A Unification of Mediator, Confounder, and Collider Effects

Abstract

Third-variable effects, such as mediation and confounding, are core concepts in prevention science, providing the theoretical basis for investigating how risk factors affect behavior and how interventions change behavior. Another third variable, the collider, is not commonly considered but is also important for prevention science. This paper describes the importance of the collider effect as well as the similarities and differences between these three third-variable effects. The single mediator model in which the third variable (T) is a mediator of the independent variable (X) to dependent variable (Y) effect is used to demonstrate how to estimate each third-variable effect. We provide difference in coefficients and product of coefficients estimators of the effects and demonstrate how to calculate these values with real data. Suppression effects are defined for each type of third-variable effect. Future directions and implications of these results are discussed.

A Unification of Mediator, Confounder, and Collider Effects

Over the last 30 years, there have been great strides in the definition and estimation of causal effects, as summarized by statistician Gary King (2015): “More has been learned about causal inference in the last few decades than the sum total of everything that had been learned about it in all prior recorded history.” Similar pronouncements have been made in epidemiology and public health (Glymour & Hamad, 2018; Hernán, 2018), but applications of modern causal inference remain rare in prevention research with some exceptions (MacKinnon et al., 2020; Musci & Stuart, 2020; Stuart et al., 2015). These advancements have established the role a third variable plays in causal inference, as a mediator, confounder, or collider (Pearl, 2009). In particular, colliders are an important new contribution of modern causal inference not often considered in prevention science. Understanding third-variable effects is critical for drawing accurate conclusions from prevention research, such as knowing when it is necessary to adjust¹statistically for a variable and when adjustment for a variable creates spurious effects.

Twenty-one years ago in this journal, MacKinnon et al. (2000) described the equivalence of the mediation and confounder models and showed how suppression occurs for each model. That paper did not describe colliders nor how mediators, confounders, and colliders fit within a general third-variable model. Thus, the purpose of this paper is fourfold. First, we describe the three different effects a third variable can have in causal inference (confounder, mediator, and collider). Second, we generalize statistical methods from mediation analysis for estimating confounder and collider effects. Third, we describe methods to investigate statistical suppression and interaction effects for each model. Fourth, we illustrate each effect using real-world examples and discuss the implications of third-variable effects for prevention. The overall goal of the paper is to help readers understand third-variable concepts and therefore be better prepared for causal inference.

Third‑Variable Effects

The case where a third variable acts as a mediator (X → Mediator → Y) has received extensive application and development over the last 30 years in prevention and other fields because it addresses a fundamental question of how a mediator transmits the causal effect of an independent variable (X) to a dependent variable (Y). Mediation analysis has been applied in research exploring how poverty affects behavior, how tobacco prevention programs reduce tobacco use, and how knowledge leads to behavior change, to name a few (MacKinnon, 2008; VanderWeele, 2015). Mediation is particularly important in the development and evaluation of treatment and prevention programs because the investigation of mediating mechanisms can make programs more effective and require fewer resources by designing programs to change the critical mediating mechanism that drives the change in the outcome. For example, the mediation model has provided evidence that interventions that change the social norms mediator prevent drug use (Botvin et al., 1999; Cuijpers, 2002; MacKinnon et al., 1991).

The case where a third variable is a confounder (X ← Confounder → Y) has also been useful in many disciplines, particularly in epidemiology and prevention, because of the ubiquitous confounding variables present in observational studies in those fields (Greenland & Morgenstern, 2001; James, 1980). Confounding variables interfere with the ability to make causal claims if they are not included in statistical analysis. Because a confounding variable causes both X and Y, the X to Y effect must be adjusted for the confounder to obtain an unbiased estimate of the causal effect between X and Y (Elwert & Winship, 2014). For example, age is a confounder of many effects, such as the effect between income and cancer risk. Age is associated with higher income as well as cancer risk, resulting in an apparent positive effect between income and cancer risk if age is not considered. Adjustment for age as a confounding variable yields a more accurate assessment of the effect between income and cancer rates.

The final effect a third variable can have in causal inference, that of a collider (X → Collider ← Y), is not as widely known as mediators and confounders in prevention science. A collider variable is caused by both X and Y, as opposed to a confounder which causes X and Y. Collider variables are challenging to understand because unlike adjustment for a mediator or confounder, which clarifies causal effects, adjustment for a collider obscures causal effects by inducing bias in the estimate of the X–Y relation. Collider effects occur in two main situations (Asendorpf, 2012): (1) explicitly, when a collider variable caused by X and Y is measured in the study and adjusted for in statistical analysis, or (2) implicitly, when there is not a measure of the collider variable in the study, but the sample is selected based on a collider variable that is caused by X and Y. We discuss each situation below.

For the explicit collider situation, X, Y, and the collider are all measured variables; for example, when X and Y are measured at an earlier time and cause a construct measured later. The later variable is a collider of the effect of X and Y, so adjustment for the collider will bias the estimate of the X to Y effect. For a hypothetical example, adjustment for physical fitness at age 40 will bias the effect between stress at age 14 and body mass index (BMI) at age 15. Explicit adjustment for the collider leads to bias because it combines the causal effect of stress (X) to BMI (Y) with the causal effect of stress and BMI to physical fitness (collider) at age 40. Richardson et al., (2019) describe a similar explicit collider of the relation of neuroticism and body mass index (BMI), where a small positive relation between neuroticism and BMI becomes negative when adjusted for self-reported health. Richardson et al. (2019) argued that it does not make sense to adjust for self-reported health because neuroticism and BMI cause self-reported health. Because self-reported health is an outcome of both neuroticism and BMI, adjusting for self-reported health will bias the estimate of the neuroticism to BMI relation, inducing a collider effect. Although measured at the same time, self-reported health is a consequent of neuroticism and BMI in the collider model. For the most part, researchers are aware of the problems introduced by adjusting for an outcome of the two variables they study, especially when the outcome occurs later in time. Researchers are less aware of how sample selection can also introduce implicit collider bias.

Samples are often selected based upon some criteria (Cresswell & Clark, 2017), e.g., selecting patients at a hospital, selecting clients in a clinical population, selecting middle school students, selecting a high-risk group, etc. When the criteria used to select a sample is a collider variable, the restricted scores on the collider make the sample non-representative of the population and cause spurious effects between X and Y (Cole et al., 2010; Elwert & Winship, 2014). Restriction of samples by a collider is an example of implicit collider bias. For example, the population effect of impulsivity on delinquency is inaccurately assessed in a sample restricted to youth at high risk of drug use for a causal model in which impulsivity and delinquency both cause drug use. High-risk status is a potential collider of the effect of impulsivity on delinquency, so adjusting for high-risk status would lead to a spurious relation between impulsivity and delinquency.

Collider bias is also known as Berkson’s bias or Berkson’s fallacy because Berkson (1946) demonstrated the effect in research on risk factors for disease. Berkson demonstrated how two diseases, such as diabetes and gall bladder inflammation, that are unrelated in the population can become spuriously associated when a sample is restricted to individuals in a hospital sample. Having diabetes and gall bladder inflammation both caused people to go to the hospital, thereby restricting the range of the sample to hospital patients. Going to the hospital was a collider that implicitly restricted the sample. A hospital patient who does not have diabetes was much more likely to have another disease like gall bladder inflammation than someone from the general population. Thus, the selection of hospital patients created a spurious positive association between gall bladder inflammation and diabetes.

Other examples of collider effects related to prevention science include the obesity paradox and the low birth weight paradox. Both the obesity paradox and the low birth weight paradox involve more complicated models with more variables than what is present in the typical three-variable collider model, but the presence of a collider forms the basis for these seemingly paradoxical results. In the obesity paradox, obesity is negatively related to mortality in samples of patients with cardiovascular disease (Banack & Kaufman, 2014). By restricting the sample to people with cardiovascular disease, it appears that obesity has the beneficial effect of reducing mortality, when in the general population obesity increases mortality. The pattern emerges in part because of an unmeasured fourth variable, such as genetic factors, that cause cardiovascular disease and mortality. Selection by the collider, cardiovascular disease, distorts the relation between obesity and mortality because genetic factors are related to mortality and cardiovascular disease. The low birth weight paradox refers to the finding that low birth weight infants born to mothers who smoke cigarettes appear to have a lower mortality rate than other low birth weight babies (Hernández-Díaz et al., 2006; Whitcomb et al., 2009). By selecting a sample of only low birth weight babies, a biased effect of maternal smoking on infant mortality emerges because smoking, along with genetic factors, is likely a cause of low birth weight and mortality in newborns. Selection by low birth weight alters the relation between smoking and mortality because genetic factors also cause mortality and low birth weight.

Most collider examples are implicit, featuring restriction of range in which two variables cause an outcome variable, and the sample is restricted based on the values of the outcome variable. Despite the lack of published examples of colliders in prevention science, it is likely that these effects are present in many studies due to common sampling techniques in the field. It is important for prevention scientists to consider possible collider effects whenever samples are restricted in some way, such as samples of clinical populations restricted by their clinical status, samples selected because of their high-risk status, or samples of college students restricted by their academic achievement. If the variable that restricts the sample is a collider of the two variables studied, there will be collider bias.

In summary, mediators, confounders, and colliders are important for planning and interpreting prevention research. Colliders, though relatively new to prevention science, are an important third variable to consider. Verbal description of mediators, confounders, and colliders is an important step in understanding the different effects a third variable can have on causal inference. Diagrams of the mediation (i.e., a chain; X → Mediator → Y), confounding (i.e., a fork; X ← Confounder → Y), and collider (i.e., an inverted fork; X → Collider ← Y) models also clarify the underlying structure of these effects (Morgan & Winship, 2015, p. 81). Translation of the models to regression equations and estimation of effects with real data further clarifies the third variable effects, as done in the next section.

Estimation of the Third‑Variable Effects

In this section, we describe how the same three regression equations are used to estimate each third-variable effect. Equations (1), (2), and (3) below are the three equations necessary for estimation of the mediated effect that can be extended to estimate confounder and collider effects. Intercepts are represented by i₁, i₂, and i₃, and residuals are represented by e₁, e₂, and e₃. For the sake of explanation, we assume that there are no other observed or unobserved variables that affect the three variables. Point and interval estimation of each third-variable effect uses information from three regression equations. The subscripts for the coefficients represent the causal direction of the effects. An effect adjusted for another variable is represented in the subscript, where the independent and dependent variables are followed by a dot and then the variable that is adjusting the primary relationship. For example, b_YX is the effect to Y from X, and b_YX⋅T is the effect to Y from X adjusted for T. Equation (1) shows the equation where Y is regressed on X and Eq. (2) shows the equation where Y is regressed on X and the third variable (T). The third equation regresses T on X, estimating the value of the coefficient b_TX , a coefficient needed for a product of coefficients method to estimate the third-variable effect. These three equations are the same for the estimation of each third-variable effect because each effect involves adjusting a bivariate effect for the third variable.

$$Y=i_1+b_{YX}X+e_1$$ (1)

$$Y=i_2+b_{YX\cdot T}X+b_{YT\cdot X}T+e_2$$ (2)

$$T=i_3+b_{TX}X+e_3$$ (3)

Estimates of the parameters in each regression equation are used to estimate third-variable effects in sample data (above coefficients such as ${\widehat{b}}_{YX}$ represent sample estimates). In each model, the difference in coefficients estimator, ${\widehat{b}}_{YX}-{\widehat{b}}_{YX\cdot T}$ is the third-variable effect using Eqs. (1) and (2). The difference in coefficients estimator estimates the X to Y effect before and after adjusting for the mediator, the X to Y effect before and after adjusting for the confounder, and the X to Y effect before and after adjusting for the collider. For mediation, including the mediator third variable in Eq. (2) provides a way to estimate mediated (indirect), ${\widehat{b}}_{YX}-{\widehat{b}}_{YX\cdot T}$, and direct effects, ${\widehat{b}}_{YX\cdot T}$ of X to Y. For confounding, including the confounder in the model adjusts the X–Y effect for the confounder’s effect, making the adjusted X–Y effect more accurate. The confounder effect, ${\widehat{b}}_{YX}-{\widehat{b}}_{YX\cdot T}$, provides an estimate of how failure to adjust for the confounder would affect the estimate (Moldonado & Greenland, 1993; Selvin, 2004). For the collider model, the correct model should not adjust for the collider, but it is possible to obtain an estimate of the collider effect by comparing the effect of X on Y adjusted and unadjusted for the collider, ${\widehat{b}}_{YX}-{\widehat{b}}_{YX\cdot T}$ The collider effect provides a numerical estimate of how much the collider will bias the estimate if it is mistakenly included in the statistical analysis.

In addition to the difference in coefficients estimator, there is an equivalent product of coefficients estimator, ${\widehat{b}}_{TX}{\widehat{b}}_{YT\cdot X}$, for each third variable effect in the linear model using Eqs. (2) and (3). The equivalence of the difference and product of coefficients estimators was shown for mediation for linear models in MacKinnon et al. (1995), where for notation in the mediation literature c–c′ corresponds to ${\widehat{b}}_{YX}-{\widehat{b}}_{YX\cdot T}$, and ab corresponds to ${\widehat{b}}_{TX}{\widehat{b}}_{YT\cdot X}$. The supplemental material for this article demonstrates the equivalence of the difference in coefficients and product of coefficients for the confounder and collider effects. This equality between the product and difference estimators is not present for nonlinear models such as logistic regression (MacKinnon & Dwyer, 1993).

Standard errors are available for the product of coefficients and difference of coefficients estimators using methods developed for mediation analysis such as those based on the multivariate delta method (Clogg et al., 1995; MacKinnon & Dwyer, 1993). The equations for the standard error of the product and difference estimators are shown in the supplemental material for this paper. As established in the mediation research literature, the mediated effect estimator does not have a normal distribution, and so the most accurate significance tests and confidence intervals are obtained using methods that do not require normal distributions such as the distribution of the product and resampling methods such as the bootstrap method (MacKinnon et al., 2002, 2004). Results for bootstrap resampling of the third-variable effect are reported in this paper.

Suppression for Each Third‑Variable Effect

Suppression may be present for each third-variable effect. An effect between the independent and dependent variables may be reduced or may be enhanced when adjusted by a third variable, whether that third variable is a mediator, confounder, or collider. MacKinnon et al. (2000) describe suppression for models with mediation and confounding effects. In each of these models, suppression occurs when the effect between X and Y increases in magnitude or changes signs after adjusting for a third variable (Conger, 1974). For example, introducing a mediator usually reduces the strength of the relationship between X and Y, but there are some situations where the magnitude of the X to Y effect is actually larger when adjusting for a mediator. Similarly, adjustment for a confounder may increase or decrease the effect between two variables (Breslow, Day, & Heseltine, 1980). Although we are not aware of empirically demonstrated suppression effects for collider bias in the research literature (though see Richardson et al., 2019, for a possible example), adjustment for a collider may also increase or decrease the effect between the independent and dependent variable in a similar manner as mediation and confounding.

The most popular definition of suppression is when the magnitude of a bivariate relation gets larger when adjusted for a third variable (Conger, 1974). For this definition, suppression occurs when the magnitude of the adjusted effect is greater than the unadjusted effect, $\left|b_{YX\cdot T}\right|>\mid b_{YX}$, and this can occur for each third variable effect. A more general definition of suppression for mediation compares the sign of the direct and indirect effect from the product of coefficient estimator, e.g., $b_{TX}{b}_{YT\cdot X}$ versus $b_{YX\cdot T}$ (MacKinnon et al., 2000). The same quantity indicates suppression for confounding and collision. The difference in sign between $b_{TX}{b}_{YT\cdot X}$ and $b_{YX\cdot T}$ method is more general than the increase in magnitude method because adjustment for a third variable may change the sign of the adjusted coefficient but may not increase the magnitude of the absolute value of the adjusted coefficient. Using the same logic, suppression occurs if the product of the three coefficients, $b_{TX}{b}_{YT\cdot X}{b}_{YX\cdot T}$, has a negative sign for each third-variable effect (Muniz & MacKinnon, 2021). Suppression is present if the product of the three coefficients is negative and is not present if the product is positive.

Interactions for Each Third‑Variable Effect

Mediation, confounding, and collider effects may also have an interaction effect such that an effect differs across the levels of the third variable. Several sources describe interactions in confounder and mediation models (Fairchild & MacKinnon, 2009; Selvin, 2004). Interactions for collider variables are possible (see Richardson et al., 2019, for an example).

Adding an interaction to Eq. (2) is shown below in Eq. (4). The interaction coefficient, h_YXT∙X,T , represents whether the effect of T on Y differs at different levels of X and whether the effect of X on Y differs at different levels of T. The intercept is represented by i₄ and the residual by e₄. For these models, the third-variable effect may differ at separate values of the third variable. With interaction effects, there may be situations where there is not a third-variable effect at certain values but there is a third-variable effect at other values. Similarly, with interactions, it may be possible that there is a suppressor effect at certain values but not a suppressor effect at other values defined by the interaction.

$$Y=i_4+b_{YX\cdot T}X+b_{YT\cdot X}T+h_{YXT\cdot X,T}XT+e_4$$ (4)

Illustrative Examples of Mediation, Confounding, and Collision

To make mediator, confounder, and collider effects more concrete, we describe three-variable real-data examples of each effect. The examples illustrate the mediation, confounding, and collision models, assuming that the underlying causal models are true. We do not include additional covariates to simplify the presentation. The results are not definitive until a proper causal analysis is performed including specifying model assumptions, identification of estimators, sensitivity analyses, and consideration of additional mediators, confounders, and colliders. Table 1 shows estimates for each model. More details about the regression equations and the calculation of each effect are provided in the supplemental material.

Table 1.

Parameter estimates and confidence intervals for the three data examples

Third Variable	Effect	Parameter	Estimate	Standard error	Est/SE	LCL	UCL
Mediation	Group on Recall	bYX	2.517	1.084	2.32	.329	4.705
	Group on Recall	bYX⋅T	.332	1.292	.26	-2.277	2.940
	Adjusted for imagery
	Imagery on Recall	bYT⋅X	.614	.226	2.72	.158	1.071
	Adjusted for group
	Group on imagery	bTX	3.558	.689	5.16	2.167	4.950
	Mediated effect	bTXbYT⋅X = bYX – bYX⋅T	2.185	.922	2.40	.631	3.847
Confounding	BMI on bench press	bYX	12.746	1.697	7.51	9.414	16.078
	BMI on bench press	bYT⋅T	10.916	1.649	6.62	7.678	14.153
	Adjusted for age
	Age on bench press	bYT⋅X	53.233	6.706	7.94	40.069	66.397
	Adjusted for BMI
	BMI on age	bTX	.034	.009	3.92	.017	.052
	Confounder effect	bTXbYT⋅X = bYX – bYX⋅T	1.830	.524	3.49	.854	2.933
Collision	Conscientiousness on Income	bYX	.022	.006	3.70	.010	.033
	Conscientiousness on	bYT⋅T	.017	.006	2.90	.006	.029
	Income adjusted for health
	Health on income	bYT⋅X	.041	.007	6.00	.028	.055
	Adjusted for conscientiousness
	Conscientiousness on	bTX	.112	.012	9.59	.089	.134
	health
	Collider effect	bTXbYT⋅X = bYX – bYX⋅T	.005	.001	5.00	.003	.007

Bootstrap samples (1000 samples) were used to compute confidence limits for mediated, confounder, and collider effects. Other confidence limits were based on the normal distribution computed with estimates and standard errors. Table entries are rounded estimates

Est estimate, SE standard error, LCL lower confidence limit, UCL upper confidence limit

Imagery as a Mediator of Rehearsal Instructions on Memory for Word Recall

Our first example is for mediation, using experimental data from a study that hypothesized that using images to encode words leads to greater recall of the words than word repetition (MacKinnon et al., 2018). Forty-four participants were randomly assigned to one of two instructions for memorizing twenty words and were asked to recall as many words as they could. Participants also reported the degree of mental imagery they engaged in while memorizing the words. Instructions (X) were hypothesized to increase imagery (T), which would increase word recall (Y).

Any linear regression statistical software package can provide the estimates of Eqs. (1), (2), and (3) for the calculation of the mediated effect. For (1), perform a regression analysis from X (group) to Y (word recall). For (2), perform a regression analysis from X (group) and the mediator imagery (T) to Y (word recall), with X and T as two predic- tor variables of Y. Equation (3) follows the same procedure, except that the analysis is from X (group) to T (imagery). These three steps provide the coefficients needed to calculate the mediated effect of the manipulation to imagery to word recall.

The difference of coefficients estimate is calculated by taking the difference between the unadjusted effect between X and Y and the adjusted effect between X and Y for T. This difference provides an estimate of the mediated effect, ${\widehat{b}}_{YX}-{\widehat{b}}_{YX\cdot T}=2.517-0.332=2.185$. The product of coefficients estimate is calculated by taking the product of the two path coefficients, ${\widehat{b}}_{TX}{\widehat{b}}_{YT\cdot X}$, (3.558) (0.614) = 2.185, which gives the same estimate as the difference in coefficient estimator. The mediated effect through imagery on word recall equals 2.185 words which corresponds to a d-effect size (mediated effect divided by the standard deviation of Y) of $0.58(.58=\frac{2.185}{3.7594})$ so the mediated effect is about half a standard deviation (Miočević et al., 2018). The bootstrap confidence interval (LCL = 0.631, UCL = 3.847) does not contain zero, leading to the conclusion that the mediated effect is larger than expected by chance. The model is not a suppression model because ${\widehat{b}}_{TX}{\widehat{b}}_{YT\cdot X}=2.185$ and ${\widehat{b}}_{YX\cdot T}=0.332$ have the same sign and, correspondingly, the product of the estimated coefficients ${\widehat{b}}_{TX}{\widehat{b}}_{YT\cdot X}{\widehat{b}}_{YX}\cdot T$ is positive. There is no evidence for a statistically significant interaction between X and T on Y, ${\widehat{h}}_{YXT\cdot X,Y}=0.749(0.715)$, t (40) = 1.05. Randomization of X in this example clarifies the causal interpretation of the X to the mediator (T) and X to Y effects. However, randomization of X does not ensure causal interpretation of the mediator (T) to Y effect, because there is no randomization of the mediator (T). One way to enhance the causal interpretation of the T to Y effect would be to include additional confounding variables related to T and Y in the statistical analysis.

Age as a Confounder of the Effect of Bench Press Performance and Body Mass Index

The next example illustrates the calculation of a confounding effect using data from a study of a steroid prevention program known as ATLAS (Adolescents Teaching and Learning to Avoid Steroids) with high-school athletes aged 13–19 (Goldberg et al., 1996). The example uses each athlete’s age, body mass index (BMI), and bench press performance (measured as pounds lifted multiplied by the numbers of repetitions). There were 773 athletes with complete information for all three variables used for this analysis.

Age is a likely confounder of the effect between BMI and bench press performance because age affects both variables (e.g., we would expect that a 13-year-old would have a lower BMI than a 19-year-old, which will affect their respective bench press performance). The steps to calculate the confounding effect are the same as for the mediated effect using Eqs. (1), (2), and (3). Here, T (age) is the confounder of the BMI (X) to bench press performance (Y) effect. To calculate the confounding effect, the unadjusted effect ${\widehat{b}}_{YX}$ is subtracted by the adjusted effect, ${\widehat{b}}_{YX\cdot T}$, 12.746 − 10.916 = 1.83. The product of coefficients method with ${\widehat{b}}_{TX}$ and ${\widehat{b}}_{YT\cdot X}$ provides the same result, (53.233) (0.034) = 1.83. The confounder effect equals 1.83 bench press by weight units which corresponds to a 0.0375 standard deviation change in bench press performance for a one standard deviation change in BMI, $(0.0375=1.83(\frac{4.521}{270758}))$ so the confounder effect is about 4% of a standard deviation. The bootstrap confidence interval (LCL = 0.854, UCL = 2.933) led to the conclusion that the confounder effect is larger than expected by chance. The model is not a suppression model because ${\widehat{b}}_{TX}{\widehat{b}}_{YT•X}=1.83$ have the same sign and the product of ${\widehat{b}}_{YM\cdot X}=10.916$ is positive. There is not a statistically significant interaction of age and BMI on bench press performance, ${\widehat{h}}_{\text{XXT}•\text{X}\cdot \text{Y}}=0.9945(1.4914),t(769)=0.67$ using Eq. (4).

In summary, the confounder effect was larger than expected by chance, but this result is tentative until all relevant variables are included in a thorough causal analysis. For example, there are likely additional important variables related to BMI, bench press performance, and age not considered in this illustrative example.

General Health as a Collider of the Effect of Conscientiousness and Family Income

We use the National Longitudinal Survey of Youth 1997 cohort data (Bureau of Labor Statistics, 2019) to demonstrate how to calculate a collider effect. We extract information from 5404 participants about their general health, levels of conscientiousness, and family income. For this example, we treat general health as a continuous collider variable and assume that conscientiousness causes family income. Theoretically, higher levels of conscientiousness and greater family income may independently lead to better overall health, making health a potential collider of the conscientiousness-income effect (Murray et al., 2014). In a real study, a researcher would not want to adjust for this collider variable, as it will bias the results. The purpose of this demonstration is to show the extent to which adjusting for the collider variable can bias the effect of interest, and to encourage researchers to think carefully about the causal structure of their models before including certain variables in analysis.

Conscientiousness scores and family income were measured in 2008, and ratings of general health were collected in 2017. Family income was transformed to normalize the distribution of the scores by taking the logarithm of each reported income plus 1000. Adding 1000 prior to taking logarithms allowed for participants that reported having no income to obtain a log-transformed score. The original scores for family income ranged from $0 to $450,000+.

The calculations proceed in the same manner as for mediated and confounding effects but using the collider third variable. The effect between conscientiousness (X) and family income (Y) is adjusted for the collider general health (T). The difference in coefficient estimate, ${\widehat{b}}_{YX}-{\widehat{b}}_{YX\cdot T}=0.022-0.017$, and product of coefficients estimate, ${\widehat{b}}_{TX}{\widehat{b}}_{YT\cdot X}=\left(0.041\right)\left(0.112\right)$, both equal 0.005. The collider effect is a 0.005 change in family income for a one-unit change in conscientiousness. Another interpretation for this value is that adjusting for the collider variable of general health biases the conscientiousness-income relationship by a value of 0.005. Putting this value in terms of a d-effect size gives a change in the effect of conscientiousness and general health of $0.0106(0.0106=0.0050(\frac{1.1162}{4828}))$ standard deviations for a one standard deviation change in general health, so the collider effect corresponds to about a one hundredth of a standard deviation. The bootstrap confidence interval (LCL = 0.003, UCL = 0.007) led to conclusion that the collider effect is larger than expected by chance. The model is not a suppression model because ${\widehat{b}}_{TX}{\widehat{b}}_{YT\cdot X}=0.005$ and ${\widehat{b}}_{YX\cdot T}=0.017$ have the same sign, and the product of the three coefficients ${\widehat{b}}_{TX}{\widehat{b}}_{YT\cdot X}{\widehat{b}}_{YX\cdot T}$ is positive. There is no evidence for a statistically significant interaction of conscientiousness and general health on income, ${\widehat{h}}_{YXT\cdot X,Y}=-0.00041(0.0058)$, t (5400) =−0.07.

As for the other illustrative examples, there are additional important variables related to income, health, and conscientiousness not considered in this example. We emphasize that a collider should not be adjusted in the statistical analysis for an accurate regression of Y on X unless a researcher is interested in testing how a collider variable could influence their results. Including the collider in the analysis illustrates the estimation of the collider effect, which represents the change in the effect if a collider is mistakenly adjusted in the statistical analysis.

Discussion

Third-variable effects are not distinguishable solely by statistical methods. Each third-variable effect can be fit to the same data, and if the relations between the variables are substantial, there will be evidence for each effect. In this sense, the confounder, mediator, and collider models are equivalent, providing an equal representation of the information contained in the data for three variables (Stelzl, 1986). Although mediation, confounding, and collision may equally explain the statistical associations among three variables, they describe different causal relations among those variables. Like much recent research on causal analysis, this paper highlights the centrality of the causal model underlying a research study and the important distinction between the causal model and the statistical model. The appropriate causal model is determined by prior empirical research and theory. The statistical analysis provides estimates for the proposed causal model.

Mediator and confounder models have received considerable attention in modern causal inference (Morgan & Winship, 2015; Pearl, 2009). The causal estimators for mediation have been defined in a counterfactual framework (Imai et al., 2010; Valeri & VanderWeele, 2013) and represent causal effects in a world in which all participants were in the treatment group or all participants were in a control group. Adjustment for a confounder gives the correct causal estimate because the confounder removes spurious effects from the relation of two variables (Greenland & Morgenstern, 2001; Pearl, 2009). That is, failing to adjust for the confounder introduces spurious effects. Although described in modern causal inference, there are fewer examples and applications of colliders than confounders or mediators. For the collider, the correct causal effect does not adjust for the collider because adjusting for the collider introduces bias that forms a spurious effect that masks the true causal effect (Elwert & Winship, 2014).

This paper described third-variable models relevant for prevention science. It is important to consider each third-variable model in planning studies and as a possible explanation of observed statistical analysis results. We described the corresponding equations and methods of estimation for mediation, confounding, and collision and illustrated these three-variable models in three datasets. We emphasized the need for further causal analysis in order to draw causal conclusions from these examples, including the consideration of additional mediators, confounders, and colliders that may be relevant for each example. Adding replications extended to new situations further clarify the veracity of the causal model. We acknowledge that prevention science models may contain many more than three variables, but mediators, confounders, and colliders form the building blocks of these models.

In summary, the linear regression models described in this paper describe a general model that is useful for investigating each third-variable effect, given the correct causal model. We wish to note that the correct causal model is an exacting criterion, requiring a program of research with precise definition of causal effects, specification of assumptions, and sensitivity analysis for how violating assumptions affects results. The statistical procedures for estimating each third-variable effect provide no information about which causal effect is being estimated, highlighting that the statistical model is different from the causal model. The correct causal model is an exacting qualification,requiring a program of research with precise definition of causal effects, specification of assumptions, and sensitivity analysis for how violatingassumptions affects results. Statistical analysis is useful for demonstrating associations between variables that are consistent or inconsistent with a causal model. The primary challenge of research is to identify the status of different variables before the study in order to test a model. Colliders in addition to mediators and confounders are core conceptual variables for specifying and testing these models in prevention science.