Controlling for a variable

Last updated March 09, 2024

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 limitation of controlling for variables is that a causal model is needed to identify important confounders (backdoor criterion is used for the identification). Without having one, a possible confounder might remain unnoticed. Another associated problem is that if a variable which is not a real confounder is controlled for, it may in fact make other variables (possibly not taken into account) become confounders while they were not confounders before. In other cases, controlling for a non-confounding variable may cause underestimation of the true causal effect of the explanatory variables on an outcome (e.g. when controlling for a mediator or its descendant).^[2]^[3] Counterfactual reasoning mitigates the influence of confounders without this drawback.^[3]

Experiments

Experiments attempt to assess the effect of manipulating one or more independent variables on one or more dependent variables. To ensure the measured effect is not influenced by external factors, other variables must be held constant. The variables made to remain constant during an experiment are referred to as control variables.

For example, if an outdoor experiment were to be conducted to compare how different wing designs of a paper airplane (the independent variable) affect how far it can fly (the dependent variable), one would want to ensure that the experiment is conducted at times when the weather is the same, because one would not want weather to affect the experiment. In this case, the control variables may be wind speed, direction and precipitation. If the experiment were conducted when it was sunny with no wind, but the weather changed, one would want to postpone the completion of the experiment until the control variables (the wind and precipitation level) were the same as when the experiment began.

In controlled experiments of medical treatment options on humans, researchers randomly assign individuals to a treatment group or control group. This is done to reduce the confounding effect of irrelevant variables that are not being studied, such as the placebo effect.

Observational studies

In an observational study, researchers have no control over the values of the independent variables, such as who receives the treatment. Instead, they must control for variables using statistics.

Observational studies are used when controlled experiments may be unethical or impractical. For instance, if a researcher wished to study the effect of unemployment (the independent variable) on health (the dependent variable), it would be considered unethical by institutional review boards to randomly assign some participants to have jobs and some not to. Instead, the researcher will have to create a sample which includes some employed people and some unemployed people. However, there could be factors that affect both whether someone is employed and how healthy he or she is. Part of any observed association between the independent variable (employment status) and the dependent variable (health) could be due to these outside, spurious factors rather than indicating a true link between them. This can be problematic even in a true random sample. By controlling for the extraneous variables, the researcher can come closer to understanding the true effect of the independent variable on the dependent variable.

In this context the extraneous variables can be controlled for by using multiple regression. The regression uses as independent variables not only the one or ones whose effects on the dependent variable are being studied, but also any potential confounding variables, thus avoiding omitted variable bias. "Confounding variables" in this context means other factors that not only influence the dependent variable (the outcome) but also influence the main independent variable.^[3]

OLS Regressions and control variables

The simplest examples of control variables in regression analysis comes from Ordinary Least Squares (OLS) estimators. The OLS framework assumes the following:

Linear relationship - OLS statistical models are linear. Hence the relationship between explanatory variables and the mean of Y must be linear.
Homoscedasticity - This requires homogeneity of variances, that is equal or similar variances across these data.
Independence/No Autocorrelation - Error terms from one (or more) observation can not be influenced by error terms of other observations.
Normality of Errors - The errors are jointly normal and uncorrelated, this implies that $(\epsilon _{i})_{i\in N}$ i.e. that the error terms are an independently and identically distributed set (iid). This implies that the unobservables between different groups or observations are independent.
No multicollinearity - Independent variables must not be highly correlated with each other. For regressions using matrix notation, the matrix must be full rank i.e. $X^{'}X$ is invertible.

Accordingly, a control variable can be interpreted as a linear explanatory variable that affects the mean value of Y (Assumption 1), but which does not present the primary variable of investigation, and which also satisfies the other assumptions above.^[4]

Example

Consider a study about whether getting older affects someone's life satisfaction. (Some researchers perceive a "u-shape": life satisfaction appears to decline first and then rise after middle age.^[5]) To identify the control variables needed here, one could ask what other variables determine not only someone's life satisfaction but also their age. Many other variables determine life satisfaction. But no other variable determines how old someone is (as long as they remain alive). (All people keep getting older, at the same rate, no matter what their other characteristics.) So, no control variables are needed here.^[6]

To determine the needed control variables, it can be useful to construct a directed acyclic graph.^[3]

Related Research Articles

Econometrics is an application of statistical methods to economic data in order to give empirical content to economic relationships. More precisely, it is "the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference." An introductory economics textbook describes econometrics as allowing economists "to sift through mountains of data to extract simple relationships." Jan Tinbergen is one of the two founding fathers of econometrics. The other, Ragnar Frisch, also coined the term in the sense in which it is used today.

Analysis of covariance (ANCOVA) is a general linear model that blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of one or more categorical independent variables (IV) and across one or more continuous variables. For example, the categorical variable(s) might describe treatment and the continuous variable(s) might be covariates or nuisance variables; or vice versa. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. Intuitively, ANCOVA can be thought of as 'adjusting' the DV by the group means of the CV(s).

<span class="mw-page-title-main">Dependent and independent variables</span> Concept in mathematical modeling, statistical modeling and experimental sciences

A variable is considered dependent if it depends on an independent variable. Dependent variables 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.

In statistics, an interaction may arise when considering the relationship among three or more variables, and describes a situation in which the effect of one causal variable on an outcome depends on the state of a second causal variable. Although commonly thought of in terms of causal relationships, the concept of an interaction can also describe non-causal associations. Interactions are often considered in the context of regression analyses or factorial experiments.

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.

In statistics, multicollinearity or collinearity is a situation where the predictors in a regression model are linearly dependent.

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 (endogenous), 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 and is not correlated with the error term, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable.

In statistics, omitted-variable bias (OVB) occurs when a statistical model leaves out one or more relevant variables. The bias results in the model attributing the effect of the missing variables to those that were included.

Internal validity is the extent to which a piece of evidence supports a claim about cause and effect, within the context of a particular study. It is one of the most important properties of scientific studies and is an important concept in reasoning about evidence more generally. Internal validity is determined by how well a study can rule out alternative explanations for its findings. It contrasts with external validity, the extent to which results can justify conclusions about other contexts. Both internal and external validity can be described using qualitative or quantitative forms of causal notation.

In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the input dataset and the output of the (linear) function of the independent variable.

In econometrics, endogeneity broadly refers to situations in which an explanatory variable is correlated with the error term. The distinction between endogenous and exogenous variables originated in simultaneous equations models, where one separates variables whose values are determined by the model from variables which are predetermined. Ignoring simultaneity in the estimation leads to biased estimates as it violates the exogeneity assumption of the Gauss–Markov theorem. The problem of endogeneity is often ignored by researchers conducting non-experimental research and doing so precludes making policy recommendations. Instrumental variable techniques are commonly used to mitigate this problem.

In causal inference, a confounder 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. 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.

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

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, which is often false, a mediation model proposes that the independent variable influences the 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.

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 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.

In statistics and regression analysis, moderation occurs when the relationship between two variables depends on a third variable. The third variable is referred to as the moderator variable or simply the moderator. The effect of a moderating variable is characterized statistically as an interaction; that is, a categorical or continuous variable that is associated with the direction and/or magnitude of the relation between dependent and independent variables. Specifically within a correlational analysis framework, a moderator is a third variable that affects the zero-order correlation between two other variables, or the value of the slope of the dependent variable on the independent variable. In analysis of variance (ANOVA) terms, a basic moderator effect can be represented as an interaction between a focal independent variable and a factor that specifies the appropriate conditions for its operation.

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 study of why things occur is called etiology, and can be described using the language of scientific causal notation. Causal inference is said to provide the evidence of causality theorized by causal reasoning.

Causal research, is the investigation of cause-relationships. To determine causality, variation in the variable presumed to influence the difference in another variable(s) must be detected, and then the variations from the other variable(s) must be calculated (s). Other confounding influences must be controlled for so they don't distort the results, either by holding them constant in the experimental creation of evidence. This type of research is very complex and the researcher can never be completely certain that there are no other factors influencing the causal relationship, especially when dealing with people's attitudes and motivations. There are often much deeper psychological considerations that even the respondent may not be aware of.

In statistics, linear regression is a statistical model which estimates the linear relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable. If the explanatory variables are measured with error then errors-in-variables models are required, also known as measurement error models.

References

↑ Frost, Jim. "A Tribute to Regression Analysis | Minitab" . Retrieved 2015-08-04.
↑ Streiner, David L (February 2016). "Control or overcontrol for covariates?". Evid Based Ment Health. 19 (1): 4–5. doi:10.1136/eb-2015-102294. PMC 10699339 . PMID 26755716. S2CID 11155639.
1 2 3 4 Pearl, Judea; Mackenzie, Dana (2018). The Book of Why: The New Science of Cause and Effect. London: Allen Lane. ISBN 978-0-241-24263-6.
↑ WEISBERG, SANFORD (2021). APPLIED LINEAR REGRESSION. JOHN WILEY. ISBN 978-1-119-58014-0. OCLC 1225621417.
↑ Blanchflower, D.; Oswald, A. (2008). "Is well-being U-shaped over the life cycle?" (PDF). Social Science & Medicine. 66 (8): 1733–1749. doi:10.1016/j.socscimed.2008.01.030. PMID 18316146.
↑ Bartram, D. (2020). "Age and Life Satisfaction: Getting Control Variables under Control". Sociology. 55 (2): 421–437. doi: 10.1177/0038038520926871 .