# Variable importance with coarsened data

## Introduction

In some settings, we don’t have access to the full data unit on each observation in our sample. These “coarsened-data” settings (see, e.g., Van der Vaart (2000)) create a layer of complication in estimating variable importance. In particular, the efficient influence function (EIF) in the coarsened-data setting is more complex, and involves estimating an additional quantity: the projection of the full-data EIF (estimated on the fully-observed sample) onto the variables that are always observed (Chapter 25.5.3 of Van der Vaart (2000); see also Example 6 in Williamson, Gilbert, Simon, et al. (2021)).

## Coarsened data in vimp

vimp can handle coarsened data, with the specification of several arguments:

• C: and binary indicator vector, denoting which observations have been coarsened; 1 denotes fully observed, while 0 denotes coarsened.
• ipc_weights: inverse probability of coarsening weights, assumed to already be inverted (i.e., ipc_weights = 1 / [estimated probability of coarsening]).
• ipc_est_type: the type of procedure used for coarsened-at-random settings; options are "ipw" (for inverse probability weighting) or "aipw" (for augmented inverse probability weighting). Only used if C is not all equal to 1.
• Z: a character vector specifying the variable(s) among Y and X that are thought to play a role in the coarsening mechanism. To specify the outcome, use "Y"; to specify covariates, use a character number corresponding to the desired position in X (e.g., "1" or "X1" [the latter is case-insensitive]).

Z plays a role in the additional estimation mentioned above. Unless otherwise specified, an internal call to SuperLearner regresses the full-data EIF (estimated on the fully-observed data) onto a matrix that is the parsed version of Z. If you wish to use any covariates from X as part of your coarsening mechanism (and thus include them in Z), and they have different names from X1, …, then you must use character numbers (i.e., "1" refers to the first variable, etc.) to refer to the variables to include in Z. Otherwise, vimp will throw an error.

## Example with missing outcomes

In this example, the outcome Y is subject to missingness. We generate data as follows:

set.seed(1234)
p <- 2
n <- 100
x <- replicate(p, stats::rnorm(n, 0, 1))
# apply the function to the x's
y <- 1 + 0.5 * x[, 1] + 0.75 * x[, 2] + stats::rnorm(n, 0, 1)
# indicator of observing Y
logit_g_x <- .01 * x[, 1] + .05 * x[, 2] - 2.5
g_x <- exp(logit_g_x) / (1 + exp(logit_g_x))
C <- rbinom(n, size = 1, prob = g_x)
obs_y <- y
obs_y[C == 0] <- NA
x_df <- as.data.frame(x)
full_df <- data.frame(Y = obs_y, x_df, C = C)

Next, we estimate the relevant components for vimp:

library("vimp")
library("SuperLearner")
# estimate the probability of missing outcome
ipc_weights <- 1 / predict(glm(C ~ V1 + V2, family = "binomial", data = full_df),
type = "response")

# set up the SL
learners <- c("SL.glm", "SL.mean")
V <- 2

# estimate vim for X2
set.seed(1234)
est <- vim(Y = obs_y, X = x_df, indx = 2, type = "r_squared", run_regression = TRUE,
SL.library = learners, alpha = 0.05, delta = 0, C = C, Z = c("Y", "1"),
ipc_weights = ipc_weights, cvControl = list(V = V))
## Warning: All algorithms have zero weight
## Warning: All metalearner coefficients are zero, predictions will all be equal to
## 0

## Warning: All metalearner coefficients are zero, predictions will all be equal to
## 0

## Example with two-phase sampling

In this example, we observe outcome Y and covariate X1 on all participants in a study. Based on the value of Y and X1, we include some participants in a second-phase sample, and further measure covariate X2 on these participants. This is an example of a two-phase study. We generate data as follows:

set.seed(4747)
p <- 2
n <- 100
x <- replicate(p, stats::rnorm(n, 0, 1))
# apply the function to the x's
y <- 1 + 0.5 * x[, 1] + 0.75 * x[, 2] + stats::rnorm(n, 0, 1)
# make this a two-phase study, assume that X2 is only measured on
# subjects in the second phase; note C = 1 is inclusion
C <- rbinom(n, size = 1, prob = exp(y + 0.1 * x[, 1]) / (1 + exp(y + 0.1 * x[, 1])))
tmp_x <- x
tmp_x[C == 0, 2] <- NA
x <- tmp_x
x_df <- as.data.frame(x)
full_df <- data.frame(Y = y, x_df, C = C)

If we want to estimate variable importance of X2, we need to use the coarsened-data arguments in vimp. This can be accomplished in the following manner:

library("vimp")
library("SuperLearner")
# estimate the probability of being included only in the first phase sample
ipc_weights <- 1 / predict(glm(C ~ y + V1, family = "binomial", data = full_df),
type = "response")

# set up the SL
learners <- c("SL.glm")
V <- 2

# estimate vim for X2
set.seed(1234)
est <- vim(Y = y, X = x_df, indx = 2, type = "r_squared", run_regression = TRUE,
SL.library = learners, alpha = 0.05, delta = 0, C = C, Z = c("Y", "1"),
ipc_weights = ipc_weights, cvControl = list(V = V), method = "method.CC_LS")
## Loading required package: quadprog
## Warning: package 'quadprog' was built under R version 4.0.3

# References

Van der Vaart, AW. 2000. “Asymptotic Statistics” 3.
Williamson, BD, PB Gilbert, NR Simon, et al. 2021. “A General Framework for Inference on Algorithm-Agnostic Variable Importance.” Journal of the American Statistical Association. https://doi.org/10.1080/01621459.2021.2003200.