Why Stratify for a Nonlinear Mendelian Randomization Analysis?
A conventional Mendelian randomization (MR) estimate often summarizes the causal effect of an exposure with one slope. That summary can be misleading when the effect changes across the exposure range. For example, a one-unit increase in a biomarker might have a small effect at lower levels and a larger effect at higher levels. Stratification-based nonlinear MR is designed to estimate an instrumented effect separately across ranked parts of the exposure distribution.
The central challenge is choosing strata without creating an association between the genetic instrument and factors that confound the exposure–outcome relationship. Stratifying directly on observed exposure is unsafe: exposure is affected by both the instrument and other causes, so conditioning on it can associate the instrument with those other causes. The stratification method instead ranks participants using residual exposure variation after accounting for the genetic instrument and measured covariates. Within each resulting stratum, it estimates an instrumental-variable ratio.
How the Stratification Method Works
Let \(X\) denote the exposure, \(Y\) the outcome, \(G\) a genetic instrument or genetic score, and \(C\) measured covariates. In a one-sample analysis, first model \(X\) as a function of \(G\) and \(C\), and calculate each participant’s residual exposure. Rank participants by those residuals and divide the ranking into prespecified groups, such as quartiles. Then estimate the \(G\)-to-\(X\) and \(G\)-to-\(Y\) associations separately in each group, adjusting for \(C\) as appropriate to the analysis.
For stratum \(k\), write the two estimated associations as \(\hat{\beta}_{GX,k}\) and \(\hat{\beta}_{GY,k}\). The ratio estimate is
If \(X\) is measured in millimoles per litre and \(Y\) in millimetres of mercury, for example, the ratio is in millimetres of mercury per millimole per litre. The associations in the numerator and denominator must refer to the same instrument, stratum, and compatible covariate adjustment. With several instruments, the analysis needs an appropriate multi-instrument estimator rather than treating a single ratio as though it were automatically valid.
Residual ranks are not identical to ranks of observed exposure. It is useful to report the mean or median observed exposure in each residual-defined stratum, since that shows which parts of the exposure distribution the estimates represent. The strata should be defined before examining their estimated causal slopes. Choosing cut points in response to the observed results can make the pattern appear more convincing than it is.
Worked Example: Calculating Three Stratum-Specific Estimates
Suppose a hypothetical one-sample study ranks 1,200 participants into three residual-exposure strata. In each stratum, the genetic score is associated with exposure measured in millimoles per litre and outcome measured in millimetres of mercury. The following are hypothetical adjusted association estimates.
| Stratum | Mean exposure (mmol/L) | \(\hat{\beta}_{GX,k}\) (mmol/L per score unit) | \(\hat{\beta}_{GY,k}\) (mm Hg per score unit) |
|---|---|---|---|
| Low residual rank | 2.1 | 0.080 | 0.040 |
| Middle residual rank | 4.8 | 0.100 | 0.110 |
| High residual rank | 8.2 | 0.075 | 0.135 |
For the low-rank stratum, the ratio is \(0.040/0.080=0.500\) mm Hg per mmol/L. For the middle stratum, it is \(0.110/0.100=1.10\) mm Hg per mmol/L. For the high-rank stratum, it is \(0.135/0.075=1.80\) mm Hg per mmol/L. These hypothetical point estimates increase across strata. That pattern is consistent with a larger causal slope at higher exposure levels, but the estimates alone do not establish that the causal effect truly changes.
This calculation assumes that each within-stratum genetic association is a valid instrument for the exposure, and that the exposure association is strong enough to support a stable ratio. The example reports no uncertainty intervals, so it does not establish whether the apparent differences exceed sampling variation.
What the Within-Stratum Ratio Represents
The ratio resembles the Wald instrumental-variable estimate used for a single instrument: it compares how much the instrument shifts the outcome with how much it shifts the exposure. If the outcome changes with exposure at a constant rate, ratios across strata should target the same slope, apart from sampling error and differences in instrument behavior. If the causal response is nonlinear, the genetic shift can produce different outcome-to-exposure ratios in strata representing different exposure levels.
The estimate is not generally the causal effect for every individual in a stratum, nor is it necessarily the derivative at the stratum’s mean exposure. It is an instrument-weighted local slope. The weighting depends on the instrument-induced exposure changes and on how causal responses vary among the participants whose exposure is shifted. Thus, the mean exposure in a stratum helps locate the estimate but does not, by itself, define exactly which exposure contrast the estimate represents.
The following simple model explains the local-slope interpretation. Suppose the exposure follows \(X=\pi G+U\), where \(G\) is centered and independent of \(U\) within a residual-defined stratum, and \(\pi\) is the instrument’s effect on exposure. Suppose the outcome has conditional mean \(m(X,U)\), with no direct effect of \(G\). For a small instrument-induced shift, a first-order Taylor expansion gives
where \(m_x\) is the derivative with respect to exposure. Taking the covariance with \(G\) within the stratum, independence of \(G\) and \(U\) gives
Meanwhile, \(\operatorname{Cov}(G,X\mid k)=\pi\operatorname{Var}(G\mid k)\). Dividing the two quantities therefore gives
When the instrument-induced shift is sufficiently small, the ratio approximates an average causal derivative among the individuals represented in that stratum. The approximation relies on the model and independence just stated; the equation is not a guarantee that every practical stratum-specific ratio has this exact interpretation.
Assumptions and Checks
The usual instrumental-variable requirements still matter, and they must be plausible within the analysis strata. In particular, a valid interpretation depends on instrument relevance, independence from causes of the outcome other than through the exposure, and the exclusion restriction that the instrument affects the outcome only through the exposure. Nonlinear stratification does not repair invalid instruments.
- Relevance within strata. The genetic instrument must predict exposure in each stratum. A strong overall association does not ensure strong associations after the sample is divided. Weak within-stratum associations can make ratios unstable and sensitive to small estimation errors.
- Independence within strata. Conditional on measured covariates and the stratum construction, the instrument should remain independent of unmeasured causes of the outcome and exposure. This requires attention to population structure, relatedness, selection, and other sources of genetic confounding.
- Exclusion within strata. The instrument should not affect the outcome through pathways other than the exposure. Horizontal pleiotropy can violate this requirement, and dividing participants into strata does not remove it.
- Residual-based stratification behaving as intended. Residual ranks aim to avoid the collider bias that direct conditioning on observed exposure can induce. But the residual model must be specified sensibly, and the method’s conditional independence argument depends on assumptions about the exposure model and instrument. Residualization is not a general guarantee against bias.
An analyst should report first-stage estimates and uncertainty within every stratum, not only for the full sample. For a single instrument in a linear first-stage regression, the first-stage \(F\)-statistic is the square of the \(t\)-statistic for its coefficient. This is a useful diagnostic, not a universal pass/fail test; weak-instrument behavior depends on the design and estimator. Stratum-specific sample sizes and confidence intervals also matter, because dividing a sample reduces precision. If instruments are correlated, the analysis must account for that correlation when estimating their joint contribution.
Worked Example: Checking First-Stage Strength by Stratum
Consider another hypothetical study using one genetic score. In the high-residual-rank stratum, its estimated exposure association is \(0.030\) mmol/L per score unit, with standard error \(0.015\) mmol/L per score unit. The first-stage \(t\)-statistic is \(0.030/0.015=2.00\), so the corresponding one-instrument \(F\)-statistic is \(2.00^2=4.00\).
In the low-rank stratum, suppose the association is \(0.080\) mmol/L per score unit with standard error \(0.020\). Then \(t=0.080/0.020=4.00\), and \(F=4.00^2=16.0\). The high-rank estimate therefore has a substantially less precise first stage in this hypothetical example. Its ratio could be especially unstable, even if its outcome association appears large.
The calculation checks relevance; it does not establish independence or exclusion. Nor does a particular \(F\)-statistic prove that weak-instrument bias is negligible. The appropriate conclusion is that the high-rank first stage deserves caution and further assessment, not that its instrument is automatically invalid or that the low-rank instrument is proven valid.
Interpreting a Nonlinear Pattern
A sequence of stratum-specific estimates is evidence about how instrumented causal slopes may vary over the exposure range. The estimates should be presented with uncertainty intervals and with the exposure levels represented by each stratum. If estimates rise as mean exposure rises, that pattern is compatible with an increasing causal slope. If estimates are similar, the data may be consistent with a roughly constant slope, although imprecision can conceal real variation.
These are not estimates of the outcome difference between the strata. Participants in different strata can differ in many ways, and the causal interpretation comes from instrument-induced exposure variation within each stratum, not from comparing their observed outcomes directly. Likewise, a pattern of different ratios does not by itself identify a complete dose-response curve. It gives local information, and conclusions about the shape of a curve require suitable assumptions, adequate precision, and careful interpretation of the exposure levels represented.
Worked Example: Interpreting an Increasing Slope Pattern
Return to the first hypothetical study. Suppose bootstrap confidence intervals for the three ratio estimates are, respectively, 0.10 to 0.90, 0.45 to 1.75, and 0.70 to 2.90 mm Hg per mmol/L. The point estimates are 0.500, 1.10, and 1.80 mm Hg per mmol/L at mean exposures of 2.1, 4.8, and 8.2 mmol/L.
The point estimates increase with mean exposure, which is compatible with a steeper causal response at higher exposure. However, the intervals overlap substantially, and each is broad. These results do not establish that the slopes differ between strata. A formal comparison should account for the covariance between estimates and the method used to construct the strata; comparing the point estimates alone is not enough.
If the assumptions hold, the low-stratum estimate concerns instrumented exposure variation among participants represented by the low residual ranks, and similarly for the other strata. It is not a claim that lowering exposure by one mmol/L for every participant would lower the outcome by exactly the estimated amount.
Common Mistake: Stratifying on Observed Exposure
A tempting approach is to divide participants into groups by their observed exposure and then estimate the genetic association within each group. The mistaken reasoning is that these groups directly represent low, middle, and high exposure, so they must be the best groups for estimating effects at those levels. But observed exposure is affected by both the genetic instrument and other causes. Conditioning on their shared consequence can create an association between the instrument and those other causes, even when they were independent before conditioning. That induced association can bias the within-group estimate.
Residual-based strata are intended to reduce this problem by ranking residual exposure after modeling the instrument and measured covariates, rather than conditioning directly on exposure. This is a design feature, not a cure-all: analysts still need to defend the residual model and evaluate the IV assumptions. A second common mistake is to call increasing point estimates proof of a nonlinear causal relationship. The correction is to examine uncertainty, first-stage strength, and the assumptions before interpreting the pattern.
Check Your Understanding
Answer each question by referring to the instrumented within-stratum comparisons, not simply to differences in observed outcomes between groups.
- Why can dividing participants into groups based directly on observed exposure induce bias in an instrumental-variable analysis?
- A stratum has an instrument–outcome association of 0.12 outcome units per score unit and an instrument–exposure association of 0.08 exposure units per score unit. What is the ratio estimate, and what are its units?
- Why should first-stage associations be reported separately for each residual-defined stratum?
- What does an increasing sequence of stratum-specific ratio estimates suggest, and why does it not by itself establish a nonlinear causal effect?
- Which instrumental-variable assumptions remain necessary after residual-based stratification?