Why Cell Counts Matter
A propensity score can look acceptable overall while particular categories of a covariate contain very few treated or untreated units. For categorical covariates, treatment-by-category tables make this problem visible directly: they show how many observations in each category contribute information about each treatment condition. The same check can be made for combinations of categories, where a problem may be hidden by apparently balanced one-variable summaries.
The aim is not to declare positivity from a table alone. The table describes the observed sample, while positivity is a condition on treatment probabilities in the population under study. Counts are useful evidence about where those probabilities may be poorly supported and where estimates may rely on a small number of observations.
For a covariate stratum \(c\) in the target population, conditional positivity requires \(0<P(A=1\mid C=c)<1\). A sample cell with \(n_{1,c}=0\) or \(n_{0,c}=0\) therefore raises a concern: one treatment condition has no observed representation in that cell. A small positive count is also important. It means that the corresponding treatment-specific information is limited, even though the observed proportion is not exactly zero or one.
Reading a Treatment-by-Category Table
Begin with one row for each category and separate columns for treated and untreated counts. Include the total and, where useful, the observed treatment proportion. Counts should remain visible even when proportions are reported: a proportion of 0.90 based on 9 observations is not as informative as 0.90 based on 900. There is no universal count threshold that turns a cell into “sparse.” The relevant concern depends on the analysis, the number of outcomes to estimate, and how much uncertainty is tolerable.
The observed treatment proportion is a descriptive sample quantity, not the population probability \(P(A=1\mid C=c)\). Under independent sampling within a cell and a common treatment probability \(\pi_c=P(A=1\mid C=c)\), the treated count has a binomial model:
This model also explains why small cells make extreme observed proportions unsurprising. If \(N_c\) is small, the count can easily be zero in one treatment group even when the population probability is strictly between zero and one. Conversely, a large cell with no observations in one group is a more concerning signal, though it still does not logically prove the population probability is zero.
Worked Example: Comparing Categories of One Covariate
Suppose a hypothetical study records treatment status and insurance category for 90 people. The counts are:
| Insurance category | Treated, \(n_{1,c}\) | Untreated, \(n_{0,c}\) | Total, \(N_c\) | Observed treated proportion, \(\hat p_c\) |
|---|---|---|---|---|
| Public | 28 | 2 | 30 | 0.933 |
| Private | 15 | 15 | 30 | 0.500 |
| Uninsured | 1 | 29 | 30 | 0.033 |
For public insurance, \(\hat p_c=28/(28+2)=28/30=0.933\). For private insurance, it is \(15/30=0.500\), and for uninsured people, it is \(1/30=0.0333\). Both treatment groups occur in every category, so none is an observed zero-count cell. However, the public category has only 2 untreated people and the uninsured category has only 1 treated person. Any treatment comparison within those categories would be supported by very few observations from one side.
The table suggests potential practical positivity concerns at both extremes. It does not establish that the population treatment probability is exactly 1 for public insurance or exactly 0 for uninsured people; the observed proportions are not population probabilities. Nor do the counts alone say whether the outcome analysis is sufficiently precise. They identify where further scrutiny is warranted.
Why Combinations Can Reveal What Marginal Tables Hide
A separate table for each categorical covariate checks support one variable at a time. It does not establish that both treatment groups occur within every combination of covariate levels. For example, treatment may be common among younger people overall and common among people with low severity overall, while younger people with low severity have almost no untreated observations. Such a joint cell can matter when the analysis conditions on both covariates.
A joint table uses a cell \(c\) for each observed combination, such as age group by severity group. Comparing its treatment counts checks support at that finer level. This is a targeted extension of marginal checks, not a claim that every conceivable combination must always be analyzed. The combinations to examine should reflect the covariates and conditional structure relevant to the causal analysis.
Worked Example: Adequate Margins, Sparse Joint Cells
In a hypothetical sample of 80 people, consider two binary covariates: age group (younger or older) and severity (low or high). The treatment counts are:
| Age group | Severity | Treated | Untreated | Total |
|---|---|---|---|---|
| Younger | Low | 18 | 2 | 20 |
| Younger | High | 2 | 18 | 20 |
| Older | Low | 15 | 5 | 20 |
| Older | High | 5 | 15 | 20 |
Marginalizing over severity, the younger group has \(18+2=20\) treated and \(2+18=20\) untreated people. The older group likewise has \(15+5=20\) treated and \(5+15=20\) untreated people. Marginalizing over age, low severity has \(18+15=33\) treated and \(2+5=7\) untreated people; high severity has \(2+5=7\) treated and \(18+15=33\) untreated people. Each marginal category contains observations from both treatment groups.
The joint table tells a more detailed story. The treated proportions for younger-low and younger-high are \(18/20=0.90\) and \(2/20=0.10\), respectively. For older-low and older-high they are \(15/20=0.75\) and \(5/20=0.25\). Every joint cell has both groups, but some cells have only 2 or 5 observations in one group. Thus, marginal support does not rule out sparse joint support. These data flag limited information for comparisons within those combinations; they do not alone establish a population positivity violation.
What a Zero Count Does—and Does Not—Show
A zero count is an especially clear sample warning. If \(n_{0,c}=0\), no untreated observation is available in cell \(c\); if \(n_{1,c}=0\), no treated observation is available. The observed treatment proportion is then 1 or 0, respectively. Estimating a treatment contrast within that cell from the observed data is not supported by both groups. But a sample zero cannot distinguish a structural impossibility from a treatment group that simply was not sampled.
A short probability calculation illustrates the distinction. If the population untreated probability in a cell is \(q=P(A=0\mid C=c)>0\), and \(N_c\) sampled units have independent treatment assignments with that probability, then:
The equation follows because each unit has probability \(1-q\) of being treated, and all \(N_c\) must be treated for the untreated count to be zero. When \(N_c\) is small, this event can have appreciable probability even if \(q\) is positive.
Worked Example: A Zero Count in a Small Cell
Suppose a hypothetical categorical stratum contains 12 sampled people, all treated. If the population probability of being untreated in this stratum were \(q=0.10\), then the probability of observing no untreated people would be:
Thus, under this illustrative model, a zero untreated count occurs about 28.2% of the time despite a positive population probability of being untreated. The calculation does not show that \(q\) is 0.10 in an actual study; it shows why a zero in a small sample cannot, by itself, prove structural nonpositivity. If the cell were much larger and still had zero untreated observations, the sample evidence against meaningful support would be stronger.
A Practical Diagnostic Sequence
State which categorical covariates are being checked and which joint cells are substantively relevant to the adjustment set.
For every level or combination, report \(n_{1,c}\), \(n_{0,c}\), and \(N_c\). Do not report only percentages.
Note cells missing one treatment group and cells in which either group has only a few observations. Interpret the flag relative to the analysis rather than a universal cutoff.
Compare separate one-variable tables with selected joint tables. Adequate marginal counts do not guarantee adequate counts within combinations.
Ask whether the proposed treatment comparison requires information in the flagged cell. Record the limitation and investigate before interpreting an adjusted estimate.
These diagnostics use treatment and covariate data, not outcome differences, to identify support concerns. Avoid deciding which cells to inspect only after seeing favorable or unfavorable outcomes. Also check coding and category definitions: an apparent empty cell can result from an error, a mislabeled level, or a category that is absent from the analytic sample.
How Cell Counts Complement Other Diagnostics
| Diagnostic | What it makes visible | Important limitation |
|---|---|---|
| One-variable treatment-by-category table | Counts and treatment proportions within each level of one categorical covariate | Can conceal sparse or empty combinations of covariates |
| Joint treatment-by-category table | Counts within specified combinations of categorical covariates | Cells become smaller as combinations become more detailed; not every possible cross-classification is useful |
| Propensity-score overlap diagnostic | Overlap in estimated treatment probabilities across treatment groups | A score summary can obscure which categorical cell creates a support problem and depends on the score model |
These approaches answer related but different questions. Score diagnostics summarize support through estimated probabilities; cell tables show where categorical levels and combinations are represented in the observed data. A reassuring score plot does not make a zero count disappear, and a sparse cell table does not by itself describe all support issues involving continuous covariates. Read the evidence together without treating any single diagnostic as a proof of positivity.
Limitations and Common Mistakes
The number of joint cells grows quickly as categorical covariates are crossed. Many tiny cells can arise even in a large study, so inspecting every possible combination mechanically may produce an unhelpful list. Focus on substantively relevant conditioning variables and report how the cells were defined. Conversely, collapsing categories just to remove a zero can conceal real differences or change the meaning of the covariate. Any regrouping should have a defensible substantive rationale, not merely improve the table.
A common mistake is to reason, “The observed proportion is 1, so positivity fails in the population.” The observed proportion \(n_{1,c}/N_c\) describes the sample, not the true probability \(P(A=1\mid C=c)\). The correction is to call the zero count a sample support warning, examine its cell size and context, and distinguish evidence of practical lack of overlap from a known structural restriction. Another mistake is to treat a nonzero count as proof of adequate support. One treated person and many untreated people technically populate both groups, but may still provide too little information for a stable comparison.
Sparse cells can make estimates imprecise or sensitive to individual observations and modeling choices. They also limit what a model can learn from the data in those regions. A count table diagnoses this limitation; it does not select a remedy or determine a new target population. Decisions about restrictions, trimming, or alternative estimands require separate reasoning about the analysis goal and are not implied by a particular count threshold.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In a category, 24 people are treated and 6 are untreated. Find \(N_c\) and \(\hat p_c\), and state one conclusion the table supports and one it does not.
- Two separate marginal tables show treated and untreated people in every category of each covariate. Does this guarantee both groups appear in every joint combination? Explain.
- A cell contains 8 people, all treated. If the population untreated probability were 0.20, calculate the probability of observing zero untreated people under independent assignments.
- Why is one treated person and 40 untreated people still a possible practical positivity concern, even though neither count is zero?
- A researcher merges two categories solely because one has zero treated observations. What is the main concern, and what should guide any decision to combine categories?
Solutions. (1) \(N_c=24+6=30\), and \(\hat p_c=24/30=0.800\). The table supports the statement that the observed treated proportion is 0.800 and that both groups occur in the sample; it does not establish the population treatment probability or prove adequate precision. (2) No. Marginal counts combine across the other covariate, so a joint combination may still have a zero or very small count for one treatment group. (3) The probability is \((1-0.20)^8=0.80^8\approx0.1678\), about 16.8%. A sample zero is therefore possible even with a positive population untreated probability. (4) A comparison based on one treated observation can be highly sensitive to that person and may have substantial uncertainty; nonzero counts do not ensure adequate information. (5) Merging can hide meaningful differences or change the covariate’s interpretation. Categories should be combined only when substantively defensible, not simply to remove an inconvenient count pattern.