Randomized Dose Is Not Randomized Exposure
Random assignment can make assigned dose independent of participants’ baseline characteristics, on average. It does not make the exposure each participant attains independent of those characteristics. People assigned the same dose can have different exposures because of differences in absorption, distribution, metabolism, elimination, and dose-taking behavior. Those differences matter: a comparison of participants with higher versus lower observed exposure is not automatically a randomized comparison.
The central distinction is between assignment, which is randomized, and exposure, which occurs after assignment and is shaped by individual factors. The earlier material on causal assumptions in individual-level exposure-response motivates this distinction. Here, the focus is on what follows from it for estimation and interpretation.
A randomized contrast can estimate the causal effect of assignment to one dose strategy rather than another, provided the randomization is implemented appropriately and outcomes are analyzed according to assignment. This effect includes consequences of assignment that operate through achieved exposure, adherence, and other consequences of the assigned strategy. It is not, without further assumptions, the effect of changing exposure while holding everything else constant.
Why Exposure Varies Within Dose Groups
A prescribed dose is an input to the pharmacokinetic process, not a guarantee of a particular concentration or exposure. For a fixed regimen, exposure can differ between participants because pharmacokinetic parameters and dosing behavior differ. Thus, even when everyone in an arm is assigned the same dose, that arm can contain a broad range of exposure values. Exposure distributions for different dose arms can also overlap.
Randomization balances baseline factors across assigned groups in expectation; it does not balance every factor across groups defined later by observed exposure. For example, participants can be grouped after the trial according to whether their measured exposure was high or low. If a baseline characteristic influences both exposure and outcome, the high- and low-exposure groups can differ in that characteristic even though dose assignment was randomized.
Worked Example: Overlapping Exposure After Dose Randomization
In a hypothetical trial, 100 participants are randomized to each of two dose regimens. The selected exposure metric is area under the concentration-time curve, measured in ng·h/mL. The low-dose arm has mean exposure 40 ng·h/mL and sample standard deviation 16 ng·h/mL; the high-dose arm has mean exposure 60 ng·h/mL and sample standard deviation 20 ng·h/mL. The observed ranges are 12–78 ng·h/mL and 18–112 ng·h/mL, respectively. The mean outcome is 5.0 points in the low-dose arm and 7.0 points in the high-dose arm.
There is substantial within-arm variation: exposure is not fixed by assigned dose. The observed ranges overlap from 18 to 78 ng·h/mL, so some participants in the low-dose arm have higher exposure than some in the high-dose arm.
The randomized assignment contrast in mean outcome is \(7.0-5.0=2.0\) points. Under the conditions for a valid randomized comparison, this estimates the effect of assignment to the high-dose rather than the low-dose strategy in this trial population. It does not show that increasing an individual’s exposure by a particular number of ng·h/mL causes a specified change in outcome. That is a different causal question.
How an Exposure-Response Slope Can Be Biased
Consider a simple linear model to make the source of bias explicit. Let \(Z\) be a randomized binary dose assignment, \(U\) a baseline factor that influences both exposure and outcome, and \(\varepsilon_E\) and \(\varepsilon_Y\) residual variation. Suppose:
Here, \(\beta\) is the exposure effect in the specified structural model, while \(\gamma\) represents an outcome effect of \(U\) that is not mediated by the chosen exposure metric. Assume, for this illustration, that \(Z\) is independent of \(U\), and that the residuals are independent of \(Z\) and \(U\). Randomization gives \(Z\) this independence from baseline \(U\); it does not remove the \(U\)-to-\(E\) and \(U\)-to-\(Y\) relationships.
An ordinary least-squares regression of \(Y\) on \(E\) alone has population slope \(\operatorname{Cov}(E,Y)/\operatorname{Var}(E)\). Substituting the outcome model yields:
If \(\varepsilon_Y\) is independent of \(E\), the last term is zero. But if \(U\) influences exposure, \(\operatorname{Cov}(E,U)\) need not be zero. The exposure-only slope then differs from \(\beta\) when \(\gamma\ne 0\). Randomizing dose can contribute variation in exposure, but it does not by itself eliminate this individual-level association between exposure and \(U\).
Worked Example: A Confounded Exposure Slope Despite Randomization
Suppose, in a hypothetical trial, \(Z\) is assigned with probability 0.5, \(U\) has mean 0 and variance 1, and the data follow:
where \(\varepsilon_Y\) is independent of \(Z\) and \(U\), has mean 0, and exposure is measured in ng·h/mL. The coefficient \(0.10\) is the exposure effect in this stipulated model; the coefficient \(0.8\) is an additional outcome association with \(U\).
Because \(Z\) is binary with probability 0.5, \(\operatorname{Var}(Z)=0.25\). The exposure variance and its covariance with \(U\) are:
The exposure-only regression slope is therefore:
It exceeds the stipulated exposure coefficient of 0.10 because \(U\) raises both exposure and outcome. Yet the randomized assignment contrast in mean exposure is 20 ng·h/mL, and the assignment contrast in mean outcome is \(0.10(20)=2.0\) points: randomization balances \(U\) between assignment groups in expectation. The example shows how a biased exposure-only slope can coexist with an unbiased randomized assignment contrast.
What Randomized Dose Contrasts Can Establish
The primary strength of randomized assignment is that, in expectation, assigned groups are comparable with respect to baseline causes of the outcome. A dose-arm contrast can therefore support a causal interpretation for the effect of assignment, subject to appropriate outcome ascertainment and analysis. In practice, the precise estimand matters: the contrast is between the assigned strategies as implemented, not necessarily between fixed exposure levels.
It is possible to use randomized dose as an instrument for exposure, but this requires additional assumptions. In a simple linear model with a constant exposure effect, let \(Z\) be randomized assignment and suppose:
- Relevance: assignment changes exposure, so \(\operatorname{Cov}(Z,E)\ne 0\).
- Independence: assignment is independent of causes of the outcome, as ensured in expectation by valid randomization.
- Exclusion: assignment affects the outcome only through the chosen exposure metric.
- Consistency and a suitable structural model: the exposure and outcome definitions correspond to the quantities in the model, and the constant-effect linear model is adequate for the interpretation sought.
Under these assumptions, write \(Y=c+\beta E+\varepsilon_Y\), with \(\operatorname{Cov}(Z,\varepsilon_Y)=0\). Then:
If relevance holds, division gives \(\operatorname{Cov}(Z,Y)/\operatorname{Cov}(Z,E)=\beta\). Equivalently, with two randomized assignment groups, the ratio of the mean outcome contrast to the mean exposure contrast estimates the model’s exposure effect. The derivation depends on the assumptions, especially exclusion; randomization alone establishes none of the other claims.
Worked Example: A Dose Contrast Used as an Instrument
In a hypothetical randomized trial, participants assigned to the lower regimen have mean exposure 30 ng·h/mL and mean outcome 4.0 points. Participants assigned to the higher regimen have mean exposure 50 ng·h/mL and mean outcome 5.6 points. Assume randomization was valid, the assignment produces a relevant exposure difference, the chosen exposure metric fully represents the pathway from assignment to outcome, and the linear constant-effect model is appropriate.
The first-stage mean exposure contrast is \(50-30=20\) ng·h/mL. The randomized mean outcome contrast is \(5.6-4.0=1.6\) points. The instrumental-variable ratio is:
Under the stated assumptions, this estimates the exposure effect in the specified model. The two randomized contrasts alone do not verify those assumptions. For example, if assigned dose affected the outcome through a pathway not captured by this exposure metric, exclusion would fail and the ratio would not isolate the exposure effect. If that additional pathway increased the outcome by 1.0 point, the outcome contrast would be \(0.0800(20)+1.0=2.6\) points, and the ratio would be \(2.6/20=0.130\), not the stipulated exposure effect of 0.0800. The randomized outcome contrast would still describe the assignment effect.
Diagnostics, Alternatives, and Limits
Before interpreting an exposure-response analysis, inspect how exposure is distributed within each randomized dose group and how much the distributions overlap. Summarize exposure by assignment and examine whether measured baseline covariates vary across observed exposure levels. These checks can reveal substantial within-dose variation, limited overlap, or plausible exposure-related imbalances. They cannot prove that all relevant confounding is absent, and no diagnostic can establish the instrumental-variable exclusion restriction from the observed data alone.
The main approaches answer different questions:
| Approach | What it estimates or describes | Main limitation |
|---|---|---|
| Randomized dose contrast | Effect of assignment to one dose strategy rather than another | Does not isolate the effect of achieved exposure |
| Regression of outcome on observed exposure | Conditional association, or a causal effect only under adequate control of exposure-outcome confounding and a suitable model | Random dose assignment alone does not ensure the required control |
| Instrumental-variable analysis using dose assignment | An exposure effect under relevance, independence, exclusion, and model assumptions | Exclusion is substantive and often difficult to establish |
These approaches also have practical limitations. A weak exposure difference between assigned groups gives little instrumental information. A limited range of exposure or substantial overlap may make dose-based contrasts imprecise for some comparisons. An exposure-response association can change when relevant baseline factors are included, but adjustment is useful only when those factors are measured and modeled adequately. Finally, a linear ratio or slope may not summarize effects well if the exposure-response relationship is not linear; a simple dose contrast still estimates an assignment effect, but it does not map the entire relationship between exposure and outcome.
Common Mistake
Wrong reasoning: “Dose was randomized, so participants with high exposure and participants with low exposure are exchangeable; the outcome-on-exposure slope is causal.” This confuses randomized assignment with a post-randomization variable. Participants can have different exposure because of characteristics that also predict outcome, so exposure groups need not be comparable.
Correction: Treat the randomized dose contrast as the direct randomized comparison. Interpret an individual-level exposure slope causally only with a defensible strategy for the exposure-outcome relationship, such as adequate control of its confounding under stated assumptions or a valid instrumental-variable argument. Do not assume that randomization of dose supplies those assumptions automatically.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A randomized trial has a mean exposure contrast of 15 ng·h/mL between two dose arms. Does this make exposure random within each arm? Explain.
- In the linear model above, suppose \(\beta=0.20\), \(\gamma=0.5\), \(\operatorname{Cov}(E,U)=8\), and \(\operatorname{Var}(E)=100\). What is the exposure-only regression slope if the outcome residual is independent of exposure?
- A randomized dose contrast changes mean exposure by 10 ng·h/mL and mean outcome by 1.2 points. Calculate the instrumental-variable ratio, and name one assumption needed to interpret it as an exposure effect.
- Suppose assignment affects the outcome both through the selected exposure metric and through an additional pathway. Which interpretation remains supported by randomization, and which instrumental-variable assumption fails?
Solutions. 1. No. The contrast shows that assignment shifts mean exposure; exposure can still vary between participants within either arm because it is not randomized. 2. The slope is \(0.20+0.5(8/100)=0.20+0.04=0.240\) outcome units per exposure unit. 3. The ratio is \(1.2/10=0.120\) outcome points per ng·h/mL; for example, the exclusion restriction must hold. 4. The randomized comparison still supports an effect of assignment to the dose strategy, under the conditions for a randomized comparison. The exclusion restriction fails because assignment has an outcome pathway outside the selected exposure metric.