adjusted_rmtl: Estimate Confounder-Adjusted Restricted Mean Time Lost

Description

This function can be utilized to estimate the confounder-adjusted restricted mean time lost (RMTL), possibly due to a specific cause, given previously estimated adjusted survival curves / CIFs created using the adjustedsurv or adjustedcif function.

Usage

adjusted_rmtl(adj, to, from=0, conf_int=FALSE,
              conf_level=0.95, interpolation="steps",
              difference=FALSE, ratio=FALSE,
              contrast="none", group_1=NULL,
              group_2=NULL)

Value

Returns a data.frame containing the columns group (groups in variable) and rmtl (the estimated restricted mean time lost).

If conf_int=TRUE was used it additionally contains the columns to (the supplied to values), se (the standard error of the restricted mean time lost), ci_lower (lower limit of the confidence interval), ci_upper (upper limit of the confidence interval) and n_boot (the actual number of bootstrap estimates used).

If contrast="diff" was used, it instead returns a data.frame that contains the columns to, diff (the difference between the RMTL values), se (the standard error of the difference), ci_lower (lower limit of the confidence interval), ci_upper (upper limit of the confidence interval) and p_value (the p-value of the one-sample t-test). The same results are presented when using contrast="ratio", except that the diff column is named ratio and that there is no se column.

Arguments

adj: An adjustedsurv object created using the adjustedsurv function or a adjustedcif object created using the adjustedcif function.
from: A single number specifying the left side of the time interval of interest. See details. Usually this should be kept at 0 (default) to estimate the standard RMTL. Should only be changed if there are good reasons for it.
to: One or more numbers specifying the right side of the time interval of interest. If a vector of numbers is supplied, the adjusted RMTL will be estimated for each value of to. See details.
conf_int: Whether bootstrap estimates should be used to estimate the standard errors and confidence intervals of the RMST estimates. Can only be used if bootstrap=TRUE was used in the adjustedsurv or adjustedcif call.
conf_level: A number specifying the confidence level of the bootstrap confidence intervals.
interpolation: Either "steps" (default) or "linear". This parameter controls how interpolation is performed. If this argument is set to "steps", the curves will be treated as step functions. If it is set to "linear", the curves wil be treated as if there are straight lines between the point estimates instead. Points that lie between estimated points will be interpolated accordingly. Should usually be kept at "steps". See Details.
difference: DEPRECATED. Use contrast="diff" instead.
ratio: DEPRECATED. Use contrast="ratio" instead.
contrast: A single character string, specifying which contrast should be estimated. Needs to be one of "none" (estimate no contrasts, just return the adjusted RMTL, the default), "diff" (estimate the difference) or "ratio" (estimate the ratio). When conf_int=TRUE is also specified and bootstrapping was performed in the original adjustedsurv call, this function will also estimate the corresponding standard error, the confidence interval and a p-value testing whether the difference is equal to 0 (or the ratio is equal to 1). To specify which difference/ratio should be calculated, the group_1 and group_2 arguments can be used. By default, the difference/ratio between the first and second level in variable is computed.
group_1: Optional argument to get a specific difference or ratio. This argument takes a single character string specifying one of the levels of the variable used in the original adjustedsurv or adjustedcif function call. This group will be subtracted from. For example if group_1="A" and group_2="B" and contrast="diff" the difference A - B will be used. If NULL, the order of the factor levels in the original data determines the order. Ignored if contrast="none".
group_2: Also a single character string specifying one of the levels of variable. This corresponds to the right side of the difference/ratio equation. See argument group_2. Ignored if contrast="none".

Author

Robin Denz

Details

The cause-specific adjusted restricted mean time lost (RMTL) is calculated by integrating the estimated adjusted cause-specific CIF in a specified interval. Let $Z$ be the grouping variable (corresponding to the variable argument in the adjustedcif function) with possible levels $Z \in \{0, 1, 2, ..., k\}$. $T$ is defined as the time and $\hat{F}_z^d(t)$ denotes the estimated counterfactual CIF for cause $d$. The RMTL is then defined as:

$$RMTL_{z}^d(to) = \int_{from=0}^{to} \hat{F}_z^d(t)dt$$

It can be interpreted as the mean time it takes an individual to succumb to the event of interest in group $Z = z$ in the interval [0, to]. . More information on the method itself can be found in the references. Note however that simply subtracting the estimates from each other does not give a correct estimate of the area between the CIFs if the respective curves cross at some point. The adjusted_curve_test function can be used to calculate the actual area between the curves instead. See ?adjusted_curve_test for more information.

If an adjustedsurv object is supplied in the adj argument, the CIF is calculated from the adjusted survival curves using the simple transformation: $\hat{F}_{z}(t) = 1 - \hat{S}_z(t)$. All further calculations are identical.

Confidence Intervals

If the adj object was created with bootstrap=TRUE in the corresponding function, bootstrap confidence intervals and standard errors for the RMTLs can be approximated by setting conf_int to TRUE. If bootstrap samples occur where the CIF is not estimated up to to, the bootstrap sample is discarded and not used in further calculations. Approximate variance calculations not relying on the bootstrap estimates are currently not implemented. When using contrast="diff" the standard error of the difference between the two RMST values is approximated by $SE_{group_1 - group_2} = \sqrt{SE_{group_1}^2 + SE_{group_2}^2}$. When using contrast="ratio" the confidence intervals are calculated using the approximate formula given by Fieller (1954), assuming that the values are independent.

More than Two Groups

If more than two groups are present in variable, all other comparisons except for group_1 vs. group_2 are ignored. If multiple comparisons are desired, the user needs to call this function multiple times and adjust the group_1 and group_2 arguments accordingly.

Multiple Imputation

If multiple imputation was used when creating the adj object, the analysis is carried out on all multiply imputed datasets and pooled using Rubins Rule. When bootstrapping was carried out as well, the pooled standard error over all imputed datasets is used in combination with the normal approximation to re-calculate the bootstrap confidence intervals.

Graphical Displays

A plot of the RMTL over time (with changing values for the to argument) can be produced using the plot_rmtl_curve function.

Computational Details

Instead of relying on numerical integration, this function uses exact calculations. This is achieved by using either step-function interpolation (interpolation="steps", the default) or linear interpolation (interpolation="linear"). In the former case, the integral is simply the sum of the area of the squares defined by the step function. In the second case, the integral is simply the sum of the area of the rectangles. Either way, there is no need for approximations. In some situations (for example when using parametric models with method="direct"), the curves are not step functions. In this case the interpolation argument should be set to "linear".

References

Sarah C. Conner and Ludovic Trunquart (2021). "Estimation and Modeling of the Restricted Mean Time Lost in the Presence of Competing Risks". In: Statistics in Medicine

Edgar C. Fieller (1954). "Some Problems in Interval Estimation". In: Journal of the Royal Statistical Society, Series B 16.2, pp. 175-185

Examples

Run this code

library(adjustedCurves)
library(survival)

###### when using single-event survival data

# simulate some data as example
sim_dat <- sim_confounded_surv(n=50, max_t=1.2)
sim_dat$group <- as.factor(sim_dat$group)

# estimate a cox-regression for the outcome
cox_mod <- coxph(Surv(time, event) ~ x1 + x2 + x3 + x4 + x5 + x6 + group,
                 data=sim_dat, x=TRUE)

# use it to calculate adjusted survival curves with bootstrapping
adjsurv <- adjustedsurv(data=sim_dat,
                        variable="group",
                        ev_time="time",
                        event="event",
                        method="direct",
                        outcome_model=cox_mod,
                        conf_int=FALSE,
                        bootstrap=TRUE,
                        n_boot=10) # n_boot should be much higher in reality

# calculate adjusted restricted mean survival times from 0 to 1
adjrmst <- adjusted_rmst(adjsurv, from=0, to=1, conf_int=FALSE)

# calculate adjusted restricted mean survival times from 0 to 0.5
# and from 0 to 1 simulatenously
adjrmst <- adjusted_rmst(adjsurv, from=0, to=c(0.5, 1), conf_int=FALSE)

# calculate adjusted restricted mean time lost estimates from 0 to 1,
# including standard errors and confidence intervals
adjrmst <- adjusted_rmst(adjsurv, from=0, to=1, conf_int=TRUE,
                         conf_level=0.95)

# calculate difference in adjusted restricted mean survival times from 0 to 1
adjrmst <- adjusted_rmst(adjsurv, from=0, to=1, conf_int=FALSE,
                         contrast="diff")

###### when using data with competing-risks

if (requireNamespace("riskRegression") & requireNamespace("prodlim")) {

library(riskRegression)
library(prodlim)

# simulate some data as example
set.seed(42)
sim_dat <- sim_confounded_crisk(n=100)
sim_dat$group <- as.factor(sim_dat$group)

# estimate a cause-specific cox-regression model for the outcome
csc_mod <- CSC(Hist(time, event) ~ x1 + x2 + x3 + x4 + x5 + x6 + group,
               data=sim_dat)

# calculate confounder-adjusted cause-specific CIFs for cause = 1
adjcif <- adjustedcif(data=sim_dat,
                      variable="group",
                      ev_time="time",
                      event="event",
                      method="direct",
                      outcome_model=csc_mod,
                      conf_int=FALSE,
                      bootstrap=TRUE,
                      n_boot=10,
                      cause=1)

# calculate adjusted restricted mean time lost estimates from 0 to 1
# including standard errors and confidence intervals
adjrmtl <- adjusted_rmtl(adjcif, from=0, to=1, conf_int=TRUE)

# calculate ratio of adjusted restricted mean time lost estimates from 0 to 1
# including confidence interval and p-value
adjrmtl <- adjusted_rmtl(adjcif, from=0, to=1, conf_int=TRUE, contrast="ratio")
}

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