Sensitivity Parameters in Binary Treatment-Outcome Models

Why Specify Sensitivity Parameters?

When a confounder is unmeasured, the data do not reveal how strongly it is associated with treatment or outcome. Sensitivity analysis makes those unknown relationships explicit: rather than treating the hidden confounder as if its distribution and outcome associations were known, we specify plausible values and calculate what they imply for bias. For a binary treatment, binary outcome, and binary confounder, these relationships can be summarized with interpretable probabilities and odds or risk ratios.

The parameters are assumptions, not estimates obtained from the observed treatment and outcome alone. Their value is that they make a causal conclusion conditional and transparent: a reader can see which combinations of confounder prevalence and outcome association would materially change the estimate.

Model and Sensitivity Parameters

Let \(A\in\{0,1\}\) denote treatment, \(Y\in\{0,1\}\) a binary outcome, and \(U\in\{0,1\}\) an unmeasured binary confounder. The calculations below describe one target population, with no measured covariates; they can also be applied within measured-covariate strata and then standardized across those strata. Define \(p_a=P(U=1\mid A=a)\), and define the outcome risks \(q_{au}=P(Y=1\mid A=a,U=u)\). Let \(p=P(U=1)\) be the confounder prevalence in the target population.

To interpret standardized risks causally, assume consistency, positivity, and conditional exchangeability given \(U\): \(Y(a)\perp A\mid U\) for each \(a\). Positivity requires a positive probability of each treatment level at each confounder level in the target population. Under these assumptions, the risk under treatment \(a\), standardized to the target distribution of \(U\), is \((1-p)q_{a0}+pq_{a1}\). Without the exchangeability assumption, these calculations remain a model for bias in observed associations, but the standardized quantities need not be causal risks.

Definition: The treatment-assignment sensitivity parameter is the odds ratio comparing the prevalence of \(U=1\) among treated and untreated people: \(\Gamma=\{p_1/(1-p_1)\}/\{p_0/(1-p_0)\}\). The outcome-association parameters describe how the outcome risk differs between \(U=1\) and \(U=0\) within each treatment level. They may be specified as risk ratios \(r_a=q_{a1}/q_{a0}\), or as odds ratios \(\Theta_a=\{q_{a1}/(1-q_{a1})\}/\{q_{a0}/(1-q_{a0})\}\).

These are associations within the specified model, not effects of intervening on \(U\). In particular, an outcome-association odds ratio is not a risk ratio. The parameters also need not be constant across treatment levels: \(r_0\) may differ from \(r_1\), just as \(\Theta_0\) may differ from \(\Theta_1\).

From Parameters to Confounder Prevalences and Risks

The assignment parameter \(\Gamma\), together with the untreated-group prevalence \(p_0\), determines the treated-group prevalence \(p_1\). Solving the odds-ratio definition gives the conversion used in sensitivity calculations.

\frac{p_1}{1-p_1}=\Gamma\frac{p_0}{1-p_0} \quad\Longrightarrow\quad p_1=\frac{\Gamma p_0}{1-p_0+\Gamma p_0}.

For a risk-ratio specification, \(q_{a1}=r_aq_{a0}\). The assumed value must satisfy \(0\leq r_aq_{a0}\leq1\). For an outcome odds-ratio specification, solving for the risk among people with \(U=1\) gives:

\frac{q_{a1}}{1-q_{a1}}=\Theta_a\frac{q_{a0}}{1-q_{a0}} \quad\Longrightarrow\quad q_{a1}=\frac{\Theta_a q_{a0}}{1-q_{a0}+\Theta_a q_{a0}}.

This conversion matters especially when the baseline risk is not small: multiplying a risk by an odds ratio generally gives the wrong answer. Once the assumed risks and prevalences are specified, the observed and standardized risks follow by averaging over the relevant distributions of \(U\).

R_a^{\mathrm{obs}}=(1-p_a)q_{a0}+p_aq_{a1}, \qquad R_a^{\mathrm{std}}=(1-p)q_{a0}+pq_{a1}.

Here \(R_a^{\mathrm{obs}}\) is the outcome risk actually averaged over the confounder distribution in treatment group \(a\). Under the causal assumptions stated above, \(R_a^{\mathrm{std}}\) is the target-population risk under treatment \(a\). The difference between them is an exact risk-scale bias expression.

R_a^{\mathrm{obs}}-R_a^{\mathrm{std}} =(p_a-p)(q_{a1}-q_{a0}).

Worked Applications

Worked Example: Converting an Assignment Odds Ratio

Suppose a hypothetical sensitivity analysis sets the prevalence of \(U=1\) among untreated people to \(p_0=0.12\), and assumes \(\Gamma=2.5\). What prevalence among treated people does this imply?

Substitute into the conversion formula:

p_1=\frac{2.5(0.12)}{1-0.12+2.5(0.12)} =\frac{0.30}{1.18} =0.2542.

Under this assumption, the confounder prevalence is about 25.42% among treated people, compared with 12.00% among untreated people. The parameter is an odds ratio of prevalences, not a statement that the prevalence itself is multiplied by 2.5: \(2.5(0.12)=0.30\) is only the numerator in the conversion.

Worked Example: Outcome Odds Ratios and Risk-Ratio Bias

Consider hypothetical values with \(p_0=1/6\), target prevalence \(p=0.20\), \(q_{00}=0.10\), and \(q_{10}=0.08\). Assume the treatment-assignment parameter is \(\Gamma=2\), the outcome odds ratios are \(\Theta_0=3\) and \(\Theta_1=2\), and the model assumptions above hold. Calculate the observed and standardized risk ratios.

First convert the assignment odds ratio to \(p_1\):

p_1=\frac{2(1/6)}{1-1/6+2(1/6)} =\frac{1/3}{7/6} =\frac{2}{7}.

The outcome odds-ratio conversion gives \(q_{01}=3(0.10)/(1-0.10+3(0.10))=0.30/1.20=0.25\). For treated people, it gives:

q_{11}=\frac{2(0.08)}{1-0.08+2(0.08)} =\frac{0.16}{1.08} =0.148148.

The denominator is \(1.08\), so the risk is about 0.1481, not 0.1739. Average the risks over the observed treatment-group prevalences:

R_1^{\mathrm{obs}} =0.08+\frac{2}{7}(0.148148-0.08) =0.099471, \qquad R_0^{\mathrm{obs}} =0.10+\frac{1}{6}(0.25-0.10) =0.1250.

Thus the observed risk ratio is \(RR_{\mathrm{obs}}=0.099471/0.1250=0.7958\). For standardization, use the target prevalence \(p=0.20\) for both treatment levels:

R_1^{\mathrm{std}}=0.08+0.20(0.148148-0.08)=0.093630, \qquad R_0^{\mathrm{std}}=0.10+0.20(0.25-0.10)=0.1300.

The standardized risk ratio is \(RR_{\mathrm{std}}=0.093630/0.1300=0.7202\). The observed-to-standardized ratio is \(0.7958/0.7202=1.1049\). Under this hypothetical model, the observed risk ratio is about 10.49% larger than the standardized risk ratio. This comparison is conditional on the chosen sensitivity parameters and causal assumptions; it does not establish that these assumed values describe any actual study.

Worked Example: Risk-Ratio Assumptions on the Outcome Scale

In a second hypothetical model, suppose \(p_0=0.10\), \(p_1=0.30\), and \(p=0.20\). Let the outcome risks for \(U=0\) be \(q_{00}=0.05\) and \(q_{10}=0.10\), and assume \(r_0=r_1=3\). These risk-ratio assumptions imply \(q_{01}=3(0.05)=0.15\) and \(q_{11}=3(0.10)=0.30\), both valid probabilities.

The assignment association corresponding to these prevalences is:

\Gamma =\frac{0.30/0.70}{0.10/0.90} =\frac{3/7}{1/9} =\frac{27}{7} =3.857.

The observed risks are \(R_1^{\mathrm{obs}}=0.10+0.30(0.30-0.10)=0.1600\) and \(R_0^{\mathrm{obs}}=0.05+0.10(0.15-0.05)=0.0600\). Therefore \(RR_{\mathrm{obs}}=0.1600/0.0600=2.667\). Standardizing both groups to \(p=0.20\) gives:

R_1^{\mathrm{std}}=0.10+0.20(0.30-0.10)=0.1400, \qquad R_0^{\mathrm{std}}=0.05+0.20(0.15-0.05)=0.0700, \qquad RR_{\mathrm{std}}=\frac{0.1400}{0.0700}=2.000.

The observed-to-standardized risk-ratio factor is \(2.667/2.000=1.333\). In this model, the confounder is more prevalent among treated people and is associated with higher outcome risk in both groups; the observed risk ratio is consequently farther above 1 than the standardized ratio.

Why the Bias Depends on Both Associations

The risk-scale identity shows why specifying only one association is insufficient. If \(U\) is equally prevalent in the treatment groups and target population, then \(p_a=p\) and the risk bias is zero, even if \(U\) strongly predicts the outcome. If \(U\) is associated with treatment but has no outcome association within treatment levels, then \(q_{a1}=q_{a0}\) and the risk bias is also zero. Bias arises when the distribution shift and the outcome-risk contrast combine.

For risk ratios, the relevant comparison is:

RR_{\mathrm{obs}} =\frac{(1-p_1)q_{10}+p_1q_{11}} {(1-p_0)q_{00}+p_0q_{01}}, \qquad RR_{\mathrm{std}} =\frac{(1-p)q_{10}+pq_{11}} {(1-p)q_{00}+pq_{01}}.

These expressions also show why a single universal correction factor does not follow from the sensitivity parameters without further restrictions. The result depends on the baseline outcome risks, the confounder prevalences, and how the outcome association varies by treatment. An odds-ratio specification and a risk-ratio specification can imply different \(q_{a1}\) values, especially when \(q_{a0}\) is not small; the chosen scale must therefore be stated.

Common Mistakes

Multiplying risk by an odds ratio. If \(q_{a0}=0.08\) and \(\Theta_a=2\), the implied risk is \(0.16/(0.92+0.16)=0.1481\), not \(0.16\).
Multiplying prevalence by the assignment odds ratio. The odds are multiplied by \(\Gamma\); the prevalence must then be recovered with the denominator \(1-p_0+\Gamma p_0\).
Calling an assumed association a causal effect of \(U\). These sensitivity parameters describe conditional associations. A causal interpretation of standardized risks additionally requires the stated identification assumptions.
Comparing risks standardized to different populations. To isolate the effect of changing the confounder distribution, use the same target prevalence \(p\) for both treatment levels.

The practical value of these parameters is disciplined interpretation. Report the assumed prevalence contrast, the outcome-association scale and values, and the resulting observed-versus-standardized risks. The sensitivity analysis then states exactly how the causal estimate depends on the unmeasured-confounder model, rather than hiding that dependence in a qualitative caveat.

Check Your Understanding

Use the definitions and calculations in this tutorial to answer the following questions.

If \(p_0=0.20\) and \(\Gamma=3\), what value of \(p_1\) does the assignment model imply?
For \(q_{a0}=0.10\), what risk \(q_{a1}\) follows from an outcome odds ratio of 2?
State the risk-scale bias expression for treatment level \(a\), and identify the two differences whose product determines it.
Why can a strong association between \(U\) and outcome still produce zero risk bias?
What additional assumptions are needed to interpret \(R_a^{\mathrm{std}}\) as a causal risk rather than a standardized association?