Why InSIDE Matters
MR-Egger allows genetic variants to have direct effects on the outcome that do not operate through the exposure. The fitted intercept is generally \(\hat{\alpha}_{\mathrm{Egger}}=\bar{\alpha}_w-\bar{\beta}_{X,w}\operatorname{Cov}_w(\beta_X,\alpha)/\operatorname{Var}_w(\beta_X)\), not simply the weighted mean direct effect; it equals that mean when \(\bar{\beta}_{X,w}\operatorname{Cov}_w(\beta_X,\alpha)=0\). But allowing an intercept does not, by itself, protect the slope from every pattern of pleiotropy. If variants with stronger associations with the exposure also tend to have systematically different direct effects on the outcome, the fitted slope can absorb some of that direct-effect pattern.
The key assumption is called Instrument Strength Independent of Direct Effect, abbreviated InSIDE. Here, instrument strength means a variant’s association with the exposure, and direct effect means its effect on the outcome through pathways other than the exposure. The assumption concerns how these quantities relate across the genetic variants used as instruments.
The distinction between independence and the weighted condition is important. InSIDE is often described as independence across variants. Independence in an unweighted distribution is a familiar way to express the scientific idea that instrument strength does not predict direct effects. However, the MR-Egger slope is usually obtained by weighted regression. Its exact condition is therefore about the weighted distribution induced by the regression weights. Independence before weighting does not guarantee zero weighted association for arbitrary weights.
How InSIDE Enters the Slope
Consider a weighted regression of the outcome associations on the exposure associations, with an intercept. Let \(w_j>0\) denote the regression weight for variant \(j\), and define the normalized weights \(q_j=w_j/\sum_k w_k\). The weighted means are \(\bar{\beta}_{X,w}=\sum_j q_j\beta_{Xj}\) and \(\bar{\beta}_{Y,w}=\sum_j q_j\beta_{Yj}\). The weighted slope is the weighted centered cross-product divided by the weighted variation in exposure associations.
To see the role of direct effects, substitute the association model \(\beta_{Yj}=\theta\beta_{Xj}+\alpha_j\). Centering the outcome association gives \(\beta_{Yj}-\bar{\beta}_{Y,w}=\theta(\beta_{Xj}-\bar{\beta}_{X,w})+(\alpha_j-\bar{\alpha}_w)\), where \(\bar{\alpha}_w=\sum_jq_j\alpha_j\). The numerator separates into a causal component and a direct-effect component.
This is the central justification: the fitted slope has no systematic shift from the direct effects when the weighted centered cross-product is zero, provided the weighted variation in \(\beta_{Xj}\) is positive. In the formula, \(\operatorname{Cov}_w(\beta_X,\alpha)\) is the weighted covariance across variants and \(\operatorname{Var}_w(\beta_X)\) is their weighted variance. A zero weighted covariance is the precise slope condition. Independence under the weighted distribution is sufficient when the relevant moments exist, but the reverse need not hold: zero covariance does not establish independence.
Unweighted independence can imply the desired condition if the weighting does not create strength–direct-effect association. For example, in a population where \(\beta_X\) and \(\alpha\) are independent, weighting that depends only on \(\beta_X\) reweights the distribution of strength without changing the distribution of \(\alpha\) conditional on strength. The weighted covariance remains zero. More generally, weights must be compatible with the assumed independence. Weights that depend jointly on strength and direct effect can create a nonzero weighted covariance, even if those quantities were independent before weighting.
Worked Example: A Pattern Compatible With the Slope Condition
Suppose four hypothetical variants have exposure associations \(0.02, 0.04, 0.06,\) and \(0.08\), and direct effects \(-0.01, 0.01, 0.01,\) and \(-0.01\), respectively. Use equal weights, and suppose the causal effect is \(\theta=0.50\) outcome units per exposure unit. The outcome associations follow \(\beta_{Yj}=0.50\beta_{Xj}+\alpha_j\).
The exposure mean is \(0.05\), and the direct-effect mean is \(0\). The centered cross-product is \( (-0.03)(-0.01)+(-0.01)(0.01)+(0.01)(0.01)+(0.03)(-0.01)=0.0003-0.0001+0.0001-0.0003=0\). The centered exposure sum of squares is \(0.0009+0.0001+0.0001+0.0009=0.0020\). Thus the direct-effect contribution to the slope is \(0/0.0020=0\).
The outcome associations are \(0.00, 0.03, 0.04,\) and \(0.03\). The fitted slope is therefore \(0.50\), with intercept \(0\), in this idealized calculation. This four-variant pattern demonstrates the zero-cross-product condition; it does not prove that strength and direct effects are statistically independent. Independence is a stronger distributional claim than zero covariance.
When Strength and Direct Effects Are Associated
A direct effect can be correlated with instrument strength when the genetic variants influence the outcome through biological pathways that also track their effects on the exposure. In that case, the intercept can still accommodate the average direct effect, but it cannot remove the part of the direct effect that changes systematically with \(\beta_{Xj}\). The slope then combines the causal effect with that trend.
Worked Example: Correlated Direct Effects Shift the Slope
Consider four hypothetical variants with exposure associations \(0.02, 0.04, 0.06,\) and \(0.08\). Let the causal effect be \(\theta=0.20\), and let the direct effects be \(-0.015, -0.005, 0.005,\) and \(0.015\). Use equal weights.
The exposure mean is \(0.05\), and the direct-effect mean is \(0\). The weighted (here, ordinary) centered cross-product is \( (-0.03)(-0.015)+(-0.01)(-0.005)+(0.01)(0.005)+(0.03)(0.015)=0.00045+0.00005+0.00005+0.00045=0.0010\). The exposure sum of squared deviations is \(0.0020\), so the direct-effect contribution to the slope is \(0.0010/0.0020=0.50\).
Thus the fitted MR-Egger slope is \(0.20+0.50=0.70\), rather than the causal effect \(0.20\). The outcome associations are \( -0.011, 0.003, 0.017,\) and \(0.031\), which lie exactly on the line \(\beta_Y=-0.025+0.70\beta_X\). Its fitted intercept is \(-0.025\), not the mean direct effect (which is \(0\)): it equals \(\bar{\alpha}_w-\bar{\beta}_{X,w}\operatorname{Cov}_w(\beta_X,\alpha)/\operatorname{Var}_w(\beta_X)\). The intercept therefore does not prevent the slope shift. The units of both slopes are outcome units per exposure unit.
Why the Weights Require Separate Attention
A weighted regression changes which variants contribute most to the fitted line. Consequently, the relevant InSIDE condition is not simply “independent before weighting”; it is zero weighted covariance in the distribution used by the slope calculation. The following example shows why this distinction cannot be ignored.
Worked Example: Weighting Creates a Slope Shift
For a transparent hypothetical calculation, take four variants with standardized exposure associations \(\beta_X=(-1,-1,1,1)\) and direct effects \(\alpha=(-1,1,-1,1)\). In the unweighted set, all four strength–direct-effect combinations occur once, so the unweighted means are zero and the unweighted cross-product is \(1-1-1+1=0\). Strength and direct effect are independent under the equal-weight distribution.
Now assign weight \(2\) when \(\beta_X=\alpha\), and weight \(1\) otherwise. The total weight is \(6\). The weighted means remain zero: the weighted sum of \(\beta_X\) is \(-2-1+1+2=0\), and the weighted sum of \(\alpha\) is also \(0\). But the weighted centered cross-product is \(2(1)+1(-1)+1(-1)+2(1)=2\), while the weighted sum of squared centered exposure associations is \(2+1+1+2=6\). The weighted slope shift is therefore \(2/6=1/3\).
If the true causal slope is \(0.40\), this weighted regression has slope \(0.40+1/3=0.7333\). The weight rule has induced a weighted strength–direct-effect association even though the unweighted variables were independent. The example is deliberately simple, but its lesson applies generally: inspect the weighted condition, rather than assuming arbitrary regression weights preserve unweighted InSIDE.
Assumptions, Diagnostics, and Limits
The InSIDE condition has a specific role; it is not the whole set of requirements for a causal interpretation. The MR-Egger setup also relies on valid harmonization and appropriately defined genetic associations, and on the usual instrumental-variable requirements, including instrument relevance and independence from confounders of the exposure–outcome relation. MR-Egger permits direct effects, so the exclusion restriction is not required for every variant in its simplest form. InSIDE is the additional condition that keeps systematic direct effects from shifting the slope.
The derivation above treats the associations and weights as fixed population quantities to isolate the assumption. In actual data, associations are estimated, and uncertainty or measurement error can affect the fitted slope and weights. Those issues are distinct from InSIDE: satisfying the zero weighted cross-product condition does not, by itself, resolve estimation error. Also, the weighted exposure associations must vary; if \(\operatorname{Var}_w(\beta_X)=0\), the slope is undefined.
InSIDE is generally not verifiable from the MR-Egger fit alone. Several checks can reveal problems or prompt caution, but none establishes the assumption:
- A nonzero fitted intercept may indicate average directional pleiotropy, but a zero intercept does not show that direct effects are unrelated to instrument strength. Direct effects can average to zero and still covary with strength.
- Heterogeneity among variant-specific associations can reveal departures from a simple common-effect pattern, but heterogeneity does not identify whether its source is InSIDE violation, other pleiotropic structure, or sampling variation.
- Plots of outcome association against exposure association can help reveal a trend or influential variants. A visually plausible line is not proof that the direct effects satisfy the required weighted condition.
- Biological knowledge may suggest pathways that make direct effects track exposure strength. Such evidence can undermine the plausibility of InSIDE even when statistical diagnostics are inconclusive.
The main contrast is with inverse-variance weighted estimation. IVW does not include an intercept to represent average direct effects; its standard causal interpretation relies on stronger restrictions on pleiotropy. MR-Egger relaxes those restrictions by fitting an intercept, but pays for that flexibility by relying on InSIDE for its slope interpretation. Neither method is automatically protected from invalid instruments: the assumptions differ, and scientific judgment is needed to assess their plausibility.
A common mistake is to reason, “The variants’ direct effects are independent of their exposure associations, so the weighted MR-Egger slope must be unbiased.” That conclusion skips the weights. The correction is to calculate or otherwise justify the weighted centered cross-product. Independence in an unweighted population only supports the slope condition when the weights do not create an association; the directly relevant condition is \(\operatorname{Cov}_w(\beta_X,\alpha)=0\).
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- If the weighted centered cross-product is \(0.0012\), the weighted exposure variance is \(0.0040\), and the causal effect is \(0.25\), what slope does the population decomposition give?
- Does a fitted MR-Egger intercept of zero establish InSIDE? Explain.
- Can independence of exposure strength and direct effect under equal weights guarantee zero weighted covariance for every possible set of positive regression weights?
- State one condition on weights that can preserve zero weighted covariance when strength and direct effect are independent.
Solutions. (1) The direct-effect contribution is \(0.0012/0.0040=0.30\), so the slope is \(0.25+0.30=0.55\). (2) No. A zero intercept concerns the fitted average direct effect; direct effects may average to zero while covarying with strength, shifting the slope. (3) No. The weights can depend jointly on strength and direct effect and induce nonzero weighted covariance, as in the four-variant example. (4) One sufficient condition is that the weights depend only on strength, when strength and direct effect are independent in the underlying population and the required moments exist. More generally, the weights must not induce a weighted strength–direct-effect association.