NO Measurement Error In MR-Egger

Why Measurement Error Matters in MR-Egger

MR-Egger regresses estimated SNP–outcome associations on estimated SNP–exposure associations, allowing a nonzero intercept. Its slope is intended to estimate the causal effect, subject to assumptions including InSIDE. But the regression also treats the estimated SNP–exposure associations as if they were exact. In practice, those associations are estimated from data and have sampling error. When that error is substantial relative to the differences between variants’ exposure associations, the fitted slope can be pulled toward zero.

The assumption that this source of error is negligible is called NO Measurement Error, or NOME. It concerns measurement error in the SNP–exposure association estimates used as the regression predictors. It does not mean that the exposure itself is measured perfectly for every participant, nor does it guarantee that the causal estimate is unbiased overall.

Definition: For variant \(j\), let \(\gamma_j\) be its true association with the exposure and let \(\hat{\gamma}_j\) be its estimated association, with standard error \(se_{Xj}\). NOME assumes that uncertainty in the \(\hat{\gamma}_j\) values is negligible for the MR-Egger regression, relative to the variation in the true \(\gamma_j\) values across variants. In the usual summary-data notation, \(\hat{\gamma}_j\) is the estimated SNP–exposure association and \(\hat{\Gamma}_j\) is the estimated SNP–outcome association.

A simple summary-data MR-Egger regression fits an intercept and a slope in

\hat{\Gamma}_j = \beta_0 + \theta_{\text{Egger}}\hat{\gamma}_j + \varepsilon_j,

typically using weights \(1/se_{Yj}^2\), where \(se_{Yj}\) is the standard error of the SNP–outcome association estimate. The intercept \(\beta_0\) represents the average directional pleiotropic contribution under the model; \(\theta_{\text{Egger}}\) is the MR-Egger slope. The usual weighted regression accounts for uncertainty in the outcome-association estimates through its weights, but does not, by itself, account for uncertainty in the predictor estimates \(\hat{\gamma}_j\). NOME is the condition that makes this treatment of the predictors a reasonable approximation.

The InSIDE assumption concerns the relationship between instrument strength and direct effects. NOME concerns error in the estimated instrument strengths. They are distinct: a study can have little measurement error but still violate InSIDE, or have plausible InSIDE but substantial measurement error in \(\hat{\gamma}_j\).

How Error in the Predictor Dilutes the Slope

A simplified model makes the direction of the problem clear. Suppose the true SNP–exposure associations vary across variants, and the true SNP–outcome association follows

\Gamma_j = \beta_0 + \theta\gamma_j + \alpha_j,

where \(\theta\) is the causal effect and \(\alpha_j\) is the direct effect of variant \(j\) on the outcome. InSIDE, in this simplified setting, means that the direct effects do not systematically covary with the true SNP–exposure associations. Now suppose the observed exposure estimates satisfy \(\hat{\gamma}_j=\gamma_j+u_j\), where \(u_j\) is estimation error. If \(u_j\) is independent of \(\gamma_j\), \(\alpha_j\), and outcome-association error, then the regression of the outcome association on the observed exposure association has an attenuated slope.

Justification: In the simplified, unweighted setting, the regression slope is the covariance of predictor and outcome divided by predictor variance. Under the independence conditions just stated, and with \(\operatorname{Cov}(\gamma_j,\alpha_j)=0\),

\frac{\operatorname{Cov}(\hat{\gamma}_j,\Gamma_j)} {\operatorname{Var}(\hat{\gamma}_j)} = \frac{\theta\operatorname{Var}(\gamma_j)} {\operatorname{Var}(\gamma_j)+\operatorname{Var}(u_j)} = \theta\lambda, \qquad \lambda = \frac{\operatorname{Var}(\gamma_j)} {\operatorname{Var}(\gamma_j)+\operatorname{Var}(u_j)}.

Here \(0\leq\lambda\leq1\) is a reliability ratio. When predictor error is negligible, \(\lambda\) is close to 1 and the dilution is small. When error variance is large relative to the variation in true associations, \(\lambda\) is smaller and the expected fitted slope is closer to zero than \(\theta\). This derivation illustrates the mechanism; real MR-Egger analyses are weighted, estimates can have unequal standard errors, and additional sources of error or dependence can complicate the exact amount and direction of bias.

Worked Example: A Reliability Ratio and Expected Attenuation

Suppose, for illustration, that the true SNP–exposure associations across a set of independent variants have variance \(0.0125\) in a chosen exposure unit, and their estimation errors have variance \(0.0100\) in the squared exposure-association unit. Assume the simplified independence conditions above, no covariance between true exposure associations and direct effects, and a causal effect \(\theta=0.50\) outcome units per exposure unit.

The reliability ratio is

\lambda = \frac{0.0125}{0.0125+0.0100} = \frac{0.0125}{0.0225} = 0.5556.

The expected slope under this simplified model is therefore

\theta\lambda = 0.50(0.5556)=0.2778.

Thus, predictor error alone would produce an expected slope of about \(0.278\) outcome units per exposure unit rather than the true \(0.50\), under the stated assumptions. These are hypothetical population-moment calculations, not an estimate from a particular MR data set. They show why a precise-looking regression can still be attenuated if the exposure-association estimates are noisy.

Using \(I^2_{GX}\) as a Diagnostic

The \(I^2_{GX}\) statistic summarizes how much of the observed dispersion in SNP–exposure estimates reflects between-variant differences rather than sampling error. It is calculated from a Cochran-style heterogeneity statistic for the \(\hat{\gamma}_j\) values. For independent variants, define the inverse-variance weighted mean

\bar{\gamma}_{w} = \frac{\sum_{j=1}^{K}\hat{\gamma}_j/se_{Xj}^{2}} {\sum_{j=1}^{K}1/se_{Xj}^{2}}, \qquad Q_{GX} = \sum_{j=1}^{K} \frac{(\hat{\gamma}_j-\bar{\gamma}_{w})^2}{se_{Xj}^{2}}.

Here \(K\) is the number of variants. The reference degrees of freedom are \(K-1\), because the weighted mean is estimated from the same \(K\) values. The commonly used statistic is

I^2_{GX} = \max\left\{0,\frac{Q_{GX}-(K-1)}{Q_{GX}}\right\}.

It is often reported as a percentage by multiplying the result by 100. The truncation at zero prevents a negative estimate when the observed \(Q_{GX}\) is no larger than its reference degrees of freedom. In settings with correlated variants, the covariance among estimates must be accounted for rather than using the independent-variant sum above.

A value nearer 1 (or 100%) indicates that relatively little of the observed dispersion is attributed to sampling error under this diagnostic, so regression dilution from measurement error in \(\hat{\gamma}_j\) is expected to be less severe. A low value indicates that sampling error may account for a substantial part of the observed dispersion and that the NOME approximation may be poor. There is no universal threshold that turns \(I^2_{GX}\) into a pass-or-fail test. It is a diagnostic of relative dispersion, not a direct measurement of the causal effect’s bias, a test of InSIDE, or proof that the instruments are valid.

Worked Example: Calculating a High \(I^2_{GX}\)

Consider four independent variants with exposure-association estimates \(0.10,\ 0.20,\ 0.30,\ 0.40\), each with standard error \(0.02\). Assume the estimates are on the same exposure scale and that the reported standard errors are appropriate. Because the standard errors are equal, the inverse-variance weighted mean is the ordinary mean:

\bar{\gamma}_{w} = \frac{0.10+0.20+0.30+0.40}{4} = 0.25.

Each variance is \(0.02^2=0.0004\). The deviations from the weighted mean are \(-0.15,\ -0.05,\ 0.05,\ 0.15\), so

Q_{GX} = \frac{(-0.15)^2+(-0.05)^2+(0.05)^2+(0.15)^2}{0.0004} = \frac{0.0500}{0.0004} = 125.

With \(K-1=3\) degrees of freedom,

I^2_{GX} = \frac{125-3}{125} = 0.976 = 97.6\%.

This high value suggests that sampling error accounts for relatively little of the dispersion in these exposure-association estimates, so regression dilution from this source is likely to be limited. It does not establish NOME exactly, and it says nothing by itself about InSIDE or other validity assumptions.

Worked Example: A Low \(I^2_{GX}\)

Now consider four independent variants with exposure-association estimates \(0.10,\ 0.12,\ 0.14,\ 0.16\), again each with standard error \(0.02\). Assume the same scale and appropriate standard errors. The weighted mean is

\bar{\gamma}_{w} = \frac{0.10+0.12+0.14+0.16}{4} = 0.13.

The deviations are \(-0.03,\ -0.01,\ 0.01,\ 0.03\), giving

Q_{GX} = \frac{(-0.03)^2+(-0.01)^2+(0.01)^2+(0.03)^2}{0.02^2} = \frac{0.0020}{0.0004} = 5.

Therefore,

I^2_{GX} = \frac{5-3}{5} = 0.400 = 40.0\%.

This comparatively low value flags possible measurement-error dilution: much less of the observed dispersion is clearly larger than expected from sampling variation alone. It is a warning to interpret an MR-Egger slope cautiously, not a correction factor that can simply be applied to that slope.

Assumptions, Diagnostics, and Limitations

The \(I^2_{GX}\) calculation relies on comparable SNP–exposure association estimates and valid standard errors. Its usual \(Q_{GX}\) reference uses independent variants and a common weighted-mean calculation. If variants are correlated, ignoring their covariance can misstate the dispersion; a covariance-aware calculation is required. Also inspect the estimates and standard errors themselves: a high value can coexist with a small number of influential variants, and a low value can result when exposure-association estimates are imprecise relative to their spread.

Keep separate the questions answered by different assumptions and diagnostics. \(I^2_{GX}\) addresses the extent of relative measurement error in the MR-Egger predictor estimates. InSIDE addresses whether direct effects are independent of instrument strength. Strong instruments, suitable allele harmonization, valid outcome and exposure estimates, and an appropriate regression model remain important; a favorable \(I^2_{GX}\) value does not replace those checks. Nor does a low \(I^2_{GX}\) prove that the estimated slope is attenuated by a precisely known amount.

The reliability-ratio derivation gives a useful intuition but should not be over-applied. It is a simple unweighted model with independent errors and specific covariance conditions. In practical summary-data MR, the estimates may have different standard errors, there may be sample overlap, and variants may be correlated. These features can change the behavior of the estimator. The diagnostic indicates potential regression dilution; it does not fully model every source of uncertainty or bias.

Common Mistake and Practical Interpretation

A common mistake is to read a high \(I^2_{GX}\) as proof that MR-Egger assumptions are satisfied, or to read a low value as a measured percentage of bias in the slope. Neither conclusion follows. \(I^2_{GX}=97.6\%\) does not mean there is a 97.6% probability that NOME holds, and \(I^2_{GX}=40.0\%\) does not mean the causal estimate is biased by 60%.

The correct interpretation is comparative and diagnostic: the statistic estimates the proportion of observed exposure-association dispersion beyond the amount expected from sampling error, with truncation at zero. A high value supports the view that predictor measurement error is less consequential; a low value raises concern that the MR-Egger slope may be diluted. In reporting, state the statistic, the number and dependence structure of variants, and the implication as a caution rather than a guarantee.

Compared with treating the exposure-association estimates as error-free, the \(I^2_{GX}\) check makes the NOME concern visible and quantifiable. It is not itself an alternative estimator, and it does not remedy the problem. Other approaches may address measurement error or rely on different identifying assumptions, but their choice requires examining their own assumptions and is not determined by \(I^2_{GX}\) alone.

Key takeaway: MR-Egger’s NOME assumption says that sampling error in the SNP–exposure estimates is negligible for the regression. Error in those predictors can attenuate the slope toward zero. \(I^2_{GX}\) is a useful diagnostic of relative exposure-association dispersion, but it neither validates the full MR-Egger model nor quantifies all bias.

Check Your Understanding

Answer each question, then compare with the solutions below.

In one sentence, distinguish NOME from InSIDE.
In the simplified model, what happens to the expected MR-Egger slope as the variance of exposure-association estimation error increases while the variance of true associations remains fixed?
For \(K=5\) independent variants with \(Q_{GX}=8\), calculate \(I^2_{GX}\).
Why does a high \(I^2_{GX}\) not establish that the MR-Egger causal estimate is valid?

Solutions. (1) NOME concerns negligible sampling error in the estimated SNP–exposure associations used as predictors, whereas InSIDE concerns the relationship between instrument strength and direct effects. (2) The reliability ratio \(\lambda=\operatorname{Var}(\gamma_j)/[\operatorname{Var}(\gamma_j)+\operatorname{Var}(u_j)]\) decreases, so the expected slope \(\theta\lambda\) moves closer to zero under the derivation’s assumptions. (3) The degrees of freedom are \(K-1=4\), so \(I^2_{GX}=(8-4)/8=0.500\), or 50.0%. (4) The statistic diagnoses relative measurement error in exposure-association estimates; it does not test InSIDE or establish other instrument-validity and model assumptions.