Why Calibrate Gamma With Measured Covariates?
A sensitivity analysis asks how conclusions might change if treatment assignment depended on an unmeasured factor, even after matching on observed characteristics. In Rosenbaum’s Gamma model, \(\Gamma\) bounds how much two otherwise comparable individuals’ treatment-assignment odds may differ because of hidden bias. The value \(\Gamma=1\) represents no such difference; larger values allow greater departures from equal odds.
A numerical value for \(\Gamma\) is easier to interpret when compared with the assignment association of a measured covariate. For example, if a measured severity indicator is strongly associated with treatment assignment among matched individuals, a benchmark of similar magnitude can help describe the scale of hidden bias being examined. This is a calibration aid, not evidence that an unmeasured factor has that strength or that measured and unmeasured factors operate in the same way.
The relevant comparison is usually conditional on the matched sets and the covariates already controlled by design or analysis. A crude association from the full, unmatched sample may instead reflect differences between sets, poor overlap, or the matching process. It can still be a descriptive starting point, but should not automatically be treated as the calibration target.
Estimating Assignment Odds Ratios
Let \(Z_{is}=1\) if individual \(i\) in matched set \(s\) received treatment, and \(Z_{is}=0\) otherwise. For a binary measured covariate \(X_{is}\), a conditional logistic model can be written as
Here \(\alpha_s\) represents the baseline assignment propensity for set \(s\), while \(\beta\) describes the conditional association with \(X\). Holding the set fixed, the treatment-assignment odds ratio for \(X=1\) versus \(X=0\) is \(\exp(\beta)\). For a continuous covariate, a contrast of \(d\) units corresponds to \(\exp(\beta d)\); choosing a meaningful contrast, such as one standard deviation, is essential to interpretation.
For matched sets containing one treated individual, conditioning on the fact that exactly one person in the set was treated removes the set-specific intercept. The conditional likelihood then compares the observed assignment with other allocations allowed within that set. For sets with multiple treated individuals, the conditional likelihood similarly conditions on the observed number treated in each set. This is why a conditional estimate can differ from a pooled two-by-two odds ratio.
Worked Example: A Crude Assignment Odds Ratio
Suppose a hypothetical study has 120 treated and untreated individuals. The table records whether each person had a measured risk indicator \(X=1\). Calculate the crude treatment-assignment odds ratio comparing \(X=1\) with \(X=0\).
| Treated | Untreated | |
|---|---|---|
| \(X=1\) | 36 | 18 |
| \(X=0\) | 24 | 42 |
The odds of treatment among those with \(X=1\) are \(36/18=2\). Among those with \(X=0\), they are \(24/42=4/7\). Thus,
In this hypothetical sample, the crude odds of treatment are 3.5 times as high for individuals with \(X=1\) as for those with \(X=0\). The result is a crude association: it is not adjusted for matched sets or other covariates, and by itself does not establish that \(X\) caused treatment assignment.
Why Matched-Set Conditioning Matters
Matching deliberately changes which individuals are compared. A pooled table can combine treated and untreated individuals from different regions of covariate space, even when the causal analysis compares individuals only within matched sets. In that setting, conditional estimates align more closely with the assignment contrasts relevant to the sensitivity analysis.
A simple paired-set calculation shows the distinction. Consider sets with one treated and one control, and a binary covariate. Only pairs in which the covariate differs between the two individuals inform its conditional association with treatment. Among these discordant pairs, count how often the treated person has \(X=1\) and how often the control has \(X=1\). Under the conditional logistic model, the ratio of these counts estimates the conditional odds ratio, provided both orientations occur.
Worked Example: A Matched-Pair Conditional Estimate
In a hypothetical matched study, there are 20 pairs discordant on a binary severity indicator. In 15 pairs, the treated member has \(X=1\) and the control has \(X=0\); in the other 5 pairs, the treated member has \(X=0\) and the control has \(X=1\). Assume the conditional logistic model above is appropriate and the pairs are independent.
Let \(p\) be the probability that the treated member has \(X=1\), conditional on a pair being discordant. The model gives \(p=\exp(\beta)/(1+\exp(\beta))\). The likelihood for the 20 orientations is proportional to \(p^{15}(1-p)^5\), so its maximum occurs at \(\hat p=15/20=0.75\). Converting this probability to odds gives
Thus the estimated within-pair assignment odds ratio for the severity contrast is 3. This need not equal the crude odds ratio from all individuals: the conditional estimate uses within-pair information, whereas a pooled estimate can also reflect between-pair differences.
From Measured Covariates to Gamma Benchmarks
In the hidden-bias model, \(\Gamma\) is an upper bound on the ratio of treatment-assignment odds for two individuals in the same matched set who are otherwise equivalent under the observed design, allowing for their differing hidden covariate values. A benchmark can be made more concrete by asking: how strong is the assignment association of a measured covariate over a clearly stated contrast?
For a binary covariate in a conditional logistic model, an estimated odds ratio \(\exp(\hat\beta)=2.4\) provides a natural reference point for a candidate \(\Gamma\) near 2.4. It does not imply that the hidden factor is binary, has the same distribution, or acts through the same mechanisms. Nor does it say that measured covariates are themselves unmeasured. The comparison communicates scale: the sensitivity analysis allows hidden assignment odds differences comparable to the observed contrast for that covariate.
Worked Example: Choosing Benchmarks From an Adjusted Model
Suppose a hypothetical matched analysis fits a conditional logistic model including measured severity and age. The estimated severity odds ratio for treatment assignment is 2.4 for severe versus non-severe individuals. The estimated age odds ratio is 1.6 per 10-year increase. Both are conditional on the matched sets and the other covariate in the model.
The severity contrast suggests a benchmark around \(\Gamma=2.4\). The age contrast should be stated in its 10-year units; comparing a one-year age effect to severity would obscure the scale. For a sensitivity analysis, an interpretable grid might include \(\Gamma=1,\ 1.5,\ 2,\ 2.5,\) and \(3\). The value 2.5 is close to the measured severity association, while 1.5 and 2 show weaker hidden-bias allowances and 3 a somewhat stronger one.
This grid is a design choice, not a statistical estimate of \(\Gamma\). The analysis can report the sensitivity results across these values, explaining that \(\Gamma=2.5\) permits within-set treatment-assignment odds to differ by as much as a factor of 2.5 because of hidden bias. The data do not identify whether an actual hidden factor creates that difference.
When several observed covariates are available, calibrate against more than one where practical. A weak association for a single covariate should not be presented as evidence that hidden bias must also be weak. Conversely, choosing the largest observed odds ratio as a universal benchmark can be overly conservative or misleading if that estimate is unstable, poorly adjusted, or based on a contrast unlike the relevant hidden difference.
Assumptions, Diagnostics, and Limits
The interpretation of these benchmarks depends on the model and data used to estimate them. In particular, consider the following conditions and checks before using a reported odds ratio as a reference:
- Respect the design. Use matched-set conditional estimates when the target comparison is within matched sets. State which covariates were conditioned on and whether the estimate is crude, conditional, or adjusted.
- Check overlap and informative variation. If a covariate barely varies within sets, there may be little information about its conditional assignment association. In pairs, only discordant covariate pairs inform the simple conditional estimate.
- Inspect sparse cells and separation. Very small counts can make logistic estimates unstable or infinite. Report uncertainty, consider an appropriate penalized or exact method, and avoid treating an extreme point estimate as a reliable benchmark.
- Choose defensible contrasts. For continuous variables, report the units and contrast used. Check whether a linear effect on the log-odds scale is plausible; a single per-unit coefficient can hide nonlinearity.
- Consider joint structure. Strong correlations among measured covariates, interactions, and model misspecification affect adjusted odds ratios. A one-covariate comparison may not describe the assignment strength of a realistic combination of characteristics.
Odds ratios are also non-collapsible: a conditional odds ratio can differ from a marginal odds ratio even without confounding. Therefore, the crude value 3.5 in the first example and the conditional value 3 in the second are not automatically contradictory; they refer to different comparisons and data structures. Calibration should prioritize a clearly specified contrast that resembles the within-set hidden-bias comparison, rather than selecting whichever estimate is largest.
Other approaches can complement this calibration. A propensity-score model can summarize measured predictors of treatment assignment, but its coefficients and score distributions do not directly estimate hidden bias. Standardized differences assess covariate balance, not treatment-assignment odds. Subject-matter knowledge can motivate plausible hidden factors and contrasts, while sensitivity analysis displays how conclusions respond to chosen \(\Gamma\) values. None replaces the others: the odds-ratio benchmark helps explain scale, and the sensitivity analysis quantifies robustness under its stated model.
Common Pitfall: Treating the Benchmark as a Bound on Reality
A common mistake is to say, “The largest observed covariate odds ratio is 2.4, so hidden bias cannot exceed \(\Gamma=2.4\).” That conclusion does not follow. Measured covariates are observed, may be measured with different accuracy, and may have different distributions and relationships with treatment than an unmeasured factor. Their assignment associations offer comparisons, not limits on the strength of unmeasured confounding.
A more defensible statement is: “We examined \(\Gamma\) values up to 2.5, a scale comparable to the estimated within-set assignment odds ratio for measured severity, and also considered weaker and stronger values.” This makes the benchmark transparent without claiming that the hidden factor is known or constrained by observed data.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A binary covariate has a conditional logistic coefficient \(\hat\beta=\log(2)\). What is the estimated assignment odds ratio for \(X=1\) versus \(X=0\), and what does it mean?
- In 24 discordant matched pairs, the treated member has the risk indicator in 18 pairs and the control member has it in 6. Under the simple conditional model, calculate the matched-pair estimate.
- Why might a crude treatment-assignment odds ratio differ from a conditional odds ratio estimated within matched sets?
- A continuous covariate has an estimated assignment odds ratio of 1.05 per year. What is the corresponding odds ratio for a 10-year contrast under a linear log-odds model?
- Does observing an assignment odds ratio of 4 for a measured covariate prove that \(\Gamma\) need not exceed 4? Explain.
Solutions. (1) \(\exp\{\log(2)\}=2\); conditional on the model and matched-set structure, the treatment-assignment odds are estimated to be twice as high for \(X=1\) as for \(X=0\). (2) The conditional estimate is \(18/6=3\). (3) A crude comparison can include between-set differences, while the conditional estimate uses within-set contrasts and may condition on other covariates. (4) The 10-year odds ratio is \(1.05^{10}\approx1.629\). (5) No. The value 4 can be a useful reference scale, but observed covariate associations do not impose an upper bound on hidden bias.