continuous or binary outcome variable

binary treatment indicator, 1 - treatment, 0 - control

vector, matrix, or dataframe containing baseline covariates

optional binary indicator for intermediate covariate for controlled direct effect estimation

indicator of missing outcome or treatment assignment. 1 - observed, 0 - missing

optional nx2 matrix of initial values for Q portion of the likelihood, (E(Y|A=0,W), E(Y|A=1,W))

optional nx2 matrix of initial values for Q portion of the likelihood, (E(Y|Z=1,A=0,W), E(Y|Z=1,A=1,W)). (When specified, values for E(Y|Z=0,A=0,W), E(Y|Z=0,A=1,W) are passed in using the Q argument

optional regression formula for estimation of E(Y|A,W), suitable for call to glm

vector of upper and lower bounds on Y and predicted values for initial Q. Defaults to the range of Y, widened by 10% of the min and max values.

optional vector of prediction algorithms to use for SuperLearner estimation of initial Q

logical, if TRUE, estimates cross-validated predicted values, default=TRUE

optional vector of conditional treatment assingment probabilities, P(A=1|W)

optional regression formula of the form A~W, if specified this overrides the call to SuperLearner

value between (0,1) for truncation of predicted probabilities. See Details section for more information

optionalnx2 matrix of conditional probabilities P(Z=1|A=0,W), P(Z=1|A=1,W)

optional regression formula of the form Z~A+W, if specified this overrides the call to SuperLearner

optional matrix of conditional probabilities for missingness mechanism, nx2 when Z is NULL P(Delta=1|A=0,W), P(Delta=1|A=1,W). nx4 otherwise, P(Delta=1|Z=0,A=0,W), P(Delta=1|Z=0,A=1,W), P(Delta=1|Z=1,A=0,W), P(Delta=1|Z=1,A=1,W)

optional regression formula of the form Delta~A+W, if specified this overrides the call to SuperLearner

optional vector of prediction algorithms to use for SuperLearner estimation of g1W

optional vector of prediction algorithms to use for SuperLearner estimation of pDelta1

family specification for working regression models, generally ‘gaussian’ for continuous outcomes (default), ‘binomial’ for binary outcomes

‘logistic’ (default), or ‘linear’

used to keep predicted initial values bounded away from (0,1) for logistic fluctuation

optional subject identifier

Number of cross-validation folds for estimating Q, and for super learner estimation of g

status messages printed if set to TRUE (default=FALSE)

if TRUE, discreteSL is used instead of ensemble SL. Ignored when SL not used to estimate Q

if TRUE, discreteSL is used instead of ensemble SL. Ignored when SL not used to estimate g1W

if TRUE, discreteSL is used instead of ensemble SL. Ignored when SL not used to estimate P(Delta = 1 | A,W,L)

Screen covariates before estimating g in order to retain only those associated with Stage 1 residuals

Minimum number of covariates to retain when prescreening covariates for g. Ignored when prescreenW.g=FALSE

Flag indicating whether to retain covariates associated with the outcome RESID=FALSE, or associated only with the residuals from the outcome regression. Ignored when prescreenW.g=FALSE

When TRUE, move g from denominator of clever covariate to the weight when fitting epsilon

When TRUE, all tuning parameters are set to their default values. Number of cross validation folds and truncation level for g are set data-adaptively based on sample size (see details).

list with elements EY1 (population mean), ATE (additive treatment effect), ATT (additive treatment effect among the treated), ATC (additive treatment effect among the controls), RR (relative risk), OR (odds ratio). Each element in the estimates of these is itself a list containing

• psi - parameter estimate

• pvalue - two-sided p-value

• CI - 95% confidence interval

• var.psi - Influence-curve based variance of estimate (ATE parameter only)

• log.psi - Parameter estimate on log scale (RR and OR parameters)

• var.log.psi - Influence-curve based variance of estimate on log scale (RR and OR parameters)

initial estimate of Q. Qinit$coef are the coefficients for a glm model for Q, if applicable. Qinit$Q is an nx2 matrix, where n is the number of observations. Columns contain predicted values for Q(0,W),Q(1,W) using the initial fit. Qinit$type is method for estimating Q. Qinit$Rsq is Rsq for initial estimate of Q. Qinit$Rsq.type empirical or cross-validated (depends on value of cvQinit), Rsq or pseudo-Rsq when Y is binary.

targeted estimate of Q, an nx2 matrix with predicted values for Q(0,W), Q(1,W) using the updated fit

treatment mechanism estimate. A list with four items: g$g1W contains estimates of P(A=1|W) for each observation, g$coef the coefficients for the model for g when glm used, g$type estimation procedure, g$discreteSL flag, g$AUC empirical AUC if ROCR package is available

intermediate covariate assignment estimate (when applicable). A list with four items: g.Z$g1W an nx2 matrix containing values of P(Z=1|A=1,W), P(Z=1|A=0,W) for each observation, g.Z$coef the coefficients for the model for g when glm used, g.Z$type estimation procedure, g.Z$discreteSL flag

missingness mechanism estimate. A list with four items: g.Delta$g1W an nx4 matrix containing values of P(Delta=1|Z,A,W) for each observation, with (Z=0,A=0), (Z=0,A=1), (Z=1,A=0),(Z=1,A=1). (When Z is NULL, columns 3 and 4 are duplicates of 1 and 2.) g.Delta$coef the coefficients for the model for g when glm used, g.Delta$type estimation procedure, g.Delta$discreteSL flag

bounds used to truncate g

bounds used to truncated g for ATT and ATC estimation

names of covariates used to model the components of g