CONFOUNDING (CHAPTER 7) BIOS Confounding

Transcription

1 CONFONDING (CHPTER 7) BIOS Confounding

2 Confounding ( 7) Outline 7.1 The structure of confounding 7.2 Confounding and identifiability of causal effects 7.3 Confounders 7.4 Confounding and exchangeability 7.5 How to adjust for confounding BIOS Confounding

3 conditional on its parents, is independent of its non-descendants. This latter to the statement that the density ( ) of the variables in DG satisfies t 7.1 The structure of confounding Confounding is the bias that arises when the treatment and the outcome share a common cause Consider Figure 7.1 (same as Fig. 6.1) Figure 6.1 Figure 6.2 ( ) = which indicates two sources of association between and 1. The causal effect 2. The backdoor path ( ). =1 effect not mediated through any other va individual, or that we are unwilling to a do not exist. lternatively, the lack of a willingtoassume,that has no direct ca the population. For example, in Figure 6 either we know that disease severity affec transplant or that we are not willing to diagram does not distinguish whether an a protective effect. Furthermore, if, as i two causes, the diagram does not encode Causal diagrams like the one in Figu graphs, which is commonly abbreviated edges imply a direction: because the arro, but not the other way around. cy In general, a backdoor path is a noncausal path between treatment and outcome that remains even if all arrows pointing from treatment to other variables (in graph-theoretic terms, the descendants of treatment) are removed. That is, the path has an arrow pointing into treatment. variable cannot cause itself, either direct Directed acyclic graphs have applicat we focus on causal directed acyclic graph is causal if the common causes of any p in the graph. For example, suppose in assigned to heart transplant with a pro of their disease. Then is a common included in the graph, as shown in the suppose in our study individuals are ra with the same probability regardless of a common cause of and and need n Figure 6.1 represents a conditionally ran BIOS Confounding

4 7.1 The structure of confounding If the backdoor path was absent in Figure 7.1, the only path between treatment and outcome would be, and thus the entire association between and would be due to the causal effect of on, Eg, the associational risk ratio would equal the causal risk ratio However, the common cause creates an additional source of association between and, which we refer to as confounding the effect of on Because of confounding, the associational risk ratio does not necessarily equal the causal risk ratio BIOS Confounding

5 ( ) = =1 ( ). Figure The structure of confounding Figure 6.1 Figure 6.2 effect not mediated through any other va individual, or that we are unwilling to a do not exist. lternatively, the lack of a willingtoassume,that has no direct ca the population. For example, in Figure 6 either we know that disease severity affec transplant or that we are not willing to diagram does not distinguish whether an a protective effect. Furthermore, if, as i two causes, the diagram does not encode Causal diagrams like the one in Figu graphs, which is commonly abbreviated edges imply a direction: because the arro, but not the other way around. cy variable cannot cause itself, either direct Directed acyclic graphs have applicat we focus on causal directed acyclic graph is causal if the common causes of any p in the graph. For example, suppose in assigned to heart transplant with a pro of their disease. Then is a common included in the graph, as shown in the suppose in our study individuals are ra with the same probability regardless of a common cause of and and need n Figure 6.1 represents a conditionally ran 6.2 represents a marginally randomized e Figure 6.1 may also represent an obs might depict many examples of confounding in observational research Healty worker bias: Effect of working as a firefighter on risk of death confounded by being physically fit Confounding by indication or channeling: Effect of drug (eg, aspirin) on the risk of disease (eg, stroke) confounded if drug more likely prescribed to individuals with certain condition (eg, heart disease) BIOS Confounding

6 7.1 The structure of confounding In the above example, has a direct effect on. It may be instead that there is an unmeasured variable (eg, atherosceloris) that affects both and, as in Fig. 7.2 Figure 7.2 In this case there is still confounding because and share a common cause Figure 7.3 BIOS Confounding bias, depicted in the causal diagram a healthy worker bias. Clinical decisions: The effect of d disease (say, stroke) will be con be prescribed to individuals with ce that is both an indication for treatm Heart disease is a risk factor for s effect on as in Figure 7.1 or, as are caused by atherosclerosis, a known as confounding by indicatio being reserved to describe the bias c that encourage doctors to use ce ifestyle: The effect of behavior death) will be confounded if the be havior (say, cigarette smoking) th to co-occur with. The structure picted in the causal diagram in F variable represents the sort of pe to both lack of exercise and smoki clinical disease resultsbothinlac of clinical disease. This form o reverse causation. Genetic factors: The effect of a DN certain trait will be confounded i has a causal effect on and is mor This bias, also represented by the c as linkage disequilibrium or populat being reserved to describe the bias mixture of individuals from differe can stand for ethnicity or other sequences.

7 7.1 The structure of confounding Genetic factors: Suppose DN sequence affects risk of developing trait Suppose there is another DN sequence that also affects nd and associated due to common cause such as ethnicity Figure 7.3 Figure 7.2 Figure 7.3 gain there is confounding because is a common cause of and BIOS Confounding Clinical decisions: The effect of d disease (say, stroke) will be con be prescribed to individuals with ce that is both an indication for treatm Heart disease is a risk factor for s effect on as in Figure 7.1 or, as are caused by atherosclerosis, a known as confounding by indicatio being reserved to describe the bias that encourage doctors to use ce ifestyle: The effect of behavior death) will be confounded if the be havior (say, cigarette smoking) th to co-occur with. The structure picted in the causal diagram in F variable represents the sort of pe to both lack of exercise and smoki clinical disease resultsbothinla of clinical disease. This form o reverse causation. Genetic factors: The effect of a DN certain trait will be confounded has a causal effect on and is mor This bias, also represented by the c as linkage disequilibrium or popula being reserved to describe the bias mixture of individuals from differe can stand for ethnicity or other sequences. Social factors: The effect of income at age 75 will be confounded if th both future income and disability

8 7.2 Confounding and identifiability of causal effects Two settings where causal effects are identifiable (assuming no selection bias, measurement error, or random variability) 1. No common causes, e.g., Fig Common causes but enough measured variables (which are nondescendants of treatment) to block all backdoor paths Fig. 7.1 is an example of the 2nd Figure setting, 7.2 where we can identify the effect of on by conditioning on (or standardizing by) bias, depicted in the causal diagram a healthy worker bias. Clinical decisions: The effect of d disease (say, stroke) will be con be prescribed to individuals with ce that is both an indication for treatm Heart disease is a risk factor for s effect on as in Figure 7.1 or, as are caused by atherosclerosis, a known as confounding by indicatio being reserved to describe the bias that encourage doctors to use ce ifestyle: The effect of behavior death) will be confounded if the be havior (say, cigarette smoking) th to co-occur with. The structure picted in the causal diagram in F variable represents the sort of pe to both lack of exercise and smoki clinical disease resultsbothinla of clinical disease. This form o reverse causation. has a causal effect on and is mor This bias, also represented by the c as linkage disequilibrium or popula being reserved to describe the bias mixture of individuals from differe Figure 7.3 Genetic factors: The effect of a DN certain trait will be confounded can stand for ethnicity or other BIOS sequences. 7 Confounding

9 7.2 Confounding and identifiability of causal effects Backdoor criterion: all backdoor paths are blocked if treatment and outcome are d-separated given the measured coveraties in a graph in which the arrows out of are removed The backdoor criteiron answers three questions: 1. Does confounding exist? es if there are unblocked backdoor paths. No o/w. 2. Can confounding be eliminated? es if all backdoor paths can be blocked by some subset of measured covariates. No o/w. 3. What variables are sufficient to eliminate confounding? ny set that blocks all backdoor paths. BIOS Confounding

10 7.3 Confounders The tradition definition of confounder (statistical rather than structural criterion) 1. ssociated with treatment 2. ssociated with outcome conditional on treatment 3. Not on the causal path Confounding between treatment and outcome Consider Fig Figure 7.4 The bias induced in Figure 7.4 was described by Greenland et al (1999), and referred to as M- bias (Greenland 2003) because the structure of the variables involved in it 2 1 resembles a letter between the structural and traditional d In Figure 7.3 there is also confoundi outcome share the common cause blocked by conditioning on. Therefore given, and we say that is a confo definition, is also a confounder and associated with the treatment (it share associated with the outcome conditiona effect on ), and it does not lie on the ca outcome. gain, there is no discrepancy definitions of confounder for the causal d The key figure is Figure 7.4. In this causes of treatment and outcome, a The backdoor path between and th blocked because is a collider on that p and is due to the effect of on : need to adjust for. (djustment for eit unmeasured variables.) In fact, adjustme Traditional definition suggests M is a confounder (and therefore we should adjust for M) BIOS Confounding

11 Fig. 7.4 (M-bias) 7.3 Confounders Confounding 1 2 Figure 7.4 The bias induced in Figure 7.4 was described by Greenland et al (1999), and referred to as M- bias (Greenland 2003) because the structure of the variables involved in it 2 1 resembles a letter Mlyingonitsside. between the structural and traditional d In Figure 7.3 there is also confoundi outcome share the common cause blocked by conditioning on. Therefore given, and we say that is a confo definition, is also a confounder and associated with the treatment (it share associated with the outcome conditiona effect on ), and it does not lie on the ca outcome. gain, there is no discrepancy definitions of confounder for the causal d The key figure is Figure 7.4. In this causes of treatment and outcome, a The backdoor path between and th blocked because is a collider on that p and is due to the effect of on : need to adjust for. (djustment for eit unmeasured variables.) In fact, adjustme bias because conditioning on would o path between and. This implies tha bias, there is conditional bias for at leas bias as selection bias because it arises fro in which the association between and Though there is no confounding, m founder: it is associated with the treatm with ), it is associated with the outco shares the common cause 1 with ), a In fact, there is no confounding in this example, so there is no need to adjust for M Why? Is the path 2 1 open or blocked? et the tradition definition suggests we should adjust for M Moreover, if we do adjust, then the path above becomes open, inducing non-causal association between and BIOS Confounding

12 7.3 Confounders: M-bias example (R) set.seed(123) n < # sample size u1 <- rbinom(n,1,.25) # unmeasured covariates u1 and u2 u2 <- rbinom(n,1,.75) l <- pmax(u1,u2) # measured covariate l a <- rbinom(n,1,.25+.5*u2) # treatment y0 <- rnorm(n,u1,1); y1 <- y0 + 1 # potential outcomes yobs <- y0*(1-a)+y1*a # observed outcomes # naive estimator acehat <- mean(yobs[a==1]) - mean(yobs[a==0]) # standardized estimator ace.str0 <- mean(yobs[a==1 & l==0]) - mean(yobs[a==0 & l==0]) ace.str1 <- mean(yobs[a==1 & l==1]) - mean(yobs[a==0 & l==1]) ace.str <- ace.str0*mean(l==0)+ace.str1*mean(l==1) print(paste("acehat:",acehat,"standardized",ace.str)) [1] "acehat: standardized " BIOS Confounding

13 Mlyingonitsside. 7.3 Confounders Fig. 7.5 is another example where there is no confounding Figure 7.5 et the tradition definition suggests we should adjust for Why? Figure 7.5 is another example in which, in the absence of confounding, the traditional criteria lead to bias because conditioning on would o path between and. This implies tha bias, there is conditional bias for at leas bias as selection bias because it arises fro in which the association between and Though there is no confounding, m founder: it is associated with the treatm with ), it is associated with the outco shares the common cause 1 with ), a way between treatment and outcome. definition, is considered a confounder the absence of confounding! The result of trying to adjust for th selection bias. For example, suppose r cancer, 1 a pre-cancer lesion, a diagn and 2 a health-conscious personality (m the doctor). Then, under the causal diagr activity on cancer is unconfounded nd conditioning onselection willbias open dueatonon-causal adjustment forpath But from let us say tothat one decides to adjust. The traditional criteria would not have resulted in bias had condition 3) been replaced by the condition that the variable is not caused by treatment, i.e., it is a nondescendant of. analysis to women with a negative test ( opens the backdoor path between and was previously blocked by the collider and would be a mixture of the ass and the association due to the open back causation any more. We have described an example in w founder fails because it misleads invest when adjustment for such variable is no This problem arises because the standar founder, rather than that of confounding the structural definition first establishes t causes and then identifies the confoun confounding in the analysis. Confound causes of treatment and outcome either BIOS Confounding

14 7.4 Confounding and exchangeability How do DGs related to potential outcomes/counterfactuals? How does the structural definition of confounding relate to conditional exchangeability? Exchangeability a equivalent to no confoudning Conditional exchangability a equivalent to being a sufficient set for confounding adjustment (i.e., no unblocked backdoor paths) Single World Intervention Graphs (SWIGs) make this connection between potential outcomes and graphs explicit BIOS Confounding

15 P E[ =1 ] E[ =0 ]= ual confounding bias, depicted whose in the elimination causal diagram would require in Figure adj7 E[ = =1]Pr[ = ] variables. a healthy For brevity, workerwe bias. say that there is no unmeas P E[ = =0]Pr[ = ]. If conditioning on a set of variables (that are Clinical decisions: The effect of drug (say blocks all backdoor paths, then the treated and unt disease (say, stroke) will be confounded if th formal 7.4proof SWIGs of this result was within levels of, i.e., is a sufficient set for confo be prescribed to individuals with certain condit given by Pearl (2000). the previous section). To a non-mathematician suc that is both an indication for treatment and a r magical as there appears to be no obvious relationship Heart disease is a risk factor for stroke beca Recall Fig. 7.2 independences and the absence of back door paths bec effect on as in Figure 7.1 or, as in Figure 7.2 not included as variables on a causal graph. new are caused by atherosclerosis, an unmeasure SWIGs overcome the shortcomings World Intervention Graphs (SWIGs) seamlessly unif known as confounding by indication or channe of previously proposed twin causal graphical approaches by explicitly including the cou being reserved to describe the bias created by pa diagrams (Balke and Pearl 1994). the graph. The SWIG depicts the variables and causa that encourage doctors to use certain drug observed in a hypothetical world in which all subjects. Thatis,aSWIGisagraph ifestyle: The effect of behavior that represents (say, aexercis coun by a single death) intervention. will be confounded In contrast, if the a standard behavior is causa ass Figure 7.2 variables havior and causal (say, cigarette relations smoking) that are observed that has in a cau th The corresponding SWIGs is Fig. 7.7 can betoviewed co-occur as awith function. The that structure transforms of the a given varia given intervention. picted thethe causal following diagram examples in Figure describe 7.3, in t a a Suppose variable the causal represents diagram the sort in Figure of personality 7.2 repres an data. to The both SWIG lackinoffigure exercise 7.7and is asmoking. transformation nother o sents the clinical datadisease from a hypothetical resultsbothinlackofexercise intervention w thesametreatmentlevel. of clinical disease. This The form treatment of confoundin node is semicircles. reversethe causation. right semicircle encodes the treatmen semicircle encodes the value of that would have been Genetic factors: The effect of a DN sequence 7.3 of intervention. We use semicircles simply to remind t Figure 7.7 certain trait will be confounded if there exist variables were derived by splitting the treatment node The treatment node is split into two nodes. has a causal effect on and is more common a The is notleft a cause does node isnot the have an arrow into be This bias, also represented by the causal diagram same for all subjects. The outcome is as linkage disequilibrium or population,thevalue value of that would have stratifica 1 been observed instudy. thethe absence remainingof variables an are temporally prior being reserved to describe the bias arising from ables and take the same value as in the observati intervention, i.e., the factual in an observationalmixture of individuals from different ethnic gr exchangeability study can stand q holds because, on the SWI for ethnicity or other factors that a a and are blocked after conditioning on. sequences. Replace with potential outcome a Consider now the causal diagram in Figure 7.4 a 7.8. Marginal Social factors: exchangeability The effect of qincome holds at age because, 65 between at age 75 and will are be blocked confounded (without if the level conditioni of dis Progress! Now use the d-separation 2 rule to conclude conditional both future exchangeability income and disability q does level. notthis hold the path the causal diagram 1 in Figure is open Figure 7.8 conditioned on. This is why the marginal - assoc Environmental exposures: The effect of airborne conditional - association given is not, and thus BIOS the risk of coronary heart disease will be confo for results in bias. 7 Confounding These examples show how SWIGs whose levels co-vary with those of cause cor and graphical approaches. See also Fine Point 7.2.

16 7.4 SWIGs The SWIG for the M-bias example is 1 2 Figure 7.7 a Figure 7.8 a sents the data from a hypothetical inter thesametreatmentlevel. The treatm semicircles. The right semicircle encodes semicircle encodes the value of that wo of intervention. We use semicircles simpl variables were derived by splitting the tre is not a cause does not have an arro same for all subjects. The outcome is study. The remaining variables are tem ables and take the same value as in t exchangeability q holds because, and are blocked after conditioning on Consider now the causal diagram in 7.8. Marginal exchangeability q ho between and are blocked (withou conditional exchangeability q do the path 1 2 conditioned on. This is why the margin conditional - association given is no for results in bias. These examples show and graphical approaches. See also Fine Knowledge of the causal structure is tence of confounding and label a variable which variables need to be measured and ies, investigators measure many variable treated and the untreated are conditiona covariates. The underlying assumption may exist (confounding), the measured backdoor paths (no unmeasured confoun tee that the assumption of no unmeasur causal inference from observational data Marginal exchangeability a holds because the path between and a is blocked (unconditionally). Why? In contrast, conditional exchangeability a does not hold BIOS Confounding

17 Constructing a SWIG entails two steps this example. This failure can be verified by analyzing the SWIG in Figure 7.10, which has been set to the value. In this world, the factual variable is repl that is, the value of that would have been observed if if all subjects had receiv all paths 7.4from SWIGs to we conclude that q holds, but we cannot c q holds as is not even on the graph. (nder an FFRCISTG, any ind SWIG cannot be assumed to hold.) Therefore, we cannot ensure that the averag is identified from data on ( ). 1. Split intervention node or nodes 2. Replace all descendants of split nodes with potential outcomes E.g., the DG in Fig 7.9 becomes the SWIG in Fig 7.10 a Figure 7.9 a a There is a scientific consequence to t tional studies. Suppose you conducted a effect of heart transplant on death sured confounding given disease severity inferences from this observational study confounding. The critic is not making a Since the findings from any observationa viously true that those of your study can was to provide evidence about the short failed. His criticism is completely noninf a characteristic of observational research knew before the study was conducted. Figure 7.10 dditional conditions (e.g., no bias due to selection or measurement) arerequiredforvalidcausalinfer- To appropriately criticize your study engage in a truly scientific conversation experimental or observational findings th say something along the lines of the inf may be incorrect because of potential co c a common cause through which a back latter option provides you with a testabl unmeasured confounding. The burden o move is to try and adjust for smoking. BIOS Confounding

18 Fig SWIGs a Figure 7.9 a a effect of heart transplant on death sured confounding given disease severity inferences from this observational study confounding. The critic is not making a Since the findings from any observationa viously true that those of your study can was to provide evidence about the short failed. His criticism is completely noninf a characteristic of observational research knew before the study was conducted. Figure 7.10 dditional conditions (e.g., no bias due to selection or measurement) arerequiredforvalidcausalinference from observational data. But, unlike the expectation of no unmeasured confounding, these additional conditions may fail to hold in both observational studies and randomized experiments. To appropriately criticize your study engage in a truly scientific conversation experimental or observational findings th say something along the lines of the inf may be incorrect because of potential co a common cause through which a back latter option provides you with a testabl unmeasured confounding. The burden o move is to try and adjust for smoking. This example demonstrates that confounders cannot be descendants of treatment Note there is an open backdoor path from to a Conditioning on a blocks this backdoor path, implying a a The next section reviews the methods when, as in Figures , enough conf backdoor paths between treatment and o However we cannot make the same conclusion from the SWIG about the observable, i.e., that a, because is not on the graph n important point. We have referr all or nothing issue: either bias exists is important to consider the expected dir Fine Point 7.4. BIOS Confounding

19 7.4 Confounding and exchangeability Note that this structural definition of confounding as well as the construction of DGs or SWIGs relies on strong assumptions HR assert that knowledge of causal structure is a prerequisite to determine the existence of confounding Of course the assumption of no unmeasured confounding is never guaranteed to hold in an observational setting (there may always be some unmeasured common cause which creates a backdoor path not blocked by any set of measured covariates ) Causal inference from observational data is risky business BIOS Confounding

20 7.5 How to adjust for confounding Ideally, conduct a randomized experiment. Then there will be no common causes of exposure and treatment, no confounding, association is causality, etc. In observational studies, two sets of methods to adjust for confouding 1. G-methods: standardization ( 2), IP weighting ( 2), G-estimation ( 14) 2. Stratification based-methods: stratification/restriction ( 4.4), matching ( 4.5). BIOS Confounding