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Illustration of a simple confounding factor. In other words, Z is the cause of X and Y. Simple Confounding Case.svg
Illustration of a simple confounding factor. In other words, Z is the cause of X and Y.

In statistics, a confounder (also confounding variable, confounding factor, extraneous determinant or lurking variable) 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]



Whereas a mediator is a factor in the causal chain (1), a confounder is a spurious factor incorrectly implying causation (2) Comparison confounder mediator.svg
Whereas a mediator is a factor in the causal chain (1), a confounder is a spurious factor incorrectly implying causation (2)

Confounding is defined in terms of the data generating model (as in the figure above). 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:






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:

Causal diagram of Gender as common cause of Drug use and Recovery Confounding.PNG
Causal diagram of Gender as common cause of Drug use and Recovery

We have that






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). The statistician desires the unbiased estimate , but in cases where only observational data are available, an unbiased estimate can only 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":






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 [4] [5] and requires that the chosen set Z "blocks" (or intercepts)[ clarification needed ] every path[ clarification needed ] from X to Y that ends with 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:






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. A typical counterexample occurs when Z is a common effect of X and Y, [6] 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."

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 <4,5>. Pearl's do-calculus provide additional conditions under which P(y | do(x)) can be estimated, not necessarily by adjustment. [7]


According to Morabia (2011), [8] the word derives from the Medieval Latin verb "confudere", 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. Fisher used the word "confounding" in his 1935 book "The Design of Experiments" [9] to denote any source of error in his ideal of randomized experiment. According to Vandenbroucke (2004) [10] it was Kish [11] who used the word "confounding" in the modern sense of the word, to mean "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) [12] using the counterfactual language of Neyman (1935) [13] and Rubin (1974). [14] These were later supplemented by graphical criteria such as the Back-Door condition (Pearl 1993; Greenland, Pearl and Robins, 1999). [3] [4]

Graphical criteria were shown to be formally equivalent to the counterfactual definition, [15] 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", [16] 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:

  1. Higher maternal age is directly associated with Down Syndrome in the child
  2. Higher maternal age is directly associated with Down Syndrome, regardless of birth order (a mother having her 1st vs 3rd child at age 50 confers the same risk)
  3. Maternal age is directly associated with birth order (the 2nd child, except in the case of twins, is born when the mother is older than she was for the birth of the 1st child)
  4. Maternal age is not a consequence of birth order (having a 2nd child does not change the mother's age)

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. [19] Smoking and confounding are reviewed in occupational risk assessments such as the safety of coal mining. [20] 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.

Decreasing the potential for confounding

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. [21]

Depending on the type of study design in place, there are various ways to modify that design to actively exclude or control confounding variables: [22]

All these methods have their drawbacks:

  1. The best available defense against the possibility of spurious results due to confounding is often to dispense with efforts at stratification and instead conduct a randomized study of a sufficiently large sample taken as a whole, such that all potential confounding variables (known and unknown) will be distributed by chance across all study groups and hence will be uncorrelated with the binary variable for inclusion/exclusion in any group.
  2. Ethical considerations: In double-blind and randomized controlled trials, participants are not aware that they are recipients of sham treatments and may be denied effective treatments. [23] There is a possibility that patients only agree to invasive surgery (which carry real medical risks) under the understanding that they are receiving treatment. Although this is an ethical concern, it is not a complete account of the situation. For surgeries that are currently being performed regularly, but for which there is no concrete evidence of a genuine effect, there may be ethical issues to continue such surgeries. In such circumstances, many of people are exposed to the real risks of surgery yet these treatments may possibly offer no discernible benefit. Sham-surgery control is a method that may allow medical science to determine whether a surgical procedure is efficacious or not. Given that there are known risks associated with medical operations, it is questionably ethical to allow unverified surgeries to be conducted ad infinitum into the future.

See also

Related Research Articles

Epidemiology is the study and analysis of the distribution, patterns and determinants of health and disease conditions in defined populations.

A cohort study is a particular form of longitudinal study that samples a cohort, performing a cross-section at intervals through time. It is a type of panel study where the individuals in the panel share a common characteristic.

An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently, the ratio of the odds of B in the presence of A and the odds of B in the absence of A. Two events are independent if and only if the OR equals 1, i.e., the odds of one event are the same in either the presence or absence of the other event. If the OR is greater than 1, then A and B are associated (correlated) in the sense that, compared to the absence of B, the presence of B raises the odds of A, and symmetrically the presence of A raises the odds of B. Conversely, if the OR is less than 1, then A and B are negatively correlated, and the presence of one event reduces the odds of the other event.

Dependent and independent variables Concept in mathematical modeling, statistical modeling and experimental sciences

Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand that they depend, by some law or rule, on the values of other variables. Independent variables, in turn, are not seen as depending on any other variable in the scope of the experiment in question. In this sense, some common independent variables are time, space, density, mass, fluid flow rate, and previous values of some observed value of interest to predict future values.

Spurious relationship Apparent, but false, correlation between causally-independent variables

In statistics, a spurious relationship or spurious correlation is a mathematical relationship in which two or more events or variables are associated but not causally related, due to either coincidence or the presence of a certain third, unseen factor.

Case–control study Type of observational study comparing two existing groups differing in outcome

A case–control study is a type of observational study in which two existing groups differing in outcome are identified and compared on the basis of some supposed causal attribute. Case–control studies are often used to identify factors that may contribute to a medical condition by comparing subjects who have that condition/disease with patients who do not have the condition/disease but are otherwise similar. They require fewer resources but provide less evidence for causal inference than a randomized controlled trial. A case–control study produces only an odds ratio, which is an inferior measure of strength of association compared to relative risk.

In epidemiology, a risk factor or determinant is a variable associated with an increased risk of disease or infection. Determinant is often used as a synonym, due to a lack of harmonization across disciplines, in its more widely accepted scientific meaning. Determinant, specific to community health policy, is a health risk that is general, abstract, pertains to inequalities and is difficult for an individual to control. For example, low ingestion of dietary sources of vitamin C is a known risk factor for developing scurvy. Poverty, in the discipline of health policy, is a determinant of an individual's standard of health. The main difference lies in the realm of practice, clinical practice versus public health.

In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment. Intuitively, IVs are used when an explanatory variable of interest is correlated with the error term, in which case ordinary least squares and ANOVA give biased results. A valid instrument induces changes in the explanatory variable but has no independent effect on the dependent variable, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable.

External validity is the validity of applying the conclusions of a scientific study outside the context of that study. In other words, it is the extent to which the results of a study can be generalized to and across other situations, people, stimuli, and times. In contrast, internal validity is the validity of conclusions drawn within the context of a particular study. Because general conclusions are almost always a goal in research, external validity is an important property of any study. Mathematical analysis of external validity concerns a determination of whether generalization across heterogeneous populations is feasible, and devising statistical and computational methods that produce valid generalizations.

The following is a glossary of terms used in the mathematical sciences statistics and probability.

Relative risk Measure of association used in epidemiology

The relative risk (RR) or risk ratio is the ratio of the probability of an outcome in an exposed group to the probability of an outcome in an unexposed group. Together with risk difference and odds ratio, relative risk measures the association between the exposure and the outcome.

In statistics, ignorability is a feature of an experiment design whereby the method of data collection do not depend on the missing data. A missing data mechanism such as a treatment assignment or survey sampling strategy is "ignorable" if the missing data matrix, which indicates which variables are observed or missing, is independent of the missing data conditional on the observed data.

In fields such as epidemiology, social sciences, psychology and statistics, an observational study draws inferences from a sample to a population where the independent variable is not under the control of the researcher because of ethical concerns or logistical constraints. One common observational study is about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator. This is in contrast with experiments, such as randomized controlled trials, where each subject is randomly assigned to a treated group or a control group. Observational studies, for lacking an assignment mechanism, naturally present difficulties for inferential analysis.

Causal model Conceptual model in philosophy of science

In the philosophy of science, a causal model is a conceptual model that describes the causal mechanisms of a system. Causal models can improve study designs by providing clear rules for deciding which independent variables need to be included/controlled for.

Mediation (statistics)

In statistics, a mediation model seeks to identify and explain the mechanism or process that underlies an observed relationship between an independent variable and a dependent variable via the inclusion of a third hypothetical variable, known as a mediator variable. Rather than a direct causal relationship between the independent variable and the dependent variable, a mediation model proposes that the independent variable influences the (non-observable) mediator variable, which in turn influences the dependent variable. Thus, the mediator variable serves to clarify the nature of the relationship between the independent and dependent variables.

In causal models, controlling for a variable means binning data according to measured values of the variable. This is typically done so that the variable can no longer act as a confounder in, for example, an observational study or experiment.


A quasi-experiment is an empirical interventional study used to estimate the causal impact of an intervention on target population without random assignment. Quasi-experimental research shares similarities with the traditional experimental design or randomized controlled trial, but it specifically lacks the element of random assignment to treatment or control. Instead, quasi-experimental designs typically allow the researcher to control the assignment to the treatment condition, but using some criterion other than random assignment.

In statistics, econometrics, political science, epidemiology, and related disciplines, a regression discontinuity design (RDD) is a quasi-experimental pretest-posttest design that aims to determine the causal effects of interventions by assigning a cutoff or threshold above or below which an intervention is assigned. By comparing observations lying closely on either side of the threshold, it is possible to estimate the average treatment effect in environments in which randomisation is unfeasible. However, it remains impossible to make true causal inference with this method alone, as it does not automatically reject causal effects by any potential confounding variable. First applied by Donald Thistlethwaite and Donald Campbell to the evaluation of scholarship programs, the RDD has become increasingly popular in recent years. Recent study comparisons of randomised controlled trials (RCTs) and RDDs have empirically demonstrated the internal validity of the design.

In the statistical analysis of observational data, propensity score matching (PSM) is a statistical matching technique that attempts to estimate the effect of a treatment, policy, or other intervention by accounting for the covariates that predict receiving the treatment. PSM attempts to reduce the bias due to confounding variables that could be found in an estimate of the treatment effect obtained from simply comparing outcomes among units that received the treatment versus those that did not. Paul R. Rosenbaum and Donald Rubin introduced the technique in 1983.

Causal inference is the process of determining the independent, actual effect of a particular phenomenon that is a component of a larger system. The main difference between causal inference and inference of association is that causal inference analyzes the response of an effect variable when a cause of the effect variable is changed. The science of why things occur is called etiology. Causal inference is said to provide the evidence of causality theorized by causal reasoning.


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Further reading

These sites contain descriptions or examples of confounding variables: