Causation to Unconditional Association
Contents
1. modules cause_u_assoc 0000 printable html Definition In general common causes produce association between their effects 2330 Multiple Common Causes As there can be multiple causal paths between two variables there can be multiple common causes For example consider the common causes of your own level of athletic achievement and your sibling s level of athletic achievement below PARENTS SIBLING S ATHLETIC GENES ACHIEVEMENT YOUR ATHLETIC 1 ACHIEVEMENT 4 ENVIRONMENT FIGURE 2330 1 There are at least two common causes of these variables the genetic contribution that you and your sibling share from your parents and the environment that you and your sibling both grew up in Both common causes produce association between the level of athletic achievement among siblings Definition In general each common cause produces association between its effects lt A link to exercises in the interactive version of this module gt 2400 Causal Connection 2410 The Idea We can summarize the preceding sections as follows Each causal path between X and Y produces association between X and Y and Each common cause of X and Y produces association between X and Y 14 of 20 5 18 01 2 33 PM http www phil cmu edu 8080 jcourse cont modules cause_u_assoc O000 printable html We unify these ideas with the idea of2. A link to exercises in the interactive version of this module gt To summarize each causal connection between a pair of variables produces association between those variables We assume that when there is at least one form of causal connection between a pair of variables then they are predicted to have some overall association 2500 Common Effects Common causes produce association but do common effects AGE SEX zs HEART DISEASE FIGURE 2500 1 For example in the graph above the variables AGE and SEX have a common effect HEART DISEASE Does having a common effect produce association between AGE and SEX We know that the causal connection between AGE and HEART DISEASE produces association as does the causal connection between SEX and HEART DISEASE males get it more often Thus AGE _X_ HEART DISEASE and HEART DISEASE _X_ SEX Because AGE and HEART DISEASE are associated and HEART DISEASE and SEX are associated does that mean AGE and SEX are associated Is association always transitive The answer is no to all of these questions Association is not necessarily transitive AGE and SEX are not associated because they have a common effect and in general common effects do not produce association Consider the following example using the Setbuilder Suppose we create a fantasy world in which being blond causes you to smoke and being female causes you to smoke as well but there is no causal connect
3. 4100 Causal Chains 5 18 01 2 33 PM 18 of 20 http www phil cmu edu 8080 jcourse cont modules cause_u_assoc O000 printable html We define a causal path more precisely via an idea from graph theory called a directed path An edge between A and another variable B is out of A just in case Ais the cause and B the effect A B Similarly an edge between A and another variable B is into B just in case Ais the cause and B the effect A B In a causal graph a sequence of directed edges U is a directed path from X to Y just in case U begins with an edge out of X U ends with an edge into Y and every two adjacent edges on U are head to tail i e ZI ZJ and ZJ ZK no vertex is a cause of more than one effect on U The first two clauses are obvious The third clause in the definition ensures that all the arrows in the sequence point the same way In the figure below for example the top graph contains a directed path from X to Y that satisfies this clause but the path in the bottom graph from X to Y is not a directed path because it violates this clause the arrows connected to Z2 collide they don t point in the same direction GRAPH 1 A p TO E Cd GRAPH 2 x F 2 Bt Y FIGURE 4100 1 Directed paths have a length equal to the number of edges on the path they have endpoints and they are described by writing the sequence of edges on the directed path in order from one endpoint to the oth
4. Virus No 6 No 1 Day Virus Yes An antibiotic has no effect on viral infections so in causal assignments 2 3 5 and 6 whether the antibiotic was taken makes no difference to the outcome lt A link to exercises in the interactive version of this module gt 4 of 20 5 18 01 2 33 PM 5 of 20 http www phil cmu edu 8080 jcourse cont modules cause_u_assoc O000 printable html Below is a simulation of this causal system To run a single trial in an experiment click the OK button in the single trial row and complete the steps that follow To run multiple trial click on the OK button in the multiple trials row and complete the steps that follow The results of each trial will be displayed as histograms lt A simulation in the interactive version of this module gt Run at least 200 trials before you answer these questions lt A link to exercises in the interactive version of this module gt Suppose that we were dealing with a mutated bacteria that was resistant to our antibiotic and taking the antiobiotic had no effect DISEASE 7 DAY VIRUS 1 DAY VIRUS 7 DAY BACTERIA TAKES ANTIBIOTIC 1 DAY RECOVERY YES NO YES NO FIGURE 2100 4 Below is a simulation of the mutated bacteria system To run a single trial in an experiment click the OK button in the single trial row and complete the steps that follow To run multiple trial click on the OK button in the multiple trials row and complete the steps that f
5. a causal connection Two variables X and Y are causally connected if Xis acause of Y or Yis acause of X or there is a third variable Z that is a direct or indirect cause of X and of Y lt A link to exercises in the interactive version of this module gt 2420 Multiple Causal Connections As you would suspect there can be many causal connections between a pair of variables As you might also guess each causal connection between a pair of variables produces association In the graph below for example a child s exposure to lead and his or her IQ score are causally conncted in two ways CAUSAL GRAPH HOME ENVIRONMENT LEAD EXPOSURE CAUSAL CONNECTIONS BETWEEN LEAD EXPOSURE AND IQ PARENTAL CARE FOR 10 4 SCORE PARENTAL CARE FOR HOME ENVIRONMENT LEAD Q EXPOSURE SCORE LEAD IQ EXPOSURE SCORE FIGURE 2420 1 Since LEAD EXPOSURE is a direct cause of IQ SCORE it constitutes one causal connection and since PARENTAL CARE for the HOME ENVIRONMENT is a common cause of LEAD EXPOSURE and IQ SCORE it constitutes another Except for a technical qualification that will be explained later in section 5200 each pathway from one variable to another and each pair of pathways from a common cause to the two variables counts as a different causal connection 15 of 20 5 18 01 2 33 PM http www phil cmu edu 8080 jcourse cont modules cause_u_assoc OQ000 printable html lt
6. fertilized may have X1 or X2 Which egg is fertilized is a matter of luck 11 of 20 5 18 01 2 33 PM http www phil cmu edu 8080 jcourse cont modules cause_u_assoc 0000 printable html MOTHER X1 X2 BALD S N NON BALD SON X1 Y x2 Y FATHER x Y X1 BALDNESS GENE X2 NON BALDNESS GENE FIGURE 2310 2 So if we consider it just random luck whether a mother carrying the baldness gene gives the gene to her son then the causal graph for baldness above fills out into a deterministic graph as SO LUCKY 5 BALDNESS GENE 2 LUCKY YES NO YES NO YES NO BALD SON L BALD BROTHER YES NO YES NO FIGURE 2310 3 The response structures for BALD and BALD BROTHER are as follows TABLE 2310 2 RESPONSE STRUCTURE FOR BALDNESS GENE LUCKY AND BALD SON Causal Assignment Variable 1 Variable 2 LUCKY Effect BALD SON BALDNESS GENE 1 Yes Yes No 2 Yes No Yes 3 No Yes No 4 No No No TABLE 2310 3 RESPONSE STRUCTURE FOR MOTHER LUCK AND BALD BROTHER Causal Assignment Variable 1 Variable 2 LUCKY Effect BALD BALDNESS GENE BROTHER 1 Yes Yes No 2 Yes No Yes 3 No Yes No 4 No No No 12 of 20 5 18 01 2 33 PM http www phil cmu edu 8080 jcourse cont modules cause_u_assoc O000 printable html The following simulation embodies this system Once again explore what happens when you manipulate the common cause BALDNESS GENE To run a single trial in an experiment click the OK button in the single trial row
7. 0 Common Causes 2310 The Idea Consider the variables TABLE 2310 1 VARIABLES FOR BALDNESS Variables Values BALD SON Yes No BALD BROTHER Yes No BALDNESS GENE from mother Yes No The causal graph relating these variables is BALDNESS GENE VES NOJ BALD SON BALD BROTHER YES NO YES MO FIGURE 2310 1 From the last section we know that the variables BALDNESS GENE and BALD SON are associated because BALDNESS GENE is a cause of BALD SON and that the variables BALDNESS GENE and BALD BROTHER are associated for the same reason Being a bald son and having a bald brother are also associated in virtue of being effects of a common cause having a mother with a baldness gene How does this work in detail Consider a psuedo indeterministic system of the sort we used to show how direct and indirect causes produce association A male child gets half of his 46 chromosomes from his father and half from his mother The two sets of chromosomes from the parents merge to form 23 pairs where each pair has one chromosome from the father and one from the mother For a male child one of these pairs is the XY pair The X chromosome in this pair comes from the mother and the Y from the father If the X chromosome in the son s pair has the gene for baldness then the son will be bald The mother begins with two X chromosomes say X1 for bald and X2 for not bald and then by random luck gives X1 or X2 to the son The egg that is
8. 1 Z3 X P2 Z3 Y In the case of Z1 the paths P1 and P2 are P1 Z1 Z3 X P2 Z1 Z3 Y Both causal connections contain X Z3 Y It would therefore not be appropriate to say that there are two distinct sources of association between X and Y in this graph Thus we exclude the second case with clause 3 in the definition above 20 of 20 5 18 01 2 33 PM
9. ABOLISM the frequency of low BODY WEIGHT is less than the frequency of low BODY WEIGHT given high EXERCISE So path 2 also produces negative association between EXERCISE and BODY WEIGHT In general each causal path from X to Y produces association between X and Y lt A link to exercises in the interactive version of this module gt 2230 Offsetting Causal Paths 5 18 01 2 33 PM 10 of 20 http www phil cmu edu 8080 jcourse cont modules cause_u_assoc O000 printable html In the example involving exercise and weight loss above both causal paths produced negative association between EXERCISE and BODY WEIGHT Roughly the contributions of the individual paths add to create the total association so if the graph pictured were assumed to be correct and complete and the signs of the causal influences are correct then it predicts that we would observe a negative association between EXERCISE and BODY WEIGHT Suppose however that our view of the effect on APPETITE of EXERCISE was different Suppose we believed that an increase in EXERCISE increased APPETITE instead of decreased it as we assumed above Then we would have to switch the sign on the edge from EXERCISE to APPETITE from to as so EXERCISE _ METABOLISM HIGH LOW HIGH LOW APPETITE BODY WEIGHT HIGH LOW HIGH LOW FIGURE 2230 1 Now path 1 EXERCISE APPETITE BODY WEIGHT would produce positive association between EXERCISE and
10. BODY WEIGHT but path 2 would still produce negative association lt A link to exercises in the interactive version of this module gt The total association between EXERCISE and BODY WEIGHT predicted by this model is not determined by the information given Since one causal path produces negative association and the other produces positive association the overall association depends on the strength of each It is even possible that both paths are of exactly the same strength and offset each other thus the overall association between EXERCISE and BODY WEIGHT predicted by this model might be 0 If the two paths offset each other exactly then the model will predict that EXERCISE and BODY WEIGHT are independent This is one reason why we said before that producing an association between X and Y and predicting that X and Y are associated are different ideas In general whenever there are multiple causal paths connecting two variables in a causal graph we will assume that the total association from all the paths that produce positive association do not exactly offset all the paths that produce negative association thus producing a total association of exactly zero This is one part of an assumption we call faithfulness which we will make more precise later lt A link to exercises in the interactive version of this module gt 5 18 01 2 33 PM http www phil cmu edu 8080 jcourse cont modules cause_u_assoc O000 printable html 230
11. and complete the steps that follow To run multiple trial click on the OK button in the multiple trials row and complete the steps that follow The results of each trial will be displayed as histograms Produce at least 200 trials and study the histograms that capture the association between BALD and BALD BROTHER lt A simulation in the interactive version of this module gt lt A link to exercises in the interactive version of this module gt In general if a variable C is a direct cause of X and of Y then C is a direct common cause of X and Y And in general direct common causes produce association 2320 Indirect Common Causes Like direct and indirect cause one variable can be a common causes of a pair without being a direct common cause For example consider the graph below BALD GENE GRANDFATHER YES NO BALD GENE MOTHER BALD GENE AUNT YES NO YES NO BALD SON BALD BROTHER BALD COUSIN YES NO VES NO YES NO FIGURE 2320 1 The variable GRANDFATHER is a common cause of the variable BALD and the variable BALD COUSIN It is not a direct common cause but a common cause nevertheless In general C is a common cause of X and Y if there is a path from C to X C is a cause of X there is a path from C to Y C is a cause of Y and no variable besides C is on both of these paths 13 of 20 5 18 01 2 33 PM http www phil cmu edu 8080 jcourse cont
12. causal graph to the independence relations that follow from that graph You can construct a causal graph in the Causality Lab and then ask it to compute which pairs of variables are predicted to be associated which pairs independent and which pairs are conditionally associated or conditionally independent given values for another set of variables To learn how to get the Causality Lab to make predictions read section 4400 of the Causality Lab User Manual Do that now lt A link to exercises in the interactive version of this module gt At any time in the module you can return to this page launch the Causality Lab and explore by constructing any causal graph you like among HAPPINESS EDUCATION INCOME or any subset of these variables and ask for predictions JAVA Link to the Causality Lab for experimenting 1300 Prediction vs Discovery 5 18 01 2 33 PM 3 of 20 http www phil cmu edu 8080 jcourse cont modules cause_u_assoc OQ000 printable html In this module and the next we move from theory to evidence After these modules we consider moving the other way from evidence to theory The difference is this given a causal graph we can make a unique prediction about association and independencies Given a set of association and independencies however there are many causal graphs that would make the same prediction Associational evidence underdetermines causal theories The situation is not as bad as you might think how
13. ct causes of any length leading from X to Y We will refer to indirect causal chains as either causal paths causal chains or directed paths and we define these notions precisely in section 5000 There can be more than one causal path leading from one variable to another in a causal graph 5 18 01 2 33 PM 9 of 20 http www phil cmu edu 8080 jcourse cont modules cause_u_assoc 0000 printable html EXERCISE METABOLISM HIGH LOW LOW APPETITE BODY WEIGHT HIGH LOW ee eee HIGH LOW FIGURE 2220 1 For example in the causal graph above there are two causal paths from EXERCISE to BODY WEIGHT EXERCISE METABOLISM BODY WEIGHT EXERCISE APPETITE BODY WEIGHT If we suppose that increasing EXERCISE tends to increase METABOLISM thus the on the edge from EXERCISE a loss of BODY WEIGHT thus the on the edge from METABOLISM to BODY WEIGHT then for a fixed level of APPETITE the frequency of low BODY WEIGHT will be less than the frequency of low BODY WEIGHT given high amounts of EXERCISE Because high levels of EXERCISE go with low levels of BODY WEIGHT we say that path 1 produces negative association between EXERCISE and BODY WEIGHT Similarly if we assume that EXERCISE tends to decrease APPETITE thus the on the edge from EXERCISE to APPETITE and that decreased APPETITE tends to cause a loss of BODY WEIGHT thus the on the edge from APPETITE to BODY WEIGHT then for a fixed MET
14. direct cause common causes and common effects When you finish the module you should be able to take any causal graph and write down the set of independence relations predicted to hold in statistical samples involving the variables in the graph For any two variables X and Y you should be able to ascertain whether a causal graph involving X and Y predicts that X and Y are associated or independent In the next module we extend the ideas to conditional independence Consider for example a simple diagram that connects a causal theory about SMOKING YELLOW FINGERS and LUNG CANCER to the associations that are predicted to exist if this theory is correct After this module you should be able to write down the associations and independencies among SMOKING YELLOW FINGERS and LUNG CANCER without consulting the right hand side of the figure and after the next module you should be able to write down the conditional associations and independencies 1 of 20 5 18 01 2 33 PM 2 of 20 http www phil cmu edu 8080 jcourse cont modules cause_u_assoc OQ000 printable html HYPOTHESIS CAUSAL GRAPH SMOKING YELLOW LUNG l FINGERS CANCER PREDICTS EVIDENCE INDEPENDENCE ASSOCIATION ETC SMOKING KX YELLOW FINGERS SMOKING LUNG CANCER YELLOW FINGERS LUNG CANCER YELLOW FINGERS LUNG CANCER SMOKING FIGURE 1100 1 1200 The Causality Lab The Causality Lab is programmed to make the right inference from a particular
15. er Here is a sampling of directed paths contained in both graphs above with the length given for each 5 18 01 2 33 PM 19 of 20 http www phil cmu edu 8080 jcourse cont modules cause_u_assoc 0000 printable html TABLE 4100 1 SAMPLING OF DIRECTED PATHS Graph Endpoints Directed Path Length 1 X Z2 X Z1 Z2 2 1 X Y X Z1 Z2 Z3 4 Y 1 Z3 y Z3 Y 1 2 Y Z3 Y 23 1 2 Y Z2 Y Z3 Z2 2 The second clause in the definition of a directed path is meant to prevent loops In the figure below for example the second candidate path violates this clause because Z2 is a cause of both Z3 and Y pepeg y gt FIGURE 4100 2 TABLE 4100 2 CANDIDATE PATHS Endpoints Candidate Directed Path X Y X Z1 Z2 Y Yes X Y X Z1 Z2 Z3 Z1 No Z2 Y lt A link to exercises in the interactive version of this module gt 4200 Common Causes C is acommon cause of X and Y just in case there is a directed path P1 from C to X and there is a directed path P2 from C to Y and Cis the only variable on both P1 and P2 The first two clauses of this definition are obvious and the third clause is meant to prevent cases like Z1 in the following causal graph 5 18 01 2 33 PM http www phil cmu edu 8080 jcourse cont modules cause_u_assoc OQ000 printable html FIGURE 4200 1 Z3 satisfies the definition of common cause but Z1 does not In the case of Z3 paths P1 and P2 are P
16. et s assume that the chicken pox system is indeterministic Not every child who comes in close proximity with another infected child becomes infected with the virus and not every child who has the virus in his or her bloodstream becomes symptomatic Like we did with the antibiotics and the cell phone examples lets suppose that the system is psuedo indeterministic because of hidden variables For purposes of illustration suppose this is the complete fully deterministic causal system LUCKY IMMUNE YES NO YES NO EXPOSED INFECTED m SYMPTOMS YES NOJ YES NO YES NO FIGURE 2210 4 Only children who are exposed and unlucky become infected and only children who are infected and are not immune to chicken pox become symptomatic lt A link to exercises in the interactive version of this module gt Below is a simulation of this system To run a single trial in an experiment click the OK button in the single trial row and complete the steps that follow To run multiple trial click on the OK button in the multiple trials row and complete the steps that follow The results of each trial will be displayed as histograms Run this simulation for at least 200 trials lt A simulation in the interactive version of this module gt lt A link to exercises in the interactive version of this module gt 2220 Multiple Causal Paths In general one variable X is an indirect cause of another Y if there is a chain of dire
17. ever and therein lies the appeal of the topic Stay tuned 2000 Causation and Association 2100 Direct Causation and Association Consider a causal theory involving only two variables X and Y If X is a direct cause of Y then the direct causal relation produces association between X and Y Saying that a direct cause produces an association between X and Y is not the same as predicting that X and Y are associated in a sample because the association could come about in other ways and because in some special cases to be explained in section 2300 one variable can cause another but because of other causal relationships between the two variables the theory might still predict that the variables are independent For now let s focus on how a direct causal relation produces association between X and Y HYPOTHESIS CAUSAL GRAPH PRODUCES EVIDENCE INDEPENDENCE ASSOCIATION ETC xX HY FIGURE 2100 1 To be concrete consider an example in which everyone begins an experiment feeling sick with a fever Let X be the treatment variable TAKES ANTIBIOTIC with values Yes No and Y be the variable 1 DAY RECOVERY with values Yes No The causal graph we hypothesize to hold among these two variables is 5 18 01 2 33 PM http www phil cmu edu 8080 jcourse cont modules cause_u_assoc O0000 printable html TAKES ANTIBIOTIC 1 DAY RECOVERY VES NO YE5 NO FIGURE 2100 2 By claiming that a direct cause produces associatio
18. http www phil cmu edu 8080 jcourse cont modules cause_u_assoc 0000 printable html Causation to Association l Unconditional Association 1000 Introduction 1100 Overview Science endeavors to construct theories that explain and predict the observable world Toward this end scientists formulate hypotheses make predictions from these hypotheses test their predictions on observable evidence modify their hypotheses according to the result etc Theories and hypotheses come in many forms as does observable evidence In our setting the focus is on qualitative hypotheses of a limited sort causal graphs The evidence we consider is also of a limited and qualitative sort association and independence among variables Association doesn t have to be a qualitative concept a substantial part of Statistics deals with kinds and degrees of association For our purposes here however we are only concerned with whether two variables are associated or not associated independent thus we stay at a qualitative level Our focus in this module is to connect qualitative causal theories with qualitative associational evidence lt A link to exercises in the interactive version of this module gt This module focuses on how causal graphs explain and predict patterns of unconditional association and independence among a set of variables In the sections that follow we will break down the task into many sub tasks For example we will cover direct cause in
19. ich is a direct cause of SYMPTOMS 6 of 20 5 18 01 2 33 PM 7 of 20 http www phil cmu edu 8080 jcourse cont modules cause_u_assoc O000 printable html lt A link to exercises in the interactive version of this module gt From the previous section we have the following HYPOTHESIS CAUSAL GRAPH EXPOSURE INFECTION INFECTION EXPOSURE pe _ PRODUCES PRODUCES EVIDENCE INDEPENDENCE ASSOCIATION ETC EXPOSURE INFECTION INFECTION W EXPOSURE FIGURE 2210 2 The question is does the indirect causal relation also produce association HYPOTHESIS CAUSAL GRAPH INFECTION SYMPTOMS PRODUCES EXPOSURE EVIDENCE INDEPENDENCE ASSOCIATION ETC EXPOSURE SYMPTOMS FIGURE 2210 3 In this simple case the answer is clearly yes In most any sample of 3 year olds the frequency of symptomatic 3 year olds is clearly lower than the frequency of symptomatic 3 year olds given exposure In general the answer is yes as well but there are rare cases in which indirect causation does not produce association We will discuss these cases later but for now assume that indirect causation produces association 5 18 01 2 33 PM 8 of 20 http www phil cmu edu 8080 jcourse cont modules cause_u_assoc O000 printable html How in detail does a causal chain X Y Z produce an association between X and Z Let s examine the chicken pox case in same way we did the antibiotics case L
20. ion between HAIR COLOR and SEX 16 of 20 5 18 01 2 33 PM 17 of 20 http www phil cmu edu 8080 jcourse cont modules cause_u_assoc O0000 printable html HAIR COLOR SEX oe SMOKES FIGURE 2500 2 If we are correct then according to this causal graph HAIR COLOR and SMOKING are associated SEX and SMOKING are associated but HAIR COLOR and SEX are not associated lt A link to exercises in the interactive version of this module gt 3000 Summary The connection between causal graphs and unconditional association turns out to be very simple Two variables X and Y are predicted to be associated just in case they are causally connected in the graph X and Y are causally connected in the graph if either Xis acause of Y or Yis acause of X or there is acommon cause of X and Y Variables can be causally connected in several different ways and each causal connection produces association Although it is possible we assume that when multiple causal connections exist between X and Y the overall association between X and Y is not zero That is if X and Y are causally connected we predict they are associated and if X and Y are not causally connected we predict that they are independent Section 4000 Formalities gives the formal definitions of the concepts we ve discussed in this module It is optional and provides the technical explanations of directed paths and common causes 4000 Formalities
21. n we are claiming that this graph produces an association between taking antibiotics and recovering from an illness in one day lt A link to exercises in the interactive version of this module gt How in detail does a direct cause produce an association Lets examine the Antibiotic and Recovery case in detail Even though everyone in our experiment begins with a fever not everyone who takes antibiotics will recover in one day Some people who take the antibiotic will recover in a day and some won t Some people who don t take the antiobiotic will recover and some won t So the system is indeterministic Suppose this causal system is like the cell phone system you examined in the module on determinism and indeterminism Suppose the system involving just the two variables TAKES ANTIBIOTIC and 1 DAY RECOVERY is psuedo indeterministic but underlying it is a deterministic system involving a hidden variable Suppose this graph tells the whole story DISEASE 7 DAY VIRUS 1 DAY VIRUS 7 DAY BACTERIA TAKES ANTIBIOTIC sa 1 DAY RECOVERY YES MO YES NO FIGURE 2100 3 Here is the response structure for this system TABLE 2100 1 RESPONSE STRUCTURE FOR ANTIBIOTIC DISEASE 1 DAY RECOVERY SYSTEM Causal Assignment Causal Factor Causal Factor Effect 1 DAY TAKES ANTIBIOTIC DISEASE RECOVERY 1 Yes 7 Day Bacteria Yes 2 No 7 Day Virus No 3 Yes 1 Day virus Yes 4 No 7 Day Bacteria No 5 Yes 7 Day
22. ollow The results of each trial will be displayed as histograms Run this simulation for at least 200 trials lt A simulation in the interactive version of this module gt lt A link to exercises in the interactive version of this module gt So a direct cause X Y produces association between X and Y and a direct cause Y X also produces association between Y and X The picture below summarizes the situation for direct causation 5 18 01 2 33 PM http www phil cmu edu 8080 jcourse cont modules cause_u_assoc 0000 printable html HYPOTHESIS CAUSAL GRAPH xs PRODUCES PRODUCES EVIDENCE INDEPENDENCE ASSOCIATION ETC XN Y YN X FIGURE 2100 5 Non independence like independence is symmetric so X X Y implies Y _ X 2200 Indirect Causes and Association 2210 Simple Causal Paths Consider 3 year olds and chicken pox TABLE 2210 1 VARIABLES FOR CHICKEN POX AND 3 YEAR OLDS Variables Values EXPOSED Yes No INFECTED Yes No SYMPTOMS Yes No where EXPOSED Yes means that the 3 year old has been in close proximity to another person with chicken pox in the past week INFECTED Yes means that the child has the virus in his or her bloodstream and SYMPTOMS Yes means that he or she has the typical chicken pox rash The causal graph among these variables is clear EXPOSED b INFECTED SYMPTOMS YES NO YES NO YES NO FIGURE 2210 1 EXPOSED is a direct cause of INFECTED wh
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