Why Quantile-Based Strata Need a Precise Rule
Subclassification groups units into ranges of their propensity scores. A common way to choose the ranges is to use empirical quantiles, aiming for strata that cover successive portions of the observed score distribution. This is a construction rule, not a treatment-effect estimator: it determines which units belong together, but does not by itself estimate an effect or establish that the groups are adequate for causal analysis.
The rule needs to specify what happens when scores are tied at a cutpoint. If identical scores are kept together, a stratum may contain more or fewer observations than intended. If tied units are split to force equal counts, membership no longer depends on the score alone. These choices affect the resulting groups, so the cutpoint definition and assignment convention should be reported.
This definition uses the empirical inverse-CDF quantile: the cutpoint is the first ordered score whose rank reaches the specified fraction of the sample. Other quantile conventions, including interpolation between ordered scores, can produce different numerical cutpoints. The inverse-CDF rule above makes the examples reproducible. Within that rule, the strict lower and inclusive upper boundaries ensure that a score equal to a cutpoint belongs to the lower of its two adjacent intervals.
Construction and Its Basic Properties
To construct the strata, first decide on \(K\) and a quantile convention. Then sort the scores, calculate each cutpoint rank \(\lceil jn/K\rceil\), read off the score at that rank, and assign units using the interval definitions. If more than one unit has the same score, all of them receive the same stratum under this score-only rule.
Check that there is one estimated propensity score between zero and one for each unit included in the subclassification.
Specify \(K\) before examining the final membership counts, and record the choice.
For each \(j=1,\ldots,K-1\), use the ordered score at rank \(\lceil jn/K\rceil\).
Apply the stated boundary convention, then check the cutpoints, stratum sizes, and how ties were handled.
The intervals assign each observed score to exactly one stratum: they do not overlap, and together they cover scores from the sample minimum to the sample maximum. Thus, if \(n_j\) is the number of units assigned to stratum \(j\), then \(\sum_{j=1}^{K} n_j=n\). This partition property does not guarantee equal sizes. When all scores are distinct, the rank cutpoints divide the ordered observations into groups whose sizes are close to \(n/K\). With ties, even distinct cutpoints can yield substantially unequal sizes: a large group of identical scores may all fall on the same side of a boundary.
Worked Example: Four Strata With Distinct Scores
Suppose \(n=12\) units have the following ordered estimated propensity scores:
Use \(K=4\). The cutpoint ranks are \(\lceil 12/4\rceil=3\), \(\lceil 24/4\rceil=6\), and \(\lceil 36/4\rceil=9\). Therefore,
Assign the scores using the specified interval rule. Stratum 1 contains the first three scores, through \(0.17\), so \(n_1=3\). Stratum 2 contains \(0.22,0.27,0.31\), so \(n_2=3\). Stratum 3 contains \(0.38,0.44,0.51\), so \(n_3=3\). Stratum 4 contains \(0.59,0.68,0.82\), so \(n_4=3\). The sizes sum to \(3+3+3+3=12\). Here the scores are distinct, and the rank construction gives exactly \(n/K=3\) units in each stratum.
The equal sizes in that example are not a general guarantee. The next example illustrates what changes when scores are repeated.
Worked Example: Ties Create Unequal Stratum Sizes
Suppose \(n=12\) units have these ordered scores:
For \(K=4\), the cutpoint ranks are 3, 6, and 9, giving \(c_1=0.20\), \(c_2=0.35\), and \(c_3=0.55\). The cutpoints are distinct, but five observations have a score at or below \(0.20\): the first two scores and all three observations at \(0.20\). Hence stratum 1 contains 5 units. Stratum 2 contains only the unit with score \(0.35\), so it has 1 unit. Stratum 3 contains \(0.40,0.48,0.55\), so it has 3 units. The remaining three scores belong to stratum 4.
The counts are therefore \(5,1,3,3\), which sum to 12. The tie at \(0.20\) causes the first group to exceed its nominal target of three; the cutpoint \(0.35\) then leaves just one observation in the next group. Distinct cutpoints alone do not prevent this imbalance.
Repeated Cutpoints and Empty Strata
If a score value occurs often enough, several quantile ranks can point to that same value. Under a tie-preserving assignment rule, repeated cutpoints can produce empty strata. This is not an arithmetic error: it signals that the requested number of score-only groups cannot be formed as nonempty intervals using those empirical quantiles.
Worked Example: Repeated Cutpoints Produce Empty Strata
Consider \(n=10\) ordered scores:
Let \(K=5\). The cutpoint ranks are \(\lceil 10/5\rceil=2\), \(\lceil 20/5\rceil=4\), \(\lceil 30/5\rceil=6\), and \(\lceil 40/5\rceil=8\). Each corresponding score is \(0.20\), so \(c_1=c_2=c_3=c_4=0.20\). Stratum 1, defined by scores at or below \(c_1\), contains eight units. Each middle stratum has a lower and upper boundary equal to \(0.20\), so no score can satisfy its strict lower-bound condition; those strata are empty. Stratum 5 contains the two scores above \(0.20\). The counts are \(8,0,0,0,2\), which still sum to 10.
This construction is formally well-defined, but five useful nonempty score intervals have not been obtained. A practical response is to use fewer strata or a different, clearly reported grouping rule. Simply relabelling the empty groups does not solve the problem.
Diagnostics, Alternatives, and Limitations
After assignment, inspect the actual cutpoints and counts rather than assuming that a quantile rule produced balanced groups. At minimum, check whether cutpoints are repeated, whether any strata are empty or very small, and whether scores at a boundary were assigned consistently. It is also useful to tabulate the number of treated and untreated units in each stratum as a descriptive check of the data layout. Those counts can reveal a group containing few or no units from one treatment category, but they do not by themselves assess covariate balance or establish that causal assumptions hold.
Two common alternatives make different trade-offs:
| Grouping rule | How it assigns units | Main trade-off |
|---|---|---|
| Tie-preserving empirical quantiles | Uses score cutpoints and keeps identical scores in the same stratum | Score-only membership is clear, but ties can create unequal or empty strata |
| Rank-based groups with tie-breaking | Divides the ordered observations into target-sized groups, using an extra rule to order tied units | Can make group sizes more even, but units with identical scores may be assigned to different strata |
| Fixed-width score bins | Uses preselected score intervals, such as intervals of equal numerical width | Cutpoints are easy to state, but the number of observed units per bin can vary widely |
A rank-based procedure that breaks ties—for example, by unit identifier—may achieve nearly equal counts, but the tie-breaker is unrelated to the propensity score. If the analysis requires each subclass to represent a score interval, splitting identical scores this way changes the meaning of the groups. A fixed-width rule avoids sample-quantile cutpoints, but it does not promise balanced counts either. The appropriate choice depends on the intended grouping and should be made explicit rather than selected after seeing which rule produces a preferred result.
Quantile strata are a finite-sample partition. Their boundaries depend on the observed scores, sample size, the chosen value of \(K\), and the quantile convention. They do not automatically make units within a stratum comparable on measured covariates, guarantee that both treatment groups are represented, or establish causal identification. Those questions require separate assessment. This tutorial concerns the construction and diagnosis of the groups, not the estimation or combination of effects within them.
Common Mistake
A frequent error is to argue that quantile cutpoints imply roughly equal group sizes, even when scores are tied. That reasoning treats each ordered rank as though it corresponded to a different score value. But assignment is based on score intervals, so all observations with the same score fall on the same side of a cutpoint under the tie-preserving rule. In the example with cutpoints \(0.20,0.35,0.55\), the cutpoints are distinct, yet the group sizes are \(5,1,3,3\). The correction is to calculate memberships from the actual score values after finding the cutpoints, then report the resulting counts. With untied scores, the rank-based construction gives group sizes close to \(n/K\); with ties, even distinct cutpoints can yield substantially unequal sizes.
Check Your Understanding
Answer each question using the tie-preserving empirical-quantile rule defined above. Brief solutions follow the questions.
- For \(n=8\) and \(K=4\), which ordered ranks define the cutpoints?
- For ordered scores \(0.1,0.2,0.2,0.2,0.5,0.6,0.7,0.8\), form the \(K=4\) cutpoints and give the stratum sizes.
- Why can a tie-preserving construction produce an empty stratum?
- What is the trade-off when tied scores are split using a tie-breaker to obtain more equal-sized groups?
Solutions. (1) The ranks are \(\lceil 8/4\rceil=2\), \(\lceil 16/4\rceil=4\), and \(\lceil 24/4\rceil=6\). (2) The cutpoints are \(c_1=0.2\), \(c_2=0.2\), and \(c_3=0.6\). The first stratum contains four scores at or below \(0.2\); the second is empty because its boundaries are both \(0.2\); the third contains \(0.5\) and \(0.6\); and the fourth contains \(0.7\) and \(0.8\). The counts are \(4,0,2,2\). (3) Repeated cutpoints can make an interval have equal lower and upper boundaries, leaving no score eligible for that stratum. (4) Tie-breaking can make counts more even, but units with identical propensity scores may be placed in different strata, so the grouping is no longer determined by score intervals alone.