Confounding and Interaction: Part II

Confounding and Interaction: Part II
Confounder vs. intermediary variables Factors to be considered as potential confounders in designing a study Methods to reduce confounding during study design: Randomization Restriction Matching during study analysis: Stratified analysis Interaction What is it? How to detect it? Additive vs. multiplicative interaction Comparison with confounding Statistical testing for interaction Implementation in Stata Before we begin today’s material, do we have any questions from last week? Here is our roadmap for by today. Last week, we defined and discussed what confounding is. Today, we will carry on this discussion where we left off last time by contrasting a confounding variable with an intermediary variable. We’ll then discuss which factors you should consider as potential confounders when designing a study We’ll then spend a large part of the session discussing ways we can prevent or manage confounding and we will divide things up into what can be done in the design phase and in the analysis phase. We’ll talk about randomization, restriction, matching, and stratified analysis. We will then use the remainder of the session to discuss the counterpart to confounding, what is known as interaction. We will define it and how to detect it. . . Describe the two different kinds of interaction: additive and multiplicative Describe how to do statistical testing for interaction, including how to implement this in Stata.

Exercise and CAD Exercise HDL HDL CAD
HDL is associated with exercise HDL is associated with CAD When evaluating the relationship between exercise and CAD, is HDL a confounder or an intermediary? Exercise HDL HDL CAD Remember this from last time. We know that exercise influences HDL levels and HDL is associated with coronary disease. If we wanted to evaluate the relationship between exercise and coronary disease, is HDL a confounder or an intermediary?

It depends on the pathway under investigation
If interest is in a pathway other than through HDL, then HDL is a confounder Here, HDL is extraneous to pathway under study Confounding factors are extraneous factors Exercise And again, it depends upon the pathway under investigation. If the interest is in a different pathway, one other than through HDL, then HDL is extraneous to the pathway under study. For example, consider if we speculate that exercise might influence the occurrence of CAD by another mechanism of action - something other than HDL - maybe thru control of hypertension. Confounding factors are sometimes actually called extraneous or nuisance factors. Here, HDL is extraneous, is a confounder to what we want to study and we would want to control for it. not yet specified mechanism HDL ? CAD

HDL is not a confounder here
Exercise and CAD If pathway under study goes thru HDL, then HDL is an intermediary variable. e..g., Does exercise influence CAD risk in a newly studied population (elderly Asians)? Hence, classification of HDL as confounder or intermediary depends upon the biological pathway under investigation and the research question Exercise But if the pathway you are studying does go through HDL, then it does not make sense to control for HDL. What do I mean when I say pathway goes thru HDL? Consider for example that you want to look at the assoc between exercise and CAD risk in a newly studied population, say elderly Asians? You know that in other populations that exercise affects HDL and HDL affects CAD. So, when you are studying the association between exercise and CAD in elderly Asians, you will be perfectly willing to believe that if exercise does have an effect, that it is mediated by its influence on HDL. That said, HDL is right on the possible causal pathway and hence HDL is an intermediary variable. We would not want to control for it. HDL would not be a confounder. Looked at another way, what if you were the first to study exercise and CAD in elderly Asians and say you controlled for HDL and you found no association between exercise and CAD. Would you feel comfortable concluding there is no association? Of course not. Hence, classification of HDL as confounder or intermediary depends upon the biological pathway under investigation HDL is not a confounder here HDL CAD

What is NOT a Confounder?
Variables that are the RESULT of the disease, regardless of their association with the exposure are NOT confounders Some of you may have noted a lack of symmetry in this diagram we showed before for the definition of a confounder. We saw arrows in every which direction between the confounder and the exposure, but we did not have an arrow from disease to C. In other words, factors that are caused by the disease, even though they are associated with the exposure, ARE NOT confounders. Confounder D

Cough is not a confounder.
Smoking Cough What if you wanted to look at the association between smoking and lung cancer? A naïve investigator might consider calling “cough” a confounder because it is “associated”with smoking and “associated” with lung cancer. But if you did you would end up washing away much of the effect of smoking So, variables that are the result of the disease are not confounders. Looked at another way, the issue with confounding is not just blindly looking for factors associated with the exposure and the outcome but it is instead looking at pathways that are extraneous to the pathway under study and making sure that any effect that you declare is not the result of this extraneous pathway. In this situation, where cough is a result of lung ca, there is not a clear pathway leading from smoking to cough to lung cancer. ? Cough is not a confounder. Do not adjust for it! Lung CA

When Planning a Study, Which Factors Should be Considered as Potential Confounders?
Any factor for which prior evidence indicates it is a confounder and In newer research areas: factors known to be associated with the disease and which may be associated with exposure When in doubt, plan on measuring ALL factors associated with the disease i.e. If you don’t, you may regret it later Now that we have explained the theoretical considerations of what confounding is and what it does and what variables produce confounding, what is the practical conclusion of all of this when planning studies? When we plan our studies, which factors should we be considering as potential confounders? Certainly, if you are working in an established field, you should plan on measuring any factor that has previously been identified as a confounder. For example, if you are revisiting the association between diabetes and CAD in an understudied population, you will want to control for obesity, a known confounder in this association. But often, we are working in less well understood areas. For example, we are often evaluating new risk factors and there is no prior work describing the potential confounders. If this is the case, you would for sure want to take into account (ie make sure that you measure) any factor that theoretically be a confounder; ie factors you know to be associated with the disease in question and that might be associated with the exposure. Take the stance of measuring greater, rather than fewer, variables because you don’t want to be caught in the position of not having the made the measurement when a reviewer asks you if you controlled for factor X. Make sure that you measure any factor that is known to be associated with the disease. So, as was the case with selection bias and measurement bias, there needs to a lot of thinking up front in the study design. Many of the solutions to confounding, as we will talk about next, occur in the analysis phase but they depend upon measurements taken during the study and hence planned for.

Seeking cause of high Marin cancer rates Activists canvass residents to search for trends Sunday, November 10, 2002 Thousands of volunteers scattered across Marin County under baleful skies Saturday in an unprecedented grassroots campaign against the region's soaring cancer rate. Armed with surveys, some 2,000 volunteers went door to door in every neighborhood in the county, asking people whether they or anyone in their household has ever been diagnosed with cancer in Marin. The volunteers hope to collect enough money to hire an epidemiologist to analyze the data for use in future studies. That’s what worries me about this recent Chronicle article which reports on the grassroots effort to study the reported high incidence of breast cancer in Marin. The article reports 2000 volunteers are going door to door to collect data and that they hope to put together enough money to hire an epidemiologist to analze the data. Well, unfortunately, the time to hire the epidemiologist was before the survey not after. Let’s hope that this is just sloppy reporting by the Chron.

Preventing or Managing Confounding
ANOTHER PATHWAY TO GET TO THE DISEASE Now let’s turn to the prevention or management of confounding. The basis of this is pretty straightforward when you remember our schematic diagram which required that for confounding to occur, that a factor had to be associated with the exposure under study and the outcome in question. Confounder D

Methods to Prevent or Manage Confounding
That said, all of these methods I am going to describe for managing confounding work by one of the two ways. All of the methods to manage confounding either block or interrupt the association between the confounder and the exposure (shown in the top diagram) or the association between the confounder and the disease (in the bottom diagram). D

Methods to Prevent or Manage Confounding
By prohibiting at least one “arm” of the exposure- confounder - disease structure, confounding is precluded And so in words: by blocking one of these two arms, you, by definition, preclude confounding from occurring.

Randomization to Reduce Confounding
Definition: random assignment of subjects to exposure (e.g., treatment) categories All subjects  Randomize One of the most important inventions of the 20th Century! Exposed Unexposed If one is able, randomization is the cadillac strategy to reduce confounding. As you know, in randomization, all study subjects are randomly allocated to an exposure group. This is most commonly some form of therapy, but can also be an exposure in the sense of a diet or other behavioral modification. As usual, I’m using the word “exposure” very generally here to mean predictor or independent variable. Since the exposure groups are created by a random process, therefore, the distribution of any variable you can think of is theoretically the same in the exposed group and the unexposed group. We’ll have a lot more to say about it in our course on clinical trials, but randomization is truly one of the most important inventions of the 20th century.

Therefore, there can be no association between any would-be confounder and the exposure. Therefore confounding cannot occur.

Definition: random assignment of subjects to exposure (or treatment) categories All subjects  Randomize Applicable only for intervention (experimental) studies Special strength of randomization is its ability to control the effect of confounding variables about which the investigator is unaware Does not, however, eliminate confounding! Exposed Unexposed Now, obviously randomization is only relevant when the investigator has control over the exposure such as when we are studying drugs or other interventions. In other words, when you are doing things to people or you are instructing them to do things to themselves. There are sadly limitations to this in that we cannot, for example, randomize people to smoking or air pollution or unprotected sexual behavior. The special strength of randomization, which makes it the cadillac approach, is that not only does it prohibit confounding by known confounders but also reduces confounding by unknown confounders. Remember, the distribution of any variable you can think of is theoretically the same in exposed and unexposed groups. One note: randomization in small studies may not lead to perfect balancing of potential confounding factors in the 2 exposure arms (in the case of a drug trial, treatment and placebo, for example). Just by chance alone, there can be imbalance. The bigger the study, the less of a problem this is. For those of you in the trial business, you will want to pay attention to this and know that there are also certain specialized randomization techniques that help to assure balance of potential confounders even in small studies

Restriction to Reduce Confounding
AKA Specification Definition: Restrict enrollment to only those subjects who have a specific value/range of the confounding variable e.g., when age is confounder: include only subjects of same narrow age range In observational work, we unfortunately don’t have the luxury of randomization and hence we have to live by our wits. Restriction,aka specification, is one such method, probably the most blunt, we can use. Restriction is where restrict enrollment in a study just to those subjects who have a specific value of a confounding variable. For example, when age is confounder, like it often is because age is related to many disease states, we would restrict enrollment in our study to only those persons with a specific value or range of age.

Graphically, what restriction does is to eliminate any possible association between the exposure and the confounder and between the disease and confounder. This is because there is no variability in the values of the confounding variable. It is constant.

Advantages: conceptually straightforward Disadvantages: may limit number of eligible subjects inefficient to screen subjects, then not enroll “residual confounding” may persist if restriction categories not sufficiently narrow (e.g. “decade of age” might be too broad) limits generalizability not possible to evaluate the relationship of interest at different levels of the restricted variable (i.e. cannot assess interaction) The advantage of restriction is that it is very straightforward and straightforward is good -especially when you trying to convince people who review your manuscripts. There are disadvantages, however, such as restriction reduces the number of persons who are eligible and it may take a lot of work to sift through persons to find those with the level of the confounder you are looking for. It is inefficient to have to screen a lot of subjects and only use some of them. If you don’t restrict the values of the would-be confounder narrowly enough, you may not accomplish what you thought you did. In other words you may have residual confounding. For example, say if you are studying the association between a new risk factor and coronary disease and restrict your study population to those 40 to 50 years old. This may not be enough to preclude confounding because age is associated with the disease even in this seemingly narrow range and conceivably could be associated with the new risk factor or exposure under study. Another drawback with restriction is that by limiting the study population to only a specific group with a certain value/range of a confounder, you limit the generalizability of the study (for example, if you restrict to younger persons, you will have a tough time generalizing to older persons). Related to this is that if you restrict to only one level of an extraneous variable, you cannot evaluate the relationship of interest at different levels of the extraneous variable (ie you cannot assess interaction -something we will discuss later today). A simple example from the field of HIV is that if you were evaluating in an observational study the association between pre-therapy CD4 count and ultimate CD4 rise in patients who began protease inhibitors, the amount of prior antiretroviral use would be an important effect modifier. If you limit your study population to just those participants who had never used antiretroviral agents before, you’ll be missing much of the story.

Matching to Reduce Confounding
Definition: only unexposed/non-case subjects are chosen who match those of the reference group (either exposed or cases) in terms of the confounder in question Results in the same distribution of the potential confounder as seen in the exposed/cases Matching is another approach that can be used in the design phase to prevent the occurrence of confounding. Let me start off by saying that matching is actually a pretty complex topic and we unfortunately are not going to have a lot of time to discuss it. In matching, only subjects are chosen who match those of the reference group in terms of the confounder in question. So, the result of this is that there is the same distribution of the potential confounder in the unexposed group as in the exposed group in a cohort or cross-sectional study and the same distribution of the potential confounder in the non-cases as the cases in a case-control study.

Mechanics depends upon study design: e.g. cohort study: unexposed subjects are “matched” to exposed subjects according to their values for the potential confounder. e.g. matching on race One unexposedblack enrolled for each exposedblack One unexposedasian enrolled for each exposedasian e.g. case-control study: non-diseased controls are “matched” to diseased cases e.g. matching on age One controlage 50 enrolled for each caseage 50 One controlage 70 enrolled for each caseage 70 . The mechanics depend upon the study design. For example, in a cohort study, if we were concerned about race as a confounder, you might match on race. For each white exposed person, you would choose one white unexposed person. For each asian exposed person, you would choose one asian unexposed. Similarly, in a case-control study, if, for example, we did a study of say diet and cancer and were worried about age as a potential confounder, we might match on age. For each case who was 50 years old, we would choose one control who was 50 years old. For each 70 yo case, we would select one 70 yo control.

So, back to our schematic representation, for a cohort study, matching results in prohibiting any association between the exposure and the confounder, as seen in the top panel. In a case-control study, matching precludes any association between the disease and the confounder as shown in the bottom panel (and also illustrates one of the disadvantages of matching that I will discuss presently) D

Advantages of Matching
1. Useful in preventing confounding by factors which would be difficult to manage in any other way e.g. “neighborhood” is a nominal variable with multiple values. (complex nominal variable) e.g. Cohort study of the effect of stop light cameras in preventing MVA’s Exposed: cars going thru stop lights with camera Unexposed: cars going thru stop lights without camera Potential confounder: ambient driving practices in the neighborhood Relying upon random sampling of unexposed cars without attention to neighborhood may result in (especially in a small study) choosing no unexposed cars from some of the neighborhoods seen in the exposed group Even if all neighborhoods seen in the exposed group were represented in the unexposed group, adjusting for neighborhood with “analysis phase” strategies are problematic Matching does have several advantages. It is the best way to manage certain confounding variables. For example, “neighborhood” is a nominal variable with multiple values; think of all the neighborhoods in San Francisco, for example. Say you wanted to conduct a cohort study looking at the effect of these new cameras that are posted at selected stop lights in preventing motor vehicle accidents. Here, the exposed group is cars going through stop lights with these cameras. The unexposed group is cars going thru stop lights without these cameras. You want to exclude the possible confounding influence of ambient driving practices in the neighborhood. First, you pick out the stop lights with these cameras. If you had to rely upon random sampling of unexposed cars (those going thru stop lights without cameras) you might end up choosing no unexposed cars from some of the neighborhoods seen in the exposed group. This is especially true in a small study. If you have persons in the exposed group for whom there are no persons in the unexposed group with the same value of the neighborhood, you are not going to be able to use adjustment techniques to adjust for this confounder. Matching, of course, up front would prevent this from occurring. Even if you did manage to find in your random sampling enough persons in the unexposed group to make sure that all the neighborhoods in the exposed group were covered, the actual mechanics of adjusting for these in the analysis phase is quite problematic because the variable has multiple possible values and sometimes just not doable - matching up fronts precludes this problem.

Advantages of Matching
2. By ensuring a balanced number of cases and controls (in a case-control study) or exposed/unexposed (in a cohort study) within the various strata of the confounding variable, statistical precision is increased A second advantage is that matches brings you increased statistical precision (ie smaller confidence intervals). This is because matching works towards achieving a balanced number of cases and controls, in the context of a case-control study, or exposed/unexposed in a cohort study within the various strata of the confounder. This balance means tighter precision.

Smoking, Matches, and Lung Cancer
A. Random sample of controls Crude OR crude = 8.8 Stratified Smokers Non-Smokers OR CF+ = ORsmokers = 1.0 An example of this is our familiar smoking, matches, and lung cancer analysis. Recall from last week that if one performs a case control study and randomly samples controls from the community, you’ll get the following: a crude or unadjusted odds ratio of 8.8 looking at the association between matches and lung cancer after adjustment for smoking the two smoking-specific stratum now feature OR’s of When we pool these 2 stratum, using a method we have not described, we get on OR of 1.0 with the CI you see. But what if we instead matched on smoking when we sampled the controls. Again, we can choose the controls from the population at large but for every lung cancer case that is a smoker we take a control that is a smoker. For every lung cancer case that is a non-smoker, we take a control that is not a smoker. We end with better balance in the smoking strata. Notice, in the smokers, we see 900 cases and 900 controls, in the non-smokers, we see 100 cases and 100 controls. When we pool the 2 stratum specific OR’s to get the overall adjusted OR, we get the same point estimate as when we sampled controls randomly from the population but now notice that the CI is narrower, ie better statistical precision. Not an enormous amount but enough to take notice. This is the point that few realize about matching; it improves statistical precision. Note: What about 1:K matching? Is it possible to give some sort of explanation as to why the balance results in greater precision. Maybe an examination of the MH weights may hold some clues? OR CF- = ORnon-smokers = 1.0 ORadj= 1.0 (0.69 to 1.45) B. Controls matched on smoking Smokers Non-Smokers OR CF+ = ORsmokers = 1.0 OR CF- = ORnon-smokers = 1.0 ORadj= 1.0 (0.75 to 1.34)

Disadvantages of Matching
1. Finding appropriate matches may be difficult and expensive and limit sample size (e.g., have to throw out a case if cannot find a control). Therefore, the gains in statistical efficiency can be offset by losses in overall efficiency. 2. In a case-control study, factor used to match subjects cannot be itself evaluated as a risk factor for the disease. In general, matching decreases robustness of study to address secondary questions. 3. Decisions are irrevocable - if you happened to match on an intermediary, you likely have lost ability to evaluate role of exposure in question. 4. If potential confounding factor really isn’t a confounder, statistical precision will be worse than no matching. Disadvantages, however, can sometimes outweigh advantages. First of all, it may be time-consuming to sift through people to find appropriate matches. There may even be occasions when you have to toss out a case,for example in a case control study, because you cannot find a control. In general, the inefficiences of having to find matches may outweigh the benefits gained in statistical precision, that we just discussed. Second, in a case-control study, because you have artificially precluded any assoc between the potential confounder and the outcome, you cannot directly assess in the study whether this factor is indeed related to the outcome. This illustrates how matching in general works toward reducing the robustness of your study for secondary research questions. Third,the decisions you make about matching are irrevocable. For example, say if did not understand last week’s lecture and matched upon an intermediary variable, then you have likely lost the ability to look for an effect of your exposure through that pathway. In other words, you cannot undo matching that has already occurred. [note: consider also the issues with overmatching. Text says it cause bias to null. Is this true?] Fourth, although you will have to accept this without proof, if the variable you are concerned about producing confounding really isn’t a confounder than you will actually suffer losses, not gains, in statistical precision compared to a situation where you did not match. Suffice it to say, that the heyday for matching was clearly in the past before the advent of newer mathematical modeling techniques and the computers to run them. Think carefully before you match and seek advice!

Stratification to Reduce Confounding
Goal: evaluate the relationship between the exposure and outcome in strata homogeneous with respect to potentially confounding variables Each stratum is a mini-example of restriction! CF = confounding factor Crude Let’s now turn to what we can do in the analysis phase. We have already described what stratification is in our example regarding smoking, matches and lung cancer. By creating strata that are homogeneous with respect to the different values or levels of the potential confounder, we have in each stratum created a mini-example of restriction. Everyone in the stratum has the same level of the potential confounder and therefore confounding cannot occur in that stratum. Stratified CF Level I CF Level 2 CF Level 3

Crude OR crude Stratified Smokers Non-Smokers Recall, in our initial example, when we concerned about the confounding effect of smoking on the relationship between matches and lung ca, we stratified on two value of smoking: smoking present and smoking absent. Each of these strata is now unconfounded with respect to smoking status. OR CF+ = ORsmokers OR CF- = ORnon-smokers ORcrude = 8.8 ORsmokers = 1.0 ORnon-smoker = 1.0

Stratifying by Multiple Potential Confounders
Crude Potential Confounders: Race and Smoking To control for multiple confounders simultaneously, must construct mutually exclusive and exhaustive strata: Our smoking, matches, lung cancer example was the simplest case of stratifying by one potential confounder with two levels, but in fact stratification can also be used when there are multiple potential confounders with more than two levels. To do this, we want to look at the joint effects of these potential confounders by forming mutually exclusive and exhaustive strata. Let’s say we are looking at the association between chlamydial infection and CAD and are wary of the potential confounding effects of race and smoking. To perform stratification, we would form six strata where we could put each subject only once by crossing smoking by race. This would result in strata of white smokers, white non-smokers, black smokers, black non-smokers, latino smokers, and latino non-smokers.

Stratifying by Multiple Potential Confounders
Crude Stratified white smokers black smokers latino smokers Schematically, it would look like this with six mutually exclusive and exhaustive strata based on the two potential confounders looked at jointly. Each of these strata is unconfounded; ie all members of the strata have the same value of the two confounders. white non-smokers black non-smokers latino non-smokers

Summary Estimate from the Stratified Analyses
Goal: Create an unconfounded (“adjusted”) estimate for the relationship in question e.g. relationship between matches and lung cancer after adjustment (controlling) for smoking Process: Summarize the unconfounded estimates from the two (or more) strata to form a single overall unconfounded “summary estimate” e.g. summarize the odds ratios from the smoking stratum and non-smoking stratum into one odds ratio Now that we have formed our strata and gotten rid of confounding in each stratum, how do we summarize what the unconfounded estimates from the two or more strata are telling us? In other words, how do we take the information from the different strata and come up with one number (one measure of association) for for the exposure and disease in question that is adjusted for the potential confounder.

Crude OR crude Stratified Smokers Non-Smokers In the smoking, matches, lung cancer example, the measures of association from the different strata were identical; they are both equal to 1. Here the summary of these two strata is easy: the summary is 1. But, in real life it is seldom the case that the estimates from the various strata are identical. OR CF+ = ORsmokers OR CF- = ORnon-smokers ORcrude = 8.8 (7.2, 10.9) ORsmokers = 1.0 (0.6, 1.5) ORnon-smoker = 1.0 (0.5, 2.0)

Smoking, Caffeine Use and Delayed Conception
Crude RR crude = 1.7 Stratified Heavy Caffeine Use No Caffeine Use Consider this example from a study of the effects of smoking and caffeine use in the occurrence of delayed pregnancies among women hoping to conceive. The principal exposure in question is smoking. Look what happens after we stratify by caffeine use. In those women who do not use caffeine, smoking is associated with over a two-fold risk of delayed conception. But in women who use lots of caffeine the risk of smoking is 0.7, if anything a protective effect. Would you think it is appropriate to try to summarize these two effects, 2.4 and 0.7, into one overall number? RRcaffeine use = 0.7 RRno caffeine use = 2.4

Underlying Assumption When Forming a Summary of the Unconfounded Stratum-Specific Estimates
If the relationship between the exposure and the outcome varies meaningfully (in a clinical/biologic sense) across strata of a third variable, then it is not appropriate to create a single summary estimate of all of the strata i.e. the assumption is that no statistical interaction is present No, I think most of us would agree that it does not make much sense to try to summarize these two very different numbers into one. If we did, we are very much missing an important aspect of the system under study. This illustrates the one assumption that is needed before one embarks upon attempting to form a summary estimate between the different strata in stratified analyses. If the relationship between the exposure and disease under study differs meaningfully (in a clinical or biologic sense) according to the level of a third variable, then it is not appropriate to form an overall summary estimate of the stratum. More concisely, the assumption is that STATISTICAL INTERACTION is not occurring!

Statistical Interaction
Definition when the magnitude of a measure of association (between exposure and disease) meaningfully differs according to the value of some third variable Synonyms Effect modification Effect-measure modification Heterogeneity of effect Proper terminology e.g. Smoking, caffeine use, and delayed conception Caffeine use modifies the effect of smoking on the risk ratio for delayed conception. There is interaction between caffeine use and smoking in the risk ratio for delayed conception. Caffeine is an effect modifier in the relationship between smoking and delayed conception. The example using smoking, caffeine use, and delayed conception illustrates statistical interaction which is what we call when a particular measure of assoc (between an exposure and disease, for example a risk ratio) meaningfully differs according to the level of some third variable. Synonyms for statistical interaction include effect modification, effect-measure modification and hetergeneity of effect. You will hear interaction and effect modification most commonly. What’s the proper usage in this situation? For example, we would say: . . .

Our text, like many others, uses a graphical approach to depict interaction. Let’s look at the top graph. Risk of disease (in a log scale) is shown on the y axis; exposure status (exposed vs unexposed on the x axis). In the presence of a third variable, the risk of disease in the unexposed group is 0.05 and it goes up three fold to 0.15 in the exposed group. When the third variable is absent, risk in unexposed is 0.15 which goes up to 0.45 in the exposed group, again a 3 fold increase. In other words, relative risk does not change according to the third variable. The lines are parallel; there is not statistical interaction in terms of the risk ratio. In the bottom panel, you can see that the relative risk of disease does change according to the level of the third variable. Non-parallel lines equals statistical interaction.

What’s going on there? In the presence of the third variable, exposed persons appear to be protected relative to unexposed, but in the absence of the third variable, exposed persons are at over two fold increased risk. This is what we see in the smoking, caffeine use, and delayed conception example. The effects in the two levels of the third variable are on the opposite sides of This is what we call qualitative interaction; in other words, the interaction is huge!

Interaction is likely everywhere
Susceptibility to infections e.g., exposure: sexual activity disease: HIV infection effect modifier: chemokine receptor phenotype Susceptibility to non-infectious diseases exposure: smoking disease: lung cancer effect modifier: genetic susceptibility to smoke Susceptibility to drugs effect modifier: genetic susceptibility to drug But in practice to date, difficult to document If you think about it for a moment, I think you will agree that interaction is likely everywhere. As an example from infectious diseases, if the exposure is sexual activity and the outcome is HIV infection, we know that certain persons are more apt to become infected than others. One such effect modifier that has been discovered is the presence of a particular chemokine receptor phenotype. From non-infectious diseases, we have the example of smoking and lung cancer. Although not well worked out, we can imagine that there are host genetic factors that modify the effect of smoke and make some persons much more susceptible to the harmful effects of smoke. How about the effectiveness of drugs? We all suspect there is substantial heterogeneity in terms of how people respond and that this likely due to various genetically coded susceptibilities. These are just beginning to be described. However, although we all believe that interaction is likely everywhere around us, it has been - to date- in practice actually relatively difficult to find and document these factors. This is one hope of the whole genomics revolution, that we will be able to find these different host susceptibility factors.

Smoking, Caffeine Use and Delayed Conception: Additive vs Multiplicative Interaction
Crude RR crude = 1.7 RD crude = 0.07 Stratified Heavy Caffeine Use No Caffeine Use Back to our example, you know that the relative risk is not the only measure of association we have to characterize the association between an exposure and disease. The other measure is an absolute difference between exposed and unexposed - called a risk, rate, odds, or prevalence difference. In these 2x2’s we are dealing with risk differences, RD, which is simply the risk in the exposed minus the risk in the unexposed. When there is statistical interaction in terms of the relative risks, we call this multiplicative interaction. So, just as there could be interaction in the relative risk there might also be an interaction in the risk difference. In fact, here there is. Among caffeine users, the risk difference is Smokers have a 0.06 less risk. Among non-caffeine users, smokers have 0.12 more risk Interaction in the risk difference is called additive interaction. RRcaffeine use = 0.7 RDcaffeine use = -0.06 RRno caffeine use = 2.4 RDno caffeine use = 0.12 RD = Risk Difference = Risk exposed - Risk Unexposed

Additive vs Multiplicative Interaction
Assessment of whether interaction is present depends upon the measure of association ratio measure (multiplicative interaction) or difference measure (additive interaction) Hence, the term effect-measure modification Absence of multiplicative interaction typically implies presence of additive interaction Additive interaction present So, when talking about interaction, we have to be precise about whether we are talking about interaction of ratio measures (ie multiplicative interaction) or interaction of differences measures ie (additive interaction) or both. That’s why some like to call this effect-measure modification, because whether or not interaction is occuring depends upon the measure of association in question. Let’s go thru a few scenarios. Absence of multiplicative interaction typically implies presence of additive interaction. As you can see here, although there is no interaction for the ratio of risks, there is interaction in the risk differences. When the third variable is present, the risk difference is 0.1, but when the third variable is absent the risk difference is 0.3. What this illustrates is that when we talk about interaction we really have to tie it to the measure of association.

Absence of additive interaction typically implies presence of multiplicative interaction Multiplicative interaction present Absence of additive interaction (when an effect is present) typically implies presence of multiplicative interaction. Here, the risk difference is 0.1 in both strata of the third variable but the risk ratio differs between strata - multiplicative interaction is present.

Presence of multiplicative interaction may or may not be accompanied by additive interaction No additive interaction The presence of multiplicative interaction may or may not be accompanied by additive interaction. In the top panel, we see that despite the presence of multiplicative interactin, the risk difference is 0.1 in both strata of the third variable - ie no additive interaction. In the bottom panel, there is again multiplicative interactive, but this time the risk difference in one stratum is 0.1 and 0.4 in another - i.e., additive interaction is present. Additive interaction present

Presence of additive interaction may or may not be accompanied by multiplicative interaction Multiplicative interaction present Likewise, the presence of additive interaction may or may not be accompanied by multiplicative interaction. In the top panel, we see additive interaction and multiplicative interaction. In the bottom panel, we see additive interaction but no multiplicative interaction. Multiplicative interaction absent

Presence of qualitative multiplicative interaction is always accompanied by qualitative additive interaction One thing that you can count on for sure is that the presence of qualitative multiplicative interaction is always accompanied by qualitative additive interaction.

Additive vs Multiplicative Scales
Additive measures (e.g., risk difference): readily translated into impact of an exposure (or intervention) in terms of number of outcomes prevented e.g. 1/risk difference = no. needed to treat to prevent (or avert) one case of disease or no. of exposed persons one needs to take the exposure away from to avert one case of disease gives “public health impact” of the exposure Multiplicative measures (e.g., risk ratio) favored measure when looking for causal association So, hopefully these past few slides point the importance of paying attention to what measure of association you are dealing with. For example, there’s no need to look for additive interaction if indeed the relative risk is the right measure of association for you. Knowing which measure of association to care about gets us back to the material covered by Dennis Osmond. Additive measures (like the risk difference) are most readily translated into the impact an exposure (or intervention) has in terms of actual number of actual cases of disease. For example, as I think you all know, 1/risk difference is the number of exposed persons you would have to eliminate exposure from in order to avert one case of disease. The background incidence of disease in the unexposed group is an important component of this. In clinical trials, this is known as the number needed to treat, right. Multiplicative measures (like the risk ratio) are the favored measures when looking for causal relationships. These relative measures are favored when studying causality because they don’t depend upon the background incidence of a disease in unexposed persons.

Additive vs Multiplicative Scales
Causally related but minor public health importance RR = 2 RD = = Need to eliminate exposure in 20,000 persons to avert one case of disease Causally related but major public health importance RD = = 0.1 Need to eliminate exposure in 10 persons to avert one case of disease How about some examples? In the upper panel, we are working with a disease that is very rare but nonetheless the exposure in question is associated with a two fold risk of disease. While this may be causally related, the risk difference between exposed and unexposed is very small, just That means you have to eliminate exposure in 20,000 persons just to avert one case of disease. A simple way to infer this is if you took exposed persons and then took away their exposure, you would end up 5 cases of disease (the background) and 5 cases of disease averted. In other words, take away exp from and avert 5 cases, translates into take away exp in 20,000 to avert 1 case. Contrast that with the lower panel. Here, the disease is more common in unexposed but the relative risk of the exposure is still 2, just like above. In this case, the risk difference is 0.1 which translates into only needing to eliminate exposure in 10 persons to avert one case of disease. Here the exposure has minor public health significance; here it has major PH significance.

Family History Present
Smoking, Family History and Cancer: Additive vs Multiplicative Interaction Crude Family History Present Stratified Family History Absent Let’s look at this hypothetical example. Smoking is the exposure, cancer is the outcome, and family history of smoking is the third variable in question. We see that there is not multiplicative interaction, the RR’s are the same in both strata. However, there is apparently additive interaction. RD is 0.2 among those with a family history and 0.05 among those without a family history. If your goal was simply to assess whether smoking was a risk factor, you would probably go with the RR of 2 and not bother to report interaction to your readers, afterall, it is much easier to report just one number instead of two particulary when your study may have many different risk factors. But say you already had a pretty good sense that smoking was a risk factor and now your goal is to see where you can have the most impact in terms of getting persons to stop smoking. So, if your goal is to identify subgroups of persons to target with an intervention (say a smoking cessation intervention), then you have actually found something interesting. The impact of an intervention would differ depending upon the third variable, family history. it is well worth to report the presence of interaction based upon family history. RRfamily history = 2.0 RDfamily history = 0.20 RRno family history = 2.0 RDno family history = 0.05 No multiplicative interaction but presence of additive interaction If goal is to define sub-groups of persons to target: - Rather than ignoring, it is worth reporting that only 5 persons with a family history have to be prevented from smoking to avert one case of cancer

Confounding vs Interaction
An extraneous or nuisance pathway that an investigator hopes to prevent or rule out Interaction A more detailed description of the “true” relationship between the exposure and disease A richer description of the biologic system A finding to be reported, not a bias to be eliminated Recall, it was in an attempt to prevent potential confounding of an association between an exposure and a given disease by some third variableby performing stratification that we discovered that interaction may be present. How do confounding and interaction differ? Well, confounding is an extraneous pathway that we want rule out or avoid when looking at the direct association between our exposure in question and the disease under study. Interaction, however, when present, is a more detailed description of the biological system under study. It is not extraneous but rather a richer description of the biological system. When present, it is not bias we are seeking to eliminate but rather a new finding we hope to report.

When Assessing the Association Between an Exposure and a Disease, What are the Possible Effects of a Third Variable? No Effect I In summary, when we are assessing the association between an exposure and a disease, what are the possible roles of a third variable? A third variable could be acting as a confounder, or it could be an an intermediary variable depending upon how you are conceptualizing system, or it could be an effect modifier. Or, it could be none of these and simply have no effect. C Intermediary Variable + EM _ Confounding: ANOTHER PATHWAY TO GET TO THE DISEASE Effect Modifier (Interaction): MODIFIES THE EFFECT OF THE EXPOSURE D

Smoking, Caffeine Use and Delayed Conception
Crude RR crude = 1.7 Stratified Heavy Caffeine Use No Caffeine Use Backed to our delayed conception example. If we ignored the presence of interaction and simply formed an adjusted summary estimate (ie an average) of what is going on in the two strata (using techniques I will show you next week),we would come up with an adjusted measure of 1.4 which actually is not even statistically significant by conventional limits. Would you want this to be the final answer for this question. Absolutely not! There is much more going on here - such as that smoking appears to be very important risk factor in those persons who do not use caffeine. It may, however, be protective in non-caffeine users. So, when interaction is present, the issue of confounding becomes irrelevant. You would never want to control for caffeine use given the interaction you have found. RRcaffeine use = 0.7 RRno caffeine use = 2.4 RR adjusted = 1.4 (95% CI= 0.9 to 2.1) Here, adjustment is contraindicated!

Chance as a Cause of Interaction?
Crude OR crude = 3.5 Stratified Age < 35 Age > 35 Is every time the stratum specific estimates differ indicate that we have interaction going on and that we should not adjust for the third variable but rather report all the stratum-specific estimates. This could get kind of messy especially with multiple potential effect modifiers and multiple different levels for the different effect modifiers. In this example looking at the association between spermicide use and Down’s syndrome and the influence of age on this association, there is fairly marked differences in the OR in the two stratum but look at the sample sizes. Some of the cells are rather small and therefore we know these numbers are not very statistically precise. Somehow, we need to take this possibility of random variation into account when we assess the presence of interaction. ORage <35 = 3.4 ORage >35 = 5.7

Statistical Tests of Interaction: Test of Homogeneity (heterogeneity)
Null hypothesis: The individual stratum-specific estimates of the measure of association differ only by random variation i.e., the strength of association is homogeneous across all strata i.e., there is no interaction A variety of formal tests are available with the general format, following a chi-square distribution: where: effecti = stratum-specific measure of assoc. var(effecti) = variance of stratum-specifc m.o.a. summary effect = summary adjusted effect N = no. of strata of third variable For ratio measures of effect, e.g., OR, log transformations are used: The test statistic will have a chi-square distribution with degrees of freedom of one less than the number of strata There are statistical tests available to assess the role of chance or random variation in causing apparent interaction. The null hypothesis of these tests is that there is no differences between the strata - the apparent differences are only because of random sampling error; I.e. the strength of association is homogenous across strata; there is no interaction. We won’t dwell into the mechanics of these tests only to say that they follow a chi-square distribution with the degrees of freedom equal to the no.of strata minus one.

Interpreting Tests of Homogeneity
If the test of homogeneity is “significant”, this is evidence that there is heterogeneity (i.e. no homogeneity) i.e., interaction may be present The choice of a significance level (e.g. p < 0.05) is somewhat controversial. There are inherent limitations in the power of the test of homogeneity p < 0.05 is likely too conservative One approach is to declare interaction for p < 0.20 i.e., err on the side of assuming that interaction is present (and reporting the stratified estimates of effect) rather than on reporting a uniform estimate that may not be true across strata. So, if the null hypothesis is that there are no differences, the alternative hypothesis is that there is a difference. A small p value for a test of homogeneity means that there is a substantial evidence to reject the null hypothesis of uniformity; ie there is evidence for heterogeneity What p value to use for a cutoff is somewhat controversial. Most studies don’t have enough subjects to enable to these tests to have much statistical power. Hence, p=0.05 is probably too conservative. One approach is to be a bit more conservative and err on the side of reporting interaction and use a higher p value of say, There is no right answer here and as I hope to show you in a few minutes, this is rarely only a statistical decision.

Tests of Homogeneity with Stata
1. Determine crude measure of association e.g. for a cohort study “cs outcome-variable exposure-variable” for smoking, caffeine, delayed conception: -exposure variable = smoking -outcome variable = delayed -third variable = caffeine “cs delayed smoking” 2. Determine stratum-specific estimates by levels of third variable “cs outcome-var exposure-var, by(third-variable)” e.g. cs delayed smoking, by(caffeine) Let’s implement this test in Stata, and to do this we use the epitab set of commands that you have already used. First, lets look at the crude association. You already know this command, its cs outcome exposure. Then, look at the stratum-specific. . .

Declare vs Ignore Interaction?
Let’s finish by going through several examples to get a feeling for when we should declare, rather than ignore interaction. Let’s say we are looking at the association between a given exposure and a given disease and we have to then look at the effect of a potential effect modifier that has two levels: present and absent. In these two columns you see the stratum-specific measures of association, in this column, the p value for the test of homogeneity, and here is our verdict. If the two strata give results of 2.3 and 2.6 and a p-value for the test of heterogeneity of 0.45, what should we do with it. Ignore it. What if the pvalue is 0.001, this is an example of where we should still ignore it because because this difference really is pretty small and likely not clinically meaningful. What if we got 2.0 in one stratum and 20 in another and a p value of Here, this is worth reporting. However

Confounding and Interaction: Part II

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