Okcupid Weã¢â‚¬â„¢re Having Technical Difficulties
A quick note on terminology: I use the terms confounding and selection bias below, the terms of selection in epidemiology. The terms, however, depend on the field. In some fields, confounding is referred to every bit omitted variable bias or selection bias. Selection bias also sometimes refers to variable selection bias, a related outcome that refers to misspecified models.
Directed Acyclic Graphs
A DAG displays assumptions nigh the relationship between variables (ofttimes called nodes in the context of graphs). The assumptions we brand take the form of lines (or edges) going from one node to another. These edges are directed, which means to say that they have a single arrowhead indicating their effect. Hither'south a simple DAG where nosotros assume that x affects y:
dagify(y ~ x) %>% ggdag() 
You lot also sometimes see edges that look bi-directed, like this:
dagify(y ~ ~ x) %>% ggdag() 
Simply this is actually shorthand for an unmeasured cause of the two variables (in other words, unmeasured confounding):
# canonicalize the DAG: Add the latent variable in to the graph dagify(y ~ ~ ten) %>% ggdag_canonical() 
A DAG is likewise acyclic, which means that there are no feedback loops; a variable tin can't be its own descendant. The above are all DAGs because they are acyclic, but this is not:
dagify(y ~ ten, x ~ a, a ~ y) %>% ggdag() 
Structural Causal Graphs
ggdag is more specifically concerned with structural causal models (SCMs): DAGs that portray causal assumptions about a set of variables. Beyond being useful conceptions of the problem nosotros're working on (which they are), this also allows usa to lean on the well-developed links betwixt graphical causal paths and statistical associations. Causal DAGs are mathematically grounded, but they are also consistent and easy to empathise. Thus, when we're assessing the causal effect between an exposure and an outcome, drawing our assumptions in the course of a DAG can aid u.s. selection the right model without having to know much about the math backside it. Another fashion to think virtually DAGs is as non-parametric structural equation models (SEM): we are explicitly laying out paths between variables, but in the case of a DAG, information technology doesn't matter what form the human relationship between ii variables takes, only its management. The rules underpinning DAGs are consistent whether the relationship is a simple, linear one, or a more complicated part.
Relationships between variables
Let's say we're looking at the human relationship between smoking and cardiac abort. We might assume that smoking causes changes in cholesterol, which causes cardiac abort:
smoking_ca_dag <- dagify(cardiacarrest ~ cholesterol, cholesterol ~ smoking + weight, smoking ~ unhealthy, weight ~ unhealthy, labels = c("cardiacarrest" = "Cardiac \north Abort", "smoking" = "Smoking", "cholesterol" = "Cholesterol", "unhealthy" = "Unhealthy \n Lifestyle", "weight" = "Weight"), latent = "unhealthy", exposure = "smoking", consequence = "cardiacarrest") ggdag(smoking_ca_dag, text = Faux, use_labels = "label") 
The path from smoking to cardiac arrest is directed: smoking causes cholesterol to rise, which and then increases risk for cardiac arrest. Cholesterol is an intermediate variable betwixt smoking and cardiac arrest. Directed paths are also chains, considering each is causal on the adjacent. Let's say we too presume that weight causes cholesterol to ascent and thus increases risk of cardiac arrest. Now there'due south another chain in the DAG: from weight to cardiac arrest. Even so, this chain is indirect, at to the lowest degree as far as the relationship between smoking and cardiac arrest goes.
We also presume that a person who smokes is more probable to be someone who engages in other unhealthy behaviors, such as overeating. On the DAG, this is portrayed every bit a latent (unmeasured) node, chosen unhealthy lifestyle. Having a predilection towards unhealthy behaviors leads to both smoking and increased weight. Hither, the relationship betwixt smoking and weight is through a forked path (weight <- unhealthy lifestyle -> smoking) rather than a chain; because they have a mutual parent, smoking and weight are associated (in real life, at that place's probably a more than direct human relationship between the two, but we'll ignore that for simplicity).
Forks and chains are two of the three main types of paths:
- Chains
- Forks
- Inverted forks (paths with colliders)
An inverted fork is when two arrowheads meet at a node, which we'll talk over soon.
There are also common means of describing the relationships between nodes: parents, children, ancestors, descendants, and neighbors (at that place are a few others, every bit well, just they refer to less mutual relationships). Parents and children refer to directly relationships; descendants and ancestors can be anywhere along the path to or from a node, respectively. Here, smoking and weight are both parents of cholesterol, while smoking and weight are both children of an unhealthy lifestyle. Cardiac arrest is a descendant of an unhealthy lifestyle, which is in turn an ancestor of all nodes in the graph.
So, in studying the causal effect of smoking on cardiac arrest, where does this DAG leave us? We simply want to know the directed path from smoking to cardiac arrest, just there too exists an indirect, or back-door, path. This is confounding. Judea Pearl, who developed much of the theory of causal graphs, said that confounding is like water in a piping: it flows freely in open up pathways, and we need to block it somewhere forth the fashion. We don't necessarily demand to block the h2o at multiple points along the same back-door path, although we may have to block more than than 1 path. We frequently talk about confounders, but really we should talk most misreckoning, because it is about the pathway more than any particular node along the path.
Chains and forks are open up pathways, and so in a DAG where nothing is conditioned upon, any back-door paths must be i of the two. In improver to the directed pathway to cardiac arrest, there'south also an open back-door path through the forked path at unhealthy lifestyle and on from there through the concatenation to cardiac arrest:
ggdag_paths(smoking_ca_dag, text = FALSE, use_labels = "label", shadow = Truthful) 
Nosotros need to account for this back-door path in our analysis. There are many means to get nigh that–stratification, including the variable in a regression model, matching, changed probability weighting–all with pros and cons. Just each strategy must include a decision about which variables to business relationship for. Many analysts take the strategy of putting in all possible confounders. This can be bad news, because adjusting for colliders and mediators can introduce bias, as nosotros'll discuss shortly. Instead, nosotros'll look at minimally sufficient aligning sets: sets of covariates that, when adjusted for, cake all back-door paths, but include no more or no less than necessary. That means there can be many minimally sufficient sets, and if you remove even ane variable from a given prepare, a back-door path will open. Some DAGs, similar the get-go one in this vignette (ten -> y), have no back-door paths to close, so the minimally sufficient adjustment set is empty (sometimes written as "{}"). Others, like the cyclic DAG above, or DAGs with important variables that are unmeasured, can not produce any sets sufficient to shut dorsum-door paths.
For the smoking-cardiac abort question, there is a single prepare with a single variable: {weight}. Accounting for weight will give us an unbiased estimate of the relationship between smoking and cardiac arrest, bold our DAG is correct. Nosotros practise not demand to (or want to) control for cholesterol, however, because it's an intermediate variable between smoking and cardiac arrest; controlling for it blocks the path between the two, which will and so bias our estimate (see below for more on mediation).
ggdag_adjustment_set(smoking_ca_dag, text = Faux, use_labels = "label", shadow = TRUE) 
More complicated DAGs will produce more complicated adjustment sets; assuming your DAG is correct, any given ready will theoretically close the dorsum-door path betwixt the outcome and exposure. Nonetheless, one set may be better to use than the other, depending on your data. For instance, one set may contain a variable known to accept a lot of measurement error or with a lot of missing observations. Information technology may, then, exist better to use a set that y'all recall is going to be a improve representation of the variables you need to include. Including a variable that doesn't actually represent the node well will atomic number 82 to residue misreckoning.
What about controlling for multiple variables along the back-door path, or a variable that isn't along any back-door path? Even if those variables are not colliders or mediators, information technology can still cause a problem, depending on your model. Some estimates, like take a chance ratios, work fine when not-confounders are included. This is because they are collapsible: risk ratios are abiding across the strata of non-confounders. Some mutual estimates, though, like the odds ratio and hazard ratio, are non-collapsible: they are non necessarily constant across strata of non-confounders and thus can exist biased past their inclusion. There are situations, similar when the issue is rare in the population (the and so-called rare disease assumption), or when using sophisticated sampling techniques, like incidence-density sampling, when they approximate the gamble ratio. Otherwise, including extra variables may be problematic.
Colliders and collider-stratification bias
In a path that is an inverted fork (x -> thou <- y), the node where two or more than arrowheads run into is called a collider (considering the paths collide at that place). An inverted fork is not an open up path; it is blocked at the collider. That is to say, we don't need to account for m to appraise for the causal effect of x on y; the back-door path is already blocked past k.
Let'southward consider an example. Influenza and chicken pox are independent; their causes (influenza viruses and the varicella-zoster virus, respectively) accept nil to practise with each other. In existent life, there may be some confounders that acquaintance them, like having a depressed immune arrangement, merely for this instance we'll assume that they are unconfounded. Withal, both the flu and craven pox cause fevers. The DAG looks similar this:
fever_dag <- collider_triangle(10 = "Influenza", y = "Chicken Pox", m = "Fever") ggdag(fever_dag, text = Fake, use_labels = "label") 
If nosotros want to assess the causal effect of influenza on chicken pox, we do not demand to account for anything. In the terminology used past Pearl, they are already d-separated (direction separated), because there is no result on one by the other, nor are at that place any back-door paths:
ggdag_dseparated(fever_dag, text = Fake, use_labels = "label") 
However, if we command for fever, they become associated within strata of the collider, fever. We open a biasing pathway betwixt the two, and they get d-connected:
ggdag_dseparated(fever_dag, controlling_for = "one thousand", text = False, use_labels = "label") 
This can exist counter-intuitive at first. Why does controlling for a confounder reduce bias but adjusting for a collider increase information technology? It's because whether or non you have a fever tells me something near your illness. If you have a fever, simply you lot don't have the flu, I now accept more evidence that you accept chicken pox. Pearl presents information technology like algebra: I can't solve y = ten + grand. But when I know that thousand = 1, I can solve for y.
Unfortunately, in that location'south a 2nd, less obvious form of collider-stratification bias: adjusting on the descendant of a collider. That ways that a variable downstream from the collider can also cause this form of bias. For example, with our flu-craven pox-fever example, it may exist that having a fever leads to people taking a fever reducer, similar acetaminophen. Because fever reducers are downstream from fever, controlling for information technology induces downstream collider-stratification bias:
dagify(fever ~ flu + pox, acetaminophen ~ fever, labels = c("flu" = "Influenza", "pox" = "Chicken Pox", "fever" = "Fever", "acetaminophen" = "Acetaminophen")) %>% ggdag_dseparated(from = "flu", to = "pox", controlling_for = "acetaminophen", text = FALSE, use_labels = "label") 
Collider-stratification bias is responsible for many cases of bias, and it is often not dealt with appropriately. Selection bias, missing data, and publication bias tin all be thought of as collider-stratification bias. Information technology becomes trickier in more than complicated DAGs; sometimes colliders are too confounders, and we demand to either come up with a strategy to adjust for the resulting bias from adjusting the collider, or we need to pick the strategy that's likely to issue in the to the lowest degree amount of bias. See the vignette on common structures of bias for more than.
Source: https://cran.r-project.org/web/packages/ggdag/vignettes/intro-to-dags.html
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