Diagnose extrapolation for binary exposures with Gower distances and the convex hull

check_extrapolation() implements the extrapolation diagnostics of King and Zeng (2006) for binary point exposures. It measures, for every observation, how well the opposite exposure group covers its position in covariate space, using Gower distances that handle mixed numeric and categorical covariates, and it optionally tests convex-hull membership against the opposite group.

Usage

check_extrapolation(
  .data,
  .exposure,
  .covariates,
  nearby = 1,
  hull = NULL,
  exposure_type = c("auto", "binary")
)

Arguments

.data

A data frame.

.exposure

The binary exposure column, selected with data-masking. check_extrapolation() aborts unless the exposure has exactly two distinct values, whether that type is detected or declared through exposure_type.

.covariates

The covariate columns, selected with tidyselect. Numeric and categorical covariates are both supported by the Gower distance.

nearby

The reference radius as a multiple of the geometric variability. An observation counts an opposite-group unit as nearby when their Gower distance is at most nearby times the geometric variability. A single positive number, defaulting to 1.

hull

Whether to run the convex-hull membership test. NULL (the default) runs it automatically when there is at least one and at most ten numeric covariates and lpSolve is installed, and skips it otherwise. TRUE forces it to run, aborting when lpSolve is not installed and warning when there are no numeric covariates or the dimension is high. FALSE skips it.

exposure_type

One of "auto" (detect from the data, the default) or "binary". A supplied type is authoritative and detection is not consulted, so "binary" is rejected only when the exposure column does not have exactly two distinct values. Declaring it unlocks no call that "auto" would refuse, since detection reads any two-valued column as binary; it is accepted so that every diagnostic takes the same argument.

Value

An extrapolation_result object, an S7 subclass of positivity_diagnostic. Its @results tibble has one row per observation with columns .id (the row index), exposure (the exposure value), frac_nearby, gower_min, gower_mean, in_hull (the hull membership flag, or NA when the hull test did not run), and low_support (TRUE when the nearest opposite-group unit lies farther away than one geometric variability). It also carries the scalar properties @geometric_variability and @hull_run.

summary() aggregates the results by exposure group, returning one row per exposure level with the columns exposure, n, mean_gower_min, prop_supported (the one geometric variability fraction described in Details), and prop_in_hull (the fraction inside the opposite group's hull, NA when the hull test did not run).

generics::glance() returns a one-row tibble with n (the sample size), nearby, geometric_variability, mean_frac_nearby, prop_supported, hull_run (kept logical), and prop_in_hull, which is NA when the hull test did not run. The two readings, prop_supported and prop_in_hull, answer different questions and are both reported.

Details

Estimating a causal contrast requires comparing treated and untreated units with similar covariates. Where one exposure group has no nearby members of the other, the contrast rests on extrapolation rather than data, the concern framed by Petersen and colleagues (2012) and by King and Zeng (2006).

The Gower distance between two rows is the mean over covariates of a per-covariate distance: for a numeric covariate, the absolute difference divided by the pooled sample range; for a factor or character covariate, one when the categories differ and zero when they agree. Distances therefore lie in [0, 1] and mixed covariate types combine on a common scale. Because the denominator is the sample range, a single extreme covariate value rescales every distance on that axis.

The geometric variability is one half the mean of the full Gower distance matrix, the Cuadras-Arenas quantity used by the WhatIf software. It sets the reference radius: an observation is counted as nearby-supported when an opposite-group unit lies within nearby geometric variabilities. Three per-unit statistics are reported: gower_min (distance to the nearest opposite-group unit), gower_mean (mean distance to the opposite group), and frac_nearby (the fraction of the opposite group within nearby geometric variabilities). A planted support gap raises gower_min and lowers frac_nearby in both directions. The per-group prop_supported reported by summary(), the prop_supported statistic in glance(), and the print method share one fixed definition: the fraction of units whose nearest opposite-group unit lies within one geometric variability (gower_min <= geometric_variability), independent of nearby. The per-unit low_support column is the complement of that condition, so prop_supported is the fraction of rows whose low_support is FALSE.

When every covariate is constant, all rows coincide, the geometric variability is zero, and each observation is at distance zero from the opposite group. Such an observation is fully supported: gower_min is zero and frac_nearby is one. Constant numeric covariates likewise contribute zero distance on their own axis without affecting the others.

The convex-hull test asks whether an observation lies inside the convex hull of the opposite group, solved as a linear-programming feasibility problem (weights that are non-negative, sum to one, and reproduce the point). The hull test uses the numeric covariates only, since categorical covariates carry no ordering to interpolate over. In-hull membership collapses toward zero as the covariate dimension grows even under perfect overlap, so the test is degenerate in high dimensions (D'Amour and colleagues, 2021). It therefore runs automatically only when there is at least one and at most ten numeric covariates and lpSolve is installed. With no numeric covariates, more than ten of them, or lpSolve absent, it is skipped and in_hull is NA. Setting hull = TRUE forces the test to run and warns when the dimension is high; hull = FALSE skips it.

References

King G, Zeng L (2006). The Dangers of Extreme Counterfactuals. Political Analysis, 14(2):131-159. doi:10.1093/pan/mpj004

Petersen ML, Porter KE, Gruber S, Wang Y, van der Laan MJ (2012). Diagnosing and responding to violations in the positivity assumption. Statistical Methods in Medical Research, 21(1):31-54. doi:10.1177/0962280210386207

D'Amour A, Ding P, Feller A, Lei L, Sekhon J (2021). Overlap in Observational Studies with High-Dimensional Covariates. Journal of Econometrics, 221(2):644-654. doi:10.1016/j.jeconom.2019.10.014

Examples

set.seed(1)
n <- 100
x1 <- rnorm(n)
x2 <- rnorm(n)
exposure <- rbinom(n, 1, plogis(0.5 * x1 - 0.5 * x2))
df <- data.frame(exposure = exposure, x1 = x1, x2 = x2)

# hull = FALSE keeps the example free of the optional lpSolve dependency.
result <- check_extrapolation(df, exposure, c(x1, x2), hull = FALSE)
result
#> 
#> ── Extrapolation ───────────────────────────────────────────────────────────────
#> Exposure: "exposure" (binary)
#> Observations: 100
#> Geometric variability: 0.118
#> Nearby radius (1 x gv): 0.118
#> Mean fraction nearby: 0.156
#> Nearest opposite within one geometric variability: 97 of 100
#> Convex-hull test: not run