Package 'marbounds'

Clip propensity and probability estimates away from 0 and 1

Description

Clip propensity and probability estimates away from 0 and 1

Usage

clip_probs(p, eps = 1e-06)

Arguments

p	Numeric vector of probabilities.
eps	Small positive number (default 1e-6).

Value

Clipped vector.

---

Coefficient vectors b such that bound = mean(b' phi)

Description

theta = (E(mu0), E(mu1), E(pi0), E(pi1), E(mu0*pi0), E(mu1*pi1)) = (m1_0, m1_1, m2_0, m2_1, m3_0, m3_1) Order of phi columns: phi_1_0, phi_1_1, phi_2_0, phi_2_1, phi_3_0, phi_3_1

Usage

coef_general_lower_ate()

Details

All bounds/estimands are linear: value = b

---

Compute all requested bound/point estimates from influence matrix and parameters

Description

Compute all requested bound/point estimates from influence matrix and parameters

Usage

compute_bounds(
  phi,
  estimand = c("ate", "psi1", "psi2"),
  assumption = c("general", "bounded_delta", "monotonicity_pos", "monotonicity_neg",
    "bounded_risk", "bounded_risk_unbounded_tau", "point_ate", "point_psi1",
    "point_psi2"),
  delta_0u = 1,
  delta_1u = 1,
  delta_0l = 0,
  delta_1l = 0,
  delta_0 = NULL,
  delta_1 = NULL,
  tau_0 = NULL,
  tau_1 = NULL,
  tau = NULL,
  mu0 = NULL,
  mu1 = NULL,
  pi0 = NULL,
  pi1 = NULL,
  smooth_approximation = FALSE,
  epsilon = 0.001
)

Arguments

phi	n x 6 influence matrix.
estimand	"ate", "psi1" (composite ATE), "psi2" (SDE).
assumption	"general", "bounded_delta", "monotonicity_pos", "monotonicity_neg", "bounded_risk", "point_ate", "point_psi1", "point_psi2".
delta_0u, delta_1u	Upper bounds on proportion informative missingness (for bounded/mono bounds).
delta_0l, delta_1l	Lower bounds (for Psi_1 bounds).
delta_0, delta_1	Point values (for point identification).
tau_0, tau_1	Bounded risk ratio params (for bounded_risk).
tau	Single sensitivity parameter for point_ate (risk ratio in both arms).
mu0, mu1	Outcome nuisance functions (n-vectors).
pi0, pi1	Missingness probability nuisance functions (n-vectors).
smooth_approximation	Logical; for psi2 bounded_delta, use smooth approximation (TRUE) or indicator method (FALSE).
epsilon	Smoothing parameter for psi2 smooth approximation.

Value

List with elements estimate, lower, upper (as appropriate), and optional se_* for asymptotic SEs.

---

Estimate nuisance functions using SuperLearner with cross-fitting

Description

Estimates e(X)=P(A=1|X), pi_0(X)=P(C=1|X,A=0), pi_1(X)=P(C=1|X,A=1), mu_0(X)=E(Y|X,A=0,C=0), mu_1(X)=E(Y|X,A=1,C=0). Uses V-fold cross-fitting: for each fold, fits nuisances on the training part and predicts on the held-out part.

Usage

estimate_nuisance(
  X,
  A,
  C,
  Y,
  V = 2,
  sl_lib_prop = "SL.glm",
  sl_lib_miss = "SL.glm",
  sl_lib_outcome = "SL.glm",
  stratify_mu = TRUE,
  family_Y = "gaussian",
  seed = NULL
)

Arguments

X	Matrix of covariates.
A	Treatment indicator (0/1).
C	Missingness indicator (1=missing, 0=observed).
Y	Outcome (NA or value when C=0); only used when C=0 for outcome models.
V	Number of cross-fitting folds (default 2; the calling function may use V=1 when a simple single-library SuperLearner such as SL.glm is used).
sl_lib_prop	Character vector of SuperLearner library for propensity e (default "SL.glm").
sl_lib_miss	Character vector of SuperLearner library for missingness pi_0, pi_1 (default "SL.glm").
sl_lib_outcome	Character vector of SuperLearner library for outcome mu_0, mu_1 (default "SL.glm").
stratify_mu	Logical; if TRUE (default), estimate separate outcome models mu_0 and mu_1 stratified by treatment A. If FALSE, estimate a single pooled model E(Y|X,C=0) and predict for both A=0 and A=1.
family_Y	Character; family for outcome model ("gaussian" or "binomial"). Default "gaussian".
seed	Optional seed for fold splits.

Value

List with components: e, pi0, pi1, mu0, mu1 (each length n), and fold_id (fold index per row).

---

Bounds and point estimates for causal effects under mixed missingness

Description

Implements the methods from the paper: bounds on ATE, composite ATE (Psi_1), and separable direct effect (Psi_2), with user-specified estimand, assumptions, and sensitivity parameters. Nuisance functions are estimated via SuperLearner with cross-fitting.

Usage

mar_bounds(
  data,
  Y,
  A,
  C,
  X,
  estimand = c("ate", "psi1", "psi2"),
  assumption = c("general", "bounded_delta", "monotonicity_pos", "monotonicity_neg",
    "bounded_risk", "bounded_risk_unbounded_tau", "point_ate", "point_psi1",
    "point_psi2"),
  delta_0u = 1,
  delta_1u = 1,
  delta_0l = 0,
  delta_1l = 0,
  delta_0 = NULL,
  delta_1 = NULL,
  tau_0 = NULL,
  tau_1 = NULL,
  tau = NULL,
  V = 2,
  sl_lib = "SL.glm",
  sl_lib_prop = NULL,
  sl_lib_miss = NULL,
  sl_lib_outcome = NULL,
  stratify_mu = TRUE,
  family_Y = NULL,
  smooth_approximation = FALSE,
  epsilon = 0.001,
  seed = NULL,
  param_grid = NULL,
  grid_bounds = c("both", "lower", "upper"),
  alpha = 0.05,
  B = 1000,
  multiplier = c("rademacher", "gaussian")
)

Arguments

data	A data.frame containing the analysis data.
Y	Character; column name of the outcome (numeric; use NA when missing).
A	Character; column name of treatment (0/1).
C	Character; column name of missingness indicator (1 = missing, 0 = observed).
X	Character vector; column names of covariates.
estimand	One of "ate" (average treatment effect), "psi1" (composite ATE), "psi2" (separable direct effect).
assumption	One of "general", "bounded_delta", "monotonicity_pos", "monotonicity_neg", "bounded_risk", "bounded_risk_unbounded_tau", "point_ate", "point_psi1", "point_psi2". general No extra assumptions; widest bounds (ATE/Psi_1). bounded_delta Proportion of informative missingness bounded by delta_0u, delta_1u (and delta_0l, delta_1l for Psi_1). monotonicity_pos Informative missingness increases outcome risk; use with delta_0u, delta_1u. monotonicity_neg Informative missingness decreases outcome risk. bounded_risk Outcome risk ratio in informatively missing bounded by tau_0, tau_1, with an upper bound that uses the minimum of 1-mu and mu*(tau-1) (as in the paper). bounded_risk_unbounded_tau Original bounded-risk formula without applying the minimum-based tau constraint (provides backward-compatible behavior). point_ate Point identification under known tau and delta_0, delta_1 (Assumption known-risk + known-missingness). point_psi1 Point identification of Psi_1 under known delta_0, delta_1. point_psi2 Point identification of Psi_2 under known delta_0.
delta_0u, delta_1u	Upper bounds on proportion of missingness that is informative (in [0,1]). Default 1.
delta_0l, delta_1l	Lower bounds (for Psi_1 bounds). Default 0.
delta_0, delta_1	Point values for point identification (when assumption is point_*).
tau_0, tau_1	Bounded risk parameters (>= 1) for "bounded_risk". Single tau for "point_ate" (same in both arms).
tau	Single sensitivity parameter for "point_ate" (risk ratio in both arms).
V	Number of cross-fitting folds for nuisance estimation (default 2; uses 1 when SL.glm is the only library for all nuisance functions).
sl_lib	A single SuperLearner library specification used for all nuisance models by default (propensity, missingness, outcome); defaults to "SL.glm".
sl_lib_prop, sl_lib_miss, sl_lib_outcome	Optional SuperLearner libraries for propensity, missingness, and outcome models. If NULL, they fall back to sl_lib.
stratify_mu	Logical; if TRUE (default), estimate separate outcome models for A=0 and A=1. If FALSE, estimate a single pooled model E(Y|X,C=0).
family_Y	Character; family for outcome model ("gaussian" or "binomial"). If NULL (default), automatically detects: "binomial" for binary Y (only 0/1 values), otherwise "gaussian". Set explicitly to override auto-detection.
smooth_approximation	Logical; for Psi_2 bounds under bounded_delta, use smooth approximation (TRUE) or indicator function method (FALSE, default).
epsilon	Smoothing parameter for Psi_2 smooth approximation (default 0.001). Only used when smooth_approximation = TRUE.
seed	Optional seed for cross-fitting splits.
param_grid	Optional list or data.frame describing a grid of sensitivity parameters for which to compute bounds and/or point estimates and confidence bands. For estimand = "ate": delta_0u, delta_1u, tau_0, tau_1, tau, delta_0, delta_1 (by assumption). For estimand = "psi1" (bounded_delta): delta_0u, delta_1u, delta_0l, delta_1l; (point_psi1): delta_0, delta_1. For estimand = "psi2" (point_psi2): delta_0. Column names may also be provided without underscores (e.g., delta0u).
grid_bounds	Which bounds to compute over the grid: "both" (default), "lower", or "upper". For point-identified grids (point_ate, point_psi1, point_psi2) this is ignored and a single estimate_grid is returned.
alpha	Significance level for pointwise and uniform (multiplier bootstrap) confidence intervals over the grid (default 0.05).
B	Number of multiplier bootstrap replications for uniform bands over the grid (default 1000).
multiplier	Multiplier distribution for the bootstrap: "rademacher" (default) or "gaussian".

Value

A list with components: naive (naive estimate assuming MAR), lower, upper (for bounds), or estimate (for point ID), with se_* or se and optional nuisance, phi for downstream use. result contains a convenient dataframe summary: for scalar results, it includes the estimates and confidence intervals; for grid results (param_grid provided), it contains the combined grid output in long format with bound_type column for bounds. When param_grid is provided, the list also contains lower_grid, upper_grid, and/or estimate_grid as separate list objects with pointwise and uniform inference.

---

Multiplier bootstrap for simultaneous inference over a parameter grid

Description

For a grid of sensitivity parameters, each defining a linear functional b' theta (e.g. a bound or point estimate), computes point estimates and then uses the multiplier bootstrap to construct simultaneous confidence intervals or bands. Let psi_i = b' phi_i be the influence function for the functional. The bootstrap draws G_1,...,G_n iid (e.g. Rademacher or N(0,1)) and forms T* = sqrt(n) * mean(G_i * psi_i). Repeating B times approximates the distribution of sqrt(n)(thetahat - theta).

Usage

multiplier_bootstrap_grid(
  mar_result,
  param_grid,
  bound_spec = c("lower", "upper", "point"),
  assumption = c("bounded_delta", "monotonicity_pos", "monotonicity_neg", "bounded_risk",
    "point_ate"),
  B = 1000,
  alpha = 0.05,
  multiplier = c("rademacher", "gaussian")
)

Arguments

mar_result	Output from mar_bounds() containing phi and optionally nuisance.
param_grid	Data.frame or list; each row (or combination) gives a parameter set. Columns must include those needed for the coefficient vector (e.g. delta_0u, delta_1u, tau, etc.). Alternatively, a list of vectors with names delta_0u, delta_1u, etc., which will be expanded via expand.grid.
bound_spec	Character: "lower", "upper", or "point". For "lower"/"upper" the grid rows define the sensitivity parameters for that bound (e.g. delta_0u, delta_1u for monotonicity_pos). For "point" the grid gives (delta_0, delta_1, tau) for point_ate.
assumption	Same as in mar_bounds: "bounded_delta", "monotonicity_pos", "monotonicity_neg", "bounded_risk", "point_ate".
B	Number of multiplier bootstrap replications (default 1000).
alpha	Significance level for simultaneous CIs (default 0.05).
multiplier	"rademacher" (default) or "gaussian".

Value

List with grid (data.frame with columns from param_grid plus estimate, se, ci_lower, ci_upper, simultaneous_lower, simultaneous_upper), boot_replicates (B x nrow(grid) matrix of bootstrap T* values), and critical_value (estimated 1-alpha quantile of max over grid of |T*|).

---

Empirical variance matrix of phi (for asymptotic variance of b' theta)

Description

Empirical variance matrix of phi (for asymptotic variance of b' theta)

Usage

phi_var(phi)

Arguments

phi	n x 6 matrix.

Value

6 x 6 variance matrix.

---

Naive estimate of the ATE (Psi_tilde) assuming MAR

Description

Returns the influence-function-based estimate and variance using the EIF for the naive estimand Psi_tilde = E[mu_1(X) - mu_0(X)], where mu_a(X) = E[Y | X, A = a, C = 0] (outcome mean among observed). Package convention: C = 0 denotes observed (non-missing), C = 1 denotes missing. The EIF is I(A=1,C=0)/(P(C=0|X,A=1)*P(A=1|X)) * (Y - mu_1(X)) - I(A=0,C=0)/(P(C=0|X,A=0)*P(A=0|X)) * (Y - mu_0(X)) + mu_1(X) - mu_0(X), with variance estimated by (1/n) * sample variance of the EIF.

Usage

psi_naive(data, Y, A, C, X, nuis = NULL)

Arguments

data	Data.frame.
Y, A, C, X	Column names as in mar_bounds.
nuis	Optional pre-computed nuisance list (e, pi0, pi1, mu0, mu1).

Value

List with estimate (EIF-based point estimate), var (variance estimate (1/n)*Var(EIF)), se (sqrt(var)), and optionally eif (length-n vector of EIF values).

---

Compute empirical mean of influence functions (point estimates of theta components)

Description

Compute empirical mean of influence functions (point estimates of theta components)

Usage

theta_hat(phi)

Arguments

phi	n x 6 matrix from influence_functions().

Value

Vector of length 6: (m1_0, m1_1, m2_0, m2_1, m3_0, m3_1).

---