What a Conditional Estimate and Standard Error Mean
A treatment-effect estimate obtained in a sensitivity analysis answers a question under a specified assumption about outcomes that are not directly determined by the observed data. Changing that assumption can change the estimate, its estimated standard error, or both. The key is to report and interpret each pair together: the estimate is conditional on the chosen assumption, and its standard error describes sampling variability under that same fixed assumption.
Write \(\gamma\) for a sensitivity value and \(\theta(\gamma)\) for the treatment effect defined under that value. The estimator is \(\hat{\theta}(\gamma)\). Here “conditional on \(\gamma\)” means that the calculation fixes \(\gamma\) at the stated value; it does not mean that the treatment effect is conditioned on a random event. The standard error \(SE\{\hat{\theta}(\gamma)\}\) estimates how much the estimator would vary across repeated samples if the analysis method and the value of \(\gamma\) were held fixed.
This distinction matters because a small standard error does not make the sensitivity assumption credible. It says the estimate would be relatively precise across repeated samples if that assumption were maintained. Conversely, estimates under two assumptions can differ substantially even when each has a small standard error. The two quantities describe different sources of variation and should not be conflated.
Calculating a Conditional Standard Error
Suppose a treatment effect is a contrast between two estimated means, \(\hat{\theta}(\gamma)=\hat{\mu}_T(\gamma)-\hat{\mu}_C(\gamma)\). At a fixed \(\gamma\), the variance of a difference includes the covariance between the estimated means:
The covariance is not automatically zero. It can arise, for example, when model parameters are estimated jointly or when the estimator uses shared information. The variance estimates and covariance must correspond to the estimator fitted under the fixed sensitivity assumption, not to a different analysis scenario.
More generally, suppose the estimator can be written as \(g(\hat{\boldsymbol{\beta}};\gamma)\), where \(\hat{\boldsymbol{\beta}}\) is a vector of estimated model parameters and \(\gamma\) is held fixed. If the estimator is sufficiently smooth and the fitted parameter estimates have an estimated covariance matrix \(\hat{\Sigma}\), the delta method gives:
The gradient is taken with respect to the estimated parameters while \(\gamma\) is fixed. This is the important condition in the formula: the calculation propagates sampling variation in the estimated parameters through the treatment-effect estimator. It does not add a variance term for \(\gamma\). If an analysis procedure estimates model parameters, their relevant uncertainty still needs to be represented in \(\hat{\Sigma}\); “fixed assumption” is not a reason to treat estimated model parameters as known.
Worked Example: Comparing Two Fixed Assumptions
Consider a hypothetical randomized trial with a treatment effect measured as points of improvement on an outcome score; positive values favor treatment. Under each of two fixed sensitivity assumptions, a fitted analysis produces arm-mean estimates and their estimated covariance. The following numbers are hypothetical analysis outputs, not results from a real trial.
| Fixed assumption | \(\hat{\mu}_T\) | \(\hat{\mu}_C\) | Estimated variance of \(\hat{\mu}_T\) | Estimated variance of \(\hat{\mu}_C\) | Estimated covariance |
|---|---|---|---|---|---|
| A | 7.2 | 4.1 | 0.36 | 0.25 | 0.04 |
| B | 6.6 | 4.0 | 0.49 | 0.25 | 0.05 |
For assumption A, the estimated treatment contrast is \(7.2-4.1=3.1\) score points. Its estimated variance is \(0.36+0.25-2(0.04)=0.53\), so its standard error is \(\sqrt{0.53}=0.728\) score points, to three significant figures.
For assumption B, the estimated contrast is \(6.6-4.0=2.6\) score points. Its estimated variance is \(0.49+0.25-2(0.05)=0.64\), giving a standard error of \(\sqrt{0.64}=0.800\) score points.
Under assumption A, the estimated treatment benefit is 3.1 score points with a conditional standard error of 0.728 points. Under assumption B, it is 2.6 points with a conditional standard error of 0.800 points. These results show that both the estimate and its sampling precision differ across the two fitted scenarios. They do not, by themselves, say which assumption is more plausible.
Why the Standard Error Can Depend on the Assumption
A fixed sensitivity value is not necessarily just a constant added to the final estimate. It can change fitted means, the estimated model parameters, or the way those parameters enter the estimator. As a result, the gradient in the delta-method calculation or the estimated covariance matrix can change with \(\gamma\). In some analyses the conditional standard error will be the same across assumptions; in others it will differ. Neither pattern is required in general.
A short derivation makes the role of a fixed value explicit. Suppose an estimator is linear in a fitted intercept and slope: \(\hat{\theta}(\gamma)=\hat{\beta}_0+\gamma\hat{\beta}_1\). For fixed \(\gamma\), the coefficients of the random estimates \(\hat{\beta}_0\) and \(\hat{\beta}_1\) are constants. Expanding the variance of the sum gives:
The cross-product term follows from \(\operatorname{Var}(U+V)=\operatorname{Var}(U)+\operatorname{Var}(V)+2\operatorname{Cov}(U,V)\). The formula includes covariance because the coefficient estimates need not be independent. It has no term for variation in \(\gamma\), since \(\gamma\) is fixed in this calculation.
Worked Example: Applying the Delta Method at Two Values
In a hypothetical analysis, the conditional treatment-effect estimator has the form \(\hat{\theta}(\gamma)=\hat{\beta}_0+\gamma\hat{\beta}_1\). The fitted values are \(\hat{\beta}_0=2.4\) score points and \(\hat{\beta}_1=-0.35\) score points per unit of the sensitivity scale. Their estimated variances are 0.16 and 0.04, and their estimated covariance is \(-0.03\). Thus the standard error at a fixed value is found from \(0.16+\gamma^2(0.04)+2\gamma(-0.03)\).
At \(\gamma=0\), the estimate is \(2.4+0(-0.35)=2.4\) points. The estimated variance is \(0.16+0+0=0.16\), so the conditional standard error is \(\sqrt{0.16}=0.400\) points.
At \(\gamma=2\), the estimate is \(2.4+2(-0.35)=1.7\) points. The estimated variance is \(0.16+2^2(0.04)+2(2)(-0.03)=0.16+0.16-0.12=0.20\). The conditional standard error is \(\sqrt{0.20}=0.447\) points.
The estimate and standard error are both conditional on the stated value of \(\gamma\). The covariance term reduces the variance here; omitting it would give \(0.32\), not \(0.20\), at \(\gamma=2\). This example also shows why one should calculate the standard error for each fixed value rather than assume it must be unchanged when the estimate changes.
Checks Before Interpreting a Result
A conditional estimate is only meaningful when the scenario, estimator, and uncertainty calculation match. In practice, check the following before comparing results:
- State the fixed value or scenario. A number without its sensitivity assumption is incomplete because another assumption can define a different estimate.
- Use the covariance for that scenario. If the estimator changes across assumptions, do not automatically reuse a standard error from a different fit.
- Retain covariance terms. For a contrast or other function of multiple estimated quantities, dropping covariance is justified only when it is known to be zero or the analysis explicitly supports that simplification.
- Check units and direction. The estimate and its standard error have the units of the treatment effect. Explain what a positive or negative value means for the selected outcome scale.
- Verify what the uncertainty calculation includes. It should account for sampling variation in estimated parameters used by the estimator. Treating model-based imputed values as fixed after fitting can omit relevant uncertainty.
A useful diagnostic is to calculate the estimate and its variance from the same fitted quantities and the same fixed assumption, then check that the variance is nonnegative and that the standard error is its square root. A negative computed variance often indicates a sign error, an incorrectly assembled covariance matrix, or an incompatible set of inputs; it is not a meaningful standard error.
A common mistake is to read a conditional standard error as a measure of whether the sensitivity assumption itself is uncertain. For example, “the standard error is small, so this assumption is reliable” is not justified. The standard error addresses repeated-sampling variation conditional on that assumption. The assumption’s plausibility requires separate scientific reasoning.
How This Reporting Choice Differs From Alternatives
One alternative is to report a result under only a reference assumption. That can be clear and concise, but it does not show whether the estimate changes under other specified assumptions. Reporting estimates and standard errors for multiple fixed scenarios makes the dependence on assumptions visible, while preserving the sampling interpretation of each standard error.
A different alternative would be to treat a sensitivity value as random and include its variation in an uncertainty calculation. That answers a different question from the conditional calculation taught here: it requires a model or distribution for the sensitivity value and a method that propagates that variation. A fixed-assumption standard error should not be presented as if it included that extra uncertainty.
The conditional approach also has limits. It does not establish that the chosen sensitivity assumption is correct, and it cannot make an assumption-driven difference disappear. It can be difficult to compare standard errors if different scenarios use incompatible estimators or variance procedures. Reporting the scenario, effect scale, estimate, and conditional standard error together helps readers see precisely what is being compared.
Worked Example: The Effect of Ignoring Covariance
Suppose a hypothetical analysis estimates a treatment mean of 5.0 points and a comparator mean of 3.0 points under one fixed assumption. Their standard errors are 0.50 and 0.40 points, and their estimated correlation is 0.30. Find the treatment contrast and its standard error. Then compare with the incorrect calculation that treats the estimates as uncorrelated.
The estimated contrast is \(5.0-3.0=2.0\) points. The covariance is the correlation multiplied by the two standard errors: \(0.30(0.50)(0.40)=0.060\) points squared. Therefore, the estimated variance of the contrast is \(0.50^2+0.40^2-2(0.060)=0.25+0.16-0.12=0.29\) points squared. The conditional standard error is \(\sqrt{0.29}=0.539\) points.
If covariance were incorrectly omitted, the calculated standard error would be \(\sqrt{0.25+0.16}=\sqrt{0.41}=0.640\) points. That is not the standard error implied by the stated joint estimates. The correct calculation uses the covariance because the contrast is formed from two estimates whose sampling variation is correlated.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In one sentence, what does \(SE\{\hat{\theta}(\gamma)\}\) describe when \(\gamma\) is fixed?
- Under a fixed assumption, two estimated means have variances 0.49 and 0.36 and covariance 0.06. Calculate the standard error of their difference.
- For \(\hat{\theta}(\gamma)=\hat{\beta}_0+\gamma\hat{\beta}_1\), what variance term accounts for dependence between the coefficient estimates?
- Why does a small conditional standard error not establish that the sensitivity assumption is plausible?
Solutions. (1) It estimates the repeated-sampling standard deviation of the treatment-effect estimator when the analysis and the value of \(\gamma\) are held fixed. (2) The variance of the difference is \(0.49+0.36-2(0.06)=0.73\); therefore the standard error is \(\sqrt{0.73}=0.854\), in the units of the estimated means. (3) The term is \(2\gamma\widehat{\operatorname{Cov}}(\hat{\beta}_0,\hat{\beta}_1)\). (4) The standard error quantifies sampling variability conditional on the assumption; plausibility of the assumption is a separate scientific question.