Why MR-Egger Uses an Intercept
In Mendelian randomization (MR), genetic variants can affect an outcome through pathways other than the exposure of interest. Such effects are horizontal pleiotropy, and they violate the exclusion restriction for those variants. A conventional inverse-variance weighted (IVW) regression through the origin does not explicitly allow a nonzero average direct effect. MR-Egger instead fits an intercept, allowing a particular form of average directional pleiotropy, and uses a slope whose causal interpretation depends on an additional assumption.
The central qualification is important: MR-Egger is not a general correction for any pattern of pleiotropy. Its slope can be interpreted as a causal effect when direct effects are unrelated to instrument strengths in the way required by the weighted regression. Independence in an unweighted sense does not automatically establish that weighted condition.
Model, Regression, and Assumptions
For each variant, the model says the outcome association has two components: the association mediated by the exposure, \(\theta\beta_{Xj}\), and the direct effect, \(\alpha_j\). The model permits \(\alpha_j\ne0\); it does not require every variant to satisfy the exclusion restriction. In practice, the associations are estimated. Write these estimates as \(\hat{\beta}_{Xj}\) and \(\hat{\beta}_{Yj}\), and let \(SE(\hat{\beta}_{Yj})\) be the standard error of the outcome association estimate.
The usual summary-data MR-Egger regression is a weighted linear regression with an intercept, using weights \(w_j=1/SE(\hat{\beta}_{Yj})^2\). In its basic form, it treats the exposure-association estimates as fixed. This approximation is often called the NOME assumption: “no measurement error” in the estimated variant–exposure associations. The next tutorial addresses what happens when this approximation is violated. The basic regression also assumes variants are independent, or that any dependence has been appropriately accounted for. Allele harmonization is needed so that the exposure and outcome associations refer to the same effect allele.
Here \(b_1\) is the MR-Egger slope estimate and \(b_0\) is the intercept estimate. In the population model, these correspond to a weighted regression of \(\beta_{Yj}\) on \(\beta_{Xj}\). The slope represents \(\theta\) only under conditions that prevent the direct effects from generating a slope. The intercept is informative about average directional pleiotropy only when the slope is correctly identified.
Why the Weighted Covariance Matters
For any set of weights \(w_j\), define the total weight \(W=\sum_j w_j\), the weighted mean \(\bar{x}_w=\sum_j w_jx_j/W\), and the weighted covariance \(\operatorname{Cov}_w(x,a)=\sum_jw_j(x_j-\bar{x}_w)(a_j-\bar{a}_w)/W\). The weighted variance is defined in the same way, with \(a_j=x_j\). These definitions make explicit which relationship the regression uses.
The following derivation gives the key result. It concerns the true associations and fixed weights; sample estimates add sampling uncertainty. In the weighted regression of \(\beta_{Yj}=\theta\beta_{Xj}+\alpha_j\) on \(\beta_{Xj}\), the weighted least-squares slope is the weighted covariance of \(\beta_X\) and \(\beta_Y\), divided by the weighted variance of \(\beta_X\). Substituting the model and expanding the covariance gives:
The first term is the causal effect; the second is the slope contribution from the direct effects. Provided the weighted variance of instrument strengths is nonzero, the condition needed for the slope to equal \(\theta\) in this argument is therefore \(\operatorname{Cov}_w(\beta_X,\alpha)=0\). The weighted intercept is \(b_0=\bar{\beta}_{Y,w}-b_1\bar{\beta}_{X,w}\). If the slope condition holds, then \(b_0=\bar{\alpha}_w\), the weighted mean direct effect.
InSIDE is commonly used to describe the substantive assumption underlying MR-Egger: variants with stronger exposure associations do not systematically have larger or smaller direct effects. The weighted covariance condition spells out what is needed for the particular weighted regression argument. In an application, the weights are often determined by outcome-association precision, and their relation to the variant associations and direct effects matters. InSIDE is not directly verified by observing a nonsignificant intercept or by looking at the fitted line.
Worked Examples
Worked Example: Deriving the Slope and Intercept
Suppose three hypothetical, independent variants have the following true associations. Let the causal effect be \(\theta=0.5\), and suppose their direct effects are \(0.02\), \(-0.01\), and \(0.00\). Take equal regression weights for this illustration.
| Variant | \(\beta_{Xj}\) | \(\alpha_j\) | \(\beta_{Yj}=0.5\beta_{Xj}+\alpha_j\) |
|---|---|---|---|
| 1 | 0.10 | 0.02 | 0.07 |
| 2 | 0.20 | -0.01 | 0.09 |
| 3 | 0.30 | 0.00 | 0.15 |
The means are \(\bar{\beta}_X=0.20\) and \(\bar{\beta}_Y=0.10333\). The centered sums of squares and cross-products are $$ S_{XX}=(-0.10)^2+0^2+(0.10)^2=0.020, \qquad S_{XY}=(-0.10)(-0.03333)+0( -0.01333)+(0.10)(0.04667)=0.008. $$ Thus the regression slope is \(S_{XY}/S_{XX}=0.008/0.020=0.40\), and the intercept is \(0.10333-(0.40)(0.20)=0.02333\). The slope differs from the causal effect \(0.5\) because the direct effects are not orthogonal to the instrument strengths in this set. Indeed, their centered cross-product is \((-0.10)(0.01667)+(0)(-0.01333)+(0.10)(-0.00333)=-0.002\), so the slope contribution is \(-0.002/0.020=-0.10\). The intercept is not the unweighted mean direct effect, which is \(0.00333\), because the slope is not \(\theta\).
Worked Example: Independence Does Not Guarantee Weighted Orthogonality
Consider a hypothetical set of four variants with all four combinations of instrument strength \(x\in\{-1,1\}\) and direct effect \(a\in\{-1,1\}\), each occurring once. Across these four equally represented combinations, \(x\) and \(a\) are independent: every pairing occurs equally often. Now assign weight 2 when \(x=a\), and weight 1 when \(x\ne a\).
| \(x\) | \(a\) | Weight \(w\) | \(wxa\) |
|---|---|---|---|
| -1 | -1 | 2 | 2 |
| -1 | 1 | 1 | -1 |
| 1 | -1 | 1 | -1 |
| 1 | 1 | 2 | 2 |
The total weight is \(W=6\). The weighted means of both \(x\) and \(a\) are zero: the weighted sums for each are \(-2-1+1+2=0\). Hence $$ \operatorname{Cov}_w(x,a)=\frac{\sum_j w_jx_ja_j}{6} =\frac{2-1-1+2}{6}=\frac{1}{3}. $$ Also, \(\operatorname{Var}_w(x)=\sum_j w_jx_j^2/6=6/6=1\). If the outcome associations obey \(y_j=\theta x_j+a_j\), the weighted regression slope is therefore \(\theta+(1/3)/1=\theta+1/3\), not \(\theta\). This example demonstrates the distinction: the unweighted pairings express independence, but the specified weights produce nonzero weighted covariance. It is the weighted condition, not independence alone, that makes the regression identity yield the causal slope.
Worked Example: Interpreting an MR-Egger Fit
Suppose a hypothetical analysis of independent, harmonized variants reports an MR-Egger slope of \(0.42\) with \(SE=0.15\), and an intercept of \(0.030\) with \(SE=0.012\). Assume the exposure and outcome are measured on scales for which the slope is interpreted as outcome units per one-unit increase in exposure. For a large-sample, two-sided normal approximation, the slope’s 95% confidence interval is $$ 0.42\pm1.96(0.15)=0.42\pm0.294, \qquad\text{or}\qquad (0.126,\;0.714). $$
Subject to the MR-Egger assumptions, this interval gives a range of values compatible with the data and model for the causal effect: 0.126 to 0.714 outcome units per exposure unit. It is not a 95% probability statement about a fixed parameter. For the intercept, the test statistic for \(H_0:b_0=0\) is \(0.030/0.012=2.50\), giving a two-sided normal-approximation \(p\)-value of about \(0.0124\). The corresponding interval is \(0.030\pm1.96(0.012)=(0.00648,0.0535)\).
The intercept result is evidence against a zero fitted intercept under the regression model, consistent with average directional pleiotropy on the scale of the weighted regression. It does not establish that the InSIDE condition holds, nor does it show that every variant is pleiotropic. Conversely, a nonsignificant intercept would not prove that direct effects are absent: direct effects can vary in both directions and average out, and the test can have limited power.
What the Intercept Can and Cannot Tell You
When the slope condition holds, the intercept estimates the weighted mean of the direct effects. A nonzero intercept is evidence that their weighted average differs from zero, which is the sense in which MR-Egger can accommodate directional pleiotropy. The intercept is not a measure of the total amount of pleiotropy. For example, direct effects can be large but balanced around zero, yielding an intercept near zero. The intercept also does not test whether the direct effects are independent of instrument strengths.
The slope’s causal interpretation requires more than fitting a line with an intercept. It relies on adequate instrument-strength variation, correct harmonization and model specification, appropriate treatment of variant dependence, the NOME approximation for the usual regression, and the weighted relation between strengths and direct effects. As with other MR estimates, the result is an association-based estimate interpreted causally only under its assumptions; it does not establish causation by itself.
Common Mistake
- Wrong reasoning: “InSIDE means instrument strengths and direct effects are independent, so their weighted covariance must be zero and the MR-Egger slope is unbiased.”
- Correction: Independence alone does not ensure zero covariance under arbitrary weights. The weighted-regression result requires \(\operatorname{Cov}_w(\beta_X,\alpha)=0\). Independence supports that condition only when the weights have suitable properties. State the weighting condition explicitly when using the regression argument.
- Wrong reasoning: “A nonsignificant MR-Egger intercept proves that there is no horizontal pleiotropy.”
- Correction: The intercept concerns the weighted average direct effect, not every direct effect. Balanced pleiotropy can yield a near-zero intercept, and a test may have low power.
Check Your Understanding
Answer each question by referring to the regression model and the conditions for its interpretation.
- In \(\beta_{Yj}=\theta\beta_{Xj}+\alpha_j\), what does \(\alpha_j\) represent?
- Write the weighted-regression expression for the slope and identify the term that can make it differ from \(\theta\).
- Why does independence between instrument strengths and direct effects not automatically guarantee zero weighted covariance?
- Under what condition does the weighted MR-Egger intercept equal the weighted mean direct effect?
- What does a nonzero intercept suggest, and what can it not establish about individual variants or InSIDE?