Good Mendelian Randomization, Bad Mendelian Randomization

1. The Seductive Promise: "Nature’s Clinical Trial" and the Observational Crisis

Every observational epidemiologist lives under the shadow of a recurring public humiliation. In the 1980s and 1990s, observational cohort studies passionately declared that hormone replacement therapy (HRT) slashed coronary heart disease in postmenopausal women by 50%. Vitamin E was heralded as a potent cardioprotective antioxidant. Beta-carotene was proclaimed an anti-cancer miracle. High HDL cholesterol ("the good cholesterol") was revered as the cardiovascular guardian angel.

Then came the randomized controlled trials (RCTs). The Women's Health Initiative revealed that HRT slightly increased heart attacks, strokes, and breast cancer. Beta-carotene increased lung cancer incidence in smokers. Massive trials raising HDL via CETP inhibitors (torcetrapib, dalcetrapib) spent billions of dollars only to show zero reduction in myocardial infarction—and in one trial, excess cardiovascular deaths.

The explanation was banal: healthy user bias. Women who took HRT were wealthier, exercised more, visited prestigious physicians, and ate better diets. The correlation was real; the causality was an illusion.

Enter Mendelian Randomization (MR). Formulated conceptually by Martijn Katan in 1986 and popularized as an epidemiological paradigm by George Davey Smith and Shah Ebrahim in 2003, MR presented itself as the salvation of observational medicine:

"Why wait decades and spend billions of dollars running impractical or unethical clinical trials when nature has already executed a randomized controlled trial for every human at the moment of conception?"

The rhetoric is intoxicating. In standard instrumental variables (IV), we seek an exogenous lever $Z$ that perturbs treatment $D$ without directly touching outcome $Y$. In MR, our instrument $Z$ is a genetic variant—typically a single-nucleotide polymorphism (SNP) coded as the count of minor alleles $Z_i \in \{0, 1, 2\}$.

Proponents claimed that genetic variants are immune to the twin plagues of observational science:

1. No Confounding ($Z \perp\!\!\perp U$): Because alleles segregate at meiosis and unite at fertilization, genetic assignment is alleged to be as clean as a cryptographic coin toss.
2. No Reverse Causation ($Y \not\to Z$): Germline DNA sequences are fixed at conception. Developing coronary artery disease at age 55 cannot travel backwards in time to mutate your inherited germline nucleotide sequence.

Over the past fifteen years, this promise sparked an industrial publishing boom. Armed with open-access GWAS summary statistics from the UK Biobank, FinnGen, and the Million Veteran Program, thousands of researchers ran automated R packages (like TwoSampleMR) to publish tens of thousands of causal claims: that genetically predicted coffee consumption cures Alzheimer’s, that genetically predicted insomnia causes poverty, that genetically predicted educational attainment prevents stroke.

I look at this avalanche with profound skepticism. When an econometrician or statistician investigates how Mendelian Randomization is practiced in the wild, the reality is sobering. Much of the literature relies on fundamental statistical misunderstandings, treats biological fairy tales as identification proofs, and confuses sample size with causal validity.

To understand why, we have to strip away the epidemiological marketing and look at the physical molecular biology through the lens of mathematical statistics.

2. The Biology from a Statistician's Lens: Meiosis vs. Population Biobanks

When statisticians evaluate any randomized experiment, our first question is always the same: What is the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and what is the exact physical assignment mechanism?

In a true randomized controlled trial, the assignment mechanism is an artifact engineered by the experimenter:

Where does the "randomization" in Mendelian Randomization come from? It is attributed to Gregor Mendel's two foundational biological principles:

• Mendel’s First Law (Law of Segregation): During gametogenesis (the formation of sperm and egg cells via meiosis), an individual's pairs of alleles at a diploid locus separate so that each haploid gamete carries only one allele with probability exactly $1/2$.
• Mendel’s Second Law (Law of Independent Assortment): Genes located on different chromosomes assort into gametes independently of one another during metaphase I of meiosis.

The Anatomy of Meiosis: Crossing Over and Linkage Disequilibrium

Even within an individual, the meiotic coin flip is not a sequence of independent Bernoulli trials across the three billion base pairs of the human genome.

During prophase I of meiosis, homologous chromosomes pair up and undergo homologous recombination (crossing over). Chromosomal segments are cut and re-ligated by cellular machinery, exchanging physical blocks of DNA. The human genome experiences only about 30 to 40 crossover events per meiosis per chromosome pair.

Because crossovers occur at discrete recombination hotspots, genetic variants located physically near one another on the same chromosome are inherited together as dense contiguous haplotypes. This non-random association of alleles at different loci is called Linkage Disequilibrium (LD).

If allele $A$ at locus 1 is physically linked to allele $B$ at locus 2 ($D' = \mathbb{P}(AB) - p_A p_B \neq 0$), you cannot perturb locus 1 without perturbing locus 2. If locus 1 influences LDL cholesterol and locus 2 influences insulin sensitivity, nature has packaged two distinct biological levers into a single unbreakable delivery mechanism.

The Epidemiological Leaping Error: We Do Not Observe Meiosis

Here is the central epistemic blunder of population-based Mendelian Randomization:

Epidemiologists do not run MR on the products of a single meiotic event. They run MR on adult cross-sections drawn from biobanks (such as 500,000 participants in the UK Biobank).

When you pull blood from a 62-year-old volunteer in Manchester, you are not observing a clean meiotic draw. You are observing the somatic genome of a surviving human being whose existence is the cumulative consequence of:

1. Ten thousand generations of evolutionary natural selection, pathogen resistance, and drift.
2. Centuries of geographic migration, physical barriers (rivers, mountain ranges), and regional isolation.
3. Violently non-random mating patterns (assortative mating on religion, class, wealth, height, and education).
4. The lifelong parental environment and wealth bestowed by the mother and father who transmitted those alleles.

The assumption that drawing genotypes from a population registry replicates an RCT is like sampling cars from a city parking garage and asserting that whether a car is a Mercedes or a Honda was decided by an unconfounded coin toss.

3. The Hardy-Weinberg Equilibrium Fallacy: Why $p^2 + 2pq + q^2 = 1$ Is Not a Proof of Exogeneity

When challenged on whether genetic variants are truly exogenous in population data, MR proponents often retreat to a favorite biological talisman: Hardy-Weinberg Equilibrium (HWE).

The argument usually goes like this: "We tested our instruments for departures from Hardy-Weinberg equilibrium, and all variants passed with $p > 10^{-6}$. Therefore, genotypes are distributed at random according to binomial expansion, proving that our instruments are clean."

As a statistician, I find this defense intellectually indefensible. It represents a fundamental confusion between a mathematical identity under neutrality and an identification assumption in causal inference.

What Hardy-Weinberg Actually Proves

Formulated independently by British mathematician G.H. Hardy and German physician Wilhelm Weinberg in 1908, HWE states that in an idealized population, allele and genotype frequencies remain constant from generation to generation in the absence of evolutionary influences.

For a biallelic locus with alleles $A_1$ and $A_2$ having population frequencies $p$ and $q = 1 - p$, the expected genotype frequencies under random union of gametes are:

HWE requires five strict mathematical assumptions:

1. An infinitely large population ($N \to \infty$, no random genetic drift).
2. Completely random mating (panmixia across the entire population).
3. No mutation (alleles do not spontaneously transform).
4. No migration / gene flow (closed population with zero immigration or emigration).
5. No natural selection (all genotypes possess identical reproductive fitness).

The Three Epistemic Failures of the HWE Defense

1. Category Error: Binomial Marginals $\neq$ Joint Orthogonality

Suppose an individual's socioeconomic background is a binary confounder $U \in \{0, 1\}$ (poor vs. wealthy). In an observational cohort, 40% of the sample is poor ($U=0$) and 60% is wealthy ($U=1$).

Suppose that at locus $Z$, the allele frequency in the poor subpopulation is $p_0 = 0.30$, while in the wealthy subpopulation it is $p_1 = 0.70$.

Within each subpopulation, random mating occurs, so genotypes conform exactly to Hardy-Weinberg proportions:

• Poor group ($U=0$): $\mathbb{P}(Z=2 \mid U=0) = (0.3)^2 = 0.09$, $\mathbb{P}(Z=1 \mid U=0) = 2(0.3)(0.7) = 0.42$.
• Wealthy group ($U=1$): $\mathbb{P}(Z=2 \mid U=1) = (0.7)^2 = 0.49$, $\mathbb{P}(Z=1 \mid U=1) = 2(0.7)(0.3) = 0.42$.

The overall allele frequency is $\bar{p} = 0.4(0.3) + 0.6(0.7) = 0.54$. In the pooled population, the proportion of homozygotes is:

This slight excess of homozygotes is the classical Wahlund effect: $\text{Var}(p) = 0.4(0.3 - 0.54)^2 + 0.6(0.7 - 0.54)^2 = 0.0384$.

Here, the instrument $Z$ is violently confounded by wealth: $\text{Cov}(Z, U) \neq 0$. Genotype directly predicts whether you were born into poverty or luxury. Yet, within every stratum, HWE holds with mathematical perfection!

2. The Power Illusion: HWE Tests Cannot Detect Subtle Confounding

In typical GWAS quality control pipelines, the HWE test is implemented as a Pearson goodness-of-fit $\chi^2$ test with 1 degree of freedom:

Genomics consortia routinely apply a filter such as $p_{\text{HWE}} > 10^{-6}$ or $p_{\text{HWE}} > 10^{-10}$. Why is the threshold set so low?

Because biobank bioinformaticians are not using HWE to test for causal confounding. They are using HWE to detect spectroscopic genotyping artifacts—such as fluorescent probe failure, hybridisation errors, or copy number duplications where the genotyping machine fails to distinguish heterozygous clusters from homozygous clusters.

If subtle geographic stratification or assortative mating induces an inbreeding coefficient of $F_{\text{IS}} = 0.005$, the test statistic in a sample of $N = 50,000$ is $\chi^2 = 50,000 \times (0.005)^2 = 1.25$, which yields a $p$-value of $0.26$.

The variant passes the HWE test with flying colors. Yet, an inbreeding coefficient or ancestral correlation of $0.005$ is more than enough to completely swamp a delicate instrumental variable estimate where the first stage $R^2$ is only $0.001$!

3. The Ratio Distortion: The Asymptotic Bias of Weak Confounding

Remember the asymptotic bias of the 2SLS / Wald estimator:

Because most complex trait SNPs explain a pathetic fraction of phenotypic variance (often $R^2 < 0.1\%$, meaning $\text{Cov}(Z, D) \approx 0.03$), a tiny violation in exogeneity—say, $\text{Cov}(Z, U) = 0.01$, completely invisible to HWE tests—is magnified by a factor of 30!

4. The Fatal Epistemic Breach: Why Population MR Fails Without Parental Genotypes

This brings us to the core thesis of this essay:

Without conditioning on the genotypes of both biological parents, population-based Mendelian Randomization cannot guarantee an exogenous instrument.

The reason is mathematically unassailable: Mendel's meiotic lottery is an assignment mechanism that is defined strictly conditional on parental genotypes.

The universe does not randomly assign alleles to a child from an urn containing the British population. A child receives exactly one chromosome from their mother and one chromosome from their father.

When you fail to condition on $(Z_{\text{father}}, Z_{\text{mother}})$—which is true for 99% of published Mendelian Randomization papers that analyze unrelated individuals in biobanks—your instrument $Z_{\text{child}}$ is hopelessly entangled with three structural backdoor pathways.

Pathway 1: Genetic Nurture and Dynastic Effects

In a landmark 2018 paper in Science, Augustine Kong and colleagues at deCODE genetics in Iceland uncovered a devastating phenomenon they christened "genetic nurture".

Consider a child's educational attainment or adult health outcome $Y_c$. The parents possess genotypes $Z_p = (Z_m, Z_f)$. These parental genotypes causally influenced the parents' own cognitive development, education, income, emotional stability, nutritional choices, smoking habits, and neighbourhood selection ($U_{\text{dynasty}}$).

This parental environment $U_{\text{dynasty}}$ directly shapes the child's life:

By elementary Mendelian inheritance, the child inherits on average half of their parents' alleles:

Therefore:

Kong et al. provided the smoking gun by analyzing non-transmitted alleles. In any meiotic event, a parent possesses two alleles and transmits only one. The other allele sits untransmitted in the parental body. The child never inherits it.

If population MR were valid, the non-transmitted parental alleles should have an estimated causal effect on the child of exactly zero.

Instead, Kong et al. discovered that a polygenic score constructed purely from non-transmitted parental alleles predicted the child's educational attainment with an effect size nearly 30% to 50% as large as the transmitted score!

Let that sink in. Alleles that never physically entered the child’s body still "caused" the child's educational and health outcomes, mediated through the parents' wealth, bookshelf count, stress levels, and zip code.

When you use $Z_{\text{child}}$ as an instrument without conditioning on parental genotypes, your instrument is directly picking up the causal effect of having wealthy, educated, stable parents. The exclusion restriction is blown to pieces.

Pathway 2: Assortative Mating and Gametic Phase Disequilibrium

Human beings do not choose reproductive partners at random. They sort passionately on education, height, intelligence, socioeconomic status, political leanings, and body mass index.

What does phenotypic assortative mating do to the genome? It violates Mendel's Second Law across generations.

Suppose trait $A$ is influenced by SNPs on chromosome 2, and trait $B$ is influenced by SNPs on chromosome 19. If individuals with high trait $A$ systematically mate with individuals with high trait $B$, their offspring will inherit positive alleles for trait $A$ on chromosome 2 and positive alleles for trait $B$ on chromosome 19.

This induces cross-trait, cross-chromosome correlation (gametic phase disequilibrium). Variants that have zero physical or biological connection become statistically entangled throughout the population.

When an MR researcher takes a polygenic instrument for BMI and uses it to study depression, they are not just measuring the metabolic effects of adiposity. They are measuring the fact that across generations, individuals with higher BMI sorted into social environments and partnerships correlated with psychiatric distress.

Pathway 3: Fine-Scale Population Stratification

Standard GWAS protocols attempt to adjust for ancestral confounding by computing the top 10, 20, or 40 Principal Components (PCs) of the genetic covariance matrix.

For years, epidemiologists treated PCA adjustment as a magic wand: "We controlled for 40 PCs in white British UK Biobank participants, so population stratification is eliminated."

This was shown to be decisively false in a series of blockbuster papers (Haworth et al., 2019; Barton et al., 2019; Howe et al., 2022).

Human genetic variation in Britain mirrors geography with exquisite sensitivity. North-to-south and east-to-west genetic clines trace the migration routes of Celts, Romans, Anglo-Saxons, and Norse Vikings.

Crucially, the Industrial Revolution created massive internal migration: coal mining, steel mills, and heavy industry concentrated in the north of England, Scotland, and Wales, while financial capital concentrated in London and the Southeast.

When an individual’s ancestors moved to northern industrial cities, they were exposed to generations of coal dust, de-industrialization, dietary differences, and economic decline. As Haworth et al. demonstrated, polygenic scores for height and education correlate strongly with geographic birth location in the UK, even after 40-PC adjustment!

The Within-Family Antidote: Sibling Contrasts and Trios

How do we destroy genetic nurture, assortative mating, and population stratification in one stroke?

By running within-family Mendelian Randomization.

Consider two biological full siblings, $i=1$ and $i=2$, born to the same parents. Both siblings grew up in the same home, had the same parents, breathed the same air, and shared the same dynastic wealth.

If we difference their genotypes and outcomes:

Because both siblings share parental genotypes, all parental genetic nurture cancels out: $U_{\text{parents}} - U_{\text{parents}} = 0$. All population stratification cancels out. Assortative mating between their parents is held constant.

$\Delta Z$ is the only physical instantiation of Mendel’s laws that qualifies as a genuine, unconfounded randomized trial!

5. The Causal Architecture: Interactive DAG & Credibility Matrix

To formalize these distinctions, we present two interactive exploratory tools: an interactive Causal DAG explorer contrasting valid cis-MR with structural failure modes, and a 2D credibility matrix mapping where MR studies sit on the evidential frontier.

6. The Empirical Spectrum: Bad MR vs. Good MR (In-Depth Case Studies)

To move from mathematical theory to applied reality, let us examine the seminal empirical case studies that demonstrate how Mendelian Randomization succeeds—and how it spectacularly collapses.

Tier 1: The Bad & The Fatal (Structural Identification Collapse)

Tier 2: The Good & The Gold Standard (Biochemically Targeted Cis-MR)

7. Methodological Band-Aids: Why Statistical Fixes Fall Short of Parental Genotypes

Confronted with the specter of horizontal pleiotropy, the statistical genetics community has spent the last decade producing an impressive arsenal of mathematical sensitivity tests:

• MR-Egger Regression (Bowden et al., 2015): Allows for directional pleiotropy by fitting an unconstrained intercept in a weighted linear regression of variant-outcome effects against variant-exposure effects.
• Weighted Median Estimator (Bowden et al., 2016): Yields a consistent estimate as long as at least 50% of the weight comes from valid instruments.
• Mode-Based Estimators (MBE, Hartwig et al., 2017): Assumes the most common causal effect estimate across instruments represents the true effect (the zero modal pleiotropy assumption).
• MR-PRESSO (Verbanck et al., 2018): Detects and removes outlier instruments via residual sum of squares tests.
• CAUSE (Morrison et al., 2020): Uses Bayesian mixture modeling to distinguish correlated horizontal pleiotropy from true causal effects.

These methods represent elegant mathematical statistics. But every single one of them relies on a foundational assumption known as InSIDE (Instrument Strength Independent of Direct Effect):

Why the InSIDE Assumption Routinely Fails

InSIDE is an economist's or statistician's fantasy that ignores how biology actually works.

If horizontal pleiotropy is driven by an unobserved shared biological confounder—say, systemic cellular senescence, basal mitochondrial function, or neuro-developmental canalization—then genetic variants that have larger effects on your exposure will naturally have larger pleiotropic effects on the confounder!

Worse, genetic nurture and assortative mating create direct violations of InSIDE. If parents with higher polygenic scores create superior developmental environments, the magnitude of the environmental benefit is directly proportional to the variant's effect size on the trait.

8. Master Diagnostic Matrix & The Practitioner's Forensic Playbook

To evaluate the credibility of any Mendelian Randomization paper, we summarize the structural taxonomy in the master diagnostic matrix below:

The Practitioner’s 6-Point Forensic Checklist

Whenever you review a Mendelian Randomization manuscript or news headline claiming that "Trait X causally causes Disease Y," walk through this 6-point forensic checklist:

• 1

  Are the instruments cis-acting or trans-polygenic?

  If the authors constructed a polygenic score from hundreds of trans-SNPs scattered across the genome for a complex behavioral trait (education, sleep, exercise, income), stop reading. The exclusion restriction is dead on arrival.

• 2

  Did the study condition on parental genotypes or analyze siblings?

  If the sample consists entirely of unrelated adults, the estimate is confounded by genetic nurture and assortative mating. Demand within-sibship or trio validation before accepting any causal claim for social or behavioral traits.

• 3

  Do the authors lazily hide behind Hardy-Weinberg Equilibrium?

  If the authors cite $p_{\text{HWE}} > 10^{-6}$ as proof that their instruments are unconfounded, recognize it as a red flag. HWE is a quality-control check for genotyping chip fluorescent cluster failures, not an identification proof.

• 4

  Could the genetic variant exert direct biochemical toxicity?

  Even single-gene instruments can harbor direct pleiotropy. Ask whether the variant alters an enzyme that generates toxic intermediates (like ALDH2 and acetaldehyde) or influences multi-protein complexes.

• 5

  Does the biological exposure possess construct clarity?

  Is the exposure a distinct physical molecule (LDL cholesterol, IL-6, Lp(a)) or a diffuse sociological index (socioeconomic status, years of schooling)? MR works brilliantly for molecular targets; it collapses for social constructs.

• 6

  How do the results compare to within-family benchmarks?

  Has the exposure been tested in large sibling cohorts (such as the Howe et al. consortia)? If the effect shrinks by 70% in siblings, the population MR estimate was an artifact of familial privilege and population structure.

9. References & Foundational Literature

1. Davey Smith, G., & Ebrahim, S. (2003). 'Mendelian randomization': can genetic epidemiology contribute to understanding environmental determinants of disease? International Journal of Epidemiology, 32(1), 1–22.
2. Kong, A., Thorleifsson, G., Frigge, M. L., et al. (2018). The nature of nurture: Effects of parental genotypes. Science, 359(6374), 424–428.
3. Howe, L. J., Nivard, M. G., Morris, T. T., et al. (2022). Within-sibship genome-wide association analyses decrease bias in genetic effect estimates and mendelian randomization. Nature Communications, 13(1), 2082.
4. Brumpton, B., Sanderson, E., Heilbron, K., et al. (2020). Avoiding dynastic, assortative mating, and population stratification biases in Mendelian randomization through within-family analyses. Nature Human Behaviour, 4(10), 1022–1031.
5. Voight, B. F., Peloso, G. M., Orho-Melander, M., et al. (2012). Plasma HDL cholesterol and risk of myocardial infarction: a mendelian randomisation study. The Lancet, 380(9841), 572–580.
6. Ference, B. A., Robinson, J. G., Brook, R. D., et al. (2016). Variation in PCSK9 and HMGCR and risk of cardiovascular disease and diabetes. Journal of the American College of Cardiology, 68(23), 2419–2428.
7. Bowden, J., Davey Smith, G., & Burgess, S. (2015). Mendelian randomization with invalid instruments: effect estimation and bias detection through Egger regression. International Journal of Epidemiology, 44(2), 512–525.
8. Bowden, J., Davey Smith, G., Haycock, P. C., & Burgess, S. (2016). Consistent estimation in Mendelian randomization with some invalid instruments using a weighted median estimator. Genetic Epidemiology, 40(4), 304–314.
9. Haworth, S., Mitchell, R., Corbin, L., et al. (2019). Apparent latent structure in UK Biobank: Apparent natural selection reflects geographic population structure. Nature Communications, 10(1), 382.
10. Barton, N., Hermisson, J., & Nordborg, M. (2019). Why structure matters. eLife, 8, e45380.
11. Millwood, I. Y., Walters, R. G., Mei, X. W., et al. (2019). Conventional and genetic evidence on alcohol and vascular disease: a prospective study of 500,000 men and women in China. The Lancet, 393(10183), 1831–1842.
12. Ridker, P. M., Everett, B. M., Thuren, T., et al. (2017). Antiinflammatory therapy with canakinumab for atherosclerotic disease (CANTOS). New England Journal of Medicine, 377(12), 1119–1131.
13. Zheng, J., Haberland, V., Baird, D., et al. (2020). Phenome-wide Mendelian randomization mapping the influence of the plasma proteome on complex diseases. Nature Genetics, 52(10), 1122–1131.
14. Katan, M. B. (1986). Apolipoprotein E isoforms, serum cholesterol, and cancer. The Lancet, 327(8479), 507–508.