2 The idea: inheritance as a natural experiment

The trouble in Chapter 1 was that the confounder $U$ contaminated the exposure. If we could find some other handle on $X$, one that $U$ does not touch, we could learn about the causal effect through that handle instead. In Mendelian randomisation the handle is a genetic variant. This chapter explains why a variant can play that role, and states precisely what has to hold for the argument to be valid.

2.1 Why genotype resembles randomisation

Two facts about inheritance do the work.

First, at conception the two alleles a parent could pass on are separated more or less at random (Mendel’s law of segregation). So which version of a variant you carry is, to a good approximation, a coin flip that happened before you were born, before your diet, your income, your health, or any other confounder could have influenced it.

Second, that coin flip is fixed for life and cannot be changed by disease. A biomarker can rise because you are getting sick; your genotype cannot. This rules out reverse causation, the other headache of observational work.

Put together: a variant that nudges the exposure gives us variation in $X$ that arrives stamped with a randomised, pre-determined label. That is the natural experiment (Davey Smith and Ebrahim 2003). The formal name for such a handle is an instrumental variable (IV), and MR is instrumental-variable estimation with genetic variants as the instruments.

2.2 The picture

The whole method is easiest to hold in your head as a diagram of what causes what. Let $G$ be a genetic variant, $X$ the exposure, $Y$ the outcome, and $U$ the unmeasured confounder.

Figure 2.1: A valid instrument. G affects the outcome ONLY by way of the exposure, and shares no back-door path with it.

For the instrument to earn its keep, three conditions must hold. They are the whole ballgame; every MR method, diagnostic, and sensitivity analysis in later chapters is about defending or repairing one of them.

The three instrument assumptions

Relevance. $G$ is genuinely associated with the exposure $X$. A variant that barely moves $X$ is a weak instrument and gives unstable, biased estimates. This one we can check directly in the data.
Independence. $G$ shares no common cause with $Y$: no back-door path $G \leftarrow \cdots \rightarrow Y$. Random inheritance is what buys us this, though population structure can threaten it.
Exclusion restriction. $G$ affects $Y$ only through $X$. There is no arrow straight from $G$ to $Y$. This is the one we can never fully verify, and the one pleiotropy violates.

Assumptions 2 and 3 are the fragile ones, because neither can be confirmed from data alone. Their failures look like this:

Figure 2.2: Two ways an instrument goes bad. Left: pleiotropy, a direct G to Y path, breaks exclusion. Right: a G to U link (e.g. population structure) breaks independence.

2.3 From assumptions to an estimate

If the three conditions hold, the causal effect falls out of a simple ratio. A variant that changes the exposure by some amount, and changes the outcome by some amount, can only be changing the outcome through the exposure (that is the exclusion restriction talking). So

\[ \text{causal effect} \;=\; \frac{\text{effect of } G \text{ on } Y}{\text{effect of } G \text{ on } X}. \]

This is the Wald ratio, the atom every MR estimator is built from. With one instrument it is literally a ratio of two regression coefficients; with many, the methods in Chapters 4–6 are increasingly clever ways of averaging those ratios while guarding against the assumptions failing.

Before we estimate anything, we need data with a known answer to estimate from. That is Chapter 3.

# The idea: inheritance as a natural experiment The trouble in Chapter 1 was that the confounder $U$ contaminated the exposure. If we could find some *other* handle on $X$, one that $U$ does not touch, we could learn about the causal effect through that handle instead. In Mendelian randomisation the handle is a genetic variant. This chapter explains why a variant can play that role, and states precisely what has to hold for the argument to be valid. ## Why genotype resembles randomisation Two facts about inheritance do the work. First, at conception the two alleles a parent could pass on are separated more or less at random (Mendel's law of segregation). So which version of a variant you carry is, to a good approximation, a coin flip that happened before you were born, before your diet, your income, your health, or any other confounder could have influenced it. Second, that coin flip is fixed for life and cannot be changed by disease. A biomarker can rise *because* you are getting sick; your genotype cannot. This rules out reverse causation, the other headache of observational work. Put together: a variant that nudges the exposure gives us variation in $X$ that arrives stamped with a randomised, pre-determined label. That is the natural experiment [@daveysmith2003]. The formal name for such a handle is an **instrumental variable (IV)**, and MR is instrumental-variable estimation with genetic variants as the instruments. ## The picture The whole method is easiest to hold in your head as a diagram of what causes what. Let $G$ be a genetic variant, $X$ the exposure, $Y$ the outcome, and $U$ the unmeasured confounder. ```{dot} //| label: fig-dag-valid //| fig-cap: "A valid instrument. G affects the outcome ONLY by way of the exposure, and shares no back-door path with it." //| fig-width: 5.2 digraph valid { graph [rankdir = LR, bgcolor = "transparent"] node [shape = circle, style = filled, fontname = "Helvetica", fillcolor = "#eef3f9", color = "#2c6fbb", fontcolor = "#1b1b1b", width = 0.5] U [fillcolor = "#f3eee7", color = "#9aa5b1"] G -> X X -> Y U -> X U -> Y { rank = same; X; U } } ``` For the instrument to earn its keep, three conditions must hold. They are the whole ballgame; every MR method, diagnostic, and sensitivity analysis in later chapters is about defending or repairing one of them. ::: {.callout-note title="The three instrument assumptions"} 1. **Relevance.** $G$ is genuinely associated with the exposure $X$. A variant that barely moves $X$ is a *weak instrument* and gives unstable, biased estimates. This one we can check directly in the data. 2. **Independence.** $G$ shares no common cause with $Y$: no back-door path $G \leftarrow \cdots \rightarrow Y$. Random inheritance is what buys us this, though population structure can threaten it. 3. **Exclusion restriction.** $G$ affects $Y$ *only through* $X$. There is no arrow straight from $G$ to $Y$. This is the one we can never fully verify, and the one **pleiotropy** violates. ::: Assumptions 2 and 3 are the fragile ones, because neither can be confirmed from data alone. Their failures look like this: ```{dot} //| label: fig-dag-violations //| fig-cap: "Two ways an instrument goes bad. Left: pleiotropy, a direct G to Y path, breaks exclusion. Right: a G to U link (e.g. population structure) breaks independence." //| fig-width: 7 digraph violations { graph [rankdir = LR, bgcolor = "transparent", nodesep = 0.5] node [shape = circle, style = filled, fontname = "Helvetica", fillcolor = "#eef3f9", color = "#2c6fbb", fontcolor = "#1b1b1b", width = 0.5] subgraph cluster_p { label = "Pleiotropy (breaks exclusion)"; color = "#d98c1f"; fontname = "Helvetica"; fontcolor="#4c5b6b" Gp [label = "G"]; Xp [label = "X"]; Yp [label = "Y"]; Up [label = "U", fillcolor = "#f3eee7", color = "#9aa5b1"] Gp -> Xp; Xp -> Yp; Up -> Xp; Up -> Yp Gp -> Yp [color = "#c0392b", penwidth = 2, constraint = false] { rank = same; Xp; Up } } subgraph cluster_c { label = "Confounded instrument (breaks independence)"; color = "#d98c1f"; fontname="Helvetica"; fontcolor="#4c5b6b" Gc [label = "G"]; Xc [label = "X"]; Yc [label = "Y"]; Uc [label = "U", fillcolor = "#f3eee7", color = "#9aa5b1"] Gc -> Xc; Xc -> Yc; Uc -> Xc; Uc -> Yc Uc -> Gc [color = "#c0392b", penwidth = 2, constraint = false] { rank = same; Xc; Uc } } } ``` ## From assumptions to an estimate If the three conditions hold, the causal effect falls out of a simple ratio. A variant that changes the exposure by some amount, and changes the outcome by some amount, can only be changing the outcome *through* the exposure (that is the exclusion restriction talking). So $$ \text{causal effect} \;=\; \frac{\text{effect of } G \text{ on } Y}{\text{effect of } G \text{ on } X}. $$ This is the **Wald ratio**, the atom every MR estimator is built from. With one instrument it is literally a ratio of two regression coefficients; with many, the methods in Chapters 4–6 are increasingly clever ways of averaging those ratios while guarding against the assumptions failing. Before we estimate anything, we need data with a known answer to estimate *from*. That is [Chapter 3](03-data.qmd).