Targeted Maximum Likelihood Estimation for Longitudinal Data

Description

Targeted Maximum Likelihood Estimation (TMLE) of treatment/censoring specific mean outcome or marginal structural model for point-treatment and longitudinal data. Also provides Inverse Probability of Treatment/Censoring Weighted estimate (IPTW) and maximum likelihood based G-computation estimate (G-comp). Can be used to calculate additive treatment effect, risk ratio, and odds ratio.

Author(s)

Joshua Schwab, Samuel Lendle, Maya Petersen, and Mark van der Laan, with contributions from Susan Gruber

Maintainer: Joshua Schwab jschwab77@berkeley.edu

References

Bang, Heejung, and James M. Robins. "Doubly robust estimation in missing data and causal inference models." Biometrics 61.4 (2005): 962-973.

Lendle SD, Schwab J, Petersen ML and van der Laan MJ (2017). "ltmle: An R Package Implementing Targeted Minimum Loss-Based Estimation for Longitudinal Data." _Journal of Statistical Software_, *81*(1), pp. # ' 1-21. doi: 10.18637/jss.v081.i01 doi:10.18637/jss.v081.i01

Petersen, Maya, Schwab, Joshua and van der Laan, Mark J, "Targeted Maximum Likelihood Estimation of Marginal Structural Working Models for Dynamic Treatments Time-Dependent Outcomes", Journal of Causal Inference, 2014 https://pubmed.ncbi.nlm.nih.gov/25909047/

Robins JM, Sued M, Lei-Gomez Q, Rotnitsky A. (2007). Comment: Performance of double-robust estimators when Inverse Probability weights are highly variable. Statistical Science 22(4):544-559.

van der Laan, Mark J. and Gruber, Susan, "Targeted Minimum Loss Based Estimation of an Intervention Specific Mean Outcome" (August 2011). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 290. https://biostats.bepress.com/ucbbiostat/paper290/

van der Laan, Mark J. and Rose, Sherri, "Targeted Learning: Causal Inference for Observational and Experimental Data" New York: Springer, 2011.

See Also

ltmle

Examples

## For examples see examples(ltmle) and \url{http://joshuaschwab.github.io/ltmle/}

BinaryToCensoring

Description

Helper function for creating censoring columns as factors.

Usage

BinaryToCensoring(is.censored, is.uncensored)

Arguments

Details

Exactly one of is.censored and is.uncensored must be specified as a named argument. All elements of the input vector must be 0, 1, or NA

Value

an object of class "factor" with levels "censored" and "uncensored"

Author(s)

Joshua Schwab jschwab77@berkeley.edu

See Also

factor

Examples

 BinaryToCensoring(is.censored=c(0, 1, 1, 0, NA))
 BinaryToCensoring(is.uncensored=c(1, 0, 0, 1, NA))   #the same
 
 ## Not run: 
 BinaryToCensoring(c(0, 1))   #error because the input must be named
 
## End(Not run)

Deterministic g/Q functions - examples and templates

Description

Template for the deterministic.g.function argument to ltmle or ltmleMSM.

Usage

deterministic.g.function_template(data, current.node, nodes)

deterministic.Q.function_template(
  data,
  current.node,
  nodes,
  called.from.estimate.g
)

Arguments

Details

MaintainTreatment and MaintainControl are two commonly used deterministic.g.functions.

The intended use of the templates is for the user to copy and paste the function arguments and body and then fill in the required sections. They will not run as-is. Note that there are no comments in the functions as saved. Versions with comments may be found in Examples section below.

MaintainTreatment and MaintainControl may be passed as-is for the deterministic.g.function argument to ltmle or ltmleMSM

Note that censoring nodes in data may be passed as binaries but they are converted to the preferred format of factors with levels "censored" and "uncensored" before deterministic functions are called. Also note that nodes may be passed to ltmle as either the names of nodes or numerical column indicies, but they are all converted to numerical indicies before deterministic functions are called. If the survivalFunction argument to ltmle or ltmleMSM is TRUE, the package automatically assumes that once Y jumps to 1, all future Y nodes stay 1 and treatment does not change. It is not necessary to specify this in deterministic functions.

Value

A deterministic.g.function should return a list with components:

A deterministic.Q.function should return a list with components:

NOTE: The Q.value component is not used or required when called.from.estimate.g is TRUE

Functions

• deterministic.Q.function_template(): Template for the deterministic.Q.function argument to ltmle or ltmleMSM.

Author(s)

Joshua Schwab jschwab77@berkeley.edu

See Also

ltmle, ltmleMSM

Examples

# Show template for a deterministic.g.function (comments will not be
# shown, see below for comments)
deterministic.g.function_template

# Show template for a deterministic.Q.function (comments will not be
# shown, see below for comments)
deterministic.Q.function_template

# Use MaintainTreatment
set.seed(1)
rexpit <- function(x) rbinom(n = length(x), size = 1, prob = plogis(x))
n <- 100
W <- rnorm(n)
A1 <- rexpit(W)
A2 <- as.numeric(rexpit(W) | A1)  #treatment at time 1 implies treatment at time 2
Y <- rexpit(W + A1 + A2 + rnorm(n))
data <- data.frame(W, A1, A2, Y)

result <- ltmle(data, Anodes = c("A1", "A2"), Ynodes = "Y", abar = c(1, 1), 
    deterministic.g.function = MaintainTreatment)

# deterministic.g.function_template with comments:

deterministic.g.function_template <- function(data, current.node, nodes) {
    # data: the 'data' data.frame passed to ltmle/ltmleMSM current.node: the
    # column index of data corresponding to the A or C node (see
    # is.deterministic below) nodes: list of column indicies, components: A,
    # C, L, Y, AC (Anodes and Cnodes combined and sorted), LY (Lnodes and
    # Ynodes combined, sorted, 'blocks' removed - see ?ltmle) Note that nodes
    # may be passed to ltmle as either the names of nodes or numerical column
    # indicies, but they are all converted to numerical indicies before
    # deterministic.g.function is called
    
    # deterministic.g.function will be called at all Anodes and Cnodes
    # return(NULL) is equivalent to return(list(is.deterministic=rep(FALSE,
    # nrow(data)), prob1=numeric(0)))
    
    # define is.deterministic here: vector of logicals, length=nrow(data)
    # define prob1 here: the probability that data[is.deterministic,
    # current.node] == 1, vector of length 1 or
    # length(which(is.deterministic))
    is.deterministic <- stop("replace me!")
    prob1 <- stop("replace me!")
    return(list(is.deterministic = is.deterministic, prob1 = prob1))
}

# deterministic.Q.function_template with comments:

deterministic.Q.function_template <- function(data, current.node, nodes, 
    called.from.estimate.g) {
    # data: the 'data' data.frame passed to ltmle/ltmleMSM current.node: the
    # column index of data corresponding to the A or C node (see
    # is.deterministic below) nodes: list of column indicies, components: A,
    # C, L, Y, AC (Anodes and Cnodes combined and sorted), LY (Lnodes and
    # Ynodes combined, sorted, 'blocks' removed - see ?ltmle)
    # called.from.estimate.g: TRUE or FALSE - your function will be called
    # with called.from.estimate.g=TRUE during estimation of g and
    # called.from.estimate.g=FALSE during estimation of Q. During estimation
    # of g, only the is.deterministic element of the return list will be
    # used.  Note that nodes may be passed to ltmle as either the names of
    # nodes or numerical column indicies, but they are all converted to
    # numerical indicies before deterministic.Q.function is called
    
    # It is not necessary to specify that deterministic Y events (Y==1)
    # indicate a deterministic Q value of 1; this is automatic 
    # if the survivalFunction input to ltmle/ltmleMSM is TRUE.
    # deterministic.Q.function will be called at all Lnodes and Ynodes (after
    # removing 'blocks') and Anodes and Cnodes (see called.from.estimate.g
    # above) return(NULL) is equivalent to
    # return(list(is.deterministic=rep(FALSE, nrow(data)),
    # Q.value=numeric(0)))
    
    # define is.deterministic here: vector of logicals, length=nrow(data)
    # define Q.value here: the iterated expectation of the final Y, vector of
    # length 1 or length(which(is.deterministic))
    is.deterministic <- stop("replace me!")
    Q.value <- stop("replace me!")
    return(list(is.deterministic = is.deterministic, Q.value = Q.value))
}

Longitudinal Targeted Maximum Likelihood Estimation

Description

ltmle is Targeted Maximum Likelihood Estimation (TMLE) of treatment/censoring specific mean outcome for point-treatment and longitudinal data. ltmleMSM adds Marginal Structural Models. Both always provide Inverse Probability of Treatment/Censoring Weighted estimate (IPTW) as well. Maximum likelihood based G-computation estimate (G-comp) can be obtained instead of TMLE. ltmle can be used to calculate additive treatment effect, risk ratio, and odds ratio.

Usage

ltmle(
  data,
  Anodes,
  Cnodes = NULL,
  Lnodes = NULL,
  Ynodes,
  survivalOutcome = NULL,
  Qform = NULL,
  gform = NULL,
  abar,
  rule = NULL,
  gbounds = c(0.01, 1),
  Yrange = NULL,
  deterministic.g.function = NULL,
  stratify = FALSE,
  SL.library = "glm",
  SL.cvControl = list(),
  estimate.time = TRUE,
  gcomp = FALSE,
  iptw.only = FALSE,
  deterministic.Q.function = NULL,
  variance.method = "tmle",
  observation.weights = NULL,
  id = NULL
)

ltmleMSM(
  data,
  Anodes,
  Cnodes = NULL,
  Lnodes = NULL,
  Ynodes,
  survivalOutcome = NULL,
  Qform = NULL,
  gform = NULL,
  gbounds = c(0.01, 1),
  Yrange = NULL,
  deterministic.g.function = NULL,
  SL.library = "glm",
  SL.cvControl = list(),
  regimes,
  working.msm,
  summary.measures,
  final.Ynodes = NULL,
  stratify = FALSE,
  msm.weights = "empirical",
  estimate.time = TRUE,
  gcomp = FALSE,
  iptw.only = FALSE,
  deterministic.Q.function = NULL,
  variance.method = "tmle",
  observation.weights = NULL,
  id = NULL
)

Arguments

Details

The estimates returned by ltmle are of a treatment specific mean, E[Y_{\bar{a}}], the mean of the final treatment node, where all treatment nodes, A, are set to \bar{a} (abar) and all censoring nodes C are set to 1 (uncensored). The estimates returned by ltmleMSM are similar but are the parameters in a working marginal structural model.

data should be a data frame where the order of the columns corresponds to the time-ordering of the model.

• in censoring columns (Cnodes): factor with two levels: "censored" and "uncensored". The helper function BinaryToCensoring can be used to create these factors.

• in treatment columns (Anodes): 1 = treated, 0 = untreated (must be binary)

• in event columns (Ynodes): If survivalOutcome is TRUE, then Y nodes are treated as indicators of a one-time event. See details for survivalOutocme. If survivalOutcome is FALSE, Y nodes are treated as binary if all values are 0 or 1, and are treated as continuous otherwise. If Y nodes are continuous, they may be automatically scaled. See details for Yrange.

• time-dependent covariate columns (Lnodes): can be any numeric data

• Data in Cnodes, Anodes, Lnodes and Ynodes are not used after (to the right of) censoring (or an event when survivalOutcome==TRUE) and may be coded as NA or any other value.

• Columns in data that are before (to the left of) the first of Cnodes or Anodes are treated as baseline variables, even if they are specified as Lnodes.

• After the first of Cnodes, Anodes, Ynodes, or Lnodes, every column must be in one of Cnodes, Anodes, Ynodes, or Lnodes.

If survivalOutcome is TRUE, all Y values are indicators of an event (e.g. death) at or before the current time, where 1 = event and 0 = no event. The events in Ynodes must be of the form where once Y jumps to 1, Y remains 1 at subsequent nodes.

For continuous outcomes, (survivalOutcome==FALSE and some Y nodes are not 0 or 1,) Y values are truncated at the minimum and maximum of Yrange if specified, and then transformed and scaled to be in [0,1]. That is, transformed to (Y-min(Yrange))/(max(Yrange)-min(Yrange)). If Yrange is NULL, it is set to the range of all Y nodes. In that case, Y nodes are only scaled if any values fall outside of [0,1]. For intervention specific means (ltmle), parameter estimates are transformed back based Yrange.

Qform should be NULL, in which case all parent nodes of each L and Y node will be used as regressors, or a named character vector that can be coerced to class "formula". The length of Qform must be equal to length(Lnodes) + length(Ynodes)** and the names and order of the formulas must be the same as the names and order of the L and Y nodes in data. The left hand side of each formula should be "Q.kplus1". If SL.library is NULL, glm will be called using the elements of Qform. If SL.library is specified, SuperLearner will be called after a design matrix is created using Qform.

** If there is a "block" of L and Y nodes not separated by A or C nodes, only one regression is required at the first L/Y node in a block. You can pass regression formulas for the other L/Y nodes, but they will be ignored (with a message). See example 5.

gform should be NULL, in which case all parent nodes of each L and Y node will be used as regressors, or a character vector that can be coerced to class "formula", or a matrix/array of Prob(A=1). If gform is a character vector, the length of gform must be equal to length(Anodes) + length(Cnodes) and the order of the formulas must be the same as the order the A and C nodes appear in data. The left hand side of each formula should be the name of the Anode or Cnode. If SL.library is NULL, glm will be called using the elements of gform. If SL.library is specified, SuperLearner will be called after a design matrix is created using gform.

In ltmle, gform can also be a n x numACnodes matrix where entry (i, j) is the probability that the ith observation of the jth A/C node is 1 (if an Anode) or uncensored (if a Cnode), conditional on following abar up to that node. In ltmleMSM, gform can similarly be a n x numACnodes x numRegimes array, where entry (i, j, k) is the probability that the ith observation of the jth A/C node is 1 (if an Anode) or uncensored (if a Cnode), conditional on following regime k up to that node. If gform is a matrix/array, deterministic.g.function will not be used and should be NULL.

abar specifies the counterfactual values of the Anodes, using the order they appear in data and should have the same length (if abar is a vector) or number of columns (if abar is a matrix) as Anodes.

rule can be used to specify a dynamic treatment rule. rule is a function applied to each row of data which returns a numeric vector of the same length as Anodes.

abar and rule cannot both be specified. If one of them if a list of length 2, additive treatment effect, risk ratio, and odds ratio can be computed using summary.ltmleEffectMeasures.

regimes can be a binary array: n x numAnodes x numRegimes of counterfactual treatment or a list of 'rule' functions as described above for the rule argument for the ltmle function

deterministic.g.function can be a function used to specify model knowledge about value of Anodes and/or Cnodes that are set deterministically. For example, it may be the case that once a patient starts treatment, they always stay on treatment. For details on the form of the function and examples, see deterministic.g.function_template

deterministic.Q.function can be a function used to specify model knowledge about the final event state. For example, it may be the case that a patient can complete the study at some intermediate time point, in which case the probability of death is 0 (assuming they have not died already). For details on the form of the function and examples, see deterministic.Q.function_template

SL.library may be a character vector of libraries (or 'glm' or 'default'), in which case these libraries are used to estimate both Q and g OR a list with two components, Q and g, where each is a character vector of libraries (or 'glm' or 'default'). 'glm' indicates glm should be called instead of SuperLearner If SL.library is the string 'default', SL.library is set to list("SL.glm", "SL.stepAIC", "SL.bayesglm", c("SL.glm", "screen.corP"), c("SL.step", "screen.corP"), c("SL.step.forward", "screen.corP"), c("SL.stepAIC", "screen.corP"), c("SL.step.interaction", "screen.corP"), c("SL.bayesglm", "screen.corP"). Note that the default set of libraries consists of main terms models. It may be advisable to include squared terms, interaction terms, etc in gform and Qform or include libraries that consider non-linear terms.

If attr(SL.library, "return.fit") == TRUE, then fit$g and fit$Q will return full SuperLearner or glm objects. If not, only a summary matrix will be returned to save memory.

The print method for ltmle objects only prints the tmle estimates.

Value

ltmle returns an object of class "ltmle" (unless abar or rule is a list, in which case it returns an object of class ltmleSummaryMeasures, which has the same components as ltmleMSM.) The function summary (i.e. summary.ltmle) can be used to obtain or print a summary of the results. An object of class "ltmle" is a list containing the following components:

• tmle - vector of influence curve values for Targeted Maximum Likelihood Estimate [NULL if gcomp is TRUE]

• iptw - vector of influence curve values for Inverse Probability of Treatment/Censoring Weighted estimate

• gcomp - vector of influence curve values for Targeted Maximum Likelihood Estimate without updating [NULL if gcomp is FALSE]

• g - list of length numACnodes - glm or SuperLearner (see Details) return objects from fitting g regressions

• Q - list of length numLYnodes - glm or SuperLearner (see Details) return objects from fitting Q regressions

• Qstar - list of length numLYnodes - glm (or numerical optimization if glm fails to solve the score equation) return objects from updating the Q fit

ltmleMSM returns an object of class "ltmleMSM" The function summary (i.e. summary.ltmleMSM) can be used to obtain or print a summary of the results. An object of class "ltmleMSM" is a list containing the following components:

• g - list of length numRegimes of list of length numACnodes - glm or SuperLearner (see Details) return objects from fitting g regressions

• Q - list of length numLYnodes - glm or SuperLearner (see Details) return objects from fitting Q regressions

• Qstar - list of length numLYnodes - glm (or numerical optimization if glm fails to solve the score equation) return objects from updating the Q fit

Functions

• ltmleMSM(): Longitudinal Targeted Maximum Likelihood Estimation for a Marginal Structural Model

Author(s)

Joshua Schwab jschwab77@berkeley.edu, Samuel Lendle, Maya Petersen, and Mark van der Laan

See Also

summary.ltmle, summary.ltmleMSM, SuperLearner, deterministic.g.function_template, deterministic.Q.function_template

Examples

# See \url{http://joshuaschwab.github.io/ltmle/} for more examples.

rexpit <- function(x) rbinom(n=length(x), size=1, prob=plogis(x))

# Single time point Example
n <- 1000
W <- rnorm(n)
A <- rexpit(-1 + 2 * W)
Y <- rexpit(W + A)
data <- data.frame(W, A, Y)

result1 <- ltmle(data, Anodes="A", Ynodes="Y", abar=1)
summary(result1)
summary(result1, estimator="iptw")
# MSM Example
# Given data over 3 time points where A switches to 1 once and then stays 1. We want to know
# how death varies as a function of gender, time and an indicator of whether a patient's
# intended regime was to switch before time.
# Note that working.msm includes time and switch.time, which are columns of
# summary.measures; working.msm also includes male, which is ok because it is a baseline
# covariate (it comes before any A/C/L/Y nodes).
data(sampleDataForLtmleMSM)
Anodes <- grep("^A", names(sampleDataForLtmleMSM$data))
Lnodes <- c("CD4_1", "CD4_2")
Ynodes <- grep("^Y", names(sampleDataForLtmleMSM$data))
msm.weights <- matrix(1:12, nrow=4, ncol=3) #just an example (can also use a 200x3x4 array),
                                            #or NULL (for no weights), or "empirical" (the default)

result2 <- ltmleMSM(sampleDataForLtmleMSM$data, Anodes=Anodes, Lnodes=Lnodes, Ynodes=Ynodes,
                   survivalOutcome=TRUE,
                   regimes=sampleDataForLtmleMSM$regimes,
                   summary.measures=sampleDataForLtmleMSM$summary.measures, final.Ynodes=Ynodes,
                   working.msm="Y ~ male + time + I(pmax(time - switch.time, 0))",
                   msm.weights=msm.weights, estimate.time=FALSE)
print(summary(result2))

Sample data, regimes, and summary measures

Description

Sample data for use with ltmleMSM. Data: n=1000: male age CD4_1 A1 Y1 CD4_2 A2 Y2 CD4_3 A3 Y3 A1..A3 are treatment nodes, Y1..Y3 are death, CD4_1..CD4_3 are time varying covariates. We are interested in static regimes where a patient switches at some time. In summary.measures, switch.time is first time where At is 1 (4 if never switch), time is the horizon.

Format

List with three components: data, regimes, summary.measures

Details

regimes: 200 x 3 x 4 [n x numACnodes x numRegimes] summary.measures: 4 x 2 x 3 [numRegimes x numSummaryMeasures x numFinalYnodes]

Source

simulated data

Examples

data(sampleDataForLtmleMSM)

Get standard error, p-value, and confidence interval for one ltmle object Summarizing results from Longitudinal Targeted Maximum Likelihood Estimation (ltmle)

Description

These functions are methods for class ltmle or summary.ltmle objects.

Usage

## S3 method for class 'ltmle'
summary(object, estimator = ifelse(object$gcomp, "gcomp", "tmle"), ...)

## S3 method for class 'ltmleEffectMeasures'
summary(object, estimator = ifelse(object$gcomp, "gcomp", "tmle"), ...)

## S3 method for class 'ltmleMSM'
summary(object, estimator = ifelse(object$gcomp, "gcomp", "tmle"), ...)

## S3 method for class 'summary.ltmleMSM'
print(
  x,
  digits = max(3, getOption("digits") - 3),
  signif.stars = getOption("show.signif.stars"),
  ...
)

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

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

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

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

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

Arguments

Details

summary.ltmle returns the parameter value of the estimator, the estimated variance, a 95 percent confidence interval, and a p-value.

summary.ltmleEffectMeasures returns the additive treatment effect for each of the two objects in the abar list passed to ltmle. Relative risk, and odds ratio are also returned, along with the variance, confidence interval, and p-value for each.

summary.ltmleMSM returns a matrix of MSM parameter estimates.

Value

summary.ltmle returns an object of class "summary.ltmle", a list with components

summary.ltmleEffectMeasures returns an object of class "summary.ltmleEffectMeasures", a list with same components as summary.ltmle above, but also includes:

summary.ltmleMSM returns an object of class "summary.ltmleMSM", a matrix with rows for each MSM parameter and columns for the point estimate, standard error, 2.5percent confidence interval, 97.5percent confidence interval, and p-value.

See Also

ltmle, summary

Examples

rexpit <- function(x) rbinom(n = length(x), size = 1, prob = plogis(x))

# Compare the expected outcomes under two counterfactual plans: Treatment plan:
# set A1 to 1 if W > 0, set A2 to 1 if W > 1.5, always set A3 to 1 Control plan:
# always set A1, A2, and A3 to 0
W <- rnorm(1000)
A1 <- rexpit(W)
A2 <- rexpit(W + 2 * A1)
A3 <- rexpit(2 * A1 - A2)
Y <- rexpit(W - A1 + 0.5 * A2 + 2 * A3)
data <- data.frame(W, A1, A2, A3, Y)
treatment <- cbind(W > 0, W > 1.5, 1)
control <- matrix(0, nrow = 1000, ncol = 3)
result <- ltmle(data, Anodes = c("A1", "A2", "A3"), Ynodes = "Y", abar = list(treatment,
    control))
print(summary(result))

## For examples of summary.ltmle and summary.ltmleMSM, see example(ltmle)