confounding is a major concern in epidemiology . since the publication of the seminal paper by greenland and robins , many epidemiologists have explained the concept of confounding by examining risk measures under a simple potential - outcome ( or counterfactual ) model for a cohort of individuals . exchangeability of potential outcomes between the exposed and unexposed groups is one of the most fundamental assumptions in making causal inference , and confounding is a common source of lack of exchangeability . despite its significance , the different notions of confounding have not been fully appreciated in the literature , leading to confusion of causal concepts in epidemiology . this article aims to highlight the importance of differentiating between the subtly different notions of confounding from the perspective of counterfactual reasoning . we also show that directed acyclic graphs ( dags ) provide a simple algorithm to identify a sufficient set of confounders if the underlying causal structure is properly reflected . to achieve these goals the concept of response type is an essential foundation of causal inference because the causal effect of exposure on disease frequency in a population depends on the distribution of the response types of individuals in that population , not necessarily on the population distribution of the covariates . this point , however , has been relatively underappreciated because , despite its sophistication and usefulness , the response type of each individual is unobservable . to consider the effect of smoking cessation on lung cancer during a defined time period , we use an example of four subjects ( table  1 ) . in an epidemiologic study of these subjects , let us suppose that subjects # 1 ( male ) and # 3 ( female ) were actually exposed ( i.e. , quit smoking ) and subjects # 2 ( male ) and # 4 ( female ) were actually unexposed ( i.e. , did not quit smoking ) . during the follow - up , one of the exposed and both of the unexposed subjects suffered from lung cancer . consequently , the observed risk difference ( rd ) estimate for the effect of smoking cessation on lung cancer can be calculated as : 1/2 2/2  =  1/2 . likewise , the observed risk ratio ( rr ) estimate can be calculated as : ( 1/2)/(2/2 )  =  1/2 . when we consider a binary exposure and a binary outcome , individuals can be classified into the following four different response types.type 1 or  doomed  persons : exposure is irrelevant because outcome occurs with or without exposuretype 2 or causal  persons : outcome occurs if and only if they are exposedtype 3 or preventive  persons : outcome occurs if and only if they are unexposedtype 4 or immune  persons : exposure is irrelevant because outcome does not occur with or without exposure type 1 or  doomed  persons : exposure is irrelevant because outcome occurs with or without exposure type 2 or causal  persons : outcome occurs if and only if they are exposed type 3 or preventive  persons : outcome occurs if and only if they are unexposed type 4 or immune  persons : exposure is irrelevant because outcome does not occur with or without exposure response types of the four subjects are shown in table  1 . no subjects are classified as a  causal  response type , implying that the effect of smoking cessation is in the same direction for all four subjects . although monotonicity assumptions may be biologically plausible in some situations , they can never be empirically verified with data because they make reference to all individuals in the population . here the conditions presented in our paper can be used even when the monotonicity assumption is violated . in the following sections , we illustrate a typology of four notions of confounding by exploring and extending this simple example ( box 1 and fig .  1 ) . for simplicity , we use deterministic counterfactuals for each subject and assume that no random error attributable to sampling variability exists . confounding in distribution : we say that there is no confounding in distribution of the effect of exposure on outcome if the group that actually had a particular exposure is representative of what would have occurred had the entire target population been exposed to the same level of exposure.confounding in measure : we say that there is no confounding in measure of the effect of exposure on outcome if a particular measure of interest is equivalent to the corresponding causal measure in the target population.confounding in expectation : we say that there is no confounding in expectation of the effect of exposure on outcome if the exposure assignment mechanism results in balance.realized confounding : we say that there is no realized confounding of the effect of exposure on outcome if a particular exposure assignment results in balance , irrespective of the exposure assignment mechanism . confounding in distribution : we say that there is no confounding in distribution of the effect of exposure on outcome if the group that actually had a particular exposure is representative of what would have occurred had the entire target population been exposed to the same level of exposure . confounding in measure : we say that there is no confounding in measure of the effect of exposure on outcome if a particular measure of interest is equivalent to the corresponding causal measure in the target population . confounding in expectation : we say that there is no confounding in expectation of the effect of exposure on outcome if the exposure assignment mechanism results in balance . realized confounding : we say that there is no realized confounding of the effect of exposure on outcome if a particular exposure assignment results in balance , irrespective of the exposure assignment mechanism . the causal effect of exposure on disease frequency in a population depends on the distribution of response types of individuals in that population . table  2 shows the distribution of response types in the exposed and unexposed groups of the abovementioned example . we also show the distribution in the total population ; let pi , qi , and ri , i  =  14 , be proportions of response type i in the exposed group , the unexposed group , and the total population , respectively . note that ri can be calculated as pi/2  +  qi/2 because the numbers of the exposed and unexposed groups are balanced ( table  1 ) . among the exposed group , only type 1 and type 2 persons will develop the outcome , and the risk , or incidence proportion , of lung cancer in the exposed group is p1  +  p2 . among the unexposed group , only type 1 and type 3 persons will develop the outcome , and the corresponding risk is q1  +  q3 . therefore , the associational rd can be obtained using the proportions of response types as : ( p1  + p2 )    ( q1 + q3 )  =  1/2   2/2  =  1/2 , which is equivalent to the abovementioned observed rd estimate under the assumption of no sampling variability . effect of smoking cessation on lung cancer ( 1  =  diseased , 0  =  non - diseased ) . the associational risk difference is calculated as : ( p1 + p2 )    ( q1 + q3 )  =  1/2   2/2  =  1/2 . note that the distribution in this table applies to scenario # 2 in table  3 . causal risk difference in the exposed group is defined as : ( p1 + p2 )    ( p1 + p3 )  =  p2    p3  =  0   0  =  0 . causal risk difference in the unexposed group is defined as : ( q1 + q2 )    ( q1 + q3 )  =  q2  q3  =  0 as shown in table  1 , numbers of the exposed and unexposed groups are balanced , so a proportion of response type i in the total population , ri , can be calculated as : pi/2  +  qi/2 . the causal risk difference in the total population is defined as : ( r1 + r2 )    ( r1 + r3 )  =  r2    r3  =  0   2/4  =  2/4 . it has been well established that the target population concept plays a key role in discussions of causal inference in epidemiology . based on the distribution of these response types , we can define the true value of the causal parameters of interest in the corresponding target populations . for example , if exposure had been absent in the exposed group , only type 1 and type 3 persons would have developed the outcome , and the counterfactual risk of lung cancer would  have been p1 + p3 . therefore , the causal rd in the exposed  group ( i.e. ,  subjects # 1 and # 3 ) can be calculated as : ( p1  +  p2 )    ( p1 + p3 )  =  p2    p3  =  0   0  =  0 . likewise , the causal rds in the unexposed group ( i.e. , subjects # 2 and # 4 ) and the total population  can be calculated as : ( q1 + q2 )    ( q1 + q3 )  =  q2    q3  =  0   2/2  =  2/2 and ( r1  + r2 )    ( r1 + r3 )  =  r2    r3  =  0   2/4  =  2/4 , respectively . these descriptions in terms of response types are valuable in highlighting the significance that the notion of confounding can be defined with respect to both the distribution of potential outcomes ( i.e. , confounding in distribution ) and a specific effect measure ( i.e. , confounding in measure ) . if we use rd as a measure of interest , when the exposed group is the target population , we say that there is confounding in measure if the causal rd in the exposed group ( i.e. , ( p1 + p2 )    ( p1 + p3 ) ) is not equivalent to the associational rd ( i.e. , ( p1 + p2 )    ( q1 + q3 ) ) . therefore , a sufficient and necessary condition for no confounding in measure is given by:(1)(p1+p2)(p1+p3)=(p1+p2)(q1+q3)(p1+p3)=(q1+q3),which is violated in table  2 . conversely , when the unexposed group is the target population , a sufficient and necessary condition for no confounding in measure is given by:(2)(q1+q2)(q1+q3)=(p1+p2)(q1+q3)(p1+p2)=(q1+q2),which is also violated in table  2 . note that these conditions can be also derived when the notion of confounding is defined with respect to the distribution of potential outcomes , rather than with respect to a specific effect measure . for example , when the exposed group is the target population , we say that there is confounding in distribution if the actual unexposed group is not representative of what would have occurred in the actual exposed group had they been unexposed . therefore , a sufficient and necessary condition for no confounding in distribution is identical to equation ( 1 ) . when the exposed and unexposed groups are used as target populations , distinguishing between the notions of confounding in distribution and confounding in measure becomes a subtle issue , and there is confounding in distribution and confounding in measure in table  2 . when the target is the total population , however , it is crucial to distinguish between these because conditions for no confounding vary according to the two notions of confounding . if we use the notion of confounding in distribution , a sufficient and necessary condition for no confounding is given by:(3){(r1+r2)=(p1+p2)}{(r1+r3)=(q1+q3)}{(p1+p2)=(q1+q2)}{(p1+p3)=(q1+q3)},which is obviously violated in table  2 . if equation ( 3 ) holds , the groups that are actually exposed and unexposed are representative of what would have occurred had the total population been exposed and unexposed , respectively . meanwhile , if we use rd as a measure of interest , a sufficient and necessary condition for no confounding in measure is given by:(4)(r1+r2)(r1+r3)=(p1+p2)(q1+q3)(p2+q2)/2(p3+q3)/2=(p1+p2)(q1+q3)(p1+p3)+(p1+p2)=(q1+q3)+(q1+q2),which is weaker than equation ( 3 ) and is met in table  2 . for example , if we use rr as a measure of interest , a sufficient and necessary condition for no confounding in measure is given by:(5)(r1+r2)/(r1+r3)=(p1+p2)/(q1+q3)[{(p1+q1)+(p2+q2)}/2]/[{(p1+q1)+(p3+q3)}/2]=(p1+p2)/(q1+q3)(p1+p2)(p1+p3)=(q1+q3)(q1+q2),which is , though weaker than equation ( 3 ) , violated in table  2 . consequently , even if the distribution of response types in the exposed and unexposed groups are not comparable ( i.e. , ( p1,p2,p3,p4 )    ( q1,q2,q3,q4 ) ) , as in table  2 , we may obtain an unconfounded estimate for the target population . complete comparability of response types between the exposed and unexposed groups ( i.e. , ( p1,p2,p3,p4 )  =  ( q1,q2,q3,q4 ) ) is a sufficient , but not a necessary , condition for no confounding in the three target populations , irrespective of whether one uses the notion of confounding in distribution or confounding in measure . in conclusion , confounding depends not only on the population chosen as the target of inference , but also on the notion of confounding itself , when the target is the total population . this point has been relatively underappreciated , partly because some books ( including modern epidemiology and encyclopedia of epidemiology ) discuss the concept of confounding using only the exposed and unexposed groups as the target populations . as examples of the confusion surrounding the concept of confounding , a recent review article by gatto et  al . explains the conditions for no confounding that are implicitly derived from the notion of confounding in distribution , whereas maldonado implicitly used the notion of confounding in measure when discussing the concept of confounding . to clarify the significance of differentiating the two notions of confounding , it would be helpful to use not only the exposed and unexposed groups , but also the total population as the target populations . as discussed in an update of the classic methodology paper by greenland and robins , a further distinction can be drawn between confounding  in expectation  and  realized  confounding . epidemiologic literature has typically addressed the issue of  no confounding  in the notion of realized confounding ; however , the notion of confounding in expectation is also valuable . the concept of bias is defined by comparing the expected value of an estimator and the true value of the parameter , and confounding is a common source of bias . in an ideal randomized controlled trial , the randomized groups will be comparable in their potential outcomes on average , over repeated experiments . for any given experiment , however , the particular randomization may result in imbalances by chance , because of the particular allocation or exposure assignment . such a scenario would result in no confounding in expectation but there would be realized confounding for that particular trial . confounding in expectation and realized confounding have been also referred to as confounded mechanisms and confounded assignments , respectively . to grasp the explicit distinction between these notions of confounding , we need to understand the mechanism that generates exposure events , not the product of that mechanism . in the example of four smokers , the lack of comparability of response types between the exposed and unexposed groups in the example could be because the configuration was observed randomly , with the estimator being unconfounded in expectation , or there could be a reason for the lack of comparability that is not revealed in the above explanation . to obtain the expected value of an estimator in the target population , we need to consider the distribution of exposure status in that population . to illustrate this , the following discussion focuses on the situation in which the target population is the total population . note that table  1 shows only one pattern of exposure status of the four subjects . given a total of four subjects , we can consider a maximum of six patterns of exposure status , when the numbers of the exposed and unexposed groups are balanced ( table  3 ) ( the exposure status shown in table  1 corresponds to scenario # 2 in table  3 . ) . in each scenario scenarios # 1 and # 3 are identical from the perspective of counterfactual reasoning , because the distributions of response types are the same in these scenarios . similarly , scenarios # 4 and # 6 are identical from the perspective of counterfactual reasoning . consequently , these six scenarios are grouped into a total of four patterns in terms of the distributions of response types . exposure status shown in table  1 corresponds to scenario # 2 . from the perspective of counterfactual reasoning , scenarios # 1 and # 3 in table  3 are essentially identical ; the distributions of response types in the exposed and unexposed groups are the same . therefore , the observed rd estimates are identical in these scenarios ( i.e. , 0 ) . similarly , scenarios # 4 and # 6 are essentially identical , because the distributions of response types in these scenarios are the same . as a result , the six scenarios can be grouped into a total of four patterns in terms of the distributions of response types . to highlight the distinction between confounding in expectation and realized confounding , we discuss two situations below , according to different mechanisms that generate the exposure events . in situation 1 , the treatment assignment is randomly determined , whereas , in situation 2 , the treatment assignment is causally influenced by the subject 's sex . see eappendix 1 and etable  1 , etable  2 , etable  3 , and etable  4 for a technical discussion of these situations . when the treatment assignment of each subject is randomly determined , we assume the probability of the two males quitting smoking is equal to the probability of the two females quitting smoking ( i.e. , p[quitting | male ]  =  p[quitting | female ]  =  1/2 ) . in this situation , the six scenarios are induced randomly , with a probability of 1/6 ( table  4 ) . probability of the four subjects quitting smoking is 1/2 , so the six scenarios are induced randomly . probabilities of the two males and the two females quitting smoking are 2/3 and 1/3 , respectively , so scenario # 1 is expected to occur 16 times ( i.e. , 2 ) as often as scenario # 6 . likewise , each of the scenarios # 25 is expected to occur four times ( i.e. , 2 ) as often as scenario # 6 . the expected values of the estimators for risk in the exposed and unexposed groups are calculated as : 3    ( 1/6   1/2 )  +  3 ( 1/6  0/2 )  =  1/4 and 3    ( 1/6   1/2 )  +  3    ( 1/6    2/2 )  =  3/4 , respectively ( table  4 ) . consequently , the expected value of the estimator for rd in the four subjects is calculated as : 1/4   3/4  =  1/2 , which is equivalent to the causal rd in the four subjects . in the notion of confounding in expectation , we say that there is no confounding in measure of the effect of smoking cessation on lung cancer in the four subjects ( i.e. , the second quadrant in fig .  1 ) . in this case , we describe the estimator as being an unbiased estimator and the realized value is referred to as an unbiased estimate . using the notion of confounding in expectation , each of the observed rd estimates in the six scenarios is referred to as an unbiased estimate for the four subjects , even if it is not equivalent to the causal rd in the four subjects . conversely , if the notion of realized confounding is used , we simply compare the observed rd estimate in each scenario with the causal rd in the four subjects . although there is no realized confounding in measure in scenarios # 2 and # 5 , there is realized confounding in measure in scenarios # 1 , # 3 , # 4 , and # 6 ( i.e. , the third quadrant in fig .  1 ) . this clearly demonstrates that realized confounding can be present even if the exposure assignment mechanism is completely random , especially if the study size is small . note that the expected values of the  estimators for risk in the exposed and unexposed groups ( i.e. , 1/4 and 3/4 , respectively ) are equivalent to the risk if all four subjects had been exposed and unexposed , respectively ( table  4 ) . in the notion of confounding in expectation , there is no confounding in distribution of the effect of smoking cessation on lung cancer in the four subjects ( i.e. , the first quadrant in fig .  1 ) . conversely , if the notion of realized confounding is used in table  1 , the observed risk in the exposed group ( i.e. , 1/2 ) is not equivalent to the risk if all four subjects had been exposed ( i.e. , 1/4 ) . the observed risk of the unexposed group ( i.e. , 2/2 ) is not equivalent to the risk if all four subjects had been unexposed ( i.e. , 3/4 ) . in the notion of realized confounding , there is confounding in distribution ( i.e. , the fourth quadrant in fig , we consider a situation in which the treatment assignment of each subject is influenced by his or her sex . for example , the probability of the two males quitting smoking is twice as high as the probability of the two females quitting smoking ( i.e. , p[quitting  | male ]  =  2/3 and p[quitting | female ]  =  1/3 ) . consequently , scenario # 1 is expected to occur 16 times ( i.e. , 2 ) as often as scenario # 6 . likewise , each of the scenarios # 25 is expected to occur four times ( i.e. , 2 ) as often as scenario # 6 . in other words , scenario # 1 is expected to occur with a probability of 16/33 ; each of the scenarios # 25 is expected to occur with a probability of 4/33 ; and scenario # 6 is expected to occur with a probability of 1/33 ( table  4 ) . the expected values of the estimators for risk in the exposed and unexposed groups are calculated as : ( 16/33   1/2 )  +  2    ( 4/33  1/2 )  +  2   ( 4/33   0/2 )  +  ( 1/33   0/2 )  =  4/11 and ( 16/33   1/2 )  +  2   ( 4/33   2/2 )  +  2    ( 4/33 1/2 )  +  ( 1/33   2/2 )  =  7/11 , respectively ( table  4 ) . consequently , the expected value of the estimator for rd in the four subjects are calculated as : 4/11   7/11  =  3/11 , which is not equivalent to the causal rd in the four subjects ( i.e. ,  1/2 ) . in this case , the estimator is a biased estimator , and the differential between the two values ( i.e. ,  3/11   ( 1/2 )  =  5/22 ) is called bias . in situation 2 , the observed rd estimate in table  1 ( i.e. ,  1/2 ) is confounded in the notion of confounding in expectation ( i.e. , the second quadrant in fig .  1 ) , although it is equivalent to the causal rd in the four subjects . in contrast , in the notion of realized confounding , there is no confounding in measure of the effect of smoking cessation on lung cancer in the four subjects ( i.e. , the third quadrant in fig .  1 ) . when examining the presence of confounding in distribution , the expected values of the estimators for risk in the exposed and unexposed groups ( i.e. , 4/11 and 7/11 , respectively ) are not equivalent to the risk had all four subjects been exposed and unexposed , respectively ( table  4 ) . in the notion of confounding in expectation , there is confounding in distribution of the effect of smoking cessation on lung cancer in the four subjects ( i.e. , the first quadrant in fig .  1 ) . using the notion of realized confounding in table  1 , there is also confounding in distribution of the effect of smoking cessation on lung cancer in the four subjects ( i.e. , the fourth quadrant in fig .  1 ) . a simple example to clarify the distinction between confounding in expectation and realized confounding is a useful educational tool to explain the concept of confounding more clearly . however , it is important to note that the exposed and unexposed groups are , by definition , determined by the specific pattern of exposure status . if these subpopulations are used as target populations , the discussion is logically restricted to the notion of realized confounding for the two target populations in that particular pattern . therefore , to teach a generalized concept of bias , it is important to use the total population as the target population , as this facilitates the differentiation between notions of confounding in expectation and realized confounding . as pointed out in a recent review by schwartz et  al . , there are discrepancies in the literature regarding whether random error should be included as a bias . part of the confusion stems from lack of appreciation of the distinction between the notions of confounding in expectation and realized confounding . similarly , the notion of confounding in expectation is essential to understand the relationship between accuracy , validity , and precision . see eappendix 2 for further discussion , including a numerical description of the relationships in situations 1 and 2 . in epidemiologic research , dags have been used extensively to determine the variables for which it is sufficient to control for confounding to estimate causal effects . in this section , we graphically illustrate this point , highlighting the usefulness of dags in examining the presence of confounding in distribution in the notion of confounding in expectation ( i.e. , the first quadrant in fig .  1 ) . the presence of a dashed arrow from  male  to  smoking cessation  is determined by comparing the probability of the two males quitting smoking ( i.e. , p[quitting | male ] ) and the probability of the two females quitting smoking ( i.e. , p[quitting | female ] ) . this is explained simply , using the two previously discussed situations . when the treatment assignment of the four subjects is randomly determined ( i.e. , p[quitting | male ]  =  p[quitting | female ]  =  1/2 ) ,  male  and  smoking cessation  are independent ; there is no association between these two events . the dashed arrow from  male  to  smoking cessation  in fig . 2 is absent , and there is no backdoor path ( i.e. , a non - causal path that has an arrow pointing into the exposure ) from  smoking cessation  to  lung cancer  . consistent with our explanation in the previous section , in the notion of confounding in expectation , when the target population is the total population , bias does not occur in situation 1 ; there is no confounding in distribution . when the treatment assignment of the four subjects are influenced by their sex ( i.e. , p[quitting | male ]  p[quitting | female ] ) ,  male  and  smoking cessation  are not independent ; there is a causal association between these two events . the dashed arrow from  male  to  smoking cessation  in fig . 2 is present , resulting in a backdoor path from  smoking cessation  to  lung cancer  . consistent with our explanation in the previous section , in the notion of confounding in expectation , when the target population is the total population , bias occurs in situation 2 ; there is confounding in distribution . dags are primarily useful to examine the presence of confounding in distribution in the notion of confounding in expectation ( i.e. , the first quadrant in fig .  1 ) . they are also practical tools to identify realized confounding , if the size of population is large enough . furthermore , it has been shown that dags with signed edges , or signed dags , are useful in drawing conclusions about the direction of bias because of unmeasured confounders ( fig .  2 ) . finally , although the causal effect of exposure on disease frequency in a population depends on the distribution of response types of individuals in that population , and dags have been used to provide visual summaries of hypothetical relationships among variables , they do not describe the relationships between underlying response types . to overcome this , suzuki et  al . discussed how dags can be extended by integrating response types and observed variables , showing the conceptual link between unobservable response types and observed ( or observable ) data frequencies in the population . as an example of their usefulness , the principal stratification approach can be illustrated using extended dags . in this paper , we provide a typology of the four notions of confounding in epidemiology from the perspective of counterfactual reasoning . in etable  5 , we summarize sufficient and necessary conditions for no confounding in terms of response types , when the target population is the total population . confounding in expectation is dependent on each mechanism that generates exposure events , whereas realized confounding is dependent on each product of that mechanism ( or scenario ) . in both the notions of confounding in expectation and realized confounding , no confounding in distribution is a sufficient condition for no confounding in measure , irrespective of whether rd or rr are used as a measure of interest . both confounding and selection bias result in a lack of exchangeability , and comparable conditions apply to selection bias . the different notions of confounding have not been fully appreciated in the literature , which has led to confusion of causal concepts in epidemiology . this lack of clear understanding could lead to inappropriate use of and underappreciation of dags , which provide a simple algorithm for examining the presence of confounding in distribution in the notion of confounding in expectation . for simplicity , throughout this paper we have ignored that causal effects estimated in one population are often intended for use in making decisions in another population . transportability  of the causal effect , and this is a question of external validity . when transporting the causal effect estimates to external targets , we need to consider the distributions of effect modifiers , interference patterns , and versions of treatment across populations . although we have focused on the three usual internal target populations ( i.e. , the exposed group , the unexposed group , and the total population ) , our discussion about the notions of confounding analogously applies when using external targets . the potential outcomes and causal graph frameworks are also relevant , as they are for internal targets . simple examples can be powerful tools for understanding complex causal concepts ; the power of simplicity , however , should be carefully exercised to achieve this goal . recent developments in epidemiologic methods have demonstrated that substantial insight can be obtained from simplified representations of complex biological reality . none of us would deny the value of an old dictum ( often referred to as occam 's razor ) that one should not introduce needless complexity for purposes of prediction or explanation . undeniably , however , all of us should also keep in mind einstein 's razor , the warning against being too simplistic :  everything should be made as simple as possible , but not simpler  . we hope that our simple example can serve as an effective tool to illustrate a typology of notions of confounding .
confounding is a major concern in epidemiology . despite its significance 
 , the different notions of confounding have not been fully appreciated in the literature , leading to confusion of causal concepts in epidemiology . in this article 
 , we aim to highlight the importance of differentiating between the subtly different notions of confounding from the perspective of counterfactual reasoning . by using a simple example 
 , we illustrate the significance of considering the distribution of response types to distinguish causation from association , highlighting that confounding depends not only on the population chosen as the target of inference , but also on the notions of confounding in distribution and confounding in measure . 
 this point has been relatively underappreciated , partly because some literature on the concept of confounding has only used the exposed and unexposed groups as the target populations , while it would be helpful to use the total population as the target population . 
 moreover , to clarify a further distinction between confounding  in expectation  and  realized  confounding , we illustrate the usefulness of examining the distribution of exposure status in the target population . to grasp the explicit distinction between confounding in expectation and realized confounding , we need to understand the mechanism that generates exposure events , not the product of that mechanism . 
 finally , we graphically illustrate this point , highlighting the usefulness of directed acyclic graphs in examining the presence of confounding in distribution , in the notion of confounding in expectation .