Why Check Support in More Than One Dimension?
Treated and control units can have similar covariate averages, similar marginal ranges, or overlapping propensity-score distributions and still occupy different regions of the joint covariate space. For example, two groups may have the same range for each of two covariates, but treated units may occur mostly when both covariates are high or both are low, while controls occur when one is high and the other is low. A comparison based only on separate ranges would miss that pattern.
Multivariate support diagnostics examine where covariate vectors occur, not just where each covariate occurs separately. They help identify whether the observed data contain plausible treated and control comparators across the covariate combinations relevant to an analysis. These diagnostics are descriptive evidence about support; they do not by themselves prove that causal effects are identified.
For continuous covariates, a single point can have probability zero and still belong to the population support. That is why support must not be defined as the set of values having positive probability individually. The common population support is the intersection of the treated and control supports. In a finite sample, overlap of empirical support is not a direct test of population overlap: two samples from the same continuous distribution will generally have no exactly repeated vectors.
The earlier material on conditional positivity and common score support distinguishes a condition on treatment probabilities from overlap in a one-dimensional score. Here the focus is the geometry of the measured covariates themselves. A practical assessment assumes that the covariates being compared are measured consistently, that the groups are defined correctly, and that the chosen covariates are scientifically relevant to the comparison. No plot can detect an important covariate that was not measured or included.
Pair Plots: Inspecting Joint Patterns
A pair plot displays each covariate against each other covariate, often with points colored by treatment group. With two covariates, it is a scatterplot; with several, it is a matrix of scatterplots. Look for regions containing one group but few or no observations from the other, separated clusters, curved patterns, or combinations that occur only in one group. Marginal histograms or separate ranges can supplement the plot, but they do not replace the joint panels.
Pair plots are most informative when the number of covariates is modest and axes are labeled in meaningful units. If scales differ greatly, the visual impression of closeness can be misleading. Plots also become crowded as the number of covariates grows, and pairwise panels can miss patterns involving three or more variables at once.
Worked Example: Pairwise Overlap That Hides a Joint Gap
Consider hypothetical data with three binary covariates, \(X_1,X_2,X_3\). The treated group has the vectors \((0,0,0),(0,1,1),(1,0,1),(1,1,0)\). The control group has \((0,0,1),(0,1,0),(1,0,0),(1,1,1)\). What would pair plots show?
In every two-covariate projection, each group has all four possible pairs. For example, the treated and control groups each contain \((X_1,X_2)=(0,0),(0,1),(1,0),(1,1)\). The same is true for the \((X_1,X_3)\) and \((X_2,X_3)\) panels. Thus, each pair plot panel shows identical empirical support for the pair.
Yet the complete three-dimensional vectors do not coincide. The treated vectors have an even number of 1s, while the control vectors have an odd number. The empirical supports in three dimensions are disjoint. Pair plots therefore provide useful projections, not a guarantee that the full joint support overlaps. In this small example, inspecting the full covariate combinations reveals what the pairwise views conceal.
Cross-Group Distances
A distance summary quantifies how far an observation is from the nearest observation in the opposite treatment group. Before calculating distances, specify how covariates are scaled. Euclidean distance in raw units can be dominated by a covariate measured on a numerically large scale. One common descriptive choice is to center and scale each continuous covariate using a stated reference scale, such as its pooled sample standard deviation, and then calculate Euclidean distances in the resulting standardized coordinates. This makes distances dimensionless, but does not make all directions equally informative or determine a universally meaningful cutoff.
Worked Example: Interpreting Nearest-Neighbor Distances
Suppose two covariates have already been standardized, so the coordinates below are in standard-deviation units. The hypothetical treated vectors are \(T_1=(0,0)\) and \(T_2=(0.2,0.1)\); the controls are \(C_1=(0.1,0)\) and \(C_2=(2,2)\). Use Euclidean distance to calculate each unit’s nearest cross-group distance.
For \(T_1\), the distance to \(C_1\) is \(\sqrt{(0-0.1)^2+(0-0)^2}=0.100\). The distance to \(C_2\) is \(\sqrt{(0-2)^2+(0-2)^2}=2.828\), so \(T_1\)'s nearest distance is \(0.100\). For \(T_2\), the distance to \(C_1\) is \(\sqrt{(0.2-0.1)^2+(0.1-0)^2}=\sqrt{0.02}=0.141\), while the distance to \(C_2\) is \(\sqrt{1.8^2+1.9^2}=\sqrt{6.85}=2.617\). Its nearest distance is \(0.141\).
For the control direction, \(C_1\)'s nearest treated distance is \(0.100\). For \(C_2\), the distance to \(T_1\) is \(2.828\), and the distance to \(T_2\) is \(\sqrt{1.8^2+1.9^2}=2.617\); its nearest distance is \(2.617\). Thus the treated-to-control nearest distances are \(0.100\) and \(0.141\), whereas the control-to-treated distances are \(0.100\) and \(2.617\). The large control-side distance signals that one control has no close treated counterpart in this sample. It does not establish that the underlying population has no treated support near that vector.
The direction-specific summaries matter. A group may be concentrated in a region represented by the other group, while the other group also contains observations far from any counterpart. A single pooled mean distance can hide this asymmetry. Distance summaries also depend on the metric and scaling; a Mahalanobis distance, for example, accounts for covariance among coordinates but can be unstable when the covariance estimate is poorly conditioned or the dimension is large. State the metric and scaling whenever reporting distances.
Convex-Hull Checks
The convex hull of a group’s observed vectors is the smallest convex set containing them. A point lies in that hull if it can be written as a weighted average of the group’s observed vectors, with nonnegative weights that sum to one. Hull checks ask whether observed vectors from one treatment group lie inside the other group’s hull. They provide a geometric screen for extrapolation beyond the other group’s observed envelope.
Worked Example: Testing Membership in a Two-Dimensional Hull
Suppose the hypothetical control covariate vectors are \((0,0),(2,0),(0,2)\). Their convex hull is a triangle. For a point \((x,y)\), membership requires \(x\geq0\), \(y\geq0\), and \(x+y\leq2\): these inequalities describe the triangle’s three sides and its interior.
Consider two treated vectors. The point \((0.5,0.5)\) satisfies all three inequalities, since \(0.5\geq0\), \(0.5\geq0\), and \(0.5+0.5=1\leq2\). It is inside the control hull. The point \((1.5,1.5)\) also has nonnegative coordinates, but \(1.5+1.5=3>2\), so it lies outside the control hull. The first vector can be expressed as a convex combination of control vertices; for example, \((0.5,0.5)=0.5(0,0)+0.25(2,0)+0.25(0,2)\). The second cannot be such a combination.
This check is directional: it tests treated vectors against the control hull. A full diagnostic should also test control vectors against the treated hull. Even when a point is inside the other group’s hull, it may be near a boundary or far from actual observations. A hull fills all convex combinations, including potentially large empty regions between observed points.
Combining Diagnostics and Reading Failure Signals
No one diagnostic captures every feature of multivariate support. Pair plots reveal visible patterns in selected projections; nearest-neighbor distances summarize local proximity; convex hulls assess whether vectors fall inside an observed group envelope. Using them together gives a more informative picture than treating any one as definitive.
Specify the covariates, units, transformations, missing-data handling, and distance scaling before comparing groups. Keep the representation fixed across treatment groups.
Use pair plots and marginal summaries to locate visible separation, unusual clusters, and regions represented by only one group. Treat pairwise overlap as incomplete evidence about higher-order combinations.
Compute cross-group nearest-neighbor distances in both directions, report the metric and scaling, and examine the distribution for long-distance observations.
Where computationally feasible, assess hull membership in both directions. Note points outside the opposite group’s hull and points that are inside but poorly represented locally.
A warning sign is a consistent pattern across diagnostics: for example, one group has long cross-group nearest-neighbor distances and many of its points lie outside the other group’s hull, with plots showing a separated region. A less dramatic result is not proof of adequate support. Sparse samples can leave holes undetected, and a broad hull can conceal empty or thinly populated regions. In dimensions beyond a few, plots become difficult to inspect, distances can concentrate, and hull calculations become computationally demanding or uninformative. These limitations are reasons to interpret diagnostics cautiously, not reasons to assume overlap.
Covariate-space diagnostics complement, rather than replace, propensity-score overlap plots, subgroup checks, and covariate balance summaries. Score plots compress covariates into one estimated quantity and can look reassuring while joint covariate patterns differ. Balance summaries address differences in selected moments, but similar means do not establish common support. Conversely, support diagnostics do not show that covariate distributions are balanced after a particular design. The questions are related but distinct.
Common Mistakes and Limitations
- Calling observed overlap proof of population overlap. A finite sample shows only its observed vectors. It cannot establish that every relevant population neighborhood has positive probability in both groups.
- Calling a large distance a positivity violation. A large value is relative to the chosen metric, scaling, and sample. It is evidence of weak observed proximity, not a formal population conclusion by itself.
- Equating hull inclusion with dense support. A convex hull includes empty space between its vertices. Membership does not mean that nearby opposite-group observations are plentiful.
- Checking only marginal ranges or pairwise panels. All one- and two-dimensional projections can overlap even when the full joint configurations differ, as the three-covariate example demonstrates.
- Changing scales without saying so. Distances can change substantially when covariates are rescaled. Report how the metric was constructed, and do not interpret a cutoff as universal.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Why can a point belong to the population support of a continuous covariate distribution even if its probability as an exact value is zero?
- In a distance diagnostic, why should nearest-neighbor distances be calculated in both treatment directions?
- A point lies inside the control group’s convex hull but has no nearby observed controls. What does hull membership establish, and what does it fail to establish?
- What can identical pairwise projections conceal about support in three covariates?
Solutions. (1) Support is defined by neighborhoods: a point is in the support when every open neighborhood around it has positive conditional probability. The probability of the point itself can still be zero. (2) The groups can be asymmetric: one group’s observations may all have close counterparts while some observations in the other group are far from any counterpart. (3) Hull membership establishes that the point is a convex combination of observed control vectors, so it lies inside their convex envelope. It does not establish that controls are locally dense near the point; the hull may contain empty regions. (4) Pairwise projections can agree even when the complete combinations of all three covariates differ, so a higher-order joint gap can remain hidden.