Loading AI tools
Variable or factor in causal inference From Wikipedia, the free encyclopedia
In causal inference, a confounder[lower-alpha 1] is a variable that influences both the dependent variable and independent variable, causing a spurious association. Confounding is a causal concept, and as such, cannot be described in terms of correlations or associations.[1][2][3] The existence of confounders is an important quantitative explanation why correlation does not imply causation. Some notations are explicitly designed to identify the existence, possible existence, or non-existence of confounders in causal relationships between elements of a system.
![]() | This article may be too technical for most readers to understand. (September 2019) |
Confounds are threats to internal validity.[4]
Let's assume that a trucking company owns a fleet of trucks made by two different manufacturers. Trucks made by one manufacturer are called "A Trucks" and trucks made by the other manufacturer are called "B Trucks." We want to find out whether A Trucks or B Trucks get better fuel economy. We measure fuel and miles driven for a month and calculate the MPG for each truck. We then run the appropriate analysis, which determines that there is a statistically significant trend that A Trucks are more fuel efficient than B Trucks. Upon further reflection, however, we also notice that A Trucks are more likely to be assigned highway routes, and B Trucks are more likely to be assigned city routes. This is a confounding variable. The confounding variable makes the results of the analysis unreliable. It is quite likely that we are just measuring the fact that highway driving results in better fuel economy than city driving.
In statistics terms, the make of the truck is the independent variable, the fuel economy (MPG) is the dependent variable and the amount of city driving is the confounding variable. To fix this study, we have several choices. One is to randomize the truck assignments so that A trucks and B Trucks end up with equal amounts of city and highway driving. That eliminates the confounding variable. Another choice is to quantify the amount of city driving and use that as a second independent variable. A third choice is to segment the study, first comparing MPG during city driving for all trucks, and then run a separate study comparing MPG during highway driving.
Confounding is defined in terms of the data generating model. Let X be some independent variable, and Y some dependent variable. To estimate the effect of X on Y, the statistician must suppress the effects of extraneous variables that influence both X and Y. We say that X and Y are confounded by some other variable Z whenever Z causally influences both X and Y.
Let be the probability of event Y = y under the hypothetical intervention X = x. X and Y are not confounded if and only if the following holds:
(1) |
for all values X = x and Y = y, where is the conditional probability upon seeing X = x. Intuitively, this equality states that X and Y are not confounded whenever the observationally witnessed association between them is the same as the association that would be measured in a controlled experiment, with x randomized.
In principle, the defining equality can be verified from the data generating model, assuming we have all the equations and probabilities associated with the model. This is done by simulating an intervention (see Bayesian network) and checking whether the resulting probability of Y equals the conditional probability . It turns out, however, that graph structure alone is sufficient for verifying the equality .
Consider a researcher attempting to assess the effectiveness of drug X, from population data in which drug usage was a patient's choice. The data shows that gender (Z) influences a patient's choice of drug as well as their chances of recovery (Y). In this scenario, gender Z confounds the relation between X and Y since Z is a cause of both X and Y:
We have that
(2) |
because the observational quantity contains information about the correlation between X and Z, and the interventional quantity does not (since X is not correlated with Z in a randomized experiment). It can be shown[5] that, in cases where only observational data is available, an unbiased estimate of the desired quantity , can be obtained by "adjusting" for all confounding factors, namely, conditioning on their various values and averaging the result. In the case of a single confounder Z, this leads to the "adjustment formula":
(3) |
which gives an unbiased estimate for the causal effect of X on Y. The same adjustment formula works when there are multiple confounders except, in this case, the choice of a set Z of variables that would guarantee unbiased estimates must be done with caution. The criterion for a proper choice of variables is called the Back-Door[5][6] and requires that the chosen set Z "blocks" (or intercepts) every path between X and Y that contains an arrow into X. Such sets are called "Back-Door admissible" and may include variables which are not common causes of X and Y, but merely proxies thereof.
Returning to the drug use example, since Z complies with the Back-Door requirement (i.e., it intercepts the one Back-Door path ), the Back-Door adjustment formula is valid:
(4) |
In this way the physician can predict the likely effect of administering the drug from observational studies in which the conditional probabilities appearing on the right-hand side of the equation can be estimated by regression.
Contrary to common beliefs, adding covariates to the adjustment set Z can introduce bias.[7] A typical counterexample occurs when Z is a common effect of X and Y,[8] a case in which Z is not a confounder (i.e., the null set is Back-door admissible) and adjusting for Z would create bias known as "collider bias" or "Berkson's paradox." Controls that are not good confounders are sometimes called bad controls.
In general, confounding can be controlled by adjustment if and only if there is a set of observed covariates that satisfies the Back-Door condition. Moreover, if Z is such a set, then the adjustment formula of Eq. (3) is valid.[5][6] Pearl's do-calculus provides all possible conditions under which can be estimated, not necessarily by adjustment.[9]
According to Morabia (2011),[10] the word confounding derives from the Medieval Latin verb "confundere", which meant "mixing", and was probably chosen to represent the confusion (from Latin: con=with + fusus=mix or fuse together) between the cause one wishes to assess and other causes that may affect the outcome and thus confuse, or stand in the way of the desired assessment. Greenland, Robins and Pearl[11] note an early use of the term "confounding" in causal inference by John Stuart Mill in 1843.
Fisher introduced the word "confounding" in his 1935 book "The Design of Experiments"[12] to refer specifically to a consequence of blocking (i.e., partitioning) the set of treatment combinations in a factorial experiment, whereby certain interactions may be "confounded with blocks". This popularized the notion of confounding in statistics, although Fisher was concerned with the control of heterogeneity in experimental units, not with causal inference.
According to Vandenbroucke (2004)[13] it was Kish[14] who used the word "confounding" in the sense of "incomparability" of two or more groups (e.g., exposed and unexposed) in an observational study. Formal conditions defining what makes certain groups "comparable" and others "incomparable" were later developed in epidemiology by Greenland and Robins (1986)[15] using the counterfactual language of Neyman (1935)[16] and Rubin (1974).[17] These were later supplemented by graphical criteria such as the Back-Door condition (Pearl 1993; Greenland, Robins and Pearl 1999).[11][5]
Graphical criteria were shown to be formally equivalent to the counterfactual definition[18] but more transparent to researchers relying on process models.
In the case of risk assessments evaluating the magnitude and nature of risk to human health, it is important to control for confounding to isolate the effect of a particular hazard such as a food additive, pesticide, or new drug. For prospective studies, it is difficult to recruit and screen for volunteers with the same background (age, diet, education, geography, etc.), and in historical studies, there can be similar variability. Due to the inability to control for variability of volunteers and human studies, confounding is a particular challenge. For these reasons, experiments offer a way to avoid most forms of confounding.
In some disciplines, confounding is categorized into different types. In epidemiology, one type is "confounding by indication",[19] which relates to confounding from observational studies. Because prognostic factors may influence treatment decisions (and bias estimates of treatment effects), controlling for known prognostic factors may reduce this problem, but it is always possible that a forgotten or unknown factor was not included or that factors interact complexly. Confounding by indication has been described as the most important limitation of observational studies. Randomized trials are not affected by confounding by indication due to random assignment.
Confounding variables may also be categorised according to their source. The choice of measurement instrument (operational confound), situational characteristics (procedural confound), or inter-individual differences (person confound).
Say one is studying the relation between birth order (1st child, 2nd child, etc.) and the presence of Down Syndrome in the child. In this scenario, maternal age would be a confounding variable:[citation needed]
In risk assessments, factors such as age, gender, and educational levels often affect health status and so should be controlled. Beyond these factors, researchers may not consider or have access to data on other causal factors. An example is on the study of smoking tobacco on human health. Smoking, drinking alcohol, and diet are lifestyle activities that are related. A risk assessment that looks at the effects of smoking but does not control for alcohol consumption or diet may overestimate the risk of smoking.[22] Smoking and confounding are reviewed in occupational risk assessments such as the safety of coal mining.[23] When there is not a large sample population of non-smokers or non-drinkers in a particular occupation, the risk assessment may be biased towards finding a negative effect on health.
A reduction in the potential for the occurrence and effect of confounding factors can be obtained by increasing the types and numbers of comparisons performed in an analysis. If measures or manipulations of core constructs are confounded (i.e. operational or procedural confounds exist), subgroup analysis may not reveal problems in the analysis. Additionally, increasing the number of comparisons can create other problems (see multiple comparisons).
Peer review is a process that can assist in reducing instances of confounding, either before study implementation or after analysis has occurred. Peer review relies on collective expertise within a discipline to identify potential weaknesses in study design and analysis, including ways in which results may depend on confounding. Similarly, replication can test for the robustness of findings from one study under alternative study conditions or alternative analyses (e.g., controlling for potential confounds not identified in the initial study).
Confounding effects may be less likely to occur and act similarly at multiple times and locations.[citation needed] In selecting study sites, the environment can be characterized in detail at the study sites to ensure sites are ecologically similar and therefore less likely to have confounding variables. Lastly, the relationship between the environmental variables that possibly confound the analysis and the measured parameters can be studied. The information pertaining to environmental variables can then be used in site-specific models to identify residual variance that may be due to real effects.[24]
Depending on the type of study design in place, there are various ways to modify that design to actively exclude or control confounding variables:[25]
All these methods have their drawbacks:
Artifacts are variables that should have been systematically varied, either within or across studies, but that were accidentally held constant. Artifacts are thus threats to external validity. Artifacts are factors that covary with the treatment and the outcome. Campbell and Stanley[27] identify several artifacts. The major threats to internal validity are history, maturation, testing, instrumentation, statistical regression, selection, experimental mortality, and selection-history interactions.
One way to minimize the influence of artifacts is to use a pretest-posttest control group design. Within this design, "groups of people who are initially equivalent (at the pretest phase) are randomly assigned to receive the experimental treatment or a control condition and then assessed again after this differential experience (posttest phase)".[28] Thus, any effects of artifacts are (ideally) equally distributed in participants in both the treatment and control conditions.
Seamless Wikipedia browsing. On steroids.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.