Estimate Counterfactual Survival or Failure Probabilities for Levels of a Continuous Variable

This function can be utilized to estimate counterfactual survival curves or cumulative incidence functions (CIF) for specific values of a continuous covariate.

Usage

curve_cont(data, variable, model, horizon,
           times, group=NULL, cause=1, cif=FALSE,
           contrast="none", reference="km", ref_value=NULL,
           event_time=NULL, event_status=NULL,
           conf_int=FALSE, conf_level=0.95,
           n_boot=300, n_cores=1,
           na.action=options()$na.action,
           return_boot=FALSE, ...)

Arguments

data: A data.frame containing all required variables.
variable: A single character string specifying the continuous variable of interest, for which the survival curves should be estimated. This variable has to be contained in the data.frame that is supplied to the data argument.
model: A model describing the time-to-event process (such as an coxph model). Needs to include variable as an independent variable. It also has to have an associated predictRisk method. See ?predictRisk for more details.
horizon: A numeric vector containing a range of values of variable for which the survival curves should be calculated.
times: A numeric vector containing points in time at which the survival probabilities should be calculated.
group: An optional single character string specifying a factor variable in data. When used, the regression standardization is performed conditional on this factor variable, meaning that one estimate is returned for each level of the factor variable. See details for a better description. Set to NULL (default) to use no grouping variable.
cause: The cause of interest. In standard survival data with only one event type, this should be kept at 1. For data with multiple failure types, this argument should be specified. In addition, the cif argument should be set to TRUE in those cases.
cif: Whether to calculate the cumulative incidence (CIF) instead of the survival probability. If multiple failure types are present, the survival probability cannot be estimated in an unbiased way. In those cases, this argument should always be set to TRUE.
contrast: Defines what kind of estimate should be returned. Can be either "none" (default), "diff" or "ratio". When "none" is used, it simply returns the counterfactual survival probabilities (or CIF if cif=TRUE) without any further calculations. If "diff" or "ratio" is used instead, the difference or ratio to some reference value will be returned. See argument below.
reference: Defines what kind of reference value to use when estimating causal contrasts. Only used if contrast!="none", ignored otherwise. Can be either "km" (using standard Kaplan-Meier estimates as reference) or "value" (using the g-computation estimates of a specific value of the variable as reference). To specify which value to use when using reference="value", the ref_value argument should be used. See details for more information.
ref_value: A single number corresponding to the reference value used when estimating causal contrasts using reference="value". See details.
event_time: A single character string specifying the time until the occurrence of the event of interest or NULL (default). Only used when estimating causal contrasts with reference="km", ignored otherwise. If specified, this variable has to be contained in the data.frame that is supplied to the data argument.
event_status: A single character string specifying the status of the event of interest or NULL (default). Only used when estimating causal contrasts with reference="km", ignored otherwise. If specified, this variable has to be contained in the data.frame that is supplied to the data argument.
conf_int: Whether to calculate point-wise confidence intervals or not. If TRUE, n_boot bootstrap samples are drawn, the g-computation step is performed on each bootstrap sample and the confidence intervals are calculated using the percentile method from these results. Can get very slow if the dataset is large or there are many values in horizon or times. Using n_cores can speed up the process a lot.
conf_level: A number specifying the confidence level of the bootstrap confidence intervals. Ignored if conf_int=FALSE.
n_boot: A single integer specifying how many bootstrap repetitions should be performed. Ignored if conf_int=FALSE.
n_cores: The number of processor cores to use when performing the calculations. If n_cores=1 (default), single threaded processing is used. If n_cores > 1 the foreach package and the doParallel package are used to run the calculations on n_cores in parallel. This might speed up the runtime considerably when it is initially slow.
na.action: How missing values should be handled. Can be one of: na.fail, na.omit, na.pass, na.exclude or a user-defined custom function. Also accepts strings of the function names. See ?na.action for more details. By default it uses the na.action which is set in the global options by the respective user.
return_boot: Either TRUE or FALSE (default). If TRUE (and conf_int=TRUE) this function will return the individual bootstrap estimates instead of the usual estimates. This may be useful to perform tests or calculate other type of bootstrap confidence intervals manually.
...: Further arguments passed to predictRisk.

Details

This function is used internally in all plot functions included in this R-package and generally does not need to be called directly by the user. It can however be used to get specific values or as a basis to create custom plots not included in this package. Below we give a small introduction to what this function does. A more detailed description of the underlying methodology can be found in our article on this subject (Denz & Timmesfeld 2022).

Target Estimand (default)

By default (contrast="none") this function tries to estimate the survival probability at times that would have been observed if every individual in the sample had received a value of horizon in the variable. Let $Z$ be the continuous variable we are interested in. Let $T$ be the time until the occurrence of the event of interest. Under the potential outcome framework, there is an uncountably infinite amount of potential survival times $T^{(Z=z)}$, one for each possible value of $Z$. The target estimand is then defined as:

$$S_{z}(t) = E(I(T^{(Z=z)} > t))$$

If we additionally consider a categorical group variable $D$, the target estimand is similarly defined as:

$$S_{zd}(t) = E(I(T^{(Z=z, D=d)} > t))$$

where $T^{(Z=z, D=d)}$ is the survival time that would have been observed if the individual had received both $Z = z$ and $D = d$.

Target Estimand (using contrasts)

If contrasts are used (contrast!="none"), target estimands based on $S_{z}(t)$ or $S_{zd}(t)$ are used instead. When using contrast="diff" and reference="km", the target estimand is simply the difference between the observed survival probability in the entire sample (denoted by $S(t)$, estimated using a Kaplan-Meier estimator) and the counterfactual survival probability:

$$\Delta_{KM}(t, z) = S(t) - S_z(t)$$

If contrast="ratio" is used instead, the substraction sign is simply replace by a division. If group was specified, $S(t)$ is replaced by $S_d(t)$ (a stratified Kaplan-Meier estimator) and $S_z(t)$ is replaced by $S_{zd}(t)$. Instead of using a Kaplan-Meier estimator as reference, one may also use a specific $S_z(t)$ as reference using reference="value" and setting ref_value to the $Z$ that should be used. All of these causal contrasts and their implications are described in detail in the appendix of our article on this topic (Denz & Timmesfeld 2022).

Estimation Methodology

G-Computation, also known as the Corrected Group Prognosis method, Direct-Standardization or G-Formula, is used internally to estimate the counterfactual survival probability or CIF for values of a continuous variable. This is done by setting the variable to a specific value for all rows in data first. Afterwards, the model is used to predict the survival probability of each individual at all times, given the value of variable and their other observed covariates (which are included in the model as independent variables). These estimates are then averaged for each time point. This procedure is repeated for every value in horizon. If a group is supplied, these calculations are repeated for every possible value in group, with the estimated individual survival probabilities also being conditional on that value.

To obtain valid estimates of the target estimand using this function, the fundamental causal identifiability assumptions have to be met. Those are described in detail in our article on this topic (Denz & Timmesfeld 2022). If those assumptions are not met, the estimates may only be used to showcase simple associations. They cannot be endowed with a counterfactual interpretation in this case.

Supported Models

This function relies on the predictRisk function from the riskRegression package to create the covariate and time specific estimates of the probabilities. All models with an associated predictRisk method may be used in this function. This includes a variety of models, such as the Cox proportional hazards regression model and the aalen additive hazards model. If the model that should be used has no predictRisk method, the user either needs to write their own predictRisk method or contact the maintainers of the riskRegression package.

Bootstrap Confidence Intervals

By using conf_int=TRUE, bootstrap confidence intervals may be estimated. This will draw n_boot samples of size nrow(data) with replacement from data in the first step. Afterwards, the model is fit to each of those bootstrap samples and the curve_cont function is recursively called to obtain the estimates of interests for each sample. Using the percentile approach, the bootstrap confidence intervals are calculated. This requires that the model object contains a call parameter, because the update() function is used internally.

Computational Complexity

If many values are included in times, horizon or both and/or data has a lot of rows, this function may become very slow. Since bootstrapping relies on sampling with replacement from the original data and repeating the entire procedure n_boot times, this also dramatically increases the runtime. Parallel processing may be used to speed up the computations. This can be done by simply setting n_cores to values higher than 1.

Missing Data

Currently, this function does not support advanced handling of missing data. The data.frame supplied to data should contain no missing data in the relevant columns. To achieve this, the na.action argument can be set to "na.omit", which will remove all rows that do contain missing data.

Competing Events

By supplying a model that directly takes into account competing events and using the cause argument, the user may also use the functionality offered in this package to create plots in this setting. Internally, the predictRisk method will then be used to estimate the conditional cause-specific cumulative incidence function, which will then be used to carry out the g-computation step explained above. However the underlying target estimand of this procedure is dependent on which kind of model was supplied. It is therefore not possible to define it here concisely. Future research is necessary to clarify this point. This feature should only be used with great caution.

Value

Returns a data.frame containing the columns time (the point in time), est (the estimated survival probability or CIF) and cont (the specific value of variable used). If group was used, it includes the additional group column, specifying the level of the grouping variable. If conf_int=TRUE was used, it additionally includes the columns se (bootstrap standard error of the estimate), ci_lower (lower limit of the bootstrap confidence interval) and ci_upper (upper limit of the bootstrap confidence interval).

References

Robin Denz, Nina Timmesfeld (2023). "Visualizing the (Causal) Effect of a Continuous Variable on a Time-To-Event Outcome". In: Epidemiology 34.5

Brice Ozenne, Anne Lyngholm Sorensen, Thomas Scheike, Christian Torp-Pedersen and Thomas Alexander Gerds. riskRegression: Predicting the Risk of an Event using Cox Regression Models. The R Journal (2017) 9:2, pages 440-460.

I-Ming Chang, Rebecca Gelman, and Marcello Pagano. Corrected Group Prognostic Curves and Summary Statistics. Journal of Chronic Diseases (1982) 35, pages 669-674

James Robins. A New Approach to Causal Inference in Mortality Studies with a Sustained Exposure Period: Application to Control of the Healthy Worker Survivor Effect. Mathematical Modelling (1986) 7, pages 1393-1512.

Author

Robin Denz

Examples

library(contsurvplot)
library(riskRegression)
#> riskRegression version 2023.09.08
library(survival)

# using data from the survival package
data(nafld, package="survival")

# take a random sample to keep example fast
set.seed(42)
nafld1 <- nafld1[sample(nrow(nafld1), 150), ]

# fit cox-model with age
model <- coxph(Surv(futime, status) ~ age, data=nafld1, x=TRUE)

# estimate survival probability at some points in time, for
# a range of age values
plotdata <- curve_cont(data=nafld1,
                       variable="age",
                       model=model,
                       horizon=c(50, 60, 70, 80),
                       times=c(1000, 2000, 3000, 4000))

# estimate cumulative incidences instead
plotdata <- curve_cont(data=nafld1,
                       variable="age",
                       model=model,
                       horizon=c(50, 60, 70, 80),
                       times=c(1000, 2000, 3000, 4000),
                       cif=TRUE)