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An eBook on causal inference, aimed at the beginning student or observational researcher.
Epidemiology from a Causal Perspective By Patrick F. McArdle, Ph.D. Copyright © 2013, 2016 Patrick F. McArdle, Ph.D. All Rights Reserved Table of Contents Preface Chapter 1 – To have an effect in this world you have to do something Chapter 2 – Defining a causal effect Chapter 3 – Defining an association Chapter 4 – Using associations to estimate causal effects Chapter 5 – Potential outcomes Chapter 6 – Structural causal models Chapter 7 – Causal graphs Chapter 8 - Defining bias Chapter 9 – Bias from confounding Chapter 10 – Bias from selection Chapter 11 – Bias from analysis choice Chapter 12 – Bias from measurement error Chapter 13 – Bias from sampling variability References Preface Some of my best teachers when I first learned epidemiology were statisticians. As such, I was taught epidemiology from a modelling perspective. I was taught that an effect was a comparison of exposed vs unexposed, or cases vs controls. I was taught how to interpret statistical parameters such as the beta coefficient. I was taught that a confounder is a variable that changes my parameter of interest. At the time I didn’t question the perspective. I came from a data driven field and was at home in a statistical perspective. It wasn’t until later that I realized there was another perspective, a causal perspective, to many of the issues I had been taught. The differences were at times academic and at other times led to real practical changes in how I thought about things. Fundamentally the questions I am most interested in are of a causal nature. Questions such as: Should I wake up at 6am and go swimming before going into work? How effective is my teaching? What can I do to make the city I live in a better place? Those questions are all causal in nature. They all ask questions about the outcome of some action. Epidemiological tools applied from a causal perspective can help in the design of studies and analysis of data so that those questions can be answered. This seminar series aims to introduce epidemiology from a causal perspective, rather than from a statistical perspective. It is how, in retrospect, I wish I were taught. These notes are meant as a guide to be used in conjunction with the seminars. I have attempted to introduce the concepts below without the use of statistical notation. Specific notation will be used in the assigned readings. It is encouraged to apply the concepts describe in these notes using proper notation throughout the series. This journey contains plenty of opportunity for exercises and questions. I strongly recommend you take the time to work through each on your own. As students will learn throughout the course, there can be no effect without doing. Chapter 1 – To have an effect in this world you have to do something There is a small exercise I perform on the first day of class each semester (adopted from Rothman and Greenland1). I purposely start class with part of the lighting in the room turned off. During my introduction, I casually ask one of the students to turn on all the lights. Invariably the student will put down their pen, get up from their seat, walk over to the light switch on the wall and flip the switch. I then pose a question: why did you flip the switch? I typically get a very puzzled look. I ask the rest of the students if they would have done the same thing if asked to turn on the lights. Again, puzzled looks. The question is, why would the student invariably flip the switch? Why not knock on the desk and expect the light to go on? When asked, the answer typically is because they know the switch will turn on the light and they know knocking on the desk won’t. The question that stops them and makes them think is: Yes, but how do you know the switch will turn the light on and knocking on the desk won’t? This book attempts to answer that question. How do we know, or think we know, that a certain action will have an expected outcome? Think about it yourself. Are you convinced with certainty that if you flip the light switch in a dark room that the lights will go on? You shouldn’t be. Maybe the light bulb is burnt out. In that case, flipping the switch will not have an effect (yet, interestingly, not a single student when asked to turn on the lights has attempted to change the light bulb). What if I assured you that the light bulb is working properly, the wiring is intact, the switch is installed properly, and power is available. Then would you be convinced that if you flip the light switch in a dark room the lights will go on? You should be. If you flip the switch in that condition and the light does not go on, that just means there is another necessary condition I forgot to ensure you. Maybe the circuit breaker is not allowing current to flow. Given all other necessary condition, flipping the switch to the on position will turn the lights on. 100% of the time. If you can be convinced of that, I hope you can be convinced of this: Chance does not exist in this world. Philosophers, scientists and peoples throughout the ages have debated: do we live in a deterministic world? I won’t pretend to know the answer to that question, but I will argue that by believing the world is deterministic it makes it easier to understand some fundamental concepts when learning about epidemiology from a causal perspective. I wake up each morning and the first thing I do is flip the light switch on in my bedroom. In the past year, I noted the light turned on 95% of the time. There are many words to describe that number but a popular choice is probability. That is, there is a 95% probability (or “risk”) of the light going on when the switch was flipped on any given morning in the past year. That statement could be read as a statement of chance. There is an element of “luck” or “random chance” associated with the light going on. For example, the chance that the light goes on when I flip the switch tomorrow is 95%. Except chance does not exist in this world. The only intuitive interpretation of that number that I can find is that on 95% of the mornings within the past year all of the necessary conditions for the light to be on were present. Whether the light goes on tomorrow morning is not a matter of chance, or probability, but a deterministic function of whether all the necessary conditions (e.g. working light bulb, intact wiring, and available power) are present. When one makes the statement that tomorrow there is a 95% probability the light will come on when the switch is flipped, that is not a statement of chance. In its essence, it is a statement of ignorance. What is being said is that though I can control the switch, I am ignorant of the true state of each of the other necessary factors. Since the true state of the other factors is unknown, I will guess each condition will be absent at the same frequency tomorrow as they were last year. From this perspective, a statement of probability is a statement of ignorance about not the switch, but all the other necessary conditions. If it is known that the bulb was just replaced and trees were recently trimmed around power lines, I may believe with more certainty that the light will come on tomorrow. In the previous year, instead of flipping the switch myself each morning, my wife got out of bed to flip the switch and I just observed. Each day I observed my wife, sitting up, putting on a pair of slippers and walking over to the light switch to flip the switch. I noticed an interesting pattern. Each time my wife put on her slippers, the light went on a few seconds later. Not every time mind you, sometimes she put on her slippers just to go to the bathroom and then returned to bed. But thinking about this, the light didn’t come on every time she flipped the switch either. How do I know that the slippers aren’t responsible for the light coming on and the switch is? In a nutshell, that is epidemiology. Epidemiologist each day try to determine which actions are like flipping a switch and which actions are like putting on slippers. Most other fields of science have a very powerful tool to help them differentiate the switch from the slippers: the experiment. This is analogous to the first year, when I flipped the switch myself. If I thought putting my slippers on would turn the lights on, I could try it. I could wake up one day, put my slippers on and then wait 30 seconds to see if the lights came on. I could convince myself that the slippers did not cause the lights to come on by maybe trying it for several days, or by flipping the switch first to see if the lights did indeed come on thus ensuring the other necessary conditions were present. Epidemiologists only have limited access to the experimental tool in this sense, the use of the randomized control trial. This is a setting where an epidemiologist can control the metaphorical switch or slippers. Unfortunately the use of this powerful scientific tool is limited for a variety of reasons. In the majority of cases epidemiologist are like the second year, simply observing another individual making choices, flipping switches and putting on slippers. Fortunately there are tools available to help us distinguish switches from slippers even in the context of simply observing the world around us. But make no mistake, the ultimate goal is not merely observing and noting interesting patterns. We want to determine which actions will make this world a better place. If I determine that the world is a better place when my bedroom light is on in the morning, there are actions I can take. I can flip on the switch each morning, I can ensure proper working light bulbs and maybe I can trim that tree that is dangerously close to the power lines. In order to have an effect on this world you have to do something. It is the job of the epidemiologist to figure out which actions will have what effects. Chapter 2 – Defining a causal effect Assume I am interested in estimating the effectiveness of my course as judged by the performance of students on the comprehensive exam. The program I teach in admits a variety of students. Some students are incredibly intelligent and would pass the comprehensive exam whether they took my course or not. Some students are the opposite. Some students are not bright enough to pass the exam regardless of whether they take my course or not. Other students are the ones I hope enroll in my course. These students will pass the exam if they take my course but would fail otherwise. A fourth type of student is the type I hope avoids my class. These are students bright enough to pass the exam without taking my course, but if they take my course I end up confusing them and they fail the exam. The following ten students are eligible to take my course and have not taken the comprehensive exam yet. Table 2.1 Student Exam grade if student took my course Exam grade if student did not take my course Adam Pass Pass Ben Fail Fail Chris Pass Fail Dan Pass Fail Ed Pass Pass Frank Fail Pass Greg Pass Pass Henry Pass Fail Ian Pass Fail Jason Pass Fail Does this course have an effect on my student’s ability to pass the comprehensive exam? The outcome of a student’s exam if he either takes my course or not is called a potential outcome (also called a counterfactual). Each student has two potential outcomes of the comprehensive exam, the potential outcome if he takes my course and the potential outcome if he doesn’t take my course. In some instances, for students like Adam and Ben, those two potential outcomes are the same. Adam’s two potential outcomes are both Pass. Ben’s two potential outcomes are both Fail. But in other cases the potential outcomes are different. A causal effect is a contrast of counterfactual states. A causal effect can be defined on an individual, or it can be defined on a population. If defined on a population, it is typically defined as the proportion of individuals who experience some event (e.g. pass the exam) if they were all exposed to some treatment (e.g. my course) compared with the proportion who experience the same if all were exposed to some alternative (e.g. not taking my course, they took some other course). If the outcome is continuous, such as a numeric grade rather than given a pass/fail grade, the definition of a causal effect on the population is typically defined as a comparison of average test scores in the two counterfactual states. For this reason the causal effect on a population is usually called the Average Causal Effect (ACE). The Average Causal Effect (ACE) is a contrast between a population when everyone is exposed to a treatment and when everyone in the same population is exposed to an alternative treatment. Note this is not the only causal effect that can be defined on a population. A contrast between counterfactual states under any two exposure distributions can define a causal effect. The two distributions do not necessarily have to be all exposed and all unexposed. In the case described in Table 2.1, since I know both counterfactual states for everyone in the population, I can state the causal effect for each individual as well as the entire population. The average causal effect can be stated as the difference in proportions of students who will pass the exam under the two conditions, 80% of the students would pass the exam if they all took my course and 40% of the students would pass the exam if none of them took my course. Thus I could say that 40% (80%-40%) more students will pass the exam if all of them took my course compared to if they didn’t. I could also claim that twice as many (80%/40% = 2) students will pass the exam if all took my course compared to if none did. Note that this is not the same thing as saying that my course makes an individual twice as likely to pass the exam or increases their score by 40%. All students either pass the exam or don’t, there is no chance in this world. Word spreads. Take my course and you are more likely to pass the comprehensive exam. Is that good advice for everyone? Say my course starts at 8am and generally speaking students like to sleep in, so there is a good reason to not take my course if one could avoid it. Would you recommend everyone take my course? Buoyed by the success of my course in the first year, I don’t change a thing. The next year the following ten students are eligible to enroll in my course. Table 2.2 Student Exam grad if student took my course Exam grade if student did not take my course Alice Pass Pass Barb Pass Pass Christy Pass Pass Deb Pass Pass Elise Pass Pass Fran Pass Pass Grace Fail Fail Hannah Fail Fail Isabella Fail Fail Jane Fail Fail What is the causal effect of my course in the second year? Does my course have the same effect in the second year as it did in the first? Why or why not? Many causes, many effects. Clearly, my course is not the only factor which influences whether or not a student passes the comprehensive exam. For some students, like Chris, Dan, Henry, Ian and Jason, it is a contributing factor. But for other students, like Adam, Ed, Alice and Barb; they can pass the exam whether they take my course or not. Would it be correct to say that my course is one of the factors that contribute to a student passing the comps? An action can be said to be a cause of an outcome if there is at least one unit in the population of interest which only experiences the outcome after the action and would not otherwise. The population I am interested in having an effect on is the population of students enrolled in the PhD program. I am not interested in the effect my course will have on high school students, for example. Since students like Chris, Dan, Henry, Ian and Jason enroll in PhD programs, I can rightfully claim that offering my course is a cause of passing the comps, even if similar students do not enroll every year. At the risk of oversimplifying things, let’s assume there are a limited number of ways to pass the comprehensive exam. One way is the student can be extremely intelligent, a self-learner and possess excellent studying habits. These types of students may only need a few introductory courses and would be ready to pass the exam. They would not need my course. Call these students intelligent. Second, there might be students who are bright but need my course to reinforce certain concepts. These students require my course to pass the exam. Let’s call these students bright. Are there another other types of students that can pass comps? Yes, the third type of student is like Frank. He would pass the exam if he didn’t take my course but somehow I confuse him and he would fail the exam if he took my course. Let’s call Frank and students like him confusable. Given this over simplification, there are three types of students who can pass the comprehensive exam: intelligent students regardless of whether they take my course or not, bright students as long as they take my course, and confusable students as long as they don’t take my course. Pictures can be drawn to represent these types of students. Sometimes called Sufficient Component Cause diagrams. These pictures were introduced to Epidemiology by Rothman3. Similar concepts of sufficiency and necessity have been developed, first in philosophy4 and then also the law5. Lawyers refer to this concept as NESS: Necessary Elements of Sufficient Sets. All are fundamentally the same. These pies represent collections of components which, when acting together, are sufficient to cause the outcome. A key factor in the diagrams is that each piece of the pie is necessary for the outcome. Since there is no chance in this world, we can assume that every time all the pieces come together, the outcome will definitely happen. If the outcome doesn’t happen all the time, that just means we are missing one of the pieces of the pie. Any two factors are said to interact if they are both required for the determination of the outcome. Interacting variables can be easily identified by noting their presence in a single causal pie. For example, the light switch and a working light bulb interact to produce light; remove either component and there will be no light. A cause is said to have a monotonic effect if its presence always causes the outcome or is neutral. If its presence prevents the outcome in some individuals, it does not have a monotonic effect. Another way of saying the same thing is referring to whether or not monotonicity holds for a certain exposure. My course does not have a monotonic effect. It prevents Frank from passing the exam. If and only if students like Frank are not admitted to PhD programs can we claim that the monotonicity assumption holds for my course with respect to the comprehensive exam. Note an interesting property of non-monotonic causes. Assume 10 students are accepted into the program in a given year. Five of them are confusable like Frank and 5 of them are bright like Chris. My course would have an effect on every one of the ten students. But if all ten took my course, 50% of them would pass the comps. If all ten avoided my course, again 50% of them would pass the comps. There would be no Average Causal Effect. Yet my course did have an effect on every student! Alternatively, what does it mean if a non-null Average Causal Effect exists? It is possible my course has an effect on more than just comprehensive exam scores. It is possible it also inspires, motivates, discourages or even bores some students. Assume my course discourages Dan and Ed to the point that both drop out of the program before taking the comprehensive exam. In that case, it is not clear what it means to say that Dan would pass the exam if he took my course. It is not clear precisely because my course has an effect on Ben such that he never takes the exam, thus not providing him the opportunity to either pass or fail the exam. This phenomenon of one action having multiple effects, the occurrence of one making the occurrence of a later one impossible, is commonly called the problem of competing risks. Note that many different causal questions could be asked about the effectiveness of my course. My program may simply care about students passing the exam. In that sense, students who drop out of the program are equivalent to students who fail the exam: that is, equivalent in the sense that they did not pass the exam. In that case, the counterfactual states of interests are not Pass or Fail the exam, instead Pass or Not Pass the exam. Alternatively, the program may have a new system that they have developed which they feel confident will keep students in the program (e.g. unreasonably high stipends, maybe?). In that case, the question may be a compound counterfactual: what would be the effect of dually administering my course and this new incentive program? In this case, Dan and Ed’s counterfactuals in tables 2.1 and 2.2 retain meaning. They are the counterfactuals under the joint administration of the new incentive program and either registration or not of my course. Most actions in this world have more than one effect. Even the simple flipping of the switch will bring about light as well as heat. When using counterfactuals to describe the future states of the world, it is important to remember the action under study may have an effect on more than just the outcome of primary interest. Chapter 3 – Defining an association The effects described above are sometimes called the “true” causal effect, either on the individual level or the population level because they are based on knowing each true potential outcome for each student. But in reality the true potential outcomes are not known. I only know if each student took my course and whether they passed the exam or not. Of the ten students eligible to take my course the first year, five of them did. After the comprehensive exam I request all the students’ grades and find: Table 3.1 Student Did the student take my course? Exam grade of the student Adam Yes Pass Ben Yes Fail Chris Yes Pass Dan Yes Pass Ed Yes Pass Frank No Pass Greg No Pass Henry No Fail Ian No Fail Jason No Fail What measures of associations can be made of these data? Of the five students who took my course, 4 of them passed the exam, or 80%. Of the 5 students who did not take my course, 2 of them passed the exam, or 40%. Thus 40% more students (80%-40%) who took my course passed the exam compared to those who did not take my course. Similarly, students who took my course were twice as likely (80%/40%) to pass the exam than students who did not take my course. Please note that the association between taking my course and passing the comprehensive exam is quite different than the casual effect of taking my course on passing the comprehensive exam. The association contrasts two different groups of students, those who took my course and those who did not. The causal effect compares two different counterfactual outcomes within the same person. If a causal effect is defined as a contrast between counterfactual states, an association is defined as a contrast between observations. The following table provides the registration status and comprehensive exam grade for each student in the second year I offered the course. Table 3.2 Student Did the student take my course? Exam grade of student Alice Yes Pass Barb Yes Pass Christy Yes Pass Deb Yes Pass Elise Yes Pass Fran No Pass Grace No Fail Hannah No Fail Isabella No Fail Jane No Fail What association estimates can be made of these data? Compare this association with the causal effect given in Table 2.2. Chapter 4 – Using associations to estimate causal effects Estimation of causal effects requires knowledge of counterfactual states. Yet we cannot observe all counterfactual states. Some are, by definition, counter-to-fact. This has been called the fundamental problem of causal inference2. Since we do not have access to the counterfactual state, we must rely on observations from the factual world. So how is it that an association between observations can tell us anything about the contrast of counterfactual states? Generally speaking there are two approaches. In the case of estimating the effect of my course, we can observe the same student who takes the comprehensive exam twice, once after a semester she did not take my course and once after a semester which she did take my course. Or we can observe two different students or groups of students, one who took my course and one who did not take my course who all take the exam at the same time. Both approaches rely on assumptions2: Approach 1: Temporal Stability and Causal Transiency Assume we wish to estimate the effect of my course on Chris. We know from Table 2.1 that if Chris takes my course he would pass the exam, and if he doesn’t he will fail. Chris decides not to enroll in my course the first year he can do so and subsequently takes the exam. He fails. Chris then decides to take my course the next year and passes the exam on the second try. I would be tempted to believe that I observed both counterfactual states and would declare my course as a causal factor in Chris passing the exam. That declaration relies on two assumptions. The first assumption is that the effect of my course is independent of when the student enrolls. That is the effect of my course is the same from year to year. This may or may not be true, particularly if I improve my teaching over time, update lectures or assign different readings. This is called the temporal stability assumption and assumes the action is consistent over time. The second assumption states that it doesn’t matter in what order the students takes or doesn’t take my course. This assumption is called causal transiency because it assumes that the first enrollment decision or the taking of the exam the first time does not influence the second exam score. This is unlikely to be true in this case. Chris may be much more likely to pass the exam the second time, even if he didn’t take my course. He would have a better idea of the topics covered and could prepare for the exam more efficiently. There are certain scenarios when observing the same unit at two different times may be a very effective approach. The light switch is a good example. We can convince ourselves that the light will not come on when the switch is in the off position by just staring at the light with the switch in the off position. I can then put the switch in the on position and observe the light go on. I use the temporal stability and causal transiency assumptions to convince myself the light would have remained off if I did not flip the switch. Approach 2: Unit Homogeneity The second approach to get around the fact that we cannot observe the counterfactual is to assume that Chris is similar to another student in every relevant respect and that the other student’s exam score can be used in place of Chris’ unobserved counterfactual exam score. This assumption is called unit homogeneity since it assumes the two students are homogenous in all respects relevant to taking the exam. We know from Table 3.1 that Chris did take my course and passed the exam. I can simply select another student who did not take my course and assume his test grade is equivalent to what Chris’ would have been had Chris not taken my course. One can naturally see the problem. I could choose the wrong student. If I select Henry, I observe that Henry did not take my course and failed the exam. I would conclude that Chris would have also failed the exam had he not taken my course and would declare my course a causal influence of Chris’ passing score. If I instead select Frank, Frank did not take my course and passed the exam any way. I would conclude that Chris would have also passed the exam without taking my course and declare my course a failure. The key to making valid causal inferences using this second approach is to select a student, or a group of students, with the same exam score as Chris’ missing counterfactual. In essence, Chris’ missing counterfactual exam score is exchanged with someone else’s observed score. This concept, sometimes referred to as exchangeability, is fundamental to making causal inferences from observational data and will be explored further when confounding is discussed. Chapter 5 – Potential outcomes model The potential outcome model was introduced in the 1920s by statistician Neyman and was made popular by Rubin in the 1970s. As such, it is sometimes referred to as the Neyman-Rubin models for causality. It aims to draw causal conclusions from purely observational data by drawing on an analogy to a randomized control trial. To apply the potential outcome model, one must define three objects: units, treatments and the outcome. The units are the members of the population under study. For example, I am interested in estimating the effectiveness of my course for PhD students. Therefore the units can be defined as PhD students. The treatments are the different actions I am investigating; the two treatments above are for each student to either take my course or not take my course. Each unit can only have one treatment at any given time. To say another way, each student can either be taking my course or not in a given semester. Finally the outcome is the measure of ultimate interest; in this case, the final determination of the comprehensive exam. In order to apply the potential outcome model to a problem, units, treatments and outcomes must be defined. The notion of time is very important in discussing causality. In order for the treatment to have an effect, it must come before the outcome. If the treatment is applied after the outcome, by definition, it cannot have an effect. That is not to say the treatment can never occur after the outcome. It can. For example, a lot of students register for my course after they have taken the comprehensive exam. But obviously, my course cannot have a causal effect on their exam score in that scenario. To have an effect, the treatment must occur before the outcome. Each unit can be described by many attributes. For example, each student can be described by having either good or poor study habits, a certain entrance exam score or undergraduate major. It is important to note if each attribute can be determined either before or after the treatment was assigned. Some attributes may change over time and thus can be measured both before and after with differing results. The potential outcome approach specifically assigns variables to the counterfactual states. In our example, there will be two counterfactual variables: one that holds the exam score if the student took my course and one that holds the exam score if the student did not take my course. There is also a third factual variable that indicates whether or not the student actually passed the exam or not. Consistency states that the unit’s potential outcome variable corresponding to the observed treatment is equal to the outcome actually observed. Note that this simple statement implies a lot; and potential is best read after one convinces themselves they live in a deterministic world. It implies that, for example, the causal effect of my course is the same regardless if I teach the course or another instructor. If the course could have a different causal effect if taught by different people, then the treatment was not well enough defined initially. The treatment should not be just “my course” but instead should be “my course taught by me”. Rubin incorporates consistency into an assumption that he calls SUTVA6. SUTVA is the Stable Unit Treatment Value Assumption and must be met before a question is properly phrased to have a causal answer. This assumption takes two things into consideration: consistency and non-interference. SUTVA is the assumption that the observable variable will equal the counterfactual variable when the corresponding treatment is assigned regardless of how the treatment was assigned (consistency) or what treatment other units receive (non-interference). Non-interference is an assumption that may or may not hold. It states that the application of the treatment on one unit does not influence the outcome of another unit. This would hold if, for example, the fact that Chris took my course does not influence Greg’s exam score. Causal inference models can be extended to allow for violations of non-interference7,8. A common violation of non-interference is in infectious disease, when the infection status of one individual may influence the disease outcome of another person. There is one further criteria that must be satisfied before a question is properly described using the potential outcome model: positivity. That is, every unit must have some non-zero probability of receiving the treatment. For example, it is unclear what is meant when one asks about the causal effect of my course on individuals who are not eligible to take my course. Positivity states that each unit must have a non-zero probability of receiving each level of the treatment. Chapter 6 – Structural causal model The structural causal model (SCM) approach to causality has been most notably championed by Pearl.9 The primary difference between the SCM and the potential outcome approach is that the counterfactual variables are not defined in the SCM approach. Instead the values of all variables are determined by a series of functions. Together these functions are called the Model. The model is referred to as “structural” if each of the functions is invariant to changes in other functions. Fundamentally, I want to know if my course is one of the variables that determine whether one passes the comprehensive exam or not. If it is, then I can use my course like a switch that turns on and off a light. I can apply it, and as long as certain other conditions exist, certain students will pass the exam that otherwise would not have. Of course whether a student does or does not take my course is also a function of many things. It could be a result of my reputation, the aptitude or ambition of the student and maybe the time the course is offered. Note that it is possible that some of the factors that determine whether or not a student takes my course, may be some of the same factors that determine if the student will pass the exam. The student’s aptitude may be one of those factors. The key to the “structural” model is that the function that determines one variable is invariant to changes of other functions. That means that the function that determines whether or not a student passes the exam will not change if, for example, the department makes my course mandatory. In that case, the students’ aptitude may still influence their exam score but will not influence whether or not they take my course. The SCM approach views the world as the output of these functions. To know if one variable affects another, we simply remove nature’s function and replace it with a function that we control. We gain the power of viewing each action as a light switch that we control. We can flip the switch off or on at different times or for different units and observe the results. Chapter 7 – Causal graphs Causal graphs are graphical representations of the structural equations. They come with their own terminology, described here. A causal graph is made up of nodes and edges. Each node represents a variable or a set of variables. Each edge is directed, that is, it is drawn like an arrow. The direction of the edges is determined by the functions given in the structural equations. An arrow goes from each variable that is an input into the function and points towards the output variable of the function. A causal graph is said to be a DAG if it is a Directed Acyclic Graph. Directed refers to each edge having one and only one direction. Acyclic refers to the absence of any circular paths in the graph. That is, one cannot start at a node, and follow directed edges and eventually get back to the original node. There are two types of nodes in a graph: exogenous and endogenous. Exogenous variables are those that do not have any directed edges pointing into the variable. Endogenous variables are those with variables with directed edges pointing into them. A DAG is referred to as “complete” if all common causes of any two variables are in the graph. To complete a graph, it is then possible one must include variables that are unknown, latent or simply unmeasured. A path describes a series of nodes connected by edges, regardless of the direction of the edge. A path is described as either open or blocked. There are 4 rules that can be used to determine if a path is opened or blocked. Rule 1: If there are no variables being conditioned on, a path is blocked if and only if two arrow heads on the path “collide” at some variable. That variable is called a “collider”. Rule 2: Any path that contains a non-collider that has been conditioned on is blocked. Rule 3: A collider that has been conditioned on does not block a path. Rule 4: A collider that has a descendant that has been conditioned on does not block a path. To summarize these four rules, a path is blocked if and only if it contains either (A) a non-collider that has been conditioned on or (B) a collider that has not been conditioned on and has no descendants that have been conditioned on Two variables are said to be d-separated if all paths between them are blocked. Knowing if paths are open or blocked can help to know if two variables are expected to be dependent or independent of one another. This knowledge helps to identify when associations are expected to be biased or unbiased estimates of causal effects. Chapter 8 - Defining bias In cases of statistical estimation, one uses an estimating procedure to make estimates of some target quantity. In the particular instance of estimating causal effects, the target quantity is some contracts between counterfactual states. Bias is a term used to describe the contrast between the observed estimate and the target quantity. Are the associations based on the data provided in Tables 3.1 and 3.2 biased or unbiased estimates of the Average Causal Effects presented in Table 2.1 and 2.2? Chapter 9 – Bias from confounding Confounding is the difference between doing and observing. Confounding can result in a bias even if we study every unit in the population and measure each variable without error. It is often confused with bias itself and with purely observational measures, such as collapsibility. Confounding exists when the treatment under study shares common causes with the outcome. Using the SCM/DAG approach, these common causes can be easily identified by either examining the structural model or the graph. In the potential outcome approach it is a little more difficult. One must review all attributes of units that were determined prior to the assignment of the treatment and ask if each one is a cause of the outcome. In either case, confounding cannot be identified without considering causal connections10. Confounding cannot be identified using observations alone. It is important here to distinguish confounding from collapsibility11,12. Confounding is a causal notion, collapsibility is a purely statistical one. Collapsibility refers to a situation when a stratified estimate of an association between two observable variables is the same as the non-stratified estimate. Note that collapsibility has nothing to do with a contrast between counterfactual states. Confounding on the other hand produces a bias, which occurs when the association is not a valid estimate of the contrast between counterfactual states. Thus confounding relies on knowledge of the counterfactual while collapsibility does not. Confounding can exist with or without non-collapsibility and non-collapsibility can exist with or without confounding. A treatment assignment can be said to be ignorable if the mechanism which assigns the treatment is independent of the outcome. The most common example of an ignorable treatment assignment is using a flip of a coin. The factors which influence whether or not a coin lands on heads, the distance from the ground, the amount of force used to flip it, etc., can all safely be assumed to be irrelevant to any disease under study. Exchangeability is a term used to describe a situation when one group of individuals is similar enough to another group so that one’s observed factual state can be exchanged for another’s unobservable counterfactual state. Note that an ignorable treatment assignment is expected to produce an exchangeable condition, though this may not always be the case in every instance. For example, even if course registration was done by a flip of the coin, it may be that in any one semester, more smart students take my course compared to those who do not. Thus the group of students who took my course may not be exchangeable with the group of students who did not take my course. This lack of exchangeability can lead to bias. Thus lack of exchangeability can occur from confounding, i.e. the treatment and outcome sharing a common cause, or it may occur in a given sample simply due to simple sampling variation. Chapter 10 – Bias from selection It is usually assumed that we cannot observe the entire, potentially very large, population. Instead we rely on observations of a sample of individuals from the larger population of interest. The selection of individuals upon which to estimate associations can lead to bias. Hopefully participation in a study won’t cause one to become diseased. But it is very likely that having a disease makes one more likely to be selected into a study. Indeed, this is the very defining nature of a case control study design. If the exposure is also associated with selection, a bias can result. In order to study the effectiveness of my course, I must gain access to the exam scores of the students. Assume that each student must give his permission for me to access his exam score. Since students who take my course like and trust me, students who take my course are more likely to give me permission to see their exam scores. Also students who do not do well on the exam are less likely to share their exam scores to avoid embarrassment. Given that, I attempt to study the effectiveness of my course after the first year, assume all five students who took my course give me permission to see their exam scores, but only 2 of the 5 students who did not take my course give me permission. Those two students are Frank and Greg since they have no fear of being embarrassed by their passing grade. What measure of association can be made based on the 7 students who allow me to see their exam scores? Is that association a biased estimate of the causal effect? Selection can be illustrated through the use of DAGs by including a node for selection into the study. The estimates based upon the study are thus conditional upon being selected into the study. Including a node for selection into each DAG and conditioning upon it can help one visualize potential open paths between the exposure and the outcome. Chapter 11 – Bias from analysis choice Bias can also be introduced by the choice of analysis. The most obvious example of this is a bias introduced when one conditions upon a variable in a causal pathway when attempting to estimate the total average causal effect. Generally speaking bias has the chance to be introduced whenever a collider is conditioned upon (recall the definition of a collider from Chapter 7). Conditioning on a collider will introduce an association between the collider’s causes in at least one strata of the collider. Bias from selection can be thought of as a specific case of the more general collider stratification bias if the bias occurs in the strata of subjects selected into the study. Collider stratification bias is a bias that occurs when a collider is conditioned upon, thus forming an association between the exposure and outcome that is not relevant to the causal effect. When estimates of association are to be used to make inference on causal effects, great care needs to be taken to choose a set of conditioning variables. One aims to choose those variables which help to remove the effects of confounding without introducing bias. One safe approach is to select the conditional variables from the set determined prior to the assignment of the treatment. This will ensure no variables on the causal pathway will be included. Note that it may not be appropriate to select all variables determined prior to treatment assignment, as there are cases when some of those variables can introduce bias (so called M-bias). Alternatively, there are instances when variables measured after the treatment could be useful for removing bias due to confounding. Chapter 12 – Bias from measurement error It should be assume that all measurements made come with some error, even if that error is negligible. This assumption forces one to think about how measurement error can bias association estimates. This is typically done by including a node for each measured variable with two inputs, the true state of the variable and some error associated with the measurement process. Then one should think of any effects other variables may have on the measurement process. Thought of in this way, all measured variables are descendants of variables of true interest. Therefore, if a measured variable is conditioned upon, it may not fully close a path that otherwise the true variable may have. This can result in some bias, even if the measurement error is unrelated to any other variable in the system. Chapter 13 – Bias from sampling variability As mentioned above, bias can result from a lack of exchangeability between the groups of individuals being compared, even in the absence of confounding and the presence of an ignorable treatment assignment. Fortunately, this bias is expected to decrease as the sample size grows larger. Thus an infinitely large, perfectly conducted randomized control trial is expected to produce an unbiased estimate of the causal effect. But in practice, particularly in small trials, differences in estimates can be observed. If all other bias contributing factors can be ruled out, those differences can be explained simple by the process of random sampling variability. References 1 Rothman, K. J. & Greenland, S. Causation and causal inference in epidemiology. Am J Public Health 95 Suppl 1, S144-150, doi:10.2105/AJPH.2004.059204 (2005). 2 Holland, P. W. Statistics and causal inference. J Am Stat Assoc 81, 945-960 (1986). 3 Rothman, K. J. Causes. American journal of epidemiology 104, 587-592 (1976). 4 Mackie, J. L. Causes and conditions. American Philosophical Quarterly 2, 245-264 (1965). 5 Wright, R. Causation in tort law. California Law Review 73, 1735-1828 (1985). 6 Rubin, D. B. Which ifs have causal answers. Journal of American Statistical Association 81, 961-962 (1986). 7 Rosenbaum, P. R. Interference between units in randomized experiments. J Am Stat Assoc 102, 191- 200 (2007). 8 Hudgens, M. G. H., M.E. Toward causal inference with interference. J Am Stat Assoc 103, 832-842 (2008). 9 Pearl, J. Causality : models, reasoning, and inference. (Cambridge University Press, 2000). 10 Hernan, M. A., Hernandez-Diaz, S., Werler, M. M. & Mitchell, A. A. Causal knowledge as a prerequisite for confounding evaluation: an application to birth defects epidemiology. Am J Epidemiol 155, 176-184 (2002). 11 Greenland, S. & Robins, J. M. Identifiability, exchangeability, and epidemiological confounding. International journal of epidemiology 15, 413-419 (1986). 12 Greenland, S. & Robins, J. M. Identifiability, exchangeability and confounding revisited. Epidemiol Perspect Innov 6, 4, doi:10.1186/1742-5573-6-4 (2009). lider stratification bias if the bias occurs in the strata of subjects selected into the study. Collider stratification bias is a bias that occurs when a collider is conditioned upon, thus forming an association between the exposure and outcome that is not relevant to the causal effect. When estimates of association are to be used to make inference on causal effects, great care needs to be taken to choose a set of conditioning variables. One aims to choose those variables which help to remove the effects of confounding without introducing bias. One safe approach is to select the conditional variables from the set determined prior to the assignment of the treatment. This will ensure no variables on the causal pathway will be included. Note that it may not be appropriate to select all variables determined prior to treatment assignment, as there are cases when some of those variables can introduce b