Bias Formulas for a Binary Unmeasured Confounder

Why a Binary Confounder Can Bias a Risk Difference

A crude comparison of outcome risks between treated and untreated groups can differ from the causal effect because the groups may have different compositions. If a binary factor \(U\) is more common in one treatment group and is also associated with the outcome, part of the crude risk difference can reflect that imbalance rather than treatment. A useful bias formula separates these two ingredients: how much the prevalence of \(U\) differs between treatment groups, and how much outcome risk differs by \(U\) within each treatment group.

The formulas below concern a binary treatment \(A\), binary outcome \(Y\), and binary unmeasured confounder \(U\). The target is the average treatment effect on the risk-difference scale in the study population. The formulas describe bias in the crude risk difference; they do not, by themselves, make the unmeasured quantities observable.

Definition: Let \(Y(1)\) and \(Y(0)\) be the potential outcomes under treatment and no treatment. The causal average treatment effect on the risk-difference scale is \(E[Y(1)-Y(0)]\). The crude associational risk difference is \(E[Y\mid A=1]-E[Y\mid A=0]\). Define risk-difference bias as the crude associational risk difference minus the causal average treatment effect. A positive bias means the crude contrast is larger than the causal effect on this scale.

Use the notation \(m_a(u)=P(Y=1\mid A=a,U=u)\), \(r_a=P(U=1\mid A=a)\), \(\pi=P(A=1)\), and \(q=P(U=1)\). The within-treatment outcome-risk contrast for \(U\) is \(\delta_a=m_a(1)-m_a(0)\). Thus, \(\delta_1\) compares outcome risks for \(U=1\) and \(U=0\) among treated individuals, while \(\delta_0\) makes the corresponding comparison among untreated individuals. These are risk differences, not risk ratios.

Assumptions and the Exact Decomposition

To interpret standardization over \(U\) as a causal effect, assume consistency, no interference between individuals, and conditional exchangeability given \(U\): \(Y(a)\) is independent of \(A\) conditional on \(U\), for \(a=0,1\). Also require positivity: each treatment level must have positive probability within each \(U\) stratum represented in the target population. Under these assumptions, \(m_a(u)\) is the mean potential outcome under treatment level \(a\) among individuals with \(U=u\). Standardizing these risks over the study population’s distribution of \(U\) identifies the population average treatment effect, if \(U\) were measured.

The crude risk in treatment group \(a\) instead averages \(m_a(u)\) over that group’s own distribution of \(U\). The causal mean under treatment level \(a\) averages the same stratum-specific risks over the population distribution of \(U\). Their difference is therefore \(\delta_a(r_a-q)\): only the change in the weight on \(U=1\) matters, and the impact of that change is the risk contrast \(\delta_a\).

\begin{aligned} \text{Bias} &=\{E[Y\mid A=1]-E[Y\mid A=0]\} -E[Y(1)-Y(0)]\\ &=\delta_1(r_1-q)-\delta_0(r_0-q)\\ &=(r_1-r_0)\{(1-\pi)\delta_1+\pi\delta_0\}. \end{aligned}

The last equality uses \(q=\pi r_1+(1-\pi)r_0\). It follows that \(r_1-q=(1-\pi)(r_1-r_0)\) and \(r_0-q=-\pi(r_1-r_0)\). This decomposition is exact under the stated causal assumptions; it does not require the outcome-risk contrast for \(U\) to be the same in the two treatment groups.

Key result: The exact bias is the prevalence difference \(r_1-r_0\) multiplied by a treatment-prevalence-weighted average of the two within-treatment outcome-risk contrasts. If \(\delta_1=\delta_0=\delta\), the formula simplifies to \(\text{Bias}=(r_1-r_0)\delta\). This simpler form requires a common additive risk contrast for \(U\) across treatment groups.

The simplified formula is often convenient, but its scale and assumptions matter. It is a risk-difference formula and uses the difference in prevalence, not an odds ratio or a prevalence ratio. If the association of \(U\) with outcome risk differs by treatment group, use the exact formula instead. In either case, the sign of the bias depends on both the direction of the prevalence imbalance and the direction of the outcome-risk contrast.

Worked Example: A Common Outcome-Risk Contrast

Suppose a hypothetical study has 500 treated and 500 untreated participants. Among treated participants, \(U=1\) occurs in 300 people; the outcome risks are 0.45 for \(U=1\) and 0.25 for \(U=0\). Among untreated participants, \(U=1\) occurs in 100 people; the risks are 0.30 for \(U=1\) and 0.10 for \(U=0\). Assume the causal identification conditions stated above hold.

The prevalence contrast is \(r_1-r_0=300/500-100/500=0.60-0.20=0.40\). The within-treatment risk contrasts are \(\delta_1=0.45-0.25=0.20\) and \(\delta_0=0.30-0.10=0.20\). Since they are equal, the common-contrast formula applies:

\text{Bias}=(0.60-0.20)(0.20)=0.080.

Check against standardization. The crude treated risk is \((300(0.45)+200(0.25))/500=0.37\), and the crude untreated risk is \((100(0.30)+400(0.10))/500=0.14\). Their crude risk difference is \(0.37-0.14=0.23\). In this study, \(q=400/1000=0.40\). The standardized treated risk is \(0.40(0.45)+0.60(0.25)=0.33\), and the standardized untreated risk is \(0.40(0.30)+0.60(0.10)=0.18\). The causal risk difference is therefore \(0.33-0.18=0.15\), and \(0.23-0.15=0.08\), as predicted. Here the crude comparison overstates the causal effect by 0.080, or 8.0 percentage points.

When the Confounder’s Outcome Association Differs by Treatment

A binary confounder need not have the same outcome-risk contrast in treated and untreated people. In that case, the simple product of one prevalence difference and one common outcome contrast is not justified. The exact formula accounts for both contrasts and weights them according to the proportion treated in the study population.

Worked Example: Different Risk Contrasts Across Treatment Groups

Consider a hypothetical population with \(\pi=0.40\). Suppose \(r_1=0.70\) and \(r_0=0.20\), so \(U\) is more prevalent among treated participants. The outcome risks are \(m_1(1)=0.60\), \(m_1(0)=0.30\), \(m_0(1)=0.25\), and \(m_0(0)=0.15\). Assume consistency, no interference, conditional exchangeability given \(U\), and positivity.

The prevalence difference is \(r_1-r_0=0.70-0.20=0.50\). The outcome-risk contrasts are \(\delta_1=0.60-0.30=0.30\) and \(\delta_0=0.25-0.15=0.10\). The exact bias is:

\text{Bias}=0.50\{(1-0.40)(0.30)+0.40(0.10)\} =0.50(0.18+0.04)=0.11.

The same calculation can be checked by standardizing. The population prevalence is \(q=0.40(0.70)+0.60(0.20)=0.40\). The crude treated risk is \(0.70(0.60)+0.30(0.30)=0.51\), and the crude untreated risk is \(0.20(0.25)+0.80(0.15)=0.17\), giving a crude risk difference of \(0.34\). The standardized risks are \(0.40(0.60)+0.60(0.30)=0.42\) under treatment and \(0.40(0.25)+0.60(0.15)=0.19\) without treatment. The causal risk difference is \(0.23\); the difference \(0.34-0.23=0.11\) agrees with the formula. The crude estimate is 11 percentage points larger than the causal effect.

Reading the Sign and Size of Bias

Under a common risk contrast \(\delta\), the sign is especially transparent. If \(U\) is more common among treated participants (\(r_1-r_0>0\)) and is associated with higher outcome risk (\(\delta>0\)), the bias is positive: the crude risk difference is shifted upward. If one of these contrasts is negative, the bias is negative. If either contrast is zero, the bias is zero under this model: there is no imbalance in \(U\), or \(U\) has no outcome-risk contrast on the relevant scale.

With different \(\delta_1\) and \(\delta_0\), the weighted average in the exact formula determines the direction. A positive prevalence difference does not alone establish positive bias: the weighted outcome-risk contrast could be zero or negative. Also, a zero bias does not prove that \(U\) is unrelated to treatment or outcome; contributions from the two treatment groups can cancel.

Worked Example: A Confounder Associated With Lower Risk

In a hypothetical study, let \(\pi=0.30\), \(r_1=0.25\), and \(r_0=0.55\). Suppose the risks are \(m_1(1)=0.36\), \(m_1(0)=0.28\), \(m_0(1)=0.50\), and \(m_0(0)=0.30\). Assume the same causal identification conditions as above.

Here \(r_1-r_0=0.25-0.55=-0.30\), \(\delta_1=0.36-0.28=0.08\), and \(\delta_0=0.50-0.30=0.20\). Thus:

\text{Bias}=(-0.30)\{(1-0.30)(0.08)+0.30(0.20)\} =(-0.30)(0.056+0.060)=-0.0348.

For a check, \(q=0.30(0.25)+0.70(0.55)=0.46\). The crude treated risk is \(0.25(0.36)+0.75(0.28)=0.30\); the crude untreated risk is \(0.55(0.50)+0.45(0.30)=0.41\). The crude risk difference is \(0.30-0.41=-0.11\). Standardized risks are \(0.46(0.36)+0.54(0.28)=0.3168\) and \(0.46(0.50)+0.54(0.30)=0.3920\), so the causal risk difference is \(-0.0752\). Subtracting gives \(-0.11-(-0.0752)=-0.0348\). The crude contrast is 3.48 percentage points lower than the causal effect. Although \(U\) is associated with higher risk within both treatment groups, it is less prevalent among treated participants, producing negative bias.

Using the Formula in Sensitivity Analysis

When \(U\) is unmeasured, its prevalence contrast and outcome-risk contrasts are not directly available from the observed data. The formula is therefore a sensitivity-analysis tool: specify plausible values for these quantities, calculate the implied bias, and subtract it from the crude risk difference to obtain a bias-adjusted causal estimate under the stated model. Such calculations do not show that a particular value is true. Their value is in making the assumptions and the resulting adjustment explicit.

The numerical inputs must be coherent probabilities and risk differences. Each \(r_a\) and each \(m_a(u)\) lies between 0 and 1, so each \(\delta_a\) lies between \(-1\) and 1. The implied bias is on the risk-difference scale. It should be reported in context, for example as a change of 0.05 in outcome risk, or 5 percentage points, rather than as a percent change unless a relative scale has been deliberately defined.

Specify the estimand and bias direction.
Use the population average causal risk difference and define bias as crude risk difference minus causal risk difference.

Specify the confounder imbalance.
Choose \(r_1\) and \(r_0\), then calculate \(r_1-r_0\).

Specify outcome-risk contrasts.
Choose \(\delta_1\) and \(\delta_0\). Use one common contrast only when it is defensible on the additive risk scale.

Calculate and interpret.
Apply the exact formula, then subtract the implied bias from the crude risk difference to obtain the bias-adjusted estimate under those assumptions.

Common Mistake: Multiplying the Wrong Contrasts

A frequent error is to take the prevalence contrast and multiply it by a risk contrast from only one treatment group, even when \(\delta_1\) and \(\delta_0\) differ. That calculation silently assumes a common additive risk effect of \(U\). The correction is to use \((r_1-r_0)\{(1-\pi)\delta_1+\pi\delta_0\}\) when the contrasts differ, or to justify the common-contrast simplification before using it.

Another error is to call the crude risk difference a causal effect merely because \(U\) is binary or because the bias formula has been evaluated. Causal interpretation still depends on the identification assumptions, including conditional exchangeability given \(U\), and on the chosen sensitivity values being plausible. The formula quantifies consequences of those assumptions; it does not verify them.

Key takeaway: For a binary confounder, risk-difference bias depends jointly on the confounder’s prevalence difference between treatment groups and its within-group outcome-risk contrasts. The exact formula permits those contrasts to differ; the familiar single-product formula is a special case requiring a common additive risk contrast.

Check Your Understanding

Use the definitions and formulas above to answer the following questions.

State the sign convention used for risk-difference bias. What does a negative bias say about the crude risk difference relative to the causal effect?
If \(r_1=r_0\), what does the exact formula imply about bias? State the model context in which that conclusion holds.
Suppose \(r_1-r_0=0.25\), \(\delta_1=0.20\), \(\delta_0=0.10\), and \(\pi=0.40\). Calculate the exact bias.
Why is \((r_1-r_0)\delta_0\) generally not the exact bias when the two outcome-risk contrasts differ?
What additional assumptions are needed to interpret standardization over \(U\) as a causal average treatment effect?