Causal Graphs
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1. Variable Value POWER SWITCH on the TV On Off REMOTE SWITCH On Off PICTURE Yes No The causal graph for these variables is POWER SWITCH REMOTE SWITCH ON OFF ON OFF PICTURE YE5 NO FIGURE 3400 1 Here PICTURE is a common effect of both REMOTE SWITCH and POWER SWITCH This is a case in which each of the cause variables independently causes the common effect When we have interacting causes the graph is the same in structure So the structure of the causal graph is by no means completely informative about the nature of the causal relationship Consider the following set of variables TABLE 3400 2 VARIABLES FOR AGRICULTURE SYSTEM Variable Value WATER did the crops get rain On Off FERTILIZER did the fertilizer get water Yes No GROWTH did the plants grow well Yes No Furthermore we will assume that plants grow well only if they are both watered and have fertilizer lt A link to exercises in the interactive version of this module gt 3500 Cyclic Causal Graphs Causation among variables is asymmetric That is if X is a cause of Y then it doesn t follow that Y is a cause of X Is a sibling of is an example of a symmetric relationship An example of an asymmetric relationship among people is likes Causation among variables is asymmetric but it isn t antisymmetric A relationship is antisymmetric if the fact that it holds one way precludes it holding the othe2. construct the graphs for several different kinds of relatively common causal systems You will be asked to construct the causal graphs that represent systems described to you in text To do so you will use a Java applet called the Causality Lab To help you get oriented to the Lab we have written an on line User Manual Before going to the next page in this section read sections 3100 3200 3410 and 3420 of the Causality Lab User Manual When you are done proceed to the next section 3200 Common Causes A variable C is a common cause of two or more other variables X and Y when C is a cause direct or indirect of both X and Y Consider the following three variables for TVs that function normally TABLE 3200 1 VARIABLES FOR A TV SYSTEM Variable Value SOUND Yes No POWER SWITCH On Off PICTURE Yes No 9 of 14 4 5 01 3 34 PM 10 of 14 http www phil cmu edu 8080 jcourse cont modules causal_graphs Q000 printable html Suppose the causal graph for these variables is POWER SWITCH ON OFF ee e PICTURE SOUND YES NO YES NO FIGURE 3200 1 Here POWER SWITCH is acommon cause of both PICTURE and SOUND because changing the state of POWER SWITCH changes both the value of PICTURE and the value of SOUND lt A link to exercises in the interactive version of this module gt 3300 Causal Chains and Direct vs Indirect Causation If one variable only influences another through some intermediate variable
3. from STRIKE MATCH to MATCH LIGHTS Why Apply the definition above to this case Is there a test pair of causal assignments that differ only by the value assigned to STRIKE MATCH that make a difference to the effect MATCH LIGHTS No If the TIP TEMPERATURE is above 350 then the match will light whether we strike it or not If the TIP TEMPERATURE is below 350 then the match will not light whether we strike it or not Thus there are no test pairs for STRIKE MATCH that make a difference to MATCH LIGHTS even to its probability 2300 Examples 2310 Deterministic Causation The Malaria Example Consider the case of malaria again The variables in the first causal system we considered are TABLE 2310 1 VARIABLES FOR THE MALARIA CASE Variable Value BITTEN Was bitten by an infected mosquito True False INOCULATED True False HAS GENE Has the sickle cell gene True False DRINKER Drinks gin and tonics regularly True False MALARIA Gets malaria True False The response structure for malaria was given by the following table in the module on Variable Causation 4 5 01 3 34 PM http www phil cmu edu 8080 jcourse cont modules causal_graphs Q000 printable html TABLE 2310 2 RESPONSE STRUCTURE FOR THE MALARIA CASE Assignment Variable 1 Variable 2 Variable 3 Variable 4 Effect BITTEN INNOCULATED HAS GENE DRINKER MALARIA 1 True True True True False 2 True True True False False 3 True True False True False 4 True Tr
4. then there is no arrow between the first feature and the third feature in the chain For example in this simulation the amount of water coming from the dam through the spout influences the speed of the turbine which influences whether electricity is generated to power the light bulb lt A simulation in the interactive version of this module gt The water is still a cause of the light bulb but only an indirect one If the variables and their values are TABLE 3300 1 VARIABLES FOR TURBINE SYSTEM Variable Value WATER Flowing Not flowing TURBINE Spinning Not spinning LIGHT On Off then the causal graph is as follows pi SSS WATER lji TURBINE LIGHT FLOWING NOT FLOWING SFINNING NOT SPINNING ON OFF FIGURE 3300 1 4 5 01 3 34 PM http www phil cmu edu 8080 jcourse cont modules causal_graphs Q000 printable html In the simulation you cannot actually directly control whether the turbine spins but only set a switch next to the turbine the Turbine Switch to up or down When the switch is up then the water is diverted away from the turbine but if the switch is down water flows over the turbine lt A link to exercises in the interactive version of this module gt Causal chains highlight the fact that the idea of direct causation only makes sense relative to the set of variables under consideration In the causal graph below for example the variable REFRIGERATOR DOOR is a direct cause of th
5. dules The key concepts involve causal assignments response structures and a test pair of causal assignments If you need to review these ideas go to the module on Variable Causation Here again are the key definitions Definition Test Pair of Causal Assignments lf two causal assignments C1 and C2 are identical except for the values assigned to variable X then C1 and C2 are a test pair of causal assignments for X 4 5 01 3 34 PM 4 of 14 http www phil cmu edu 8080 jcourse cont modules causal_graphs Q000 printable html Definition Direct Cause If in a system of variables S there are any test pair of causal assignments for X in which there is a difference in the effect Y then X is a direct cause of Y relative to S In the module on Determinism and Indeterminism we explained how this definition still covered cases of indeterministic causation in which the difference in the effect amounts to a change in the probability it will occur Lets examine how a causal graph represents the causal relations in a system with a few simple examples Example 1 Switches and Lights Consider the causal system among the variables BATTERY SWITCH and LIGHT lt A simulation in the interactive version of this module gt The causal graph of this system is as follows SWITCH BATTERY OFEN CLOSED CHARGED UNCHARGED LIGHT ON OFF FIGURE 2200 2 Notice that there is no arrow from the switch to the battery nor from
6. e variable LIGHT SWITCH and the variable LIGHT SWITCH is a direct cause of the variable REFRIGERATOR LIGHT but the state of the REFRIGERATOR DOOR is not a direct cause of the state of the REFRIGERATOR LIGHT relative to the system REFRIGERATOR DOOR LIGHT SWITCH REFRIGERATOR LIGHT REFRIGERATOR DOOR _ p LIGHT SWITCH LIGHT CLOSED OPEN DEPRESSED RELEASED ON OFF FIGURE 3300 2 Why Because if we fix the variable LIGHT SWITCH at either of its values then bringing about a change in the state of the REFRIGERATOR DOOR will have no influence on the REFRIGERATOR LIGHT If we were only discussing the system REFRIGERATOR DOOR REFRIGERATOR LIGHT then the door is a direct cause of the light REFRIGERATOR DOOR _ p LIGHT CLOSED OPEN ON OFF FIGURE 3300 3 lt A link to exercises in the interactive version of this module gt 3400 Common Effects A variable E is a common effect of two or more variables X and Y when both X and Y are both causes of E and at least one causal path from X to E does not involve Y and at least one causal path from Y to E does not involve X For example consider the following three variables applied to TVs that function normally 11 of 14 4 5 01 3 34 PM http www phil cmu edu 8080 jcourse cont modules causal_graphs Q000 printable html TABLE 3400 1 VARIABLES FOR A TV SYSTEM
7. ed in the system If for example there is a causal chain A gt B gt C but A has no direct influence on C that doesn t go through B then we don t include an edge from A to C If we consider a system that doesn t include B however then relative to that system we need to include an A gt C edge Although causation among variables is asymmetric it is not anti symmetric so it is possible for one variable A to be a cause of B and also for B to be a cause of A In such a case we say the causal graph has a direct cycle If there is a chain of edges leading from any variable back to itself then we say the graph has a cycle If a graph has a cycle we Say it is a cyclic graph If the graph has no cycle then we say it is acyclic 4 5 01 3 34 PM
8. etc Lets consider whether or not the battery is charged as another variable instead of a part of the background conditions The system now includes the variable PHONE BATTERY CHARGED Yes No So now the full system is TABLE 2320 5 CELL PHONE SYSTEM WITH A NEW VARIABLE Variable Value CALL PLACED Send End PHONE BATTERY CHARGED Yes No IN RANGE OF TOWER Yes No CONNECTED Yes No Now the response structure for CONNECTED is as follows TABLE 2320 6 RESPONSE STRUCTURE FOR THE CELL PHONE SYSTEM Assignment Variable 1 CALL Variable 2 Variable 3 IN Effect PLACED BATTERY RANGE OF CONNECTED CHARGED TOWER 1 Send Yes Yes Yes 2 Send Yes No No 3 Send No Yes No 4 Send No No No 5 End Yes Yes No 6 End Yes No No 7 End No Yes No 8 End No No No 8 of 14 4 5 01 3 34 PM http www phil cmu edu 8080 jcourse cont modules causal_graphs O000 printable html Now suppose we consider the pseudo indeterministic system in which we cannot observe whether or not we are in range of the tower TABLE 2320 7 PSEUDO INDETERMINISTIC CELL PHONE SYSTEM Variable Value CALL PLACED Send End PHONE BATTERY CHARGED Yes No CONNECTED Yes No Is there still a directed edge from CALL PLACED to CONNECTED lt A link to exercises in the interactive version of this module gt 3000 Representing Different Varieties of Causation 3100 Using the Causality Lab In the next four sections 8200 through 3500 you will learn how to
9. hanging the value of BITTEN changes the effect MALARIA So BITTEN is a cause of MALARIA lt A link to exercises in the interactive version of this module gt 6 of 14 4 5 01 3 34 PM http www phil cmu edu 8080 jcourse cont modules causal_graphs O000 printable html 2320 Indeterministic Causation The Cell Phone Example Consider the case of the Cell Phone again In the full deterministic causal system there are three variables TABLE 2320 1 VARIABLES FOR THE CELL PHONE SYSTEM Variable Value CALL PLACED Send End IN RANGE OF TOWER Yes No CONNECTED Yes No The response structure for CONNECTED is as follows TABLE 2320 2 DETERMINISTIC RESPONSE STRUCTURE FOR CONNECTED Assignment CALL PLACED IN RANGE OF CONNECTED TOWER 1 Send Yes Yes 2 Send No No 3 End Yes No 4 End No No Would the causal graph among these three variables have a directed edge from CALL PLACED to CONNECTED Yes because there is a test pair for CALL PLACED that makes a difference to CONNECTED lt A link to exercises in the interactive version of this module gt In causal assignments 1 and 3 where INRANGE OF TOWER is assigned Yes then changing CALL PLACED from End to Send always changes the value of CONNECTED from No to Yes Now consider the pseudo indeterministic system involving just these variables TABLE 2320 3 VARIABLES FOR THE PSEUDO INDETERMINISTIC SYSTEM Variable Value CALL PLACED Send End CONNECTED Yes No Wo
10. http www phil cmu edu 8080 jcourse cont modules causal_graphs O000 printable html Causal Graphs 1000 Introduction Causal relations between variables are often represented by diagrams We draw an arrow from a variable X to a variable Y if X is a direct cause of Y relative to the set of variables under consideration So for example the claim that water temperature influences the height of the water in a glass by making the water expand or contract can be represented by the following diagram where the boxes are variables with the possible values they might take on in square brackets WATER TEMPERATURE WATER LEVEL DECREASES INCREASES DECREASES INCREASES FIGURE 1000 1 The causal relations between the state of a light bulb a light switch and a battery can be represented by a causal graph involving three variables SWITCH BATTERY OPEN CLOSED CHARGED UNCHARGED LIGHT ON OFF FIGURE 1000 2 and the relations between the refrigerator door the light switch and the refrigerator light by another REFRIGERATOR DOOR LIGHT SWITCH LIGHT CLOSED OPEN DEPRESSED RELEASED ON OFF FIGURE 1000 3 We use the same kind of diagram no matter whether the cause tends to prevent the effect or to bring the effect about So we would represent the claim that an inoculation with the Salk polio vaccine prevents Polio by the following graph 1 of 14 4 5 01 3 34 PM http www phil cmu edu 8080 jcourse c
11. neither exclusive nor exhaustive Someone can have hair that is both blond and short so the set is not exclusive and someone can have hair that is black and long in which case no value from the set applies so it is not exhaustive 2 of 14 4 5 01 3 34 PM 3 of 14 http www phil cmu edu 8080 jcourse cont modules causal_graphs Q000 printable html In a causal graph we represent each variable as a box with the variable s name and possible values though we will sometimes leave out the variable values So if our variables are REFRIGERATOR DOOR Closed Open LIGHT SWITCH Depressed Released and LIGHT Off On then we need to have a box for each variable REFRIGERATOR DOOR LIGHT SWITCH LIGHT CLOSED OPEN DEPRESSED RELEASED ON OFF FIGURE 2100 1 2200 Directed Edges A directed edge in a causal graph is an arrow where the head of the arrow points to the effect variable and the tail comes from the cause variable We say it is a directed edge to distinguish it from an undirected edge DIRECTED EDGE REFRIGERATOR DOOR j LIGHT SWITCH OPEN CLOSED ON OFF UNDIRECTED EDGE REFRIGERATOR DOOR LIGHT SWITCH OPEN CLOSED ON OFF FIGURE 2200 1 We include a directed edge from a variable X to a variable Y in the causal graph that represents a set of variables S if and only if X is a direct cause of Y relative to S We gave an account of cause in the Variable Causation and Determinism and Indterminism mo
12. ont modules causal_graphs O000 printable html INOCULATION gt POLIO YES NO YES NO FIGURE 1000 4 We use other means to indicate whether the causal factor tends to bring about or prevent the effect for example we might place a plus or minus sign next to the arrow INOCULATION POLIO VES NO YES NO FIGURE 1000 5 This module explains how causal graphs represent in a qualitative way the causal relations among a set of variables It also introduces and defines features of causal graphs that will be crucial in understanding the connection between causal systems and statistical data for example common cause direct vs indirect causation common effect and more 2000 The Elements of Causal Graphs 2100 Variables Causal graphs represent the causal relations in a causal system Specifically causal graphs involve 1 aset of variables and 2 aset of directed edges that connect the variables Variables were introduced in the module on Variable Causation so we only give a brief overview here The values of variables are properties of an individual e g the hair color of a person the population of a country The set of values for a variable must be both exclusive and exhaustive A set of values is exclusive if no individual can have more than one value A set of values is exhaustive if every individual has one of the values For example consider the variable Hair Type with values Red Blond Short that are
13. r For example the relationship is a parent of is antisymmetric If person X is a parent of Y then Y cannot be a parent of X 12 of 14 4 5 01 3 34 PM http www phil cmu edu 8080 jcourse cont modules causal_graphs Q000 printable html It is possible for one variable X to be a cause of Y and also for Y to be a cause of Y For example losing sleep can cause anxiety and anxiety can also cause a loss of sleep Higher wages can cause inflation and inflation can cause higher wages Success causes confidence and confidence causes success In each of these cases we say that there is a direct cycle in the causal graph LOSING SLEEP WAGE INCREASES SUCCESS YES NO YES NO VES NO EY En pI F Sh ANXIETY INFLATION CONFIDENCE YES NO YES NO YES NO FIGURE 3500 1 What does it mean to say that there is a direct cycle between SUCCESS and CONFIDENCE We can just apply the defintion we have given for a direct cause twice SUCCESS CONFIDENCE There are test pairs of causal assignments for SUCCESS that make a difference to the probability over CONFIDENCE CONFIDENCE SUCCESS There are test pairs of causal assignments for CONFIDENCE that make a difference to the probability over SUCCESS So a direct cycle is just when we have two variables X and Y and X is a direct cause of Y and Y is also a direct cause of X Cycle
14. s of causality need not be direct For example in the system including the variables LOSING SLEEP and ANXIETY we might also include the variable ADRENALINE The system would now best be represented by the following causal graph where the effect of ANXIETY on LOSING SLEEP is now indirect LOSING SLEEP __ ANXIETY YES NO va YES NO ADRENALINE YES NO FIGURE 3500 2 lt A link to exercises in the interactive version of this module gt 4000 Summary 13 of 14 4 5 01 3 34 PM 14 of 14 http www phil cmu edu 8080 jcourse cont modules causal_graphs Q000 printable html Causal graphs provide a powerful qualitative representation of causal relations among variables A causal graph includes a set of variables and a set of directed edges that connect pairs of these variables The edges are directed because causation is asymmetric and has a direction If one variable X is a direct cause of another variable Y in some causal system S then we include an arrow or directed edge from X to Y in the causal graph that represents S What does it mean for one variable to be a direct cause of another relative to a system of variables S X is a direct cause of Y relative to a set of variables Sjust in case there are test pairs of causal assignments for X across which there is a difference in Y Whether or not there is an edge from one variable to another depends on what other variables we have includ
15. the battery to the switch even though they are physically connected by wire on the circuit Why Because the state of the battery has no causal influence on the state of the switch nor does the state of the switch in this idealized example have any direct influence on the state of the battery Intervening to change the state of the battery will not affect the state of the switch even though the two are physically connected Notice second that there is an arrow from the switch to the light bulb even though when the battery is uncharged changing the switch from open to closed or from closed to open will not change the state of the light bulb it will stay off There is an arrow from the switch to the light bulb because there is some state of the battery namely when it is charged for which changing the causal assignment of the switch does change the state of the light Example 2 Lighting a Match 4 5 01 3 34 PM 5 of 14 http www phil cmu edu 8080 jcourse cont modules causal_graphs Q000 printable html The definition of direct cause highlights the role of the other variables we are considering Sometimes whether an edge should be included in a graph depends on those other variables Consider the causal graph for the process of lighting a match STRIKE MATCH 9 TIPTEMPERATURE MATCH LIGHT YES NO ABOVE 350 BELOW 350 YES NO FIGURE 2200 3 There is no edge that goes directly
16. ue False False False 5 True False True True False 6 True False True False False 7 True False False True True 8 True False False False True 9 False True True True False 10 False True True False False 11 False True False True False 12 False True False False False 13 False False True True False 14 False False True False False 15 False False False True False 16 False False False False False Should there be a direct arrow from the variable BITTEN to the variable MALARIA in the causal graph representing this system How do you answer this question Not by guessing not by common sense but by applying the definition for direct cause to the response structure above The definition requires that there is at least one test pair of causal assignments for BITTEN that make a difference to MALARIA So to apply the definition first locate the test pairs and then check to see if there are any in which the value of MALARIA is different In this case those pairs are 1 and 9 2 and 10 3 and 11 4 and 12 5 and 13 6 and 14 7 and 15 and 8 and 16 Is the value of MALARIA different across any of these pairs Put another way is the value of MALARIA different across causal assignments 1 and 9 Is it different across causal assignments 2 and 10 The answer is yes In causal assignments 8 and 16 where the variables besides BITTEN take on the values TABLE 2310 3 VARIABLE VALUES Variable Value INOCULATED False HAS GENE False DRINKER False then c
17. uld the causal graph among these two variables still have a direct edge from CALL PLACED to CONNECTED How do we answer the question in this case where the causation is indeterministic First we write out the indeterministic response structure and then apply the definition of indeterministic causation for variables 7 of 14 4 5 01 3 34 PM http www phil cmu edu 8080 jcourse cont modules causal_graphs Q000 printable html TABLE 2320 4 INDETERMINISTIC RESPONSE STRUCTURE FOR CONNECTED Assignment CALL PLACED CONNECTED Yes CONNECTED No 1 Send 50 50 2 End 0 100 Here is the definition of indeterministic causation for variables we gave in the module on Determinism and Indeterminism Definition Direct Indeterministic Cause If in a system of variables S there are any test pairs of causal assignments for X in which there is a difference in the probability of the effect Y then X is an direct indeterministic cause of Y relative to S So we need to apply this definition to the indeterministic response structure above Causal assignments 1 and 2 are a test pair for CALL PLACED and there is indeed a difference in the probability over CONNECTED across these two assignments So by applying the definition to the indeterministic response structure it is clear that CALL PLACED is a cause of CONNECTED In the cell phone example we were implicitly assuming background conditions that include a functioning cell phone with a charged battery
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