Bias-Function Sensitivity for Unobserved Confounding

Why Parameterize Unobserved Confounding?

When treatment is not randomly assigned, treated and untreated units may differ in their outcomes even if neither group received treatment. The observed difference in mean outcomes therefore need not equal a treatment effect. One way to make this problem transparent is to express the confounding concern as a difference in the groups’ mean outcomes under no treatment, then examine how conclusions change across plausible values of that difference.

This approach does not make the bias parameter observable. Instead, it asks a focused question: if the untreated potential-outcome mean for treated units differed from that for untreated units by a specified amount, what treatment-effect values would be compatible with the observed means? The resulting bounds are conditional on the chosen range and on assumptions connecting the effect for treated units to the population ATE.

Setup and Bias Parameter

Let \(D\) equal 1 for a treated unit and 0 for an untreated unit. Let \(Y(1)\) and \(Y(0)\) be that unit’s potential outcomes under treatment and no treatment, respectively. The observed outcome is \(Y=D Y(1)+(1-D)Y(0)\), by consistency. Assume there is no interference between units, and that the means below exist. Write \(m_1=E[Y\mid D=1]\) and \(m_0=E[Y\mid D=0]\) for the observed treated and untreated means.

The first term in \(\delta_0\) is counterfactual for treated units, so the observed data alone generally do not determine \(\delta_0\). A sensitivity analysis specifies a range for it using substantive knowledge, design information, or an explicitly chosen scenario. The range is an assumption about plausible confounding, not an estimate delivered by the observed outcome difference.

What the Observed Difference Identifies

Let \(\tau=Y(1)-Y(0)\) be the individual treatment effect. Consistency gives \(m_1=E[Y(1)\mid D=1]\) and \(m_0=E[Y(0)\mid D=0]\). Adding and subtracting \(E[Y(0)\mid D=1]\) yields an exact decomposition:

Thus the observed difference minus \(\delta_0\) identifies the average treatment effect among the treated, or ATT:

This identity follows from consistency and the definition of \(\delta_0\); it does not require treatment assignment to be independent of potential outcomes. But it identifies the ATT, not automatically the population average treatment effect \(ATE=E[\tau]\). In general, effects for treated and untreated units can differ, and a bias parameter involving only \(Y(0)\) does not reveal the untreated units’ missing treatment outcomes.

Turning a Bias Range into Effect Bounds

Suppose the analysis specifies that \(\delta_0\) lies in the interval \([\delta_L,\delta_U]\), with \(\delta_L\leq\delta_U\). Under mean treatment-effect transportability, the ATE is the observed gap minus \(\delta_0\). Subtraction reverses the endpoints, giving the identified set:

These endpoints are the smallest and largest values of the adjusted effect as the bias varies over the specified interval. Conditional on the stated model and bias range, every value in this interval corresponds to a value of \(\delta_0\) in that range. The interval is therefore a sensitivity bound, not a confidence interval: it describes uncertainty about the effect from the specified range of unobserved confounding, not sampling variability.

Derivation and Interpretation of the Endpoints

The endpoint calculation can be justified directly. Define \(g=m_1-m_0\). For every admissible bias value, the adjusted effect is \(g-\delta_0\). Since \(\delta_L\leq\delta_0\leq\delta_U\), multiplying the inequality by \(-1\) reverses its direction. Adding \(g\) then gives:

Consequently, the lower endpoint uses \(\delta_U\), not \(\delta_L\). If the bias range has width \(\delta_U-\delta_L\), the resulting effect interval has the same width, provided no other restrictions further narrow it. Shifting both bias endpoints upward shifts both effect endpoints downward by the same amount.

Support Restrictions and a Binary Outcome

A proposed sensitivity range should also be compatible with known restrictions on the outcome. For a binary outcome, each potential-outcome mean lies between 0 and 1. In particular, if \(m_0=E[Y(0)\mid D=0]\), then \(E[Y(0)\mid D=1]\) must lie in \([0,1]\), so the bias must satisfy \(-m_0\leq\delta_0\leq1-m_0\). A user-specified range outside these limits is not feasible as stated and should be intersected with the logical range, or reconsidered.

Common Mistakes and Practical Use

• Calling the adjusted contrast an ATE without checking the effect assumption. The identity \(m_1-m_0-\delta_0=ATT\) holds directly. It equals the ATE only under an additional condition such as mean treatment-effect transportability across treatment status.
• Adding a positive bias instead of subtracting it. If treated units would have had higher outcomes even without treatment, the observed gap overstates the effect under this decomposition. A positive \(\delta_0\) must be subtracted.
• Using the wrong endpoint order. Because the bias is subtracted, the largest bias produces the smallest adjusted effect. The bounds are \(g-\delta_U\) and \(g-\delta_L\), in that order.
• Treating the sensitivity range as data-estimated uncertainty. The range represents a maintained set of plausible bias values. Sampling uncertainty in \(m_1\) and \(m_0\) is separate and is not included in these calculations.
• Ignoring units and feasible values. The bias and effect must use the outcome’s units. For bounded outcomes, proposed bias values must also respect the possible range of the missing untreated mean.

A useful report states the observed means, the outcome scale, the bias definition, the chosen bias range and its rationale, and the assumption that permits an ATE interpretation. If mean treatment-effect transportability is not credible, report the result as an ATT sensitivity analysis instead, or introduce additional assumptions that genuinely address effects for untreated units.