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Let $X,Y$ be independent random variables, each with a Bernoulli distribution of parameter $1/2$, and put $S=X+Y$. Unconditionally, the four ordered pairs $(0,0),(0,1),(1,0),(1,1)$ each have probability $1/4$. On the event $S=1$, only $(0,1)$ and $(1,0)$ remain, each with conditional probability $1/2$. ThusKnowing that exactly one trial succeeded makes observing one outcome informative about the other. This violates conditional independence given $S$, even though the original random variables are independent. Given $S=0$ or $S=2$, both variables are constant random variables and are conditionally independent there; that does not repair the failure at $S=1$, an event of positive probability.

The conditional covariance here is $\mathbb{E}[XY\mid S=1]-\mathbb{E}[X\mid S=1]\mathbb{E}[Y\mid S=1]=0-1/4=-1/4$. Both conditional variances are $1/4$, so the conditional correlation coefficient is $-1$: within the selected group, $Y=1-X$. These conclusions follow from restricting a finite sample space, without positing any physical interaction between the trials.

This example and the shared-environment example establish both logical directions: independent random variables need not be conditionally independent, and conditionally independent variables need not be independent. When $S$ is represented as the joint result of two inputs in a causal directed acyclic graph, conditioning on it has the collider structure associated with collider bias. A probability calculation by itself establishes a dependence relation, not a physical causal mechanism or an intervention.

Under the displayed causal directed acyclic graphs, with nonzero effects along the relevant arrows and no exact cancellation of the effects on $B$, the valid instrumental variables are scenarios (i), (ii), and (iv). The path arguments are:

An upstream intermediate variable need not invalidate an instrumental variable if its routes to the outcome all pass through the target exposure; an outcome intermediate reached without that exposure does invalidate it.

The administrative data include only encounters with $M=1$. Conditioning on the stop indicator selects on the colliderwhich opens the noncausal path $D\leftrightarrow U\to Y$ and creates collider bias. The equal entries estimate only the selected risks $\mathbb{P}(Y=1\mid D=d,M=1)$. They ignore racial differences in the probability of being stopped and therefore do not identify the total causal effect of race on violence. This is also selection bias, because encounters with $M=0$ never enter the dataset.

The claim is not implied. On the path $A\leftrightarrow Y\leftarrow X$, $Y$ is a collider, and conditioning on $Y$ opens that path. Equivalently, conditioning on the common effect $Y$ can induce collider bias between $A$ and $X$.

Nonrepresentativeness does not automatically destroy internal validity: a causal contrast can remain valid among Biobank participants if selection is independent of the joint exposure-outcome process after conditioning on analysis variables. It does create bias if health or affluence affects participation and is also related to birth cohort or diabetes, especially when conditioning on participation opens a collider bias path.

Neither coefficient has a causal interpretation. Conditioning on $X_3$ opens the collider bias path $X_1\to X_3\leftarrow U\to Y$ for the $X_1$ coefficient, while the $X_3$ coefficient remains confounded by $U$ and by omitted $X_2$.

Residual exposure stratification subtracts the genetically predicted component of an exposure before forming exposure strata. Under a suitable additive first-stage model, this avoids directly conditioning on the component caused by the instrument and can reduce the resulting collider bias.