Treatment effect estimation for linear models in the presence of multiple treatments and a potentially high-dimensional number of controls, i.e. \(p \gg n\) can be handled.
Confounder Importance Learning (CIL) proposes an estimation framework where the importance of the relationship between treatments and controls is factored in into the establishment of prior inclusion probabilities for each of these controls on the response model. This is combined with the use of non-local priors to obtain BMA estimates and posterior model probabilities.
cil is built on modelSelection and produces objects of type
cilfit. Use coef and postProb to obtain treatment effect
point estimates and posterior model probabilities, respectively, on this
object class.
cil(y, D, X, I = NULL, family = 'normal', familyD = 'normal',
R = 1e4, Rinit = 500, th.search = 'EB', mod1 = 'lasso_bic',
th.prior = 'unif', priorCoef, rho.min = NULL,
th.range = NULL, max.mod = 2^20, lpen = 'lambda.1se',
eps = 1e-10, bvs.fit0 = NULL, th.EP = NULL, center = TRUE, scale =
TRUE, includevars, verbose = TRUE)Object of class cilfit, which extends a list with elements
BMA estimates, 0.95 intervals and posterior inclusion
probabilities for treatment effects in D
BMA inference for treatment effects and all other covariates
matrix returning the posterior model probabilities
computed in the CIL model
numeric vector containing the estimated marginal
posterior inclusion probabilities of the featured treatments and controls
Marginal prior inclusion probabilities, as estimated by CIL
Marginal posterior inclusion probabilities that would be obtained under a uniform model prior
Values used for the hyper-parameter theta, estimated according
to the argument th.search specified
Estimated weights of the effect of the control variables
on each of the treatments, as estimated with the method specified in argument
mod1
Object returned by modelSelection (of class msfit)
of the final model estimated by CIL.
Estimated values of theta using the EP algorithm. It coincides
with theta.hat if the argument th.search is set to EB
Initial msfit object used to estimate the inital model
where all elements in theta are set to zero (used in the optimisation process
of this hyper-parameter)
one-column matrix containing the observed responses. The response must be continuous (currently the only type supported)
treatment matrix with numeric columns, continuous or discrete. Any finite
number of treatments are supported. If only one treatment is provided, supply
this object in the same format used for y
matrix of controls with numeric columns, continuous or discrete. If only
one treatment is provided, supply this object in the same format used for y
matrix with the desired interaction terms between D and
X. If not informed, i.e. supplied as the default NULL, this term
will not be included into the response model
Distribution of the outcome, e.g. 'normal', 'binomial' or
'poisson'. See help(modelSelection) for a full list of options
Distribution of the treatment(s). Only 'normal' or 'binomial' currently allowed
Number of MCMC iterations to be
run by modelSelection on each stage of CIL (see argument niter
therein)
MCMC iterations to estimate marginal posterior inclusion probabilities under a uniform model prior, needed for EP
method to estimate theta values in the marginal prior inclusion
probabilities of the CIL model. Options are: EB (Empirical Bayes, based
on maximum marginal likelihood) and EP (Expectation propagation
approximation)
method to estimate the feature parameters corresponding to the
influence of the controls on the treatments. Supported values for this
argument are 'ginv' (generalised pseudo-inverse), lasso (see
argument lpen), lasso_bic (default), and ridge)
prior associated to the thetas for the Empirical Bayes
estimation. Currently only unif (Uniform prior) is supported,
effectively making the EB approach the maximisation of the marginal likelihood
Prior on the response model parameters, see modelSelection
value of \(\rho\) in (0, 1/2) employed in the prior probability
model of CIL. If left uninformed, i.e. supplied as the default NULL,
it will be set to \(1/p^2\), where p is the dimension of the response model.
sequence of values to be considered in the grid when searching for points to initialise the search for the optimal theta parameters. If left uninformed, the function will determine a computationally suitable grid depending on the number of parameters to be estimated
Maximum number of models considered when computing the marginal
likelihood required by empirical Bayes.
If set to Inf all visited models by the enumeration/MCMC
are considered, but it might be computationally desirable to restrict this
number when the dimension of D and/or X is large
penalty type supplied to glmnet if mod1 is set to
lasso. Default is lambda.1se (see documentation corresponding to
glmnet for options on how to set this parameter)
small scalar used to avoid round-offs to absolute zeroes or ones in marginal prior inclusion probabilities.
object returned by modelSelection under \(\theta = 0\),
used as a model exploration tool to compute EB approximation on the thetas.
This argument is only supposed to be used in case of a second computation the
model on the same data where th.search has ben changed to EB,
in order to avoid repeating the computation of the initial
modelSelection fit. To use this argument, supply the object residing
in the slot init.msfit of a cilfit-class object.
Optimal theta values under the EP approximation, obtained in a
previous CIL run. This argument is only supposed to be used in case of
a second computation the model on the same data where th.search
has ben changed to EB, in order to save the cost of the EP search
to initialise the optimisation algorithm. To use this argument, supply the
object residing int the slot th.hat of a cilfit-class
object.
If TRUE, y and x are centered to have
zero mean. Dummy variables corresponding to factors are NOT centered
If TRUE, y and columns in x are
scaled to have variance=1. Dummy variables corresponding to factors are NOT scaled
Logical vector of length ncol(x) indicating variables that should always be included in the model, i.e. variable selection is not performed for these variables
Set verbose==TRUE to print iteration progress
Miquel Torrens
We estimate treatment effects for the features present in the treatment
matrix D. Features in X, which may or may not be causal
factors of the treatments of interest, only act as controls and, therefore,
are not used as inferential subjects.
Confounder importance learning is a flexible treatment effect estimation
framework that essentially determines how the role of the influence of
X on D should affect their relationship with the response,
through establishing prior inclusion probabilities on the response model
for y according to said role. This is regulated through a hyper-
parameter theta that is set according to the method supplied to
th.search. While the EB option obtains a more precise estimate
a priori, the EP alternative achieves a reasonable approximation at a
fraction of the computational cost.
See references for further details on implementation and computation.
Torrens i Dinares M., Papaspiliopoulos O., Rossell D. Confounder importance learning for treatment effect inference. https://arxiv.org/abs/2110.00314, 2021, 1--48.
postProb to obtain posterior model probabilities.
coef for inference on the treatment parameters.
# Simulate data
set.seed(1)
X <- matrix(rnorm(100 * 50), nrow = 100, ncol = 50)
beta_y <- matrix(c(rep(1, 6), rep(0, 44)), ncol = 1)
beta_d <- matrix(c(rep(1, 6), rep(0, 44)), ncol = 1)
alpha <- 1
d <- X %*% beta_d + rnorm(100)
y <- d * alpha + X %*% beta_y + rnorm(100)
# Confounder Importance Learning
fit1 <- cil(y = y, D = d, X = X, th.search = 'EP')
# BMA for treatment effects
coef(fit1)
# BMA for all covariates
head(fit1$coef)
# Estimated prior inclusion prob
# vs. treatment regression coefficients
plotprior(fit1)
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