Instrument Strength And The F Statistic

Why Instrument Strength Matters

In Mendelian randomization (MR), an instrument must predict the exposure for an instrumental-variable estimate to be informative. This requirement is called instrument relevance. A genetic variant can be associated with an exposure but explain only a very small fraction of its variation, especially in a large sample. Measuring that explained fraction and the corresponding first-stage F statistic helps describe the strength of the exposure association.

Strength is a property of the instrument’s association with the exposure in the study data. It is not, by itself, evidence that the instrument satisfies every condition needed for a causal interpretation. Here the focus is on quantifying relevance, for a single variant and for variants tested together.

Explained Exposure Variance

For a single variant, the explained exposure variance is often reported as an \(R^2\). In a simple regression with an intercept and no additional covariates, this is the proportion of sample variation in the exposure explained by the variant. With covariates, it is important to distinguish the variant’s contribution beyond those covariates from the total \(R^2\) of the full model.

Let \(R^2_{\text{cov}}\) denote the \(R^2\) from a regression of the exposure on covariates alone, and let \(R^2_{\text{full}}\) denote the \(R^2\) after adding the \(k\) genetic predictors. The partial \(R^2\) is the increase in explained variation divided by the variation not already explained by the covariates:

This is a conditional measure: it describes the fraction of the remaining exposure variation explained by the instruments after adjustment for the covariates. If there are no covariates, \(R^2_{\text{cov}}=0\), so partial \(R^2\) equals the ordinary \(R^2\).

For an additive dosage \(G\) with effect \(\beta_{XG}\) on a continuous exposure \(X\), an approximate population calculation for one variant is

Here \(f\) is the frequency of the coded effect allele. The second equality uses the approximation \(\operatorname{Var}(G)=2f(1-f)\), which holds under Hardy–Weinberg equilibrium for an allele dosage coded 0, 1, or 2. The effect estimate and variance must use compatible units. For example, if \(X\) is standardized to variance 1, \(\beta_{XG}\) is measured in standard deviations of exposure per effect allele.

From Partial \(R^2\) to the First-Stage F Statistic

The first-stage F statistic tests the joint null hypothesis that all \(k\) instrument coefficients in the exposure regression are zero, conditional on the covariates. Under the usual linear regression model with independent, normally distributed errors of constant variance, the ordinary partial F test has an exact F distribution under that null. Let \(c\) be the number of covariate columns, excluding the intercept, and assume the design matrix has full column rank. The residual degrees of freedom in the full model are \(N-c-k-1\).

The relationship between this F statistic and partial \(R^2\) is

The numerator measures the average fraction of residual exposure variation explained per tested instrument coefficient. The denominator measures the remaining unexplained variation per residual degree of freedom. A larger partial \(R^2\), a larger sample size, or fewer tested coefficients generally yields a larger F statistic, all else equal.

Why the F Formula Uses Partial \(R^2\)

The relationship follows directly from the usual partial F test. Write \(\mathrm{SSE}_{\text{cov}}\) for the residual sum of squares from the covariate-only regression and \(\mathrm{SSE}_{\text{full}}\) for that from the full regression. Adding \(k\) predictors reduces the residual sum of squares by \(\mathrm{SSE}_{\text{cov}}-\mathrm{SSE}_{\text{full}}\). The partial F statistic is this reduction per added coefficient divided by the full-model residual mean square:

The partial \(R^2\) can also be written as

Substituting these two ratios into the F expression gives the stated formula: the sum-of-squares reduction is \(R^2_{\text{partial}}\mathrm{SSE}_{\text{cov}}\), while the full-model error sum of squares is \((1-R^2_{\text{partial}})\mathrm{SSE}_{\text{cov}}\). The common factor \(\mathrm{SSE}_{\text{cov}}\) cancels.

For one instrument and no covariates, \(k=1\), \(c=0\), and the residual degrees of freedom are \(N-2\). Therefore,

Assessing a Combined Instrument

When several variants are used together, the relevant strength measure is generally the joint first-stage test of their coefficients, conditional on the covariates. The corresponding partial \(R^2\) measures how much additional exposure variation the set explains. Variants in linkage disequilibrium can carry overlapping information, so adding their separate marginal \(R^2\) values does not in general give the set’s conditional \(R^2\). The joint regression accounts for their relationships through its design matrix.

Using Summary-Data Associations

For a single variant, a commonly available summary-data approximation is \(F\approx(\hat{\beta}_{XG}/SE)^2\), where \(\hat{\beta}_{XG}\) is the estimated variant–exposure association and \(SE\) is its standard error. This is the squared Wald statistic for testing a zero coefficient. In ordinary linear regression with one tested coefficient and conventional standard errors, the corresponding partial F statistic equals that squared statistic. With summary data, differing standard-error methods, or approximations in the reported estimates, treat the equality as approximate.

Interpreting Strength Measures Carefully

The F statistic combines the amount of exposure variation explained with the information available in the sample. Consequently, the same \(R^2\) can produce different F statistics in samples of different sizes. Conversely, a large F statistic in a very large sample can coexist with a small explained fraction. Reporting both a variance measure and an F statistic gives a more complete description than reporting either alone.

An F statistic above 10 is often used as a rough screening rule in discussions of weak-instrument concerns. It is a heuristic, not a universal boundary: it does not guarantee that an instrument is strong enough for every analysis, and falling below it does not itself quantify the bias or uncertainty in a particular estimate. Interpretation depends on such features as the study design, number of instruments, sample size, and how the statistic was calculated. A strength statistic also cannot establish the other assumptions required for causal interpretation.

Common Mistakes

• Using total \(R^2\) when covariates are present. The instrument-set F statistic uses the partial \(R^2\) for the added genetic predictors, not the full model’s total \(R^2\).
• Adding marginal \(R^2\) values for correlated variants. Their explained variation can overlap. Use the joint regression’s partial \(R^2\) and joint F statistic.
• Calling a large F statistic proof of a valid instrument. It quantifies exposure prediction; by itself it does not establish the assumptions needed for a causal interpretation.
• Treating 10 as a guarantee. The F>10 rule is a rough convention, not a universal pass/fail test or a promise of negligible bias.
• Ignoring degrees of freedom. For a joint test, use the number of tested instrument coefficients in the numerator and the full-model residual degrees of freedom in the denominator.