An introduction to balnet
This vignette provides a brief introduction to the
balnet package. For additional details, see this paper.
A commonly used approach for estimating propensity scores in
observational studies is logistic regression, often combined with
regularization when the number of covariates is large or when
overfitting is a concern. The balnet package also fits
regularized logistic regression models, but replaces the traditional
maximum likelihood loss with covariate balancing loss functions
paired with a logistic link.
A key feature of this approach is that it directly produces
balancing weights, weights that approximately equalize
covariate distributions between treatment arms. These weights can be
used as plug-in components in inverse probability weighting (IPW)
estimators, or combined with outcome models in doubly robust procedures
such as AIPW and debiased machine learning.
By targeting covariate balance directly, the fitted propensity models
are explicitly tailored to the causal estimand of interest. The example
below illustrates these ideas in a simple simulated setting.
A toy example
We begin by simulating a small example in which treatment assignment
depends on a single pre-treatment covariate. In particular, units with
certain values of X_1 are less likely
to receive treatment.
n <- 100
p <- 25
X <- matrix(rnorm(n * p), n, p)
W <- rbinom(n, 1, 1 / (1 + exp(1 - X[, 1])))
Suppose we are interested in estimating an average treatment effect
(ATE). We can fit a balnet object using the default
options.
By default, this fits a lasso-regularized path of logistic models,
with tuning parameters and path construction chosen to mirror common
glmnet usage.
A few details are worth highlighting. When propensity scores are
estimated using covariate balancing loss functions, the fitted models
depend on the target estimand. For the ATE, balnet fits two
propensity score models: one for the control arm and one for the treated
arm. The control-arm model is used to estimate E[Y_i(0)], while the treated-arm model is
used to estimate E[Y_i(1)].
Printing the fitted object shows summary information for both arms.
By default, the output is truncated to display only the beginning and
end of the regularization path, the full path can be displayed by
increasing the max argument in print.
print(fit)
#> Call: balnet(X = X, W = W)
#>
#> Control (path: 57/100)
#> Nonzero Avg|SMD| Lambda
#> 1 0 0.06671 0.23010
#> 2 1 0.06612 0.21964
#> 3 1 0.06558 0.20966
#> ...
#> 55 19 0.01684 0.01866
#> 56 19 0.01623 0.01782
#> 57 21 0.01536 0.01701
#>
#> Treated (path: 24/100)
#> Nonzero Avg|SMD| Lambda
#> 1 0 0.18036 0.62211
#> 2 1 0.18199 0.59384
#> 3 1 0.18357 0.56685
#> ...
#> 22 11 0.18505 0.23422
#> 23 13 0.18283 0.22358
#> 24 16 0.17625 0.21341
The first column reports the number of nonzero coefficients and is
analogous to the output of glmnet. As in
glmnet, the regularization path starts at a value of \lambda corresponding to an intercept-only
model and proceeds in nlambda logarithmically spaced steps
down to a minimum value determined by lambda.min.ratio.
The next column reports the absolute standardized mean difference
(SMD), averaged across covariates. Importantly, balnet
always computes and reports balance metrics on the standardized
scale.
In this simulated example, it is not possible to find weights that
exactly balance the treated and control covariate means to the
overall sample means of X. As a result,
for both treatment arms the regularization path is truncated before
reaching the default path length of nlambda = 100. The
treated arm, in particular, is more difficult to balance.
The role of λ
For lasso-regularized generalized linear models, \lambda is often interpreted as a budget on
the overall magnitude of the coefficients. In the covariate balancing
framework, the interpretation is different. Covariate balancing loss
functions arise as the primal formulation of an optimization problem
that constrains imbalance. In the lasso case, \lambda directly controls balance: it equals
the maximum allowable absolute standardized mean difference
(SMD) across covariates (since balnet standardizes
covariates by default).
To illustrate this, consider \lambda^{\max}
\approx 0.62 for the treated arm in the printed output. This
value corresponds to the imbalance in the unweighted treatment arm data
and can be verified directly:
smd.baseline <- (colMeans(X[W == 1, ]) - colMeans(X)) /
(apply(X, 2, sd) * sqrt((n - 1) / n))
max(abs(smd.baseline))
#> [1] 0.622112
Since the smallest value of \lambda
attained for the treated arm is approximately \lambda_{\min} \approx 0.21, this indicates
that the closest we can bring the standardized treated covariate means
to the overall means is an absolute SMD of about 0.21.
This interpretation of \lambda
provides a convenient way to target a desired level of imbalance,
available through the option max.imbalance. For lasso
penalization, balnet then adjusts the generated \lambda sequence so that it terminates at
this value. The algorithm then attempts to compute the full
regularization path, stopping gracefully if further reductions in
imbalance are not achievable. Alternatively, users may compute \lambda^{\max} (e.g., the maximum absolute
unweighted SMD) for their dataset and then choose
lambda.min.ratio to reflect an acceptable fraction of this
maximum imbalance. For example, if \lambda^{\max} = 10, the default setting
lambda.min.ratio = 0.01 corresponds to a target maximum
absolute SMD of 10 \times 0.01 =
0.1.
Note: Setting lambda = 0 to try to achieve exact balance is
not recommended, just as glmnet advises against it.
balnet works best by using warm starts and gradually
decreasing regularization, a strategy similar to barrier methods in
convex optimization. This approach helps the algorithm converge reliably
and improves performance on real-world datasets where achieving
covariate balance can be difficult.
Plotting path diagnostics
balnet provides default plotting methods for visualizing
regularization path diagnostics. Calling plot without
additional arguments produces a summary of metrics along the path,
indexed by \lambda on the log
scale.
Two quantities are shown, both normalized to percentages. The first
is the percent bias reduction (PBR), which measures the reduction in
absolute SMD after weighting relative to the unweighted data. The second
is the effective sample size (ESS), defined as the squared sum of
weights divided by the sum of squared weights, normalized to sum to
100.
Recall that \lambda^{\max}
corresponds to the intercept-only (unweighted) fit. As \lambda decreases, covariate imbalance is
reduced, but at the cost of a smaller effective sample size, reflecting
increased concentration of weights on a subset of units.
Individual covariate SMDs can be visualized at specific values of
\lambda by supplying the
lambda argument. If the requested value is not exactly on
the fitted \lambda sequence, the
closest value is used; in particular, setting lambda = 0
selects the smallest value along the path. balnet then
predicts the corresponding propensity scores, constructs inverse
probability weights for the chosen estimand, and computes the resulting
SMDs.
The unweighted SMDs are shown at \lambda^{\max}, while colored points
correspond to the weighted SMDs at the selected \lambda. Separate panels are displayed for
the treated and control arms, reflecting the fact that distinct
propensity score models are fit for each arm in the ATE case. In
balnet, SMDs take the form (\text{weighted covariate mean} - \text{target
mean}) ~/~ \text{sd(target)}.
In this example, the plots suggest limited overlap for the treated
arm, indicating that the ATE may not be an appropriate target estimand.
Instead, we can target the average treatment effect on the treated (ATT)
by setting target = "ATT". In this case,
balnet fits a model that aims to balance control covariate
means toward those of the treated group.
fit.att <- balnet(X, W, target = "ATT")
plot(fit.att, lambda = 0)
Here, the resulting weights achieve substantially improved
balance.
For additional functionality, users are encouraged to consult the
documentation for the standard S3 methods provided by
balnet, including predict for propensity score
prediction and balweights for extracting balancing weights.
On large datasets, we recommend calling balnet with
verbose = TRUE to interactively print balance metrics
during fitting.