Causal Inference with Continuous (Multiple Time Point) Interventions

Description

This package facilitates the estimation of counterfactual outcomes for multiple values of continuous interventions at different time points, and allows plotting of causal dose-response curves.

It implements the standard g-methods approach using the (semi-)parametric g-formula, as described in the Schomaker et al. (2024) reference listed below. Weighted dose-response curves that address positivity violations, and are fitted via sequential g-computation, are currently only available on GitHub and are not (yet) integrated in this package.

The main function of the package is currently gformula.

Details

Author(s)

Michael Schomaker

Maintainer: Michael Schomaker <michael.schomaker@stat.uni-muenchen.de>

References

Schomaker M, McIlleron H, Denti P, Diaz I. (2024) Causal Inference for Continuous Multiple Time Point Interventions, Statistics in Medicine, 43:5380-5400, see also https://arxiv.org/abs/2305.06645.

Pharmacoepidemiological HIV treatment data

Description

A hypothetical, simulated dataset which is line with the data-generating process of Schomaker et al. (2024) and inspired by the data of Bienczak et al. (2017); see references below.

Usage

data(EFV)

Format

A data frame with 5000 observations on the following variables:

sex

The patient's sex

metabolic

Metabolism status (slow, intermediate, extensive) related to the single nucleotide polymorphisms in the CYP2B6 gene, which is relevant for metabolizing evafirenz and directly affects its concentration in the body.

log_age

log(age) at baseline

NRTI

Nucleoside reverse transcriptase inhibitor (NRTI) component of HIV treatment, i.e. abacavir, stavudine or zidovudine.

weight.0

log(weight) at time 0 (baseline)

efv.0

Efavirenz concentration at time 0 (baseline)

VL.0

Elevated viral load (viral failure) at time 0 (baseline)

adherence.1

Adherence at time 1 (if 0, then signs of non-adherence)

weight.1

log(weight) at time 1

efv.1

Efavirenz concentration at time 1

VL.1

Elevated viral load (viral failure) at time 1

adherence.2

Adherence at time 2 (if 0, then signs of non-adherence)

weight.2

log(weight) at time 2

efv.2

Efavirenz concentration at time 2

VL.2

Elevated viral load (viral failure) at time 2

adherence.3

Adherence at time 3 (if 0, then signs of non-adherence)

weight.3

log(weight) at time 3

efv.3

Efavirenz concentration at time 3

VL.3

Elevated viral load (viral failure) at time 3

adherence.4

Adherence at time 4 (if 0, then signs of non-adherence)

weight.4

log(weight) at time 4

efv.4

Efavirenz concentration at time 4

VL.4

Elevated viral load (viral failure) at time 4

References

Schomaker M, McIlleron H, Denti P, Diaz I. (2024) Causal Inference for Continuous Multiple Time Point Interventions, Statistics in Medicine, 43:5380-5400, see also https://arxiv.org/abs/2305.06645.

Bienczak et al. (2017) Determinants of virological outcome and adverse events in African children treated with paediatric nevirapine fixed-dose-combination tablets, AIDS, 31:905-915

Examples

data(EFV)
str(EFV)

Pharmacoepidemiological HIV treatment data

Description

A hypothetical, simulated dataset which is line with the data-generating process of Schomaker et al. (2024) and inspired by the data of Bienczak et al. (2017); see references below. Compared to the dataset EFV, it contains all variables of the DAG in Figure 3 of Schomaker et al. (2023), also those which are not needed for identification of the counterfactual quantity of interest; that is, the expected viral suppression (VL) under a specific intervention on efavirenz concentrations (efv.0, efv.1, ...).

Usage

data(EFVfull)

Format

A data frame with 5000 observations on the following variables:

sex

The patient's sex

metabolic

Metabolism status (slow, intermediate, extensive) related to the single nucleotide polymorphisms in the CYP2B6 gene, which is relevant for metabolizing evafirenz and directly affects its concentration in the body.

log_age

log(age) at baseline

NRTI

Nucleoside reverse transcriptase inhibitor (NRTI) component of HIV treatment, i.e. abacavir, stavudine or zidovudine.

weight.0

log(weight) at time 0 (baseline)

comorbidity.0

Presence of co-morbidities at time 0 (baseline)

dose.0

Dose of efavirenz administered at time 0 (basline)

efv.0

Efavirenz concentration at time 0 (baseline)

VL.0

Elevated viral load (viral failure) at time 0 (baseline)

adherence.1

Adherence at time 1 (if 0, then signs of non-adherence)

weight.1

log(weight) at time 1

comorbidity.1

Presence of co-morbidities at time 1

dose.1

Dose of efavirenz administered at time 1

efv.1

Efavirenz concentration at time 1

VL.1

Elevated viral load (viral failure) at time 1

adherence.2

Adherence at time 2 (if 0, then signs of non-adherence)

weight.2

log(weight) at time 2

comorbidity.2

Presence of co-morbidities at time 2

dose.2

Dose of efavirenz administered at time 2

efv.2

Efavirenz concentration at time 2

VL.2

Elevated viral load (viral failure) at time 2

adherence.3

Adherence at time 3 (if 0, then signs of non-adherence)

weight.3

log(weight) at time 3

comorbidity.3

Presence of co-morbidities at time 3

dose.3

Dose of efavirenz administered at time 3

efv.3

Efavirenz concentration at time 3

VL.3

Elevated viral load (viral failure) at time 3

adherence.4

Adherence at time 4 (if 0, then signs of non-adherence)

weight.4

log(weight) at time 4

comorbidity.4

Presence of co-morbidities at time 4

dose.4

Dose of efavirenz administered at time 4

efv.4

Efavirenz concentration at time 4

VL.4

Elevated viral load (viral failure) at time

References

Schomaker M, McIlleron H, Denti P, Diaz I. (2024) Causal Inference for Continuous Multiple Time Point Interventions, Statistics in Medicine, 43:5380-5400, see also https://arxiv.org/abs/2305.06645.

Bienczak et al. (2017) Determinants of virological outcome and adverse events in African children treated with paediatric nevirapine fixed-dose-combination tablets, AIDS, 31:905-915

Examples

data(EFVfull)
str(EFVfull)

Counterfactual contrast from the parametric or sequential g-formula for continuous multiple time point interventions

Description

Estimation of a contrast between counterfactual outcomes under different values of (continuous) interventions, or across different time points, using the parametric or sequential g-formula.

Usage

contrast(X, abar, nodes, contrastType = "difference", measure = mean,
         cond = NULL, cilevel = 0.95, ...)

Arguments

Details

Causal effects are defined as contrasts between the distributions of counterfactual variables under different interventions, across different time points or across different covariate strata. The counterfactual distributions to compare must be uniquely determined, by either specifying two values of abar at a single nodes or two nodes at a single intervention level abar or the natural course scenario with abar = 'natural' or two covariate strata cond. If the natural course scenario is selected and two nodes are specified, the natural intervention is compared across the two nodes. If one nodes is specified, the natural and observed scenarios are compared at a single node.

By default, the difference between the expectations of the two counterfactual outcome distributions is calculated. The difference can be exchanged for a ratio, odds ratio or custom contrast in the contrastType argument, and expectations can be exchanged for custom measures in the measure argument. Conditional measures can be specified through the cond argument. Custom contrasts, including those comparing more than two counterfactuals, can be defined by passing a function to contrastType.

Confidence intervals are based on the nonparametric bootstrap with B samples.

Value

Returns a list of class contrastResult:

See Also

gformula and sgf for estimating expected counterfactual outcomes under multiple intervention values and custom.measure for measures other than expectations.

Examples

## Not run: 
data(EFV)
gf1 <- gformula(
  X = EFV, Anodes = c("efv.0", "efv.1", "efv.2", "efv.3", "efv.4"),
  Ynodes = c("VL.0", "VL.1", "VL.2", "VL.3", "VL.4"),
  Lnodes = c("adherence.1", "weight.1", "adherence.2", "weight.2",
             "adherence.3", "weight.3", "adherence.4", "weight.4"),
  abar = seq(1, 5), B = 10, ret = TRUE
)

# compare outcomes at last time point under (1,...,1) and (5,...,5) 
contrast(gf1, abar = c(1, 5), nodes = "VL.4")

# compare outcomes at different time points, for same intervention (2,...)
contrast(gf1, abar = 2, nodes = c("VL.3", "VL.2"))

# compare own measure (rel. risk reduction) instead of mean
# ... and conditional on subset
relativeRiskReduction <- function(k, l){(k - l) / k}

contrast(
  gf1, abar = c(1, 2), nodes = "VL.4",
  contrastType = relativeRiskReduction,
  cond = quote(sex == 1)
)

# Instead of the mean, any other measure can be taken,
# and - of course - applied also to counterfactual Lnodes
contrast(
  gf1, abar = 2, nodes = c("weight.3", "weight.2"),
  measure = median
)

## End(Not run)

Custom estimands after applying gformula

Description

The default estimate returned by gformula is the expected outcome under the respective intervention strategies abar. custom.measure takes an object of class gformula and enables estimation of other estimands based on the counterfactual datasets produced by gformula (if the option ret=TRUE had been chosen), for example estimands conditional on baseline variables, quantiles instead of expectations, and others.

Usage

custom.measure(X, fun = NULL, cond = NULL, verbose = TRUE, with.se = FALSE, ...)

Arguments

Details

In settings with censoring, it will often be needed to pass on the option na.rm=T, e.g. for the mean, median, quantiles, and others.

Calculation of the bootstrap standard error (i.e., with.se=T) is typically not needed; but, for example, necessary for the calculations after multiple imputation and hence used by mi.boot.

Value

An object of class gformula. See gformula for details.

See Also

see also gformula

Examples

data(EFV)

est <- gformula(X=EFV,
                Lnodes  = c("adherence.1","weight.1",
                            "adherence.2","weight.2",
                            "adherence.3","weight.3",
                            "adherence.4","weight.4"
                ),
                Ynodes  = c("VL.0","VL.1","VL.2","VL.3","VL.4"),
                Anodes  = c("efv.0","efv.1","efv.2","efv.3","efv.4"),
                abar=seq(0,2,1), ret=TRUE
)

est
custom.measure(est, fun=prop,categ=1) # identical
custom.measure(est, fun=prop,categ=0)
custom.measure(est, fun=prop, categ=0, cond="sex==1")
# note: metabolic has been recoded internally (see output above)
custom.measure(est, fun=prop, categ=0, cond="metabolic==0") 
# does not make sense here, just for illustration (useful for metric outcomes)
custom.measure(est, fun=quantile, probs=0.1) 

Fit models after screening

Description

Fits the models that have been generated with screening using model.formulas.update.

Usage

fit.updated.formulas(formulas, X)

Arguments

Details

Fits generalized (additive) linear models based on the screened model formula list generated by model.formulas.update.

Value

Returns a list of length 2:

See Also

model.formulas.update

Examples

data(EFV)

# first: generate generic model formulas
m <- make.model.formulas(X=EFV,
                         Lnodes  = c("adherence.1","weight.1",
                                     "adherence.2","weight.2",
                                     "adherence.3","weight.3",
                                     "adherence.4","weight.4"
                                    ),
                         Ynodes  = c("VL.0","VL.1","VL.2","VL.3","VL.4"),
                         Anodes  = c("efv.0","efv.1","efv.2","efv.3","efv.4"),
                         evaluate=FALSE) 
                         
# second: update these model formulas based on variable screening with LASSO
glmnet.formulas <-  model.formulas.update(m$model.names, EFV)
glmnet.formulas 

# then: fit and inspect the updated models
fitted.models <- fit.updated.formulas(glmnet.formulas, EFV)
fitted.models$all.summaries
fitted.models$all.summaries$Ynames[1] # first outcome model

Parametric g-formula for continuous multiple time point interventions

Description

Estimation of counterfactual outcomes for multiple values of continuous interventions at different time points using the g-formula.

Usage

gformula(X, Anodes, Ynodes, Lnodes = NULL, Cnodes = NULL,
         abar = NULL, cbar = "uncensored",
         survivalY = FALSE, 
         Yform = "GLM", Lform = "GLM", Aform = "GLM", Cform = "GLM",
         calc.support = FALSE, B = 0, ret = FALSE, ncores = 1, 
         verbose = TRUE, seed = NULL, prog = NULL, ...)

Arguments

Details

By default, expected counterfactual outcomes (specified under Ynodes) under the intervention abar are calculated. Other estimands can be specified via custom.measure.

If abar is a vector, then each vector component is used as the intervention value at each time point; that is, interventions which are constant over time are defined. If abar is a matrix (of size 'number interventions' x 'time points'), then each row of the length of Anodes refers to a particular time-varying intervention strategy. The natural intervention can be picked by setting abar='natural'.

The fitted outcome and confounder models are based on generalized additive models (GAMs) as implemented in the mgcv package. Model families are picked automatically and reported in the output if verbose=TRUE (see manual for modifications, though they hardly ever make sense). The model formulas are standard GLMs or GAMs (with penalized splines for continuous covariates), conditional on the past, unless specific formulae are given. It is recommended to use customized formulae to reduce the risk of model mis-specification and to ensure that the models make sense (e.g., not too many splines are used when this is computationally not meaningful). This can be best facilitated by using objects generated through make.model.formulas, followed by model.formulas.update and/or model.update (see examples for those functions).

For survival settings, it is required that i) survivalY=TRUE and ii) after a Cnode/Ynode is 1, every variable thereafter is set to NA. See manual for an example. By default, the package intervenes on Cnodes, i.e. calculates counterfactual outcomes under no censoring.

If calc.support=TRUE, conditional and crude support measures (i.e., diagnostics) are calculated as described in Section 3.4 of Schomaker et al. (2023). Another useful diagnostic for multiple time points is the natural course scenario, which can be evaluated under abar='natural' and cbar='natural'.

To parallelize computations automatically, it is sufficient to set ncores>1, as appropriate. To make estimates under parallelization reproducible, use the seed argument. To watch the progress of parallelized computations, set a path in the prog argument: then, a text file reports on the progress, which is particularly useful if lengthy bootstrapping computations are required.

Value

Returns an object of of class ‘gformula’:

Author(s)

Michael Schomaker

See Also

plot.gformula for plotting results as (causal) dose response curves, custom.measure for evaluating custom estimands and mi.boot for using gformula on multiply imputed data.

Examples

## Not run: 
data(EFV)
est <- gformula(X=EFV,
                      Lnodes  = c("adherence.1","weight.1",
                                  "adherence.2","weight.2",
                                  "adherence.3","weight.3",
                                  "adherence.4","weight.4"
                        ),
                        Ynodes  = c("VL.0","VL.1","VL.2","VL.3","VL.4"),
                        Anodes  = c("efv.0","efv.1","efv.2","efv.3","efv.4"),
                        abar=seq(0,10,1)
)
est

## End(Not run)

Compose appropriate model formulas

Description

Function that generates generic model formulas for Y-/L-/A- and Cnodes, according to time ordering and to be used in gformula or model.formulas.update.

Usage

make.model.formulas(X, Ynodes = NULL, Lnodes = NULL, Cnodes = NULL, Anodes = NULL,
                    survival = FALSE, evaluate = FALSE)

Arguments

Details

This is a helper function to generate model formulas for Y-/L-/A- and Cnodes, according to the time ordering: i.e. to generate GLM/GAM model formulas for the respective nodes given all past variables. In survival settings, past censoring and outcome nodes are omitted from the formulae. If censoring is present without a survival setting (e.g. Cnodes describe drop-outs and Y is a continuous outcome), then survival should be set as FALSE.

Value

Returns a named list:

See Also

The generated generic model formulas can be updated manually with model.update or in an automated manner with screening using model.formulas.update.

Examples

data(EFV)

m <- make.model.formulas(X=EFV,
                         Lnodes  = c("adherence.1","weight.1",
                                     "adherence.2","weight.2",
                                     "adherence.3","weight.3",
                                     "adherence.4","weight.4"
                                    ),
                         Ynodes  = c("VL.0","VL.1","VL.2","VL.3","VL.4"),
                         Anodes  = c("efv.0","efv.1","efv.2","efv.3","efv.4"),
                         evaluate=FALSE) # set TRUE to see fitted models
                         
m$model.names # all models potentially relevant for gformula(), given full past

Obtaining estimates from multiply imputed data

Description

Combines gformula estimates obtained from multiple imputed data sets according to the MI Boot and MI Boot pooled methods decribed in Schomaker and Heumann (2018, see reference section below)

Usage

mi.boot(x, fun, cond=NULL, pooled=FALSE, ...)

Arguments

Value

An object of class gformula. See gformula for details.

Author(s)

Michael Schomaker

References

Schomaker, M., Heumann, C. (2018) Bootstrap inference when using multiple imputation, Statistics in Medicine, 37:2252-2266

Examples

data(EFV)

# suppose the following subsets were actually multiply imputed data (M=2)
EFV_1 <- EFV[1:2500,]
EFV_2 <- EFV[2501:5000,]

# first: conduct analysis on each imputed data set. Set ret=T.
m1 <- gformula(X=EFV_1,
               Lnodes  = c("adherence.1","weight.1",
                           "adherence.2","weight.2",
                           "adherence.3","weight.3",
                           "adherence.4","weight.4"
               ),
               Ynodes  = c("VL.0","VL.1","VL.2","VL.3","VL.4"),
               Anodes  = c("efv.0","efv.1","efv.2","efv.3","efv.4"),
               abar=seq(0,5,1), verbose=FALSE, ret=TRUE
        )

m2 <- gformula(X=EFV_2,
               Lnodes  = c("adherence.1","weight.1",
                           "adherence.2","weight.2",
                           "adherence.3","weight.3",
                           "adherence.4","weight.4"
               ),
               Ynodes  = c("VL.0","VL.1","VL.2","VL.3","VL.4"),
               Anodes  = c("efv.0","efv.1","efv.2","efv.3","efv.4"),
               abar=seq(0,5,1), verbose=FALSE, ret=TRUE
)

# second combine results
m_imp <- mi.boot(list(m1,m2), mean) # uses MI rules & returns 'gformula' object
plot(m_imp)

# custom estimand: evaluate probability of suppression (Y=0), among females
m_imp2 <- mi.boot(list(m1,m2), prop, categ=0, cond="sex==1")
plot(m_imp2)

Update model formulas based on variable screening

Description

Wrapper function to facilitate variable screening on all models generated through make.model.formulas and return updated formulas in the appropriate format for gformula.

Usage

model.formulas.update(formulas, X, screening = screen.glmnet.cramer,
                      with.s = FALSE, by= NA, ...)

Arguments

Details

The default screening algorithm uses LASSO for variable screening (and Cramer's V for the categorized version of all variables if LASSO fails). It is possible to provide user-specific screening algorithms. User-specific algorithms should take the data as first argument, one model formula (i.e. one entry of the list in model.formulas) as second argument and return a vector of strings, containing the variable names that remain after screening. Another screening algorithm available in the package is screen.cramersv, which categorizes all variables, calculates their association with the outcome based on Cramer's V and selects the 4 variables with strongest associations (can be changed with option nscreen). The manual provides more information.

The fitted models of the updated models can be evaluated with fit.updated.formulas.

Value

A list of length 4 containing the updated model formulas:

See Also

make.model.formulas, model.update, fit.updated.formulas

Examples

data(EFV)

# first: generate generic model formulas
m <- make.model.formulas(X=EFV,
                         Lnodes  = c("adherence.1","weight.1",
                                     "adherence.2","weight.2",
                                     "adherence.3","weight.3",
                                     "adherence.4","weight.4"
                                    ),
                         Ynodes  = c("VL.0","VL.1","VL.2","VL.3","VL.4"),
                         Anodes  = c("efv.0","efv.1","efv.2","efv.3","efv.4"),
                         evaluate=FALSE) 
                         
# second: update these model formulas based on variable screening with LASSO
glmnet.formulas <-  model.formulas.update(m$model.names, EFV)
glmnet.formulas 


# third: use these models for estimation
est <- gformula(X=EFV,
                Lnodes  = c("adherence.1","weight.1",
                            "adherence.2","weight.2",
                            "adherence.3","weight.3",
                            "adherence.4","weight.4"
                ),
                Ynodes  = c("VL.0","VL.1","VL.2","VL.3","VL.4"),
                Anodes  = c("efv.0","efv.1","efv.2","efv.3","efv.4"),
                Yform=glmnet.formulas$Ynames, Lform=glmnet.formulas$Lnames,
                abar=seq(0,2,1)
)
est

Update GAM models

Description

A wrapper to simplify the update of GAM models

Usage

model.update(gam.object, form)

Arguments

Details

The gam object needs to be fitted with the option control=list(keepData=T), otherwise the function can not access the data that is needed to update the model fit. Note that both fit.updated.formulas and make.model.formulas with option evaluate=T produce results that are based on this option.

Value

An object of class ‘gam’, ‘glm’ and ‘lm’.

Examples

data(EFV)

m <- make.model.formulas(X=EFV,
                         Lnodes  = c("adherence.1","weight.1",
                                     "adherence.2","weight.2",
                                     "adherence.3","weight.3",
                                     "adherence.4","weight.4"
                         ),
                         Ynodes  = c("VL.0","VL.1","VL.2","VL.3","VL.4"),
                         Anodes  = c("efv.0","efv.1","efv.2","efv.3","efv.4"),
                         evaluate=TRUE) # set TRUE for model.update()

# update first confounder model of weight manually
model.update(m$fitted.models$fitted.L$m_weight.1, .~s(weight.0, by=sex))

# manual update of model formula
m$model.names$Lnames[2] <- "weight.1 ~ s(weight.0, by=sex)" 

Plot dose-response curves

Description

Function to plot dose-response curves based on results returned from gformula

Usage

## S3 method for class 'gformula'
plot(x, msm.method = c("line","loess", "gam", "none"), 
                        CI = FALSE, time.points = NULL, 
                        cols = NULL, weight = NULL, xaxis=NULL,
                        variable = "psi", difference = FALSE, ...)

Arguments

Details

Time points and variable names should be specified according to the labeling of the results table returned by gformula.

Value

Draws an object of class ‘ggplot’.

Examples

data(EFV)
est <- gformula(X=EFV,
                Lnodes  = c("adherence.1","weight.1",
                            "adherence.2","weight.2",
                            "adherence.3","weight.3",
                            "adherence.4","weight.4"
                ),
                Ynodes  = c("VL.0","VL.1","VL.2","VL.3","VL.4"),
                Anodes  = c("efv.0","efv.1","efv.2","efv.3","efv.4"),
                abar=seq(0,10,1)
)


plot(est)
plot(est, time.points=c(1,5))

Printing a contrastResult object

Description

Generic S3 function to print the results of a contrastResult object created with contrast.

Usage

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

Arguments

Value

No return value, called for printing the results of an object of class contrastResult.

Printing the results of a gformula object

Description

Generic S3 function to print the results of an object created with gformula.

Usage

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

Arguments

Value

No return value, called for printing the results of an object of class class ‘gformula’.

Helper function to calculate proportions of categorical variables

Description

To be used as fun argument in custom.measure.

Arguments

Value

A scalar numeric value of the proportion of categ values in vec