Why Read a Curve Rather Than One Sensitivity Result?
A sensitivity analysis asks how conclusions about a causal effect would change if the assumptions used to address unmeasured confounding were relaxed. Reporting the result at just one assumed confounding strength can conceal how much the conclusion depends on that choice. A sensitivity curve displays results across a range of strengths, making it possible to see where estimates shift, uncertainty grows, or a substantive conclusion changes.
The horizontal axis represents the sensitivity parameter: larger values generally allow more departure from the assumptions that would identify the causal effect from observed data. The vertical axis might show an adjusted estimate, the endpoints of an identified interval, or a confidence limit. Those are different quantities. A curve is informative only if its axes, target estimand, assumptions, and uncertainty procedure are clear.
Sensitivity Parameters and Curve Types
For a concrete example, consider the marginal sensitivity model. Let \(A\) denote treatment, \(X\) measured covariates, and \(U\) unmeasured factors. The observed propensity score is \(e(X)=P(A=1\mid X)\), while the propensity conditional on both measured and unmeasured factors is \(e^*(X,U)=P(A=1\mid X,U)\). The sensitivity parameter \(\Gamma\), with \(\Gamma\geq1\), limits how much the treatment odds conditional on \(U\) can differ from the observed treatment odds.
This restriction alone does not guarantee causal identification. Interpretation also relies on the causal setup and assumptions used for the analysis, including a well-defined treatment and outcome, consistency, and the conditions needed to interpret the observed-data analysis at the reference value. The sensitivity model describes one particular way to represent possible hidden confounding; \(\Gamma\) is not a universal measure of confounding strength.
A curve can display different analysis outputs. An estimate curve shows an adjusted point estimate as the sensitivity value changes. A bound curve shows the lower and upper endpoints of the causal effects compatible with the data and the sensitivity model. A confidence interval curve displays sampling uncertainty for an estimate or for bounds, using a stated procedure. These should not be treated as interchangeable: a bound represents uncertainty about which causal effect is compatible with the model, while a confidence interval represents sampling uncertainty in estimated quantities.
How to Read Bound Curves
Suppose the effect is measured as a difference in outcome units, with positive values indicating benefit. At each \(\Gamma\), let \([\hat L(\Gamma),\hat U(\Gamma)]\) denote estimated lower and upper bounds for the causal effect. The interval is the range of effects not ruled out by the observed data and the sensitivity model at that strength. If the entire interval is positive, all effects in that estimated range favor benefit. If it contains zero, a zero effect is compatible with the range; that does not establish that the true effect is zero.
When increasing \(\Gamma\) relaxes the model by allowing additional treatment-assignment mechanisms, the set of compatible causal effects expands or stays the same. Consequently, the population lower bound cannot increase and the population upper bound cannot decrease. This nesting property is useful for reading a plot: the lower curve should stay level or move down, and the upper curve should stay level or move up. Small violations in numerically estimated curves can arise from computation or estimation and should be checked rather than interpreted as substantive reversals.
Worked Example: Finding When a Bound First Includes Zero
Suppose a hypothetical study estimates a treatment effect in additional symptom-free days. Its bound analysis reports the following estimated intervals across sensitivity strengths:
| \(\Gamma\) | Lower bound \(\hat L(\Gamma)\) | Upper bound \(\hat U(\Gamma)\) |
|---|---|---|
| 1.0 | 2.4 | 2.4 |
| 1.5 | 1.8 | 3.0 |
| 2.0 | 0.9 | 3.6 |
| 3.0 | -0.2 | 4.4 |
At \(\Gamma=1\), the estimated effect is 2.4 additional days under the reference model. At \(\Gamma=1.5\), the compatible range is 1.8 to 3.0 days, entirely above zero. At \(\Gamma=2\), the range is 0.9 to 3.6 days, still above zero. At \(\Gamma=3\), it is \(-0.2\) to 4.4 days, which includes zero. Thus, among the displayed values, the first one at which a zero effect is compatible is \(\Gamma=3\).
If a rough crossing point is useful, linearly interpolate the lower bound between \(\Gamma=2\) and \(\Gamma=3\). The lower bound decreases by \(0.9-(-0.2)=1.1\) days across one unit of \(\Gamma\). Starting at 0.9 days, the estimated crossing is therefore
This is an interpolation between plotted estimates, not an exact threshold established by the model. The substantive conclusion is that the estimated bounds support a positive effect up to about this strength under the interpolation, while larger strengths can make a zero effect compatible. The units remain days throughout; \(\Gamma\) is dimensionless.
Point Estimates and Confidence Intervals Are Different Curves
An estimate curve answers how the fitted point estimate changes under specified sensitivity assumptions. A confidence interval curve adds sampling uncertainty. At each fixed sensitivity value, a pointwise 95% confidence interval is intended to cover the corresponding parameter in repeated samples 95% of the time, under the stated sampling and model assumptions. Pointwise intervals do not automatically provide 95% coverage for the entire curve simultaneously.
For a curve of bounds, uncertainty needs additional care. A confidence interval for the lower endpoint alone is not necessarily a confidence interval for the whole identified set. To claim that a confidence region covers the identified set, use a procedure designed for that purpose. If a plot shows confidence limits at many \(\Gamma\) values, identify whether they are pointwise or simultaneous and what quantity they cover.
Worked Example: The Estimate Crosses Zero After Its Interval Does
Suppose a hypothetical analysis estimates the treatment effect in additional symptom-free days using the sensitivity curve \(\hat\tau(\Gamma)=1.20-0.55(\Gamma-1)\). For illustration, suppose its standard error is \(SE(\Gamma)=0.25+0.12(\Gamma-1)\), and use the normal-approximation interval \(\hat\tau(\Gamma)\pm1.96SE(\Gamma)\). These are specified functions for this example, not universal formulas for sensitivity analyses. Assume the standard-error calculation and normal approximation are appropriate at each fixed \(\Gamma\).
At \(\Gamma=1\), \(\hat\tau=1.20\) days and \(SE=0.25\) days. The interval is \(1.20\pm1.96(0.25)=1.20\pm0.49\), or 0.71 to 1.69 days. At \(\Gamma=1.5\), the estimate is \(1.20-0.55(0.5)=0.925\) days and the standard error is \(0.25+0.12(0.5)=0.31\) days. The interval is \(0.925\pm1.96(0.31)=0.925\pm0.6076\), or 0.3174 to 1.5326 days.
At \(\Gamma=2\), the estimate is \(1.20-0.55=0.65\) days and the standard error is \(0.25+0.12=0.37\) days. Its interval is \(0.65\pm1.96(0.37)=0.65\pm0.7252\), or \(-0.0752\) to 1.3752 days. The point estimate remains positive, but the interval now includes zero. At \(\Gamma=3\), the estimate is \(1.20-0.55(2)=0.10\) days, with \(SE=0.49\) days and interval \(0.10\pm0.9604\), or \(-0.8604\) to 1.0604 days.
The point estimate reaches zero when \(1.20-0.55(\Gamma-1)=0\), giving \(\Gamma=3.182\). The lower confidence limit reaches zero earlier: \(1.20-0.55(\Gamma-1)-1.96\{0.25+0.12(\Gamma-1)\}=0\), which gives \(\Gamma\approx1.904\). The interval includes zero at \(\Gamma=2\), even though the point estimate does not cross zero until about \(\Gamma=3.182\). This illustrates why a report should distinguish loss of statistical evidence from a point estimate changing sign. The intervals here are pointwise; they do not guarantee simultaneous coverage over every \(\Gamma\).
Substantive Thresholds and Clear Reporting
Zero is not always the only threshold that matters. If an effect smaller than a specified amount would not be practically important, mark that threshold on the plot as well. For an outcome measured in days, for example, a study team might decide in advance that at least one additional symptom-free day is a meaningful benefit. Then ask whether the entire bound interval exceeds one day, overlaps one day, or lies below it. This addresses a different question from whether the interval includes zero.
Worked Example: Robustness to a Meaningful-Benefit Threshold
Suppose a hypothetical bound curve for additional symptom-free days is \([1.6,2.4]\) at \(\Gamma=1\), \([1.1,2.8]\) at \(\Gamma=1.5\), \([0.6,3.2]\) at \(\Gamma=2\), and \([-0.2,4.0]\) at \(\Gamma=3\). Before analysis, the team defines one additional symptom-free day as the threshold for a meaningful benefit.
At \(\Gamma=1.5\), the lower bound is 1.1 days, so the entire estimated range exceeds the one-day threshold. At \(\Gamma=2\), the interval runs from 0.6 to 3.2 days: it remains positive, but it no longer guarantees a benefit exceeding one day. Linear interpolation of the lower bound between \(\Gamma=1.5\) and \(\Gamma=2\) estimates where it reaches one day. It falls by \(1.1-0.6=0.5\) days over 0.5 units of \(\Gamma\), so the crossing is \(1.5+(1.1-1.0)/(0.5/0.5)=1.6\). Thus, under this rough interpolation, the bounds cease to guarantee at least one day of benefit near \(\Gamma=1.6\), before they include zero. This is a robustness threshold for the chosen substantive criterion, not a universal cutoff.
A useful report states the estimand and outcome units, the meaning and reference value of the sensitivity parameter, and the assumptions defining the sensitivity model. It should describe what the vertical axis displays, how uncertainty was calculated, and whether confidence limits are pointwise or simultaneous. State the value or range at which the key conclusion changes, and explain that this conclusion is conditional on the specified model rather than claiming that the data have identified the actual hidden-confounding strength.
Common Mistake
A frequent error is to say, “The effect is robust up to \(\Gamma=2\), so unmeasured confounding cannot matter.” The curve does not show that hidden confounding is absent or that \(\Gamma=2\) is the actual strength. It shows what conclusions follow for the range of departures allowed by this particular model. A better statement is: “Under the specified sensitivity model, the estimated bounds remain positive through \(\Gamma=2\); at larger displayed values they include zero.” If uncertainty is included, say separately whether the confidence interval or confidence region crosses the relevant threshold.
Check Your Understanding
Answer each question using the definitions and distinctions in this tutorial.
- In the marginal sensitivity model, what does \(\Gamma=1\) mean, and what changes as \(\Gamma\) increases?
- A bound curve is \([0.4,2.1]\) at one sensitivity value and \([-0.3,2.8]\) at a larger value. What can be said about compatibility with a zero effect at each value?
- Why is a point estimate remaining positive not enough to conclude that a confidence interval excludes zero?
- What additional coverage information is needed before interpreting confidence limits drawn along an entire sensitivity curve?
- If the lower bound falls below a prespecified meaningful-benefit threshold before it reaches zero, how should the substantive robustness conclusion be described?