standardization fit

Usage

## S3 method for class 'std_surv'
print(x, ...)

## S3 method for class 'std_glm'
print(x, ...)

## S3 method for class 'std_custom'
print(x, ...)

Arguments

x	an object of class "std_glm", "std_surv" or "std_custom".
...	unused

Value

The object being printed, invisibly.

---

Print method for parametric frailty fits

Description

Print method for parametric frailty fits

Usage

## S3 method for class 'summary.parfrailty'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

Arguments

x	An object of class "parfrailty"
digits	Number of digits to print
...	Not used

Value

The object being printed, invisibly

---

Compute the sandwich variance components from a model fit

Description

Compute the sandwich variance components from a model fit

Usage

sandwich(fit, data, weights, t, fit.detail)

Arguments

fit	A fitted model object of class glm, coxph, ah, or survfit
data	The data used to fit the model
weights	Optional weights
t	Optional fixed time point for survival objects
fit.detail	For Cox models, the result of running coxph.detail on the model fit

Value

A list consisting of the Fisher information matrix (I) and the Score equations (U)

---

Get standardized estimates using the g-formula with a custom model

Description

Get standardized estimates using the g-formula with a custom model

Usage

standardize(
  fitter,
  arguments,
  predict_fun,
  data,
  values,
  B = NULL,
  ci_level = 0.95,
  contrasts = NULL,
  reference = NULL,
  seed = NULL,
  times = NULL,
  transforms = NULL,
  progressbar = TRUE
)

Arguments

fitter	The function to call to fit the data.
arguments	The arguments to be used in the fitter function as a list.
predict_fun	The function used to predict the means/probabilities for a new data set on the response level. For survival data, this should be a matrix where each column is the time, and each row the data.
data	The data.
values	A named list or data.frame specifying the variables and values at which marginal means of the outcome will be estimated.
B	Number of nonparametric bootstrap resamples. Default is NULL (no bootstrap).
ci_level	Coverage probability of confidence intervals.
contrasts	A vector of contrasts in the following format: If set to "difference" or "ratio", then $\psi(x)-\psi(x_0)$ or $\psi(x) / \psi(x_0)$ are constructed, where $x_0$ is a reference level specified by the reference argument. Has to be NULL if no references are specified.
reference	A vector of reference levels in the following format: If contrasts is not NULL, the desired reference level(s). This must be a vector or list the same length as contrasts, and if not named, it is assumed that the order is as specified in contrasts.
seed	The seed to use with the nonparametric bootstrap.
times	For use with survival data. Set to NULL otherwise.
transforms	A vector of transforms in the following format: If set to "log", "logit", or "odds", the standardized mean $\theta(x)$ is transformed into $\psi(x)=\log\{\theta(x)\}$, $\psi(x)=\log[\theta(x)/\{1-\theta(x)\}]$, or $\psi(x)=\theta(x)/\{1-\theta(x)\}$, respectively. If the vector is NULL, then $\psi(x)=\theta(x)$.
progressbar	Logical, if TRUE will print bootstrapping progress to the console

Details

Let $Y$, $X$, and $Z$ be the outcome, the exposure, and a vector of covariates, respectively. standardize uses a model to estimate the standardized mean $\theta(x)=E\{E(Y|X=x,Z)\}$, where $x$ is a specific value of $X$, and the outer expectation is over the marginal distribution of $Z$. With survival data, $Y=I(T > t)$, and a vector of different time points times ($t$) can be given, where $T$ is the uncensored survival time.

Value

An object of class std_custom. This is a list with components estimates and fit for the outcome model.

References

Rothman K.J., Greenland S., Lash T.L. (2008). Modern Epidemiology, 3rd edition. Lippincott, Williams & Wilkins.

Sjölander A. (2016). Regression standardization with the R-package stdReg. European Journal of Epidemiology 31(6), 563-574.

Sjölander A. (2016). Estimation of causal effect measures with the R-package stdReg. European Journal of Epidemiology 33(9), 847-858.

Examples

set.seed(6)
n <- 100
Z <- rnorm(n)
X <- rnorm(n, mean = Z)
Y <- rbinom(n, 1, prob = (1 + exp(X + Z))^(-1))
dd <- data.frame(Z, X, Y)
prob_predict.glm <- function(...) predict.glm(..., type = "response")

x <- standardize(
  fitter = "glm",
  arguments = list(
    formula = Y ~ X * Z,
    family = "binomial"
  ),
  predict_fun = prob_predict.glm,
  data = dd,
  values = list(X = seq(-1, 1, 0.1)),
  B = 100,
  reference = 0,
  contrasts = "difference"
)
x

require(survival)
prob_predict.coxph <- function(object, newdata, times) {
  fit.detail <- suppressWarnings(basehaz(object))
  cum.haz <- fit.detail$hazard[sapply(times, function(x) max(which(fit.detail$time <= x)))]
  predX <- predict(object = object, newdata = newdata, type = "risk")
  res <- matrix(NA, ncol = length(times), nrow = length(predX))
  for (ti in seq_len(length(times))) {
    res[, ti] <- exp(-predX * cum.haz[ti])
  }
  res
}
set.seed(68)
n <- 500
Z <- rnorm(n)
X <- rnorm(n, mean = Z)
T <- rexp(n, rate = exp(X + Z + X * Z)) # survival time
C <- rexp(n, rate = exp(X + Z + X * Z)) # censoring time
U <- pmin(T, C) # time at risk
D <- as.numeric(T < C) # event indicator
dd <- data.frame(Z, X, U, D)
x <- standardize(
fitter = "coxph",
  arguments = list(
    formula = Surv(U, D) ~ X + Z + X * Z,
    method = "breslow",
    x = TRUE,
    y = TRUE
  ),
  predict_fun = prob_predict.coxph,
  data = dd,
  times = 1:5,
  values = list(X = c(-1, 0, 1)),
  B = 100,
  reference = 0,
  contrasts = "difference"
)
x

---

Regression standardization in Cox proportional hazards models

Description

standardize_coxph performs regression standardization in Cox proportional hazards models at specified values of the exposure over the sample covariate distribution. Let $T$, $X$, and $Z$ be the survival outcome, the exposure, and a vector of covariates, respectively. standardize_coxph fits a Cox proportional hazards model and the Breslow estimator of the baseline hazard in order to estimate the standardized survival function $\theta(t,x)=E\{S(t|X=x,Z)\}$ when measure = "survival" or the standardized restricted mean survival up to time t $\theta(t, x) = E\{\int_0^t S(u|X = x, Z) du\}$ when measure = "rmean", where $t$ is a specific value of $T$, $x$ is a specific value of $X$, and the expectation is over the marginal distribution of $Z$.

Usage

standardize_coxph(
  formula,
  data,
  values,
  times,
  measure = c("survival", "rmean"),
  clusterid,
  ci_level = 0.95,
  ci_type = "plain",
  contrasts = NULL,
  family = "gaussian",
  reference = NULL,
  transforms = NULL
)

Arguments

formula	The formula which is used to fit the model for the outcome.
data	The data.
values	A named list or data.frame specifying the variables and values at which marginal means of the outcome will be estimated.
times	A vector containing the specific values of $T$ at which to estimate the standardized survival function.
measure	Either "survival" to estimate the survival function at times or "rmean" for the restricted mean survival up to the largest of times.
clusterid	An optional string containing the name of a cluster identification variable when data are clustered.
ci_level	Coverage probability of confidence intervals.
ci_type	A string, indicating the type of confidence intervals. Either "plain", which gives untransformed intervals, or "log", which gives log-transformed intervals.
contrasts	A vector of contrasts in the following format: If set to "difference" or "ratio", then $\psi(x)-\psi(x_0)$ or $\psi(x) / \psi(x_0)$ are constructed, where $x_0$ is a reference level specified by the reference argument. Has to be NULL if no references are specified.
family	The family argument which is used to fit the glm model for the outcome.
reference	A vector of reference levels in the following format: If contrasts is not NULL, the desired reference level(s). This must be a vector or list the same length as contrasts, and if not named, it is assumed that the order is as specified in contrasts.
transforms	A vector of transforms in the following format: If set to "log", "logit", or "odds", the standardized mean $\theta(x)$ is transformed into $\psi(x)=\log\{\theta(x)\}$, $\psi(x)=\log[\theta(x)/\{1-\theta(x)\}]$, or $\psi(x)=\theta(x)/\{1-\theta(x)\}$, respectively. If the vector is NULL, then $\psi(x)=\theta(x)$.

Details

standardize_coxph fits the Cox proportional hazards model

$\lambda(t|X,Z)=\lambda_0(t)\exp\{h(X,Z;\beta)\}.$

Breslow's estimator of the cumulative baseline hazard $\Lambda_0(t)=\int_0^t\lambda_0(u)du$ is used together with the partial likelihood estimate of $\beta$ to obtain estimates of the survival function $S(t|X=x,Z)$ if measure = "survival":

$\hat{S}(t|X=x,Z)=\exp[-\hat{\Lambda}_0(t)\exp\{h(X=x,Z;\hat{\beta})\}].$

For each $t$ in the t argument and for each $x$ in the x argument, these estimates are averaged across all subjects (i.e. all observed values of $Z$) to produce estimates

$\hat{\theta}(t,x)=\sum_{i=1}^n \hat{S}(t|X=x,Z_i)/n,$

where $Z_i$ is the value of $Z$ for subject $i$, $i=1,...,n$. The variance for $\hat{\theta}(t,x)$ is obtained by the sandwich formula.

If measure = "rmean", then $\Lambda_0(t)=\int_0^t\lambda_0(u)du$ is used together with the partial likelihood estimate of $\beta$ to obtain estimates of the restricted mean survival up to time t: $\int_0^t S(u|X=x,Z) du$ for each element of times. The estimation and inference is done using the method described in Chen and Tsiatis 2001. Currently, we can only estimate the difference in RMST for a single binary exposure. Two separate Cox models are fit for each level of the exposure, which is expected to be coded as 0/1.

Value

An object of class std_surv. This is basically a list with components estimates and covariance estimates in res Results for transformations, contrasts, references are stored in res_contrasts. The output contains estimates for contrasts and confidence intervals for all combinations of transforms and reference levels. Obtain numeric results in a data frame with the tidy function.

Note

Standardized survival functions are sometimes referred to as (direct) adjusted survival functions in the literature.

standardize_coxph/standardize_parfrailty does not currently handle time-varying exposures or covariates.

standardize_coxph/standardize_parfrailty internally loops over all values in the t argument. Therefore, the function will usually be considerably faster if length(t) is small.

The variance calculation performed by standardize_coxph does not condition on the observed covariates $\bar{Z}=(Z_1,...,Z_n)$. To see how this matters, note that

$var\{\hat{\theta}(t,x)\}=E[var\{\hat{\theta}(t,x)|\bar{Z}\}]+var[E\{\hat{\theta}(t,x)|\bar{Z}\}].$

The usual parameter $\beta$ in a Cox proportional hazards model does not depend on $\bar{Z}$. Thus, $E(\hat{\beta}|\bar{Z})$ is independent of $\bar{Z}$ as well (since $E(\hat{\beta}|\bar{Z})=\beta$), so that the term $var[E\{\hat{\beta}|\bar{Z}\}]$ in the corresponding variance decomposition for $var(\hat{\beta})$ becomes equal to 0. However, $\theta(t,x)$ depends on $\bar{Z}$ through the average over the sample distribution for $Z$, and thus the term $var[E\{\hat{\theta}(t,x)|\bar{Z}\}]$ is not 0, unless one conditions on $\bar{Z}$. The variance calculation by Gail and Byar (1986) ignores this term, and thus effectively conditions on $\bar{Z}$.

Author(s)

Arvid Sjölander, Adam Brand, Michael Sachs

References

Chang I.M., Gelman G., Pagano M. (1982). Corrected group prognostic curves and summary statistics. Journal of Chronic Diseases 35, 669-674.

Gail M.H. and Byar D.P. (1986). Variance calculations for direct adjusted survival curves, with applications to testing for no treatment effect. Biometrical Journal 28(5), 587-599.

Makuch R.W. (1982). Adjusted survival curve estimation using covariates. Journal of Chronic Diseases 35, 437-443.

Sjölander A. (2016). Regression standardization with the R-package stdReg. European Journal of Epidemiology 31(6), 563-574.

Sjölander A. (2018). Estimation of causal effect measures with the R-package stdReg. European Journal of Epidemiology 33(9), 847-858.

Chen, P. Y., Tsiatis, A. A. (2001). Causal inference on the difference of the restricted mean lifetime between two groups. Biometrics, 57(4), 1030-1038.

Examples

require(survival)
set.seed(7)
n <- 300
Z <- rnorm(n)
Zbin <- rbinom(n, 1, .3)
X <- rnorm(n, mean = Z)
T <- rexp(n, rate = exp(X + Z + X * Z)) # survival time
C <- rexp(n, rate = exp(X + Z + X * Z)) # censoring time
fact <- factor(sample(letters[1:3], n, replace = TRUE))
U <- pmin(T, C) # time at risk
D <- as.numeric(T < C) # event indicator
dd <- data.frame(Z, Zbin, X, U, D, fact)
fit.std.surv <- standardize_coxph(
  formula = Surv(U, D) ~ X + Z + X * Z,
  data = dd,
  values = list(X = seq(-1, 1, 0.5)),
  times = 1:5
)
print(fit.std.surv)
plot(fit.std.surv)
tidy(fit.std.surv)

fit.std <- standardize_coxph(
  formula = Surv(U, D) ~ X + Zbin + X * Zbin + fact,
  data = dd,
  values = list(Zbin = 0:1),
  times = 1.5,
  measure = "rmean",
  contrast = "difference",
  reference = 0
)
print(fit.std)
tidy(fit.std)

---

Regression standardization in conditional generalized estimating equations

Description

standardize_gee performs regression standardization in linear and log-linear fixed effects models, at specified values of the exposure, over the sample covariate distribution. Let $Y$, $X$, and $Z$ be the outcome, the exposure, and a vector of covariates, respectively. It is assumed that data are clustered with a cluster indicator $i$. standardize_gee uses fitted fixed effects model, with cluster-specific intercept $a_i$ (see details), to estimate the standardized mean $\theta(x)=E\{E(Y|i,X=x,Z)\}$, where $x$ is a specific value of $X$, and the outer expectation is over the marginal distribution of $(a_i,Z)$.

Usage

standardize_gee(
  formula,
  link = "identity",
  data,
  values,
  clusterid,
  case_control = FALSE,
  ci_level = 0.95,
  ci_type = "plain",
  contrasts = NULL,
  family = "gaussian",
  reference = NULL,
  transforms = NULL
)

Arguments

formula	A formula to be used with "gee" in the drgee package.
link	The link function to be used with "gee".
data	The data.
values	A named list or data.frame specifying the variables and values at which marginal means of the outcome will be estimated.
clusterid	An optional string containing the name of a cluster identification variable when data are clustered.
case_control	Whether the data comes from a case-control study.
ci_level	Coverage probability of confidence intervals.
ci_type	A string, indicating the type of confidence intervals. Either "plain", which gives untransformed intervals, or "log", which gives log-transformed intervals.
contrasts	A vector of contrasts in the following format: If set to "difference" or "ratio", then $\psi(x)-\psi(x_0)$ or $\psi(x) / \psi(x_0)$ are constructed, where $x_0$ is a reference level specified by the reference argument. Has to be NULL if no references are specified.
family	The family argument which is used to fit the glm model for the outcome.
reference	A vector of reference levels in the following format: If contrasts is not NULL, the desired reference level(s). This must be a vector or list the same length as contrasts, and if not named, it is assumed that the order is as specified in contrasts.
transforms	A vector of transforms in the following format: If set to "log", "logit", or "odds", the standardized mean $\theta(x)$ is transformed into $\psi(x)=\log\{\theta(x)\}$, $\psi(x)=\log[\theta(x)/\{1-\theta(x)\}]$, or $\psi(x)=\theta(x)/\{1-\theta(x)\}$, respectively. If the vector is NULL, then $\psi(x)=\theta(x)$.

Details

standardize_gee assumes that a fixed effects model

$\eta\{E(Y|i,X,Z)\}=a_i+h(X,Z;\beta)$

has been fitted. The link function $\eta$ is assumed to be the identity link or the log link. The conditional generalized estimating equation (CGEE) estimate of $\beta$ is used to obtain estimates of the cluster-specific means:

$\hat{a}_i=\sum_{j=1}^{n_i}r_{ij}/n_i,$

where

$r_{ij}=Y_{ij}-h(X_{ij},Z_{ij};\hat{\beta})$

if $\eta$ is the identity link, and

$r_{ij}=Y_{ij}\exp\{-h(X_{ij},Z_{ij};\hat{\beta})\}$

if $\eta$ is the log link, and $(X_{ij},Z_{ij})$ is the value of $(X,Z)$ for subject $j$ in cluster $i$, $j=1,...,n_i$, $i=1,...,n$. The CGEE estimate of $\beta$ and the estimate of $a_i$ are used to estimate the mean $E(Y|i,X=x,Z)$:

$\hat{E}(Y|i,X=x,Z)=\eta^{-1}\{\hat{a}_i+h(X=x,Z;\hat{\beta})\}.$

For each $x$ in the x argument, these estimates are averaged across all subjects (i.e. all observed values of $Z$ and all estimated values of $a_i$) to produce estimates

$\hat{\theta}(x)=\sum_{i=1}^n \sum_{j=1}^{n_i} \hat{E}(Y|i,X=x,Z_i)/N,$

where $N=\sum_{i=1}^n n_i$. The variance for $\hat{\theta}(x)$ is obtained by the sandwich formula.

Value

An object of class std_glm. This is basically a list with components estimates and covariance estimates in res. Results for transformations, contrasts, references are stored in res_contrasts. Obtain numeric results in a data frame with the tidy function.

Note

The variance calculation performed by standardize_gee does not condition on the observed covariates $\bar{Z}=(Z_{11},...,Z_{nn_i})$. To see how this matters, note that

$var\{\hat{\theta}(x)\}=E[var\{\hat{\theta}(x)|\bar{Z}\}]+var[E\{\hat{\theta}(x)|\bar{Z}\}].$

The usual parameter $\beta$ in a generalized linear model does not depend on $\bar{Z}$. Thus, $E(\hat{\beta}|\bar{Z})$ is independent of $\bar{Z}$ as well (since $E(\hat{\beta}|\bar{Z})=\beta$), so that the term $var[E\{\hat{\beta}|\bar{Z}\}]$ in the corresponding variance decomposition for $var(\hat{\beta})$ becomes equal to 0. However, $\theta(x)$ depends on $\bar{Z}$ through the average over the sample distribution for $Z$, and thus the term $var[E\{\hat{\theta}(x)|\bar{Z}\}]$ is not 0, unless one conditions on $\bar{Z}$.

Author(s)

Arvid Sjölander.

References

Goetgeluk S. and Vansteelandt S. (2008). Conditional generalized estimating equations for the analysis of clustered and longitudinal data. Biometrics 64(3), 772-780.

Martin R.S. (2017). Estimation of average marginal effects in multiplicative unobserved effects panel models. Economics Letters 160, 16-19.

Sjölander A. (2019). Estimation of marginal causal effects in the presence of confounding by cluster. Biostatistics doi: 10.1093/biostatistics/kxz054

Examples

require(drgee)

set.seed(4)
n <- 300
ni <- 2
id <- rep(1:n, each = ni)
ai <- rep(rnorm(n), each = ni)
Z <- rnorm(n * ni)
X <- rnorm(n * ni, mean = ai + Z)
Y <- rnorm(n * ni, mean = ai + X + Z + 0.1 * X^2)
dd <- data.frame(id, Z, X, Y)
fit.std <- standardize_gee(
  formula = Y ~ X + Z + I(X^2),
  link = "identity",
  data = dd,
  values = list(X = seq(-3, 3, 0.5)),
  clusterid = "id"
)
print(fit.std)
plot(fit.std)

---

Get regression standardized estimates from a glm

Description

Get regression standardized estimates from a glm

Usage

standardize_glm(
  formula,
  data,
  values,
  clusterid,
  matched_density_cases,
  matched_density_controls,
  matching_variable,
  p_population,
  case_control = FALSE,
  ci_level = 0.95,
  ci_type = "plain",
  contrasts = NULL,
  family = "gaussian",
  reference = NULL,
  transforms = NULL
)

Arguments

formula	The formula which is used to fit the model for the outcome.
data	The data.
values	A named list or data.frame specifying the variables and values at which marginal means of the outcome will be estimated.
clusterid	An optional string containing the name of a cluster identification variable when data are clustered.
matched_density_cases	A function of the matching variable. The probability (or density) of the matched variable among the cases.
matched_density_controls	A function of the matching variable. The probability (or density) of the matched variable among the controls.
matching_variable	The matching variable extracted from the data set.
p_population	Specifies the incidence in the population when case_control=TRUE.
case_control	Whether the data comes from a case-control study.
ci_level	Coverage probability of confidence intervals.
ci_type	A string, indicating the type of confidence intervals. Either "plain", which gives untransformed intervals, or "log", which gives log-transformed intervals.
contrasts	A vector of contrasts in the following format: If set to "difference" or "ratio", then $\psi(x)-\psi(x_0)$ or $\psi(x) / \psi(x_0)$ are constructed, where $x_0$ is a reference level specified by the reference argument. Has to be NULL if no references are specified.
family	The family argument which is used to fit the glm model for the outcome.
reference	A vector of reference levels in the following format: If contrasts is not NULL, the desired reference level(s). This must be a vector or list the same length as contrasts, and if not named, it is assumed that the order is as specified in contrasts.
transforms	A vector of transforms in the following format: If set to "log", "logit", or "odds", the standardized mean $\theta(x)$ is transformed into $\psi(x)=\log\{\theta(x)\}$, $\psi(x)=\log[\theta(x)/\{1-\theta(x)\}]$, or $\psi(x)=\theta(x)/\{1-\theta(x)\}$, respectively. If the vector is NULL, then $\psi(x)=\theta(x)$.

Details

standardize_glm performs regression standardization in generalized linear models, at specified values of the exposure, over the sample covariate distribution. Let $Y$, $X$, and $Z$ be the outcome, the exposure, and a vector of covariates, respectively. standardize_glm uses a fitted generalized linear model to estimate the standardized mean $\theta(x)=E\{E(Y|X=x,Z)\}$, where $x$ is a specific value of $X$, and the outer expectation is over the marginal distribution of $Z$.

Value

An object of class std_glm. This is basically a list with components estimates and covariance estimates in res. Results for transformations, contrasts, references are stored in res_contrasts. Obtain numeric results in a data frame with the tidy function.

References

Rothman K.J., Greenland S., Lash T.L. (2008). Modern Epidemiology, 3rd edition. Lippincott, Williams & Wilkins.

Sjölander A. (2016). Regression standardization with the R-package stdReg. European Journal of Epidemiology 31(6), 563-574.

Sjölander A. (2016). Estimation of causal effect measures with the R-package stdReg. European Journal of Epidemiology 33(9), 847-858.

Examples

# basic example
# needs to correctly specify the outcome model and no unmeasered confounders
# (+ standard causal assunmptions)
set.seed(6)
n <- 100
Z <- rnorm(n)
X <- rnorm(n, mean = Z)
Y <- rbinom(n, 1, prob = (1 + exp(X + Z))^(-1))
dd <- data.frame(Z, X, Y)
x <- standardize_glm(
  formula = Y ~ X * Z,
  family = "binomial",
  data = dd,
  values = list(X = 0:1),
  contrasts = c("difference", "ratio"),
  reference = 0
)
x
# different transformations of causal effects

# example from Sjölander (2016) with case-control data
# here the matching variable needs to be passed as an argument
singapore <- AF::singapore
Mi <- singapore$Age
m <- mean(Mi)
s <- sd(Mi)
d <- 5
standardize_glm(
  formula = Oesophagealcancer ~ (Everhotbev + Age + Dial + Samsu + Cigs)^2,
  family = binomial, data = singapore,
  values = list(Everhotbev = 0:1), clusterid = "Set",
  case_control = TRUE,
  matched_density_cases = function(x) dnorm(x, m, s),
  matched_density_controls = function(x) dnorm(x, m - d, s),
  matching_variable = Mi,
  p_population = 19.3 / 100000
)

# multiple exposures
set.seed(7)
n <- 100
Z <- rnorm(n)
X1 <- rnorm(n, mean = Z)
X2 <- rnorm(n)
Y <- rbinom(n, 1, prob = (1 + exp(X1 + X2 + Z))^(-1))
dd <- data.frame(Z, X1, X2, Y)
x <- standardize_glm(
  formula = Y ~ X1 + X2 + Z,
  family = "binomial",
  data = dd, values = list(X1 = 0:1, X2 = 0:1),
  contrasts = c("difference", "ratio"),
  reference = c(X1 = 0, X2 = 0)
)
x
tidy(x)

# continuous exposure
set.seed(2)
n <- 100
Z <- rnorm(n)
X <- rnorm(n, mean = Z)
Y <- rnorm(n, mean = X + Z + 0.1 * X^2)
dd <- data.frame(Z, X, Y)
x <- standardize_glm(
  formula = Y ~ X * Z,
  family = "gaussian",
  data = dd,
  values = list(X = seq(-1, 1, 0.1))
)

# plot standardized mean as a function of x
plot(x)
# plot standardized mean - standardized mean at x = 0 as a function of x
plot(x, contrast = "difference", reference = 0)

---

Get regression standardized doubly-robust estimates from a glm

Description

Get regression standardized doubly-robust estimates from a glm

Usage

standardize_glm_dr(
  formula_outcome,
  formula_exposure,
  data,
  values,
  ci_level = 0.95,
  ci_type = "plain",
  contrasts = NULL,
  family_outcome = "gaussian",
  family_exposure = "binomial",
  reference = NULL,
  transforms = NULL
)

Arguments

formula_outcome	The formula which is used to fit the glm model for the outcome.
formula_exposure	The formula which is used to fit the glm model for the exposure. If not NULL, a doubly robust estimator of the standardized estimator is used.
data	The data.
values	A named list or data.frame specifying the variables and values at which marginal means of the outcome will be estimated.
ci_level	Coverage probability of confidence intervals.
ci_type	A string, indicating the type of confidence intervals. Either "plain", which gives untransformed intervals, or "log", which gives log-transformed intervals.
contrasts	A vector of contrasts in the following format: If set to "difference" or "ratio", then $\psi(x)-\psi(x_0)$ or $\psi(x) / \psi(x_0)$ are constructed, where $x_0$ is a reference level specified by the reference argument. Has to be NULL if no references are specified.
family_outcome	The family argument which is used to fit the glm model for the outcome.
family_exposure	The family argument which is used to fit the glm model for the exposure.
reference	A vector of reference levels in the following format: If contrasts is not NULL, the desired reference level(s). This must be a vector or list the same length as contrasts, and if not named, it is assumed that the order is as specified in contrasts.
transforms	A vector of transforms in the following format: If set to "log", "logit", or "odds", the standardized mean $\theta(x)$ is transformed into $\psi(x)=\log\{\theta(x)\}$, $\psi(x)=\log[\theta(x)/\{1-\theta(x)\}]$, or $\psi(x)=\theta(x)/\{1-\theta(x)\}$, respectively. If the vector is NULL, then $\psi(x)=\theta(x)$.

Details

standardize_glm_dr performs regression standardization in generalized linear models, see e.g., documentation for standardize_glm_dr. Specifically, this version uses a doubly robust estimator for standardization, meaning inference is valid when either the outcome regression or the exposure model is correctly specified and there is no unmeasured confounding.

Value

An object of class std_glm. This is basically a list with components estimates and covariance estimates in res. Results for transformations, contrasts, references are stored in res_contrasts. Obtain numeric results in a data frame with the tidy function.

References

Gabriel E.E., Sachs, M.C., Martinussen T., Waernbaum I., Goetghebeur E., Vansteelandt S., Sjölander A. (2024), Inverse probability of treatment weighting with generalized linear outcome models for doubly robust estimation. Statistics in Medicine, 43(3):534–547.

Examples

# doubly robust estimator
# needs to correctly specify either the outcome model or the exposure model
# for confounding
# NOTE: only works with binary exposures
data <- AF::clslowbwt
x <- standardize_glm_dr(
  formula_outcome = bwt ~ smoker * (race + age + lwt) + I(age^2) + I(lwt^2),
  formula_exposure = smoker ~ race * age * lwt + I(age^2) + I(lwt^2),
  family_outcome = "gaussian",
  family_exposure = "binomial",
  data = data,
  values = list(smoker = c(0, 1)), contrasts = "difference", reference = 0
)

set.seed(6)
n <- 100
Z <- rnorm(n)
X <- rbinom(n, 1, prob = (1 + exp(Z))^(-1))
Y <- rbinom(n, 1, prob = (1 + exp(X + Z))^(-1))
dd <- data.frame(Z, X, Y)
x <- standardize_glm_dr(
  formula_outcome = Y ~ X * Z, formula_exposure = X ~ Z,
  family_outcome = "binomial",
  data = dd,
  values = list(X = 0:1), reference = 0,
  contrasts = c("difference"), transforms = c("odds")
)

---

Get standardized estimates using the g-formula with and separate models for each exposure level in the data

Description

Get standardized estimates using the g-formula with and separate models for each exposure level in the data

Usage

standardize_level(
  fitter_list,
  arguments,
  predict_fun_list,
  data,
  values,
  B = NULL,
  ci_level = 0.95,
  contrasts = NULL,
  reference = NULL,
  seed = NULL,
  times = NULL,
  transforms = NULL,
  progressbar = TRUE
)

Arguments

fitter_list	The function to call to fit the data (as a list).
arguments	The arguments to be used in the fitter function as a list.
predict_fun_list	The function used to predict the means/probabilities for a new data set on the response level. For survival data, this should be a matrix where each column is the time, and each row the data (as a list).
data	The data.
values	A named list or data.frame specifying the variables and values at which marginal means of the outcome will be estimated.
B	Number of nonparametric bootstrap resamples. Default is NULL (no bootstrap).
ci_level	Coverage probability of confidence intervals.
contrasts	A vector of contrasts in the following format: If set to "difference" or "ratio", then $\psi(x)-\psi(x_0)$ or $\psi(x) / \psi(x_0)$ are constructed, where $x_0$ is a reference level specified by the reference argument. Has to be NULL if no references are specified.
reference	A vector of reference levels in the following format: If contrasts is not NULL, the desired reference level(s). This must be a vector or list the same length as contrasts, and if not named, it is assumed that the order is as specified in contrasts.
seed	The seed to use with the nonparametric bootstrap.
times	For use with survival data. Set to NULL otherwise.
transforms	A vector of transforms in the following format: If set to "log", "logit", or "odds", the standardized mean $\theta(x)$ is transformed into $\psi(x)=\log\{\theta(x)\}$, $\psi(x)=\log[\theta(x)/\{1-\theta(x)\}]$, or $\psi(x)=\theta(x)/\{1-\theta(x)\}$, respectively. If the vector is NULL, then $\psi(x)=\theta(x)$.
progressbar	Logical, if TRUE will print bootstrapping progress to the console

Details

See standardize. The difference is here that different models can be fitted for each value of x in values.

Value

An object of class std_custom. This is a list with components estimates and fit for the outcome model.

References

Rothman K.J., Greenland S., Lash T.L. (2008). Modern Epidemiology, 3rd edition. Lippincott, Williams & Wilkins.

Sjölander A. (2016). Regression standardization with the R-package stdReg. European Journal of Epidemiology 31(6), 563-574.

Sjölander A. (2016). Estimation of causal effect measures with the R-package stdReg. European Journal of Epidemiology 33(9), 847-858.

Examples

require(survival)
prob_predict.coxph <- function(object, newdata, times) {
  fit.detail <- suppressWarnings(basehaz(object))
  cum.haz <- fit.detail$hazard[sapply(times, function(x) max(which(fit.detail$time <= x)))]
  predX <- predict(object = object, newdata = newdata, type = "risk")
  res <- matrix(NA, ncol = length(times), nrow = length(predX))
  for (ti in seq_len(length(times))) {
    res[, ti] <- exp(-predX * cum.haz[ti])
  }
  res
}

set.seed(68)
n <- 500
Z <- rnorm(n)
X <- rbinom(n, 1, prob = 0.5)
T <- rexp(n, rate = exp(X + Z + X * Z)) # survival time
C <- rexp(n, rate = exp(X + Z + X * Z)) # censoring time
U <- pmin(T, C) # time at risk
D <- as.numeric(T < C) # event indicator
dd <- data.frame(Z, X, U, D)
x <- standardize_level(
  fitter_list = list("coxph", "coxph"),
  arguments = list(
    list(
      formula = Surv(U, D) ~ X + Z + X * Z,
      method = "breslow",
      x = TRUE,
      y = TRUE
    ),
    list(
      formula = Surv(U, D) ~ X,
      method = "breslow",
      x = TRUE,
      y = TRUE
    )
  ),
  predict_fun_list = list(prob_predict.coxph, prob_predict.coxph),
  data = dd,
  times = seq(1, 5, 0.1),
  values = list(X = c(0, 1)),
  B = 100,
  reference = 0,
  contrasts = "difference"
)
print(x)

---

Regression standardization in shared frailty gamma-Weibull models

Description

standardize_parfrailty performs regression standardization in shared frailty gamma-Weibull models, at specified values of the exposure, over the sample covariate distribution. Let $T$, $X$, and $Z$ be the survival outcome, the exposure, and a vector of covariates, respectively. standardize_parfrailty fits a parametric frailty model to estimate the standardized survival function $\theta(t,x)=E\{S(t|X=x,Z)\}$, where $t$ is a specific value of $T$, $x$ is a specific value of $X$, and the expectation is over the marginal distribution of $Z$.

Usage

standardize_parfrailty(
  formula,
  data,
  values,
  times,
  clusterid,
  ci_level = 0.95,
  ci_type = "plain",
  contrasts = NULL,
  family = "gaussian",
  reference = NULL,
  transforms = NULL
)

Arguments

formula	The formula which is used to fit the model for the outcome.
data	The data.
values	A named list or data.frame specifying the variables and values at which marginal means of the outcome will be estimated.
times	A vector containing the specific values of $T$ at which to estimate the standardized survival function.
clusterid	An optional string containing the name of a cluster identification variable when data are clustered.
ci_level	Coverage probability of confidence intervals.
ci_type	A string, indicating the type of confidence intervals. Either "plain", which gives untransformed intervals, or "log", which gives log-transformed intervals.
contrasts	A vector of contrasts in the following format: If set to "difference" or "ratio", then $\psi(x)-\psi(x_0)$ or $\psi(x) / \psi(x_0)$ are constructed, where $x_0$ is a reference level specified by the reference argument. Has to be NULL if no references are specified.
family	The family argument which is used to fit the glm model for the outcome.
reference	A vector of reference levels in the following format: If contrasts is not NULL, the desired reference level(s). This must be a vector or list the same length as contrasts, and if not named, it is assumed that the order is as specified in contrasts.
transforms	A vector of transforms in the following format: If set to "log", "logit", or "odds", the standardized mean $\theta(x)$ is transformed into $\psi(x)=\log\{\theta(x)\}$, $\psi(x)=\log[\theta(x)/\{1-\theta(x)\}]$, or $\psi(x)=\theta(x)/\{1-\theta(x)\}$, respectively. If the vector is NULL, then $\psi(x)=\theta(x)$.

Details

standardize_parfrailty fits a shared frailty gamma-Weibull model

$\lambda(t_{ij}|X_{ij},Z_{ij})=\lambda(t_{ij};\alpha,\eta)U_iexp\{h(X_{ij},Z_{ij};\beta)\}$

, with parameterization as described in the help section for parfrailty. Integrating out the gamma frailty gives the survival function

$S(t|X,Z)=[1+\phi\Lambda_0(t;\alpha,\eta)\exp\{h(X,Z;\beta)\}]^{-1/\phi},$

where $\Lambda_0(t;\alpha,\eta)$ is the cumulative baseline hazard

$(t/\alpha)^{\eta}.$

The ML estimates of $(\alpha,\eta,\phi,\beta)$ are used to obtain estimates of the survival function $S(t|X=x,Z)$:

$\hat{S}(t|X=x,Z)=[1+\hat{\phi}\Lambda_0(t;\hat{\alpha},\hat{\eta})\exp\{h(X,Z;\hat{\beta})\}]^{-1/\hat{\phi}}.$

For each $t$ in the t argument and for each $x$ in the x argument, these estimates are averaged across all subjects (i.e. all observed values of $Z$) to produce estimates

$\hat{\theta}(t,x)=\sum_{i=1}^n \hat{S}(t|X=x,Z_i)/n.$

The variance for $\hat{\theta}(t,x)$ is obtained by the sandwich formula.

Value

An object of class std_surv. This is basically a list with components estimates and covariance estimates in res Results for transformations, contrasts, references are stored in res_contrasts. The output contains estimates for contrasts and confidence intervals for all combinations of transforms and reference levels. Obtain numeric results in a data frame with the tidy function.

Note

Standardized survival functions are sometimes referred to as (direct) adjusted survival functions in the literature.

standardize_coxph/standardize_parfrailty does not currently handle time-varying exposures or covariates.

standardize_coxph/standardize_parfrailty internally loops over all values in the t argument. Therefore, the function will usually be considerably faster if length(t) is small.

The variance calculation performed by standardize_coxph does not condition on the observed covariates $\bar{Z}=(Z_1,...,Z_n)$. To see how this matters, note that

$var\{\hat{\theta}(t,x)\}=E[var\{\hat{\theta}(t,x)|\bar{Z}\}]+var[E\{\hat{\theta}(t,x)|\bar{Z}\}].$

The usual parameter $\beta$ in a Cox proportional hazards model does not depend on $\bar{Z}$. Thus, $E(\hat{\beta}|\bar{Z})$ is independent of $\bar{Z}$ as well (since $E(\hat{\beta}|\bar{Z})=\beta$), so that the term $var[E\{\hat{\beta}|\bar{Z}\}]$ in the corresponding variance decomposition for $var(\hat{\beta})$ becomes equal to 0. However, $\theta(t,x)$ depends on $\bar{Z}$ through the average over the sample distribution for $Z$, and thus the term $var[E\{\hat{\theta}(t,x)|\bar{Z}\}]$ is not 0, unless one conditions on $\bar{Z}$. The variance calculation by Gail and Byar (1986) ignores this term, and thus effectively conditions on $\bar{Z}$.

Author(s)

Arvid Sjölander

References

Chang I.M., Gelman G., Pagano M. (1982). Corrected group prognostic curves and summary statistics. Journal of Chronic Diseases 35, 669-674.

Dahlqwist E., Pawitan Y., Sjölander A. (2019). Regression standardization and attributable fraction estimation with between-within frailty models for clustered survival data. Statistical Methods in Medical Research 28(2), 462-485.

Gail M.H. and Byar D.P. (1986). Variance calculations for direct adjusted survival curves, with applications to testing for no treatment effect. Biometrical Journal 28(5), 587-599.

Makuch R.W. (1982). Adjusted survival curve estimation using covariates. Journal of Chronic Diseases 35, 437-443.

Examples

require(survival)

# simulate data
set.seed(6)
n <- 300
m <- 3
alpha <- 1.5
eta <- 1
phi <- 0.5
beta <- 1
id <- rep(1:n, each = m)
U <- rep(rgamma(n, shape = 1 / phi, scale = phi), each = m)
X <- rnorm(n * m)
# reparameterize scale as in rweibull function
weibull.scale <- alpha / (U * exp(beta * X))^(1 / eta)
T <- rweibull(n * m, shape = eta, scale = weibull.scale)

# right censoring
C <- runif(n * m, 0, 10)
D <- as.numeric(T < C)
T <- pmin(T, C)

# strong left-truncation
L <- runif(n * m, 0, 2)
incl <- T > L
incl <- ave(x = incl, id, FUN = sum) == m
dd <- data.frame(L, T, D, X, id)
dd <- dd[incl, ]

fit.std <- standardize_parfrailty(
  formula = Surv(L, T, D) ~ X,
  data = dd,
  values = list(X = seq(-1, 1, 0.5)),
  times = 1:5,
  clusterid = "id"
)
print(fit.std)
plot(fit.std)

---

Summarizes parfrailty fit

Description

This is a summary method for class "parfrailty".

Usage

## S3 method for class 'parfrailty'
summary(
  object,
  ci_type = "plain",
  ci_level = 0.95,
  digits = max(3L, getOption("digits") - 3L),
  ...
)

Arguments

object	an object of class "parfrailty".
ci_type	string, indicating the type of confidence intervals. Either "plain", which gives untransformed intervals, or "log", which gives log-transformed intervals.
ci_level	desired coverage probability of confidence intervals, in decimal form.
digits	the number of significant digits to use when printing..
...	not used.

Value

An object of class "summary.parfrailty", which is a list that contains relevant summary statistics about the fitted model

Author(s)

Arvid Sjölander and Elisabeth Dahlqwist.

See Also

parfrailty

Examples

## See documentation for parfrailty

---

Provide tidy output from a std_glm object for use in downstream computations

Description

Tidy summarizes information about the components of the standardized regression fit.

Usage

## S3 method for class 'std_glm'
tidy(x, ...)

Arguments

x	An object of class std_glm
...	Not currently used

Value

A data.frame

Examples

set.seed(6)
n <- 100
Z <- rnorm(n)
X <- rnorm(n, mean = Z)
Y <- rbinom(n, 1, prob = (1 + exp(X + Z))^(-1))
dd <- data.frame(Z, X, Y)
x <- standardize_glm(
  formula = Y ~ X * Z,
  family = "binomial",
  data = dd,
  values = list(X = 0:1),
  contrasts = c("difference", "ratio"),
  reference = 0
)
tidy(x)

---

Provide tidy output from a std_surv object for use in downstream computations

Description

Tidy summarizes information about the components of the standardized model fit.

Usage

## S3 method for class 'std_surv'
tidy(x, ...)

Arguments

x	An object of class std_surv
...	Not currently used

Value

A data.frame

Examples

require(survival)
set.seed(8)
n <- 300
Z <- rnorm(n)
X <- rnorm(n, mean = Z)
time <- rexp(n, rate = exp(X + Z + X * Z)) # survival time
C <- rexp(n, rate = exp(X + Z + X * Z)) # censoring time
U <- pmin(time, C) # time at risk
D <- as.numeric(time < C) # event indicator
dd <- data.frame(Z, X, U, D)
x <- standardize_coxph(
  formula = Surv(U, D) ~ X + Z + X * Z,
  data = dd, values = list(X = seq(-1, 1, 0.5)), times = c(2,3,4)
)

tidy(x)

---