Total, direct and indirect effects
Confounding, mediation, and some general considerations in regression modelingMultivariable regression and its variations are currently the most frequently used type of statistical technique in behavioral medicine research. A typical example of a multivariable model in our field might be a regression model that attempts to evaluate a set of psychosocial predictors of disease, such as heart failure.? The measure indicating disease could be operationalized in a number of measurement forms: as a continuous variable, such as left ventricular ejection fraction; as two or more categories, such as the ordinal values of a symptom severity score; or even as the time elapsed between some defined occasion and the diagnosis of heart failure.? The term multivariable indicates that the model contains a single response (dependent) variable, in this case, the marker of heart failure, and at least two predictor variables in the model.? The ostensible aim is to understand the ‘independent’ association of each predictor variable with the response variable.? In the most commonly used type of model, the regression coefficient or parameter estimate for a given predictor represents the association between that predictor and the response, adjusting for all other predictors in the model.? We also might have more than one response variable, perhaps two separate indicators of heart failure, such as LVEF and a symptom severity score.? We might simply conduct a separate regression analysis for each outcome, but also might elect to use a model that contains the two response variables, referred to as a multivariate model.? Regardless of which model we choose, a number of important decisions must be made in developing the model.? Foremost is the selection of the form of probability model that best suits the response variable(s) under study.? Next, and often the most difficult part of the process, is to decide which predictors should be included in the model.? For the vast majority of work we do in behavioral medicine, an important part of the variable selection process involves our presumed causal model. If we conduct a regression model to examine the association between, say, tobacco use and heart failure, we are more often than not proposing that tobacco use is a cause of heart failure. Upon proposing this model, we must immediately set about identifying potential confounders, that is, other variables that may threaten our causal conclusion. We also may be interested in variables that carry information about mechanisms that occur in between the act of smoking and the outcome of heart failure. ?Finally, we also may be concerned, or even believe a priori, that the association between a given predictor and the response may differ depending on the level of another variable.? For example, tobacco use may be related to heart failure only for persons with a certain genotype.? Of course, there are many additional considerations in conducting a multivariable regression analysis, including testing assumptions, proper scaling or standardization of the predictors, perhaps centering, rescaling or orthogonalizing predictors, to name a few.? In the present chapter, our focus will be relatively narrow. After a few preliminaries, we will discuss 1) considerations in selecting predictor variables for a model; 2) modern approaches to mediation; 3) testing for moderation, and finally 4) the role of sample size in estimating regression models.? Preliminaries: What is a Model?What is a model and why use one? The statistical models we use in behavioral medicine typically take the general form of one of more ‘predictor’ variables and one outcome, or response variable, such as y = bx1 + bx2 + bx3 + …?where y is the response variable, the x’s are the predictor variables, and the b’s are regression weights. In the vast majority of modern modeling algorithms, the predictor variables can be of any form, including continuous, categorical, and ordinal (and as we will note again later, there is no normality requirement for variables on the predictor side of an equation).? A few words about nomenclature are appropriate here.? Techniques such as Analysis of Variance (ANOVA) and Analysis of Covariance (ANCOVA), and multivariable (often referred to as “multiple”) regression have been almost entirely displaced by more general models (e.g., the general linear model, and the generalized linear model).? This transition has created a somewhat confusing amalgam of terminology from these older techniques.? Variables on the x-side of the equation are referred to interchangeably as independent variables, predictors, covariates, covariables, or, for variables measured as categories, factors. Variables on the y-side are referred to as the response, outcome, or dependent variable.? Models per se tend to be preferred over traditional tests (e.g., t-tests, chi-square tests) nowadays for several reasons.? First, models provide not only the same information as more conventional testing approaches, that is, whether the effect of interest is “statistically significant,” but also yield information about the size of the effect of interest, along with information about the uncertainty of the effect estimate, usually in the form of a confidence interval. For example, in a clinical trial comparing a new blood pressure-lowering drug to a standard drug could legitimately be evaluated using a simple t-test that compares the treatment groups on mean blood pressure at the end of the trial.? However, the t-test would provide no information on how big the difference was. A key advantage of multivariable models is that we can include so-called adjustment variables in addition to the primary variable or variable of interest.? These adjustment variables can serve a variety of purposes in a multivariable model, and these purposes are at the heart of the remainder of this chapter. In modern practice most of the earlier techniques, such as t-tests, chi-square tests, ANOVA, ANCOVA, multiple regression, etc., have been subsumed under the a few more general algorithms. The generalized linear model ADDIN EN.CITE <EndNote><Cite><Author>McCullagh</Author><Year>1989</Year><RecNum>21</RecNum><record><rec-number>21</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">21</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>McCullagh, P.</author><author>Nelder, J.</author></authors></contributors><titles><title>Generalized Linear Models</title></titles><keywords><keyword>Linear Models</keyword><keyword>models</keyword></keywords><dates><year>1989</year></dates><pub-location>London</pub-location><publisher>Chapman and Hall</publisher><accession-num>2538</accession-num><urls></urls></record></Cite></EndNote>(1), for example, can contain one or more variables of virtually any measurement form on the predictor side, and the probability distribution of the dependent variable can take a variety of forms beyond normal.? These include the binomial, negative binomial, and gamma distributions.? Hence, multiple regression, Llogistic regression, P oisson regression, and many other conventional models models are often still estimated using dedicated logistic regression routines, but also can be accomplished estimated using the generalized linear model. For time-to-event data, the Cox regression model ADDIN EN.CITE <EndNote><Cite><Author>Cox</Author><Year>1984</Year><RecNum>734</RecNum><record><rec-number>734</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">734</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Cox, D.R.</author><author>Oakes, D.</author></authors></contributors><titles><title>Analysis of survival data</title></titles><dates><year>1984</year></dates><pub-location>London</pub-location><publisher>Chapman & Hall</publisher><urls></urls></record></Cite></EndNote>(2) is probably the most commonly used approach today, although parametric techniques also appear with relative frequency. In addition to general linear model, structural equation models (SEM) ADDIN EN.CITE <EndNote><Cite><Author>Muthen</Author><Year>2004</Year><RecNum>22</RecNum><record><rec-number>22</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">22</key></foreign-keys><ref-type name="Generic">13</ref-type><contributors><authors><author>Muthen, L. K.</author><author>Muthen, B.</author></authors></contributors><titles><title>Mplus User's Guide. 3rd ed</title></titles><dates><year>2004</year></dates><pub-location>Los Angeles, CA</pub-location><publisher>Muthen and Muthen</publisher><accession-num>1933</accession-num><urls></urls></record></Cite></EndNote>(3) have now been extended sufficiently such that they also can perform virtually all of the above functions. SEM also has the advantage of allowing so-called indirect relations to be estimated and tested, which we will discuss further in the section on mediation below. Cause. Except for the relatively rare case where a regression model is used for completely blind empirical prediction, researchers typically use regression models to help understand something substantive about the phenomena under study.? Regardless of whether we care to admit it or not, researchers are largely interested in using regression models to understand cause.? Why would we measure and model, for example, risk factors as predictors of cardiac disease if we were not interested in those risk factors as causes?? If understanding causation underlies our models, most would agree that a useful model will include as many of the casually relevant variables in the system as possible.? What makes a variable “relevant?” This question has been of great interest and debate from many decades in the statistics literature, and is often cast in terms of the problem of “variable selection.” We argue that relevance depends on the causal model underlying the analysis. Confounding. In the context of causal hypotheses, confounders represent a highly relevant type of variable.? For a variety of reasons, we know that should never be fooled into believing that an association between two variables is sufficient evidence for causation.? It may be the case, for example, that the putative cause is confounded with another variable. The term confounding derives from the Latin confundere, to pour together, or to mix ADDIN EN.CITE <EndNote><Cite><Author>Glare</Author><Year>1982</Year><RecNum>48</RecNum><record><rec-number>48</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">48</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Glare, P.G.W.</author></authors></contributors><titles><title>Oxford Latin dictionary</title></titles><dates><year>1982</year></dates><publisher>Oxford University Press</publisher><urls></urls></record></Cite></EndNote>(4). At root, confounding is the mixing of the role of two predictor variables.? Imagine you are at the bottom of a steep ravine looking up at a train trestle.? Suddenly a very small boy goes running across the train trestle, followed shortly by a much larger boy, who is shouting at the small boy.? You conclude that the large boy is chasing the small boy, that is, causing the small boy to run.? However, shortly after the boys cross the trestle, a train comes barreling across the trestle behind them.? In fact, the larger boy was not chasing the small boy at all; the train was causing both of them to run across the trestle quickly.? The causal role of the large boy and the train were mixed up, or confounded.? The presence of the large boy was really just a red herring; he just happened to be running from the train, too.? In conducting research we study one or a just a few variables that are of particular interest in order to understand something about the causal relation with between that variable and some outcome variable.? A simple example in a research context is presented in a didactic paper Rubin ADDIN EN.CITE <EndNote><Cite><Author>Rubin</Author><Year>1997</Year><RecNum>26</RecNum><record><rec-number>26</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">26</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Rubin, D. B.</author></authors></contributors><titles><title>Estimating causal effects from large data sets using propensity scores</title><secondary-title>Annals of Internal Medicine</secondary-title></titles><periodical><full-title>Annals of Internal Medicine</full-title><abbr-1>Ann. Intern. Med.</abbr-1><abbr-2>Ann Intern Med</abbr-2></periodical><pages>757-763</pages><volume>127</volume><number>8_Part_2</number><keywords><keyword>analysis</keyword><keyword>Clinical Trials</keyword><keyword>Health</keyword><keyword>methods</keyword><keyword>Observation</keyword><keyword>regression</keyword><keyword>Software</keyword><keyword>Treatment</keyword></keywords><dates><year>1997</year></dates><accession-num>2625</accession-num><urls><related-urls><url>;(5).? Rubin presents several large epidemiological studies that all seem to show that smoking tobacco in a pipe or cigarette is associated with a higher rate of cancer deaths than smoking tobacco in cigarette form.? It is often helpful to draw a diagram of our causal hypothesis.?? This result is, of course, contrary to our understanding of the relative dangers of the types of tobacco delivery.? So, we need to ask whether tobacco type (pipe/cigar vs. cigarette) might be confounded with some other variable.? More formally, we consider the general criteria for confounding, which are as follows: 1) the confounding variable is presumed to be must becausally related to the predictor under study; 2) the confounding variable must beis presumed to be causally related to the outcome; 3) the confounding variable cannot be in the causal chain betweenis either common cause or a proxy for a common cause of the predictor and the outcome.? In our tobacco example, tobacco type is the predictor of interest and cancer death is the outcome.? What variable might be associated with the tobacco type and cancer death but is not in the causal chain between the two?? One obvious candidate is age.? Older people are more likely than younger people to smoke cigars or pipes and are also more likely to die of cancer.? Although chronological age is clearly causally related to cancer death, age cannot be caused by the type of tobacco we smoke.??? In Rubin’s examples, in each of the samples it was clear that pipe/cigar smokers were much older on average than cigarette smokers, and that the death rate also was higher among older individuals.? When age was properly accounted for in the analysis, the death rate among pipe/cigar smokers was no longer higher than among cigarette smokers—in fact, it became lower.? Thus the ‘effect’ of tobacco type was confounded with age.? Causal graphs. Causal models can be easier to comprehend if presented in graphic form. Often the graphs are used as an informal heuristic tool and sometimes they are employed in a more formal way as (causal) Directed Acyclic Graphs (DAGs) or to represent a Structural Equation Model (SEM). Providing an introduction to graph theory is beyond the scope of this text, but a non-technical introduction to causal DAGs can be found in Glymour and Greenland ADDIN EN.CITE <EndNote><Cite><Author>Glymour</Author><Year>2008</Year><RecNum>707</RecNum><record><rec-number>707</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">707</key></foreign-keys><ref-type name="Book Section">5</ref-type><contributors><authors><author>Glymour, M. M.</author><author>Greenland, S.</author><author>Rothman, K. J.</author><author>Lash, T. L.</author></authors></contributors><titles><title>Causal diagrams</title><secondary-title>Modern Epidemiology</secondary-title></titles><pages>183-212</pages><volume>3rd</volume><keywords><keyword>epidemiology</keyword></keywords><dates><year>2008</year></dates><pub-location>Philadelphia</pub-location><publisher>Lippincott Williams & Wilkins</publisher><urls></urls></record></Cite></EndNote>(6) and an overview of causal analysis in the context of mediation is provided by VanderWeele and Vansteeland ADDIN EN.CITE <EndNote><Cite><Author>VanderWeele</Author><Year>2009</Year><RecNum>729</RecNum><record><rec-number>729</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">729</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>VanderWeele, T. J.</author><author>Vansteelandt, S.</author></authors></contributors><titles><title>Conceptual issues concerning mediation, interventions and composition</title><secondary-title>Stastistics and Its Interface</secondary-title></titles><periodical><abbr-1>Stastistics and Its Interface</abbr-1></periodical><pages>457-468</pages><volume>2</volume><dates><year>2009</year></dates><urls></urls></record></Cite></EndNote>(7). We will say much more about DAGs in the section on mediation below, but for now, we’ll introduce just one basic element of DAG notation. Causal DAGs use single-headed arrows to represent the hypothesized causal direction between variables. Double-headed arrows, contrast, represent associations with no specified causal direction. In the first DAG below, tobacco type and cancer mortality are associated, but with no causal direction. In the second DAG, tobacco type is posited as the cause of cancer mortality.The arrow in each of the figures, are, at this point black boxes, representing a potential host of processes, not all of which are necessarily causal. Put differently, a raw or zero-order correlation, or an unadjusted regression coefficient between two variables can be a function of a variety of different processes, some possibly causal, but many others that have nothing to do with cause. In this case, the association between tobacco type and cancer was actually generated by the presence of a “third variable,” age, which was a common cause of both tobacco type (in that age captures the cultural cohort) and cancer mortality. This confounding is depicted below.? When we use regression models to study the effect of one or a few putative causes of an outcome, we strive to identify and include other variables in the model that might confound the relations under study.? A critical step in planning a study of virtually any design is considering carefully what variables might confound the relations under study, and then being sure to measure those variables. This is particularly important when the design is observational where there is no randomization to control for confounding. By including confounding variables in the analysis of observational data, we may be at least a bit closer to being able to understand cause. Considering potential confounders is also important in randomized experiments. Except in extremely large studies, perfect baseline balance is rarely achieved across randomized arms. When there is baseline imbalance in a randomized experiment, the treatment effect under study may be confounded with the variable that is not balanced. Unless the arms are substantially unbalanced, including potential confounding variables as adjustment variables in a model will effectively reduce the threat of confounding when interpreting the treatment effect.Including variables to increase precision. Variables other than confounders may be relevant to the regression model. We also want our model to include predictors that are associated with the outcome, even if they are not associated with other predictors.? In a linear model, such as multiple regression, including additional predictors in the model (within the limits of sample size, which we will discuss below), the precision of the regression weightparameter estimates is improved and power of the tests of the regression weights is improved.? Intuitively, power is improved because additional predictors explain variance in the response, and therefore reduce the magnitude of the error term by which the individual regression estimates are evaluated.? For nonlinear models, such as logistic regression and Cox survival models, the picture is a bit more complicated.? Adding additional variables will increase the standard errors for the parameter estimates, resulting in less power.? However, the estimates will also always be larger.? Simulation studies have shown that the benefit of the increased magnitude of the estimates outweighs the problem of larger standard errors ADDIN EN.CITE <EndNote><Cite><Author>Steyerberg</Author><Year>2009</Year><RecNum>47</RecNum><record><rec-number>47</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">47</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Steyerberg, E.W.</author></authors></contributors><titles><title>Clinical Prediction Models</title></titles><dates><year>2009</year></dates><pub-location>New York</pub-location><publisher>Springer</publisher><urls></urls></record></Cite></EndNote>(8).? Thus, when the sample size is large enough, in the most frequently used models in behavioral medicine, including additional predictors is generally desirable.? MediationIn addition to addressing confounding and increasing precision, we also might include additional predictors in a model to study the possibility of mediation.? Since the early paper on mediation by Baron and Kenny ADDIN EN.CITE <EndNote><Cite><Author>Baron</Author><Year>1986</Year><RecNum>58</RecNum><record><rec-number>58</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">58</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Baron, Reuben M.</author><author>Kenny, David A.</author></authors></contributors><titles><title>The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations</title><secondary-title>Journal of Personality and Social Psychology</secondary-title></titles><periodical><full-title>Journal of Personality and Social Psychology</full-title><abbr-1>J. Pers. Soc. Psychol.</abbr-1><abbr-2>J Pers Soc Psychol</abbr-2><abbr-3>Journal of Personality & Social Psychology</abbr-3></periodical><pages>1173-1182</pages><volume>51</volume><number>6</number><dates><year>1986</year><pub-dates><date>Dec</date></pub-dates></dates><accession-num>Peer Reviewed Journal: 1987-13085-001</accession-num><urls></urls></record></Cite></EndNote>(9), analyses of mediation has become increasingly prevalent in the literature.? Its importance has grown so much, in fact, that we have elected to devote a substantial section of this chapter to it.?? The notion of mediation is used to describe a scenario where a variable affects another variable though one or more intermediary variables. In the following sections, we will review some conceptual issues involved in mediation and discuss methods that can be used to statistically model mediation. Total, direct and indirect effects. We begin with a little orientation to the nomenclature of modern mediation analysis. Recall our graphic representation of a proposed causal association between two variables: In this graph the arrow pointing from X to Y indicates that the variable X affects the variable Y. We will refer to this as the total effect of X on Y. The total effect of X on Y depicted in Figure 1 may come about through any number of intermediary variables, but these can be left out when the objective is to describe the total effect. If there are intermediary variables between X and Y, as we noted earlier, the arrow from x to y in the above graph is a black box: We know the input (X) and the output (Y), but not the mechanisms responsible for creating the association. In this graph, the variable X affects the variable Y and the variable M. Also, we can see that the variable M affects the variable Y. As a consequence we can distinguish between two different kinds of effects of X on Y: A direct effect (X → Y) and an indirect effect through the variable M (X → M → Y). The second graph suggests that there is both a direct and an indirect effect of X on Y. In other words, Figure 2 suggests that the variable M mediates some of the total effect of X on Y, but it also suggests that there is an effect of X on Y that does not involve M. It is important to note that the direct effect may in fact involve intermediary variable, just not the intermediary variable M, so the direct effect might more appropriately be termed the non-M mediated effect as the direct effect can be thought of as the sum of all pathways from X to Y that does not involve the mediator M. Establishing the relative importance of the direct and indirect effect is often a primary concern in mediation analysis. Figure 2 also illustrates the difference between confounding and mediation: M is a mediator between X and Y because it lies on the pathway from X to Y. X is a confounder of the association between M and Y because X affects M and Y.Why mediation? Before elaborating further on the technique of mediation, it may prove fruitful to examine the motivation for looking at mediation in the first place: Why is mediation important to begin with? A recent paper by Hafeman & Schwartz listed three reasons: To support the evidence of the main effect hypothesis, to examine the importance of path-specific mechanisms, and to provide targets for intervention ADDIN EN.CITE <EndNote><Cite><Author>Hafeman</Author><Year>2009</Year><RecNum>709</RecNum><record><rec-number>709</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">709</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Hafeman, D. M.</author><author>Schwartz, S.</author></authors></contributors><auth-address>Department of Epidemiology, Mailman School of Public Health, Columbia University, USA. sbs5@columbia.edu</auth-address><titles><title>Opening the Black Box: a motivation for the assessment of mediation</title><secondary-title>Int.J.Epidemiol.</secondary-title></titles><periodical><full-title>analysis</full-title><abbr-1>Int.J.Epidemiol.</abbr-1></periodical><pages>838-845</pages><volume>38</volume><number>3</number><keywords><keyword>analysis</keyword><keyword>Causality</keyword><keyword>Confounding Factors (Epidemiology)</keyword><keyword>Epidemiologic Methods</keyword><keyword>Epidemiologic Research Design</keyword><keyword>epidemiology</keyword><keyword>Evidence-Based Medicine</keyword><keyword>Health</keyword><keyword>Humans</keyword><keyword>methods</keyword><keyword>Models,Theoretical</keyword><keyword>Motivation</keyword><keyword>Public Health</keyword><keyword>PUBLIC-HEALTH</keyword><keyword>Research</keyword><keyword>Risk Assessment</keyword><keyword>Social Sciences</keyword><keyword>Universities</keyword></keywords><dates><year>2009</year></dates><urls><related-urls><url>PM:19261660</url></related-urls></urls></record></Cite></EndNote>(10). In 2005, a paper reported that women with a high level of perceived stress had a decreased risk of breast cancer ADDIN EN.CITE <EndNote><Cite><Author>Nielsen</Author><Year>2005</Year><RecNum>721</RecNum><record><rec-number>721</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">721</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Nielsen, N. R.</author><author>Zhang, Z. F.</author><author>Kristensen, T. S.</author><author>Netterstrom, B.</author><author>Schnohr, P.</author><author>Gronbaek, M.</author></authors></contributors><auth-address>National Institute of Public Health, Oster Farimagsgade 5A, DK-1399 Copenhagen K, Denmark. nrn@niph.dk</auth-address><titles><title>Self reported stress and risk of breast cancer: prospective cohort study</title><secondary-title>BMJ</secondary-title></titles><periodical><full-title>BMJ</full-title><abbr-1>BMJ</abbr-1><abbr-2>BMJ</abbr-2></periodical><pages>548</pages><volume>331</volume><number>7516</number><keywords><keyword>Adult</keyword><keyword>Aged</keyword><keyword>Aged,80 and over</keyword><keyword>Breast Neoplasms</keyword><keyword>COHORT</keyword><keyword>Cohort Studies</keyword><keyword>complications</keyword><keyword>Denmark</keyword><keyword>Epidemiologic Methods</keyword><keyword>epidemiology</keyword><keyword>Female</keyword><keyword>Health</keyword><keyword>Heart</keyword><keyword>Humans</keyword><keyword>Incidence</keyword><keyword>Middle Aged</keyword><keyword>psychology</keyword><keyword>Public Health</keyword><keyword>PUBLIC-HEALTH</keyword><keyword>Registries</keyword><keyword>Research</keyword><keyword>Risk</keyword><keyword>Stress,Psychological</keyword><keyword>therapy</keyword><keyword>Time</keyword><keyword>WOMEN</keyword></keywords><dates><year>2005</year></dates><urls><related-urls><url>PM:16103031</url></related-urls></urls></record></Cite></EndNote>(11). This finding was quite surprising to many as high levels of stress had previously been shown to have detrimental effects on various health outcomes, so could it be that the findings were due to bias and confounding rather than a causal effect of perceived stress on the risk of breast cancer? In the discussion the authors argue that the effect of stress was due to the fact that stress hormones suppress estrogen secretion, which lowers the risk of developing breast cancer. This pathway acts as a mediator between perceived stress and breast cancer. No information on estrogen levels where available in this study, but an analysis of the mediating role of estrogen would have improved the argument for a causal role of perceived stress in the development of breast cancer because it would have served to open the black box of how the exposure and outcome were connected. In fact, another research group had previously used this strategy to show that the association between BMI and breast cancer was mediated by serum estrogen levels PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5LZXk8L0F1dGhvcj48WWVhcj4yMDAzPC9ZZWFyPjxSZWNO
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ADDIN EN.CITE.DATA (12). Another use for mediation is to examine path-specific hypotheses. An association between low parental socioeconomic position and low offspring birth weight has been observed in many different populations and across different measures of socioeconomic position. A study by Mortensen et al. examined the role of two possible mediators of the relationship between maternal educational attainment and offspring birth weight in a cohort of women followed throughout pregnancy ADDIN EN.CITE <EndNote><Cite><Author>Mortensen</Author><Year>2009</Year><RecNum>720</RecNum><record><rec-number>720</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">720</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Mortensen, L. H.</author><author>Diderichsen, F.</author><author>Smith, G. D.</author><author>Andersen, A. M.</author></authors></contributors><auth-address>National Institute of Public Health, University of Southern Denmark, Oster Farimagsgade 5A, 2nd floor, 1399 Copenhagen K, Denmark. laust.mortensen@</auth-address><titles><title>The social gradient in birthweight at term: quantification of the mediating role of maternal smoking and body mass index</title><secondary-title>Hum.Reprod.</secondary-title></titles><periodical><full-title>Adult</full-title><abbr-1>Hum.Reprod.</abbr-1></periodical><pages>2629-2635</pages><volume>24</volume><number>10</number><keywords><keyword>Adult</keyword><keyword>Birth Weight</keyword><keyword>Body Mass Index</keyword><keyword>CHILDREN</keyword><keyword>COHORT</keyword><keyword>Cohort Studies</keyword><keyword>Denmark</keyword><keyword>EDUCATION</keyword><keyword>Educational Status</keyword><keyword>Female</keyword><keyword>Gestational Age</keyword><keyword>GRADIENT</keyword><keyword>Health</keyword><keyword>Humans</keyword><keyword>Maternal Exposure</keyword><keyword>methods</keyword><keyword>Mothers</keyword><keyword>Obesity</keyword><keyword>Pregnancy</keyword><keyword>Prevalence</keyword><keyword>Public Health</keyword><keyword>PUBLIC-HEALTH</keyword><keyword>Research</keyword><keyword>Smoking</keyword><keyword>Socioeconomic Factors</keyword><keyword>Universities</keyword><keyword>WOMEN</keyword></keywords><dates><year>2009</year></dates><urls><related-urls><url>PM:19535360</url></related-urls></urls></record></Cite></EndNote>(13). The two mediators were prepregnant Body Mass Index (BMI) and smoking in the third trimester. Smoking in pregnancy and high BMI is more prevalent among mothers with short education, but these two factors have different effects on birth weight: A high BMI increases birth weight, while smoking decreases it. This means that these two pathways have opposite contributions to the total effect: if all mothers had the BMI of the highest educated mothers, the educational differences would be larger because the higher prevalence of obesity among women with short educations increase their children’s birth weights. If all mothers smoked like the highest educated mothers, mothers with a shorter education would in fact give birth to the heaviest babies because of the high prevalence of overweight and obesity among this group. The total effect of education (short education is associated with a lower birth weight) reflects that the birth weight reducing influence of the smoking-pathway is stronger that the birth weight increasing BMI-pathway. The example of Mortensen et al. shows that the examination of different pathways can increase our understanding of the total effects. For example, it suggests that the educational gradient in birth weight that has been observed in numerous studies might reverse once smoking among pregnant women is eliminated. It also underscores that mediation might be worth looking at, even in the absence of a total effect. This is because a lack of association between the exposure and the outcome might occur when different pathways that pull the total effect in opposite directions balance each other out. This is sometimes referred to as a suppressor effect. In this case an analysis of the relevant mediators would help the investigator retrieve the pathway-specific effects of the exposure on the outcome. A third use of mediation is to improve and evaluate interventions. Mediation is in a certain sense an integrated part of the setup in all randomized controlled trials: The effect of randomization to treatment on the outcome is mediated by the treatment received. The intention-to-treat analysis is a measure of the effect of randomization to intervention, regardless of the intervention actually received. In mediation terms, this corresponds to the total effect of randomization. The motivation for the intention-to-treat analysis is that the results, because of the random assignment to intervention or control, are unconfounded by factors that affect the intervention received and the outcome, e.g. compliance to assigned treatment. However, the effect of the intervention on the outcome is often the quantity of substantive interest, not the effect of randomization to intervention. If this is indeed the case the intention to treat analysis can be supplemented with analyses of mediation ADDIN EN.CITE <EndNote><Cite><Author>Kraemer</Author><Year>2002</Year><RecNum>46</RecNum><record><rec-number>46</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">46</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Kraemer, H. C.</author><author>Wilson, G. T.</author><author>Fairburn, C. G.</author><author>Agras, W. S. </author></authors></contributors><titles><title>Mediators and moderators of treatment effects in randomized clinical trials</title><secondary-title>Archives of General Psychiatry</secondary-title></titles><periodical><full-title>Archives of General Psychiatry</full-title><abbr-1>Arch. Gen. Psychiatry</abbr-1><abbr-2>Arch Gen Psychiatry</abbr-2></periodical><pages>877-883</pages><volume>59</volume><dates><year>2002</year></dates><urls></urls></record></Cite></EndNote>(14).A similar use of mediation can be found in studies that uses naturally occurring experiments rather than experiments under the investigator’s control. Mendelian randomization is a strategy for causal inference that uses genetic variants as proxies for potentially modifiable factors, obesity for example ADDIN EN.CITE <EndNote><Cite><Author>Smith</Author><Year>2004</Year><RecNum>727</RecNum><record><rec-number>727</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">727</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Smith, G. D.</author><author>Ebrahim, S.</author></authors></contributors><auth-address>Department of Social Medicine, Canynge Hall, Whiteladies Road, Bristol BS8 2PR. zetkin@bristol.ac.uk</auth-address><titles><title>Mendelian randomization: prospects, potentials, and limitations</title><secondary-title>International Journal of Epidemiology</secondary-title></titles><periodical><full-title>International Journal of Epidemiology</full-title><abbr-1>Int. J. Epidemiol.</abbr-1><abbr-2>Int J Epidemiol</abbr-2></periodical><pages>30-42</pages><volume>33</volume><number>1</number><keywords><keyword>administration & dosage</keyword><keyword>Calcium,Dietary</keyword><keyword>Causality</keyword><keyword>Coronary Disease</keyword><keyword>Environmental Exposure</keyword><keyword>epidemiology</keyword><keyword>Genetic Predisposition to Disease</keyword><keyword>genetics</keyword><keyword>Genotype</keyword><keyword>Humans</keyword><keyword>Hyperlipoproteinemia Type II</keyword><keyword>Lactose Intolerance</keyword><keyword>Neoplasms</keyword><keyword>Phenotype</keyword><keyword>Risk Factors</keyword><keyword>Tuberculosis</keyword></keywords><dates><year>2004</year></dates><urls><related-urls><url>PM:15075143</url></related-urls></urls></record></Cite></EndNote>(15). In mendelian randomization the effect of the gene on the outcome is mediated by the modifiable factors. There are special statistical methods (instrumental variable methods) that can be used to recover the effect of the modifiable factors in a way that potentially avoids many of the biases in observational studies. Another use of the concept of mediation in intervention studies is that of surrogate endpoint in randomized controlled trials, where the aim typically is to examine if an intervention has an effect on one or more clinical disease endpoints such as cancer or cardiovascular disease. In order to detect effects, clinical endpoints trials often require that a large number of participants are followed for at long time. Because of this surrogate endpoints are often used ADDIN EN.CITE <EndNote><Cite><Author>Cohn</Author><Year>2004</Year><RecNum>703</RecNum><record><rec-number>703</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">703</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Cohn, J. N.</author></authors></contributors><auth-address>Cardiovascular Division, Department of Medicine, University of Minnesota Medical School, Minneapolis, MN, USA</auth-address><titles><title>Introduction to surrogate markers</title><secondary-title>Circulation</secondary-title></titles><periodical><full-title>Circulation</full-title><abbr-1>Circulation</abbr-1><abbr-2>Circulation</abbr-2></periodical><pages>IV20-IV21</pages><volume>109</volume><number>25 Suppl 1</number><keywords><keyword>analysis</keyword><keyword>Arteriosclerosis</keyword><keyword>Biological Markers</keyword><keyword>Cardiovascular Diseases</keyword><keyword>complications</keyword><keyword>diagnosis</keyword><keyword>Disease Progression</keyword><keyword>etiology</keyword><keyword>Humans</keyword><keyword>Minnesota</keyword><keyword>therapy</keyword><keyword>Thrombosis</keyword><keyword>Universities</keyword></keywords><dates><year>2004</year></dates><urls><related-urls><url>PM:15226247</url></related-urls></urls></record></Cite></EndNote>(16). Surrogate endpoints are biomarkers for disease progression and are as such mediators between the intervention and the clinical endpoints. For example, CD4 cell count can be used as a surrogate endpoint in HIV treatment trials and serum cholesterol levels as a surrogate of coronary heart disease.Because mediation allows the investigator to peak into the black box it can also provide insight into why interventions might work or fail and thus guide future interventions. The paper by Mortensen et al. suggests that inventions that target smoking will likely reduce the educational gradient in birth weight, particularly if the intervention is successful among mothers with a short education. Such analyses of randomized trials might also provide clues as to what the ‘active ingredient’ in a given intervention might be. Analyses of mediation are, however, not a free lunch: they come at the cost of a number of added assumptions. Causal knowledge as a prerequisite for mediation. The attentive reader will have noticed that we used the term ‘affect’ to describe the relationship between variables. This is because, as was the case for confounding, the notion of mediation makes little sense unless we have a causal model in mind. In the case of mediation, the variables involved must be known or at least proposed to be causally related in a way that is at least partly known to the investigator. For example, Boyle et al. reported that the association between hostility and mortality was partly mediated by a pattern of episodic excessive alcohol use (binge drinking) among hostile men ADDIN EN.CITE <EndNote><Cite><Author>Boyle</Author><Year>2008</Year><RecNum>702</RecNum><record><rec-number>702</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">702</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Boyle, S. H.</author><author>Mortensen, L.</author><author>Gronbaek, M.</author><author>Barefoot, J. C.</author></authors></contributors><auth-address>Behavioral Medicine Research Center, Duke University Medical Center, Durham, NC, USA. shboyle@duke.edu</auth-address><titles><title>Hostility, drinking pattern and mortality</title><secondary-title>Addiction</secondary-title></titles><periodical><full-title>Addiction</full-title><abbr-1>Addiction</abbr-1><abbr-2>Addiction</abbr-2></periodical><pages>54-59</pages><volume>103</volume><number>1</number><keywords><keyword>Adult</keyword><keyword>alcohol</keyword><keyword>Alcohol Drinking</keyword><keyword>Anger</keyword><keyword>Cause of Death</keyword><keyword>COHORT</keyword><keyword>epidemiology</keyword><keyword>Health</keyword><keyword>Health Surveys</keyword><keyword>Humans</keyword><keyword>Male</keyword><keyword>Mortality</keyword><keyword>psychology</keyword><keyword>Regression Analysis</keyword><keyword>Research</keyword><keyword>trends</keyword><keyword>United States</keyword><keyword>UNITED-STATES</keyword><keyword>Universities</keyword></keywords><dates><year>2008</year></dates><urls><related-urls><url>PM:17995996</url></related-urls></urls></record></Cite></EndNote>(17). If high hostility is the cause of binge drinking use (i.e. hostility → binge drinking), then the investigators’ conclusion is correct. Let us assume that (unknown to the investigators) binge drinking over time increase hostility. If binge drinking is the cause of hostility (i.e. binge drinking → hostility) then alcohol use is not a mediator between hostility and mortality, but rather a common cause of these two variables. If this was the case, binge drinking would act as a confounder of the association between hostility and mortality, not as a mediator. In order for analyses of mediation to make sense, assumptions about the nature of the relationships between variables are needed. This may at first seem like a rather strong requirement because it appears to force the investigator to make conclusions in advance about the relationships that are under investigation. However, causational direction of relationship cannot be extracted from data alone ADDIN EN.CITE <EndNote><Cite><Author>Hernan</Author><Year>2002</Year><RecNum>710</RecNum><record><rec-number>710</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">710</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Hernan, M. A.</author><author>Hernandez-Diaz, S.</author><author>Werler, M. M.</author><author>Mitchell, A. A.</author></authors></contributors><auth-address>Department of Epidemiology, Harvard School of Public Health, Boston, MA 02115, USA. miguel_hernan@post.harvard.edu</auth-address><titles><title>Causal knowledge as a prerequisite for confounding evaluation: an application to birth defects epidemiology</title><secondary-title>Am.J.Epidemiol.</secondary-title></titles><periodical><full-title>analysis</full-title><abbr-1>Am.J.Epidemiol.</abbr-1></periodical><pages>176-184</pages><volume>155</volume><number>2</number><keywords><keyword>analysis</keyword><keyword>Bias (Epidemiology)</keyword><keyword>Case-Control Studies</keyword><keyword>Causality</keyword><keyword>Confounding Factors (Epidemiology)</keyword><keyword>Epidemiologic Methods</keyword><keyword>epidemiology</keyword><keyword>Female</keyword><keyword>Folic Acid</keyword><keyword>Health</keyword><keyword>Humans</keyword><keyword>Infant,Newborn</keyword><keyword>Mental Recall</keyword><keyword>Neural Tube Defects</keyword><keyword>Odds Ratio</keyword><keyword>Ontario</keyword><keyword>Pregnancy</keyword><keyword>prevention & control</keyword><keyword>Public Health</keyword><keyword>PUBLIC-HEALTH</keyword><keyword>Research</keyword><keyword>Risk</keyword><keyword>therapeutic use</keyword><keyword>United States</keyword></keywords><dates><year>2002</year></dates><urls><related-urls><url>PM:11790682</url></related-urls></urls></record></Cite></EndNote>(18). Investigators will usually get around this by relying to existing knowledge. In the example of Boyle et al. the prospective design will ensure that the outcome (mortality) occurs after the exposures are recorded. But the relationship between hostility and binge drinking is cross sectional so there is nothing in the design of the study to help the investigator decide about the direction of the relationship. Most studies carefully consider whether the exposure in fact causes the outcome. It is probably fair to say that in general less caution is exercised when it comes to making assumptions about the causal relationship between exposure and mediator. Never the less the analysis is conducted and the findings will most often be interpreted as if the mediator is caused by the exposure. To this end, graphs are a helpful tool because they encode the investigator’s assumptions about the possible causal relationships between variables.Bearing this in mind, it may be fruitful to think of mediation in terms of (hypothetical) interventions: If we could somehow intervene and change the subjects’ hostility levels in a certain way, would we expect their alcohol use to decline? Would the association between hostility and mortality change if the investigators had forced everyone to not to drink alcohol or forced everyone to binge drink once a week? Thinking of mediation in terms of possible interventions has the added advantage of providing a non-technical interpretation of the outcome of the analysis (given that the analysis is conducted accordingly). Starting off with a vague question (“does alcohol mediate the association between hostility and mortality”) may make it difficult to interpret the results. Just as important, it will also serve to make the in many cases highly hypothetical nature of the mediation analysis apparent ADDIN EN.CITE <EndNote><Cite ExcludeYear="1"><Author>Hernan</Author><RecNum>711</RecNum><record><rec-number>711</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">711</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Hernan, M. A.</author><author>Robins, J. M.</author></authors></contributors><titles><title>Causal Inference</title></titles><dates></dates><publisher>Chapman Hall/CRC</publisher><urls></urls></record></Cite><Cite><Author>Dawid</Author><Year>2000</Year><RecNum>705</RecNum><record><rec-number>705</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">705</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Dawid, A. P.</author></authors></contributors><titles><title>Causal Inference Without Counterfactuals</title><secondary-title>Journal of the American Statistical Association</secondary-title></titles><periodical><full-title>Journal of the American Statistical Association</full-title><abbr-1>J Am Stat Assoc</abbr-1><abbr-2>J Am Stat Assoc</abbr-2></periodical><pages>407-424</pages><volume>95</volume><number>450</number><keywords><keyword>analysis</keyword><keyword>OUTCOMES</keyword></keywords><dates><year>2000</year></dates><isbn>01621459</isbn><urls><related-urls><url>;(19, 20). How to analyze mediation. There are numerous ways to statistically model mediation ("for a review, see” ADDIN EN.CITE <EndNote><Cite><Author>Mackinnon</Author><Year>2002</Year><RecNum>718</RecNum><record><rec-number>718</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">718</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Mackinnon, D. P.</author><author>Lockwood, C. M.</author><author>Hoffman, J. M.</author><author>West, S. G.</author><author>Sheets, V.</author></authors></contributors><auth-address>Department of Psychology, Arizona State University, Tempe 85287-1104, USA. David.MacKinnon@asu.edu</auth-address><titles><title>A comparison of methods to test mediation and other intervening variable effects</title><secondary-title>Psychol.Methods</secondary-title></titles><periodical><full-title>Comparative Study</full-title><abbr-1>Psychol.Methods</abbr-1></periodical><pages>83-104</pages><volume>7</volume><number>1</number><keywords><keyword>Comparative Study</keyword><keyword>Humans</keyword><keyword>methods</keyword><keyword>Models,Psychological</keyword><keyword>psychology</keyword><keyword>Research</keyword></keywords><dates><year>2002</year></dates><urls><related-urls><url>PM:11928892</url></related-urls></urls></record></Cite></EndNote>(21)}. In a much cited 1986 paper, Baron and Kenny stated that the objective of such an analysis was to “test for mediation” ADDIN EN.CITE <EndNote><Cite><Author>Baron</Author><Year>1986</Year><RecNum>58</RecNum><record><rec-number>58</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">58</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Baron, Reuben M.</author><author>Kenny, David A.</author></authors></contributors><titles><title>The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations</title><secondary-title>Journal of Personality and Social Psychology</secondary-title></titles><periodical><full-title>Journal of Personality and Social Psychology</full-title><abbr-1>J. Pers. Soc. Psychol.</abbr-1><abbr-2>J Pers Soc Psychol</abbr-2><abbr-3>Journal of Personality & Social Psychology</abbr-3></periodical><pages>1173-1182</pages><volume>51</volume><number>6</number><dates><year>1986</year><pub-dates><date>Dec</date></pub-dates></dates><accession-num>Peer Reviewed Journal: 1987-13085-001</accession-num><urls></urls></record></Cite></EndNote>(9). This led them to device a method that was based on a significance test. However, it can be argued that the question of interest is not to determine if a given mediator is a statistically significant mediator, but rather to quantify how important the mediator is. This follows the general arguments against relying only on test of statistical significance in medical research ADDIN EN.CITE <EndNote><Cite><Author>Sterne</Author><Year>2001</Year><RecNum>728</RecNum><record><rec-number>728</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">728</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Sterne, J. A.</author><author>Davey, Smith G.</author></authors></contributors><auth-address>Department of Social Medicine, University of Bristol, Bristol BS8 2PR. jonathan.sterne@bristol.ac.uk</auth-address><titles><title>Sifting the evidence-what's wrong with significance tests?</title><secondary-title>BMJ</secondary-title></titles><periodical><full-title>BMJ</full-title><abbr-1>BMJ</abbr-1><abbr-2>BMJ</abbr-2></periodical><pages>226-231</pages><volume>322</volume><number>7280</number><keywords><keyword>Bayes Theorem</keyword><keyword>Data Interpretation,Statistical</keyword><keyword>history</keyword><keyword>History,20th Century</keyword><keyword>Humans</keyword><keyword>methods</keyword><keyword>Research</keyword><keyword>Statistics as Topic</keyword><keyword>Universities</keyword></keywords><dates><year>2001</year></dates><urls><related-urls><url>PM:11159626</url></related-urls></urls></record></Cite><Cite><Author>Rothman</Author><Year>1986</Year><RecNum>725</RecNum><record><rec-number>725</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">725</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Rothman, K. J.</author></authors></contributors><titles><title>Significance questing</title><secondary-title>Ann.Intern.Med.</secondary-title></titles><periodical><full-title>Statistics as Topic</full-title><abbr-1>Ann.Intern.Med.</abbr-1></periodical><pages>445-447</pages><volume>105</volume><number>3</number><keywords><keyword>Statistics as Topic</keyword></keywords><dates><year>1986</year></dates><urls><related-urls><url>PM:3740684</url></related-urls></urls></record></Cite></EndNote>(22, 23). In the applied literature, one of two somewhat different modeling approaches to mediation is often used. The one approach is to use a Structural Equation Model (SEM) and the other is to run a series of regressions to obtain and compare the total and direct (non-mediated) effect of the exposure on the outcome. This latter approach, which is a simplified version of the method of Baron and Kenny, involves controlling for the mediator to estimate the direct effect ADDIN EN.CITE <EndNote><Cite><Author>Kaufman</Author><Year>2004</Year><RecNum>713</RecNum><record><rec-number>713</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">713</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Kaufman, J. S.</author><author>MacLehose, R. F.</author><author>Kaufman, S.</author></authors></contributors><auth-address>Department of Epidemiology, University of North Carolina School of Public Health, Chapel Hill, NC 27599-7435 USA. Jay_Kaufman@unc.edu</auth-address><titles><title>A further critique of the analytic strategy of adjusting for covariates to identify biologic mediation</title><secondary-title>Epidemiol Perspect.Innov.</secondary-title></titles><pages>4</pages><volume>1</volume><number>1</number><keywords><keyword>epidemiology</keyword><keyword>methods</keyword><keyword>North Carolina</keyword><keyword>Population</keyword><keyword>Public Health</keyword><keyword>Research</keyword></keywords><dates><year>2004</year></dates><urls><related-urls><url>PM:15507130</url></related-urls></urls></record></Cite></EndNote>(24). In some cases these two approaches will yield similar results, in other cases the results will be different. The SEM approach has the advantage that the statistical model corresponds to the graphs typically used to conceptualize mediation, so that every arrow in the graph is estimated as a parameter from one single model. SEMs are primarily used in the social sciences, whereas in the health sciences SEMs appear to be the less popular choice. This is perhaps because SEMs are somewhat limited in the sense that they are an extension of linear regression, which is not always well suited for the kinds of data encountered in medicine. However, modern SEM theory (and modern SEM software) is relatively flexible with regards to finding models that fit most problems that involve mediation. A perhaps more important reason of the lack of popularity of SEMs for mediation analyses in the medical sciences is that most investigators and scientific journals in the health sciences will be familiar with multiple regression, but may not have experience with SEMs. In the following we will concentrate on the mediator adjustment approach. For an example of an applied paper that uses both approaches, see Batty et al. PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5CYXR0eTwvQXV0aG9yPjxZZWFyPjIwMDg8L1llYXI+PFJl
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ADDIN EN.CITE.DATA (25)The mediator adjustment approach involves estimating the total effect and direct effect in two separate regressions. To estimate the total effect, we need to take account of confounders of the exposure outcome association. In the simple situation where there is no confounding, the total effect is simply the outcome regressed on the exposure. The direct effect is typically estimated as the association between exposure and outcome when conditioning on the mediator. Once we condition on the mediator, we get the controlled direct effect the association between exposure and outcome. This is called a controlled effect because it corresponds to evaluating the association between exposure and outcome in a population where the mediator is forced by intervention at a certain level. In order for this to make sense some conditions need to be met. We will discuss these conditions using the example of Boyle et al. Adjustment for path-specific confounding. In addition to adjustment for confounders of the association between exposure and outcome, all confounders of the association between the mediator and the outcome have to be controlled for. Consider again the example of Boyle et al. In this graph, Early life socioeconomic position (SEP) confounds the relationship between hostility and mortality because it affects both. It also confounds the association between binge drinking and mortality. The graphs thus suggests that we should adjust for early life SEP when estimating the total effect and when estimating the direct effect. Suppose that unemployment affects binge drinking and mortality (loss of a job → increased binge drinking, mortality), but that this variable is not affected by hostility (the dotted arrow does not exist). Then unemployment is not a confounder of the total effect, but acts as a confounder of the association between binge drinking and mortality. In this situation the investigator needs to adjust for unemployment even though unemployment does not confound the total effect. If we fail to adjust for unemployment when estimating the direct effect of hostility on mortality the results will generally be biased ADDIN EN.CITE <EndNote><Cite><Author>Cole</Author><Year>2002</Year><RecNum>692</RecNum><record><rec-number>692</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">692</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Cole, Stephen R</author><author>Hernan, Miguel A</author></authors></contributors><titles><title>Fallibility in estimating direct effects</title><secondary-title>International Journal of Epidemiology</secondary-title></titles><periodical><full-title>International Journal of Epidemiology</full-title><abbr-1>Int. J. Epidemiol.</abbr-1><abbr-2>Int J Epidemiol</abbr-2></periodical><pages>163-165</pages><volume>31</volume><number>1</number><dates><year>2002</year><pub-dates><date>February 1, 2002</date></pub-dates></dates><urls><related-urls><url>;(26). This is because we need to condition on binge drinking to estimate the non-binge drinking mediated effect of hostility on mortality: Among those who binge drink, unemployment will be more frequent and mortality will be increased as a consequence. Suppose that highly hostile men tend to fight with colleagues and management and that they consequently are more likely to become unemployed (indicated by the dotted arrow). In this case unemployment confounds the association between binge drinking and mortality. This suggests that we should condition on it when estimating the direct effect. However, if we control for unemployment we eliminate the contribution of the hostility → unemployment → mortality pathway to the indirect effect. The problem arises because the dotted arrow contributes both to the direct effect (non-binge drinking mediated) and the indirect (binge drinking mediated) effect of hostility on mortality. This problem can be solved by resorting to a SEM or by applying special methods ADDIN EN.CITE <EndNote><Cite><Author>Petersen</Author><Year>2006</Year><RecNum>723</RecNum><record><rec-number>723</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">723</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Petersen, M. L.</author><author>Sinisi, S. E.</author><author>van der Laan, M. J.</author></authors></contributors><auth-address>Division of Biostatistics, University of California, School of Public Health, Berkeley, California 94720-7360, USA. mayaliv@berkeley.edu</auth-address><titles><title>Estimation of direct causal effects</title><secondary-title>Epidemiology</secondary-title></titles><periodical><full-title>Epidemiology</full-title><abbr-1>Epidemiology</abbr-1><abbr-2>Epidemiology</abbr-2></periodical><pages>276-284</pages><volume>17</volume><number>3</number><keywords><keyword>Clinical Trials</keyword><keyword>Disease</keyword><keyword>Effect Modifiers (Epidemiology)</keyword><keyword>Epidemiologic Studies</keyword><keyword>Humans</keyword><keyword>Models,Statistical</keyword><keyword>Public Health</keyword><keyword>Research</keyword><keyword>Research Design</keyword><keyword>United States</keyword></keywords><dates><year>2006</year></dates><urls><related-urls><url>PM:16617276</url></related-urls></urls></record></Cite></EndNote>(27).Measurement error. The mediator has to be measured without error. While mismeasurement is generally something that should be avoided, studies that aim to examine mediation should pay particular attention to measurement error. This is because even random error in the measurement of the mediator will bias both the direct effect and the indirect effect, but in different directions. The actual direction and strength of this bias depends on the pattern of mismeasurement. For example, suppose that instead of measuring binge drinking in the study by Boyle et al. the investigators tossed a coin for each participant to determine whether he was a binge drinker. In this case the direct effect would most likely be overestimated to the point that it would equal the total effect. SEM software usually has build-in features for handling measurement error, whereas some work is needed to take account of this in multiple regression ADDIN EN.CITE <EndNote><Cite><Author>Gustafson</Author><Year>2003</Year><RecNum>708</RecNum><record><rec-number>708</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">708</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Gustafson, P.</author></authors></contributors><titles><title>Measurement error and misclassification in statistics and epidemiology: impacts and Bayesian adjustments</title></titles><keywords><keyword>Statistics</keyword><keyword>epidemiology</keyword><keyword>IMPACT</keyword></keywords><dates><year>2003</year></dates><pub-location>Boca Raton, FL</pub-location><publisher>CRC Press</publisher><urls></urls></record></Cite></EndNote>(28) "for solutions, see e.g. "}.No interaction between exposure and mediator. The direct effect of the exposure on the outcome must not depend on at which particular value of the mediator variable it is assessed. In statistical terms this can be viewed as an assumption of no statistical interaction between the exposure and the mediator. For example, if the effect of hostility on mortality is stronger among those who binge drink than among those who do not, we can estimate two different controlled direct effects: one for binge drinkers and one of non-binge drinkers. Unless there is a strong argument for at what value of the mediator the association between exposure and outcome should be evaluated, the controlled direct effect does not make sense in the presence of statistical interactions. This also relates to the difference between mediators and moderators. As discussed above, a mediator is a variable that lies in a causal pathway (e.g. hostility → binge drinking → mortality). Moderation has to do with how two (or more) variables alone and in combination affects a third variable. In statistical terms, this is an interaction. It is important to note that statistical interaction depends on the choice of scale: Two variables that do not interact on multiplicative scale (e.g. in a logistic regression) will interact on an additive scale (linear regression) and vice versa. Because interaction depends on the choice of effect measure, statistical interaction is often denoted effect measure modification in epidemiology ADDIN EN.CITE <EndNote><Cite><Author>Rothman</Author><Year>2002</Year><RecNum>726</RecNum><record><rec-number>726</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">726</key></foreign-keys><ref-type name="Book Section">5</ref-type><contributors><authors><author>Rothman, K. J.</author></authors></contributors><titles><title>Measuring Interactions</title><secondary-title>Epidemiology. An Introduction</secondary-title></titles><pages>168-180</pages><volume>1st</volume><keywords><keyword>epidemiology</keyword></keywords><dates><year>2002</year></dates><pub-location>New York</pub-location><publisher>Oxford University Press</publisher><urls></urls></record></Cite></EndNote>(29). The concepts of mediation and moderation are fundamentally different and not mutually exclusive, so that a given variable can act as a mediator or as a moderator or as both. A discussion of interaction and moderation is given in Hernan & Robins. ADDIN EN.CITE <EndNote><Cite ExcludeYear="1"><Author>Hernan</Author><RecNum>711</RecNum><record><rec-number>711</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">711</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Hernan, M. A.</author><author>Robins, J. M.</author></authors></contributors><titles><title>Causal Inference</title></titles><dates></dates><publisher>Chapman Hall/CRC</publisher><urls></urls></record></Cite></EndNote>(19)Decomposition of total effects and indirect effects. We have now examined how total effects and the controlled direct effects can be estimated given, but what about the indirect (mediated) effect? In an SEM context, the total effect can readily be decomposed into a direct and indirect effect, but this is more difficult when using the mediator adjustment approach. Intuitively it seems reasonable to assume that the total is the sum of the parts, so that the indirect effect can the calculated by subtracting the direct effect from the total effect. This is the case in some situations, but in many situations it is not. If linear regression is used to estimate the total and direct effect, this strategy works well although the standard error of the indirect effect is not directly estimated. But often various kinds of non-linear regression models are used. Studies that use logistic regression will often report the percent reduction in the Odds Ratio after adjustment for the mediator(s). Unfortunately this strategy will generally not work PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5EaXRsZXZzZW48L0F1dGhvcj48WWVhcj4yMDA1PC9ZZWFy
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ADDIN EN.CITE.DATA (24, 30, 31). The problem is that this approach assumes that the change in Odds Ratio from one logistic regression to another has a very specific interpretation. This trick works in linear models because a mixture of two linear regressions is a linear regression, but this is not generally the case in logistic regression for example. It is worth noting that total and direct effects can be estimated from non-linear regression, but the indirect effect cannot consistently be calculated by contrasting the total and direct effects. There are also other (non-technical) reasons for resorting to a linear model. For example, it was long believed that traditional risk factors did not explain the social gradient in cardiovascular disease. This finding was predominantly supported by studies that used the mediator adjustment approach in multiplicative, non-linear models. This was something of a paradox in so far that research on the traditional risk factors suggested that these explained 90% of the cases. A landmark paper by Lynch et al. from 2003 showed that this apparent paradox was explained by the choice of relative measures of association. After adjustment for traditional cardiovascular risk factors (the mediators) the relative differences decreased by about 25%. The absolute risk differences, however, were reduced by about 75% ADDIN EN.CITE <EndNote><Cite><Author>Lynch</Author><Year>2006</Year><RecNum>717</RecNum><record><rec-number>717</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">717</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Lynch, J.</author><author>Davey, Smith G.</author><author>Harper, S.</author><author>Bainbridge, K.</author></authors></contributors><auth-address>Department Epidemiology, Biostatistics and Occupational Health, McGill University, Purvis Hall, 1020 Pine Avenue West, Montreal, Canada QC H3A 1A2. John.Lynch@McGill.ca</auth-address><titles><title>Explaining the social gradient in coronary heart disease: comparing relative and absolute risk approaches</title><secondary-title>Journal of Epidemiology and Community Health</secondary-title></titles><periodical><full-title>Journal of Epidemiology and Community Health</full-title><abbr-1>J. Epidemiol. Community Health</abbr-1><abbr-2>J Epidemiol Community Health</abbr-2><abbr-3>Journal of Epidemiology & Community Health</abbr-3></periodical><pages>436-441</pages><volume>60</volume><number>5</number><keywords><keyword>Adult</keyword><keyword>Aged</keyword><keyword>Canada</keyword><keyword>Coronary Disease</keyword><keyword>Diabetes Mellitus</keyword><keyword>Disease</keyword><keyword>Dyslipidemias</keyword><keyword>epidemiology</keyword><keyword>Finland</keyword><keyword>Humans</keyword><keyword>Hypertension</keyword><keyword>Male</keyword><keyword>Middle Aged</keyword><keyword>Population</keyword><keyword>Prospective Studies</keyword><keyword>Research</keyword><keyword>Risk</keyword><keyword>Risk Factors</keyword><keyword>Smoking</keyword><keyword>Social Class</keyword></keywords><dates><year>2006</year></dates><urls><related-urls><url>PM:16614335</url></related-urls></urls></record></Cite></EndNote>(32). This highlights an inherent problem in calculating a relative change in a relative measure. Mediation and interaction. As noted above the problem of assessing mediation in the presence of statistical interactions is exacerbated by the fact that statistical interactions are dependent on the choice of scale. This means that the choices of statistical model and measure of association will in part determine whether mediation is a tractable problem, which is both impractical and conceptually unsatisfying. One solution to this problem is to take a close look at the relationship between the exposure and the mediator. Recall that the controlled direct effect is estimated by fixing the mediator at some value, for example eradicating all binge drinking. For many real life problems it is difficult to imagine scenarios where forcing the mediator to attain a particular value is possible. But is this is quite different from what we would expect the exposure to do to the mediator. If we could somehow manipulate the exposure by intervention a reasonable expectation would be that the distribution of the mediator would shift from the distribution it had under no exposure to the distribution it has when the exposure is present. Consider the pathway perceived stress → estrogen → breast cancer from Nielsen et al. If we somehow intervened and eliminated all perceived stress, we would expect the subjects’ levels of serum estrogen to increase. This would result in a shift in the distribution of estrogen. So, instead of fixing the mediator at certain level, we can then calculate the direct effect when the mediator has a certain distribution. If we wanted to estimate the direct (non-estrogen mediated) effect of perceived stress on breast cancer, we could use this information. For example, we could evaluate the association between perceived stress under the distribution that estrogen has under no exposure to perceived stress instead of evaluating it in an analysis where we fixed everyone’s estrogen levels to attain the exact same value. This leads to the estimation of a natural direct effect. This is done using simple standardization techniques such as those used to calculate standardized rates. It is important to note that using natural direct effects will yield results that are identical to controlled direct effect unless there is statistical interaction between exposure and outcome, but even in this case the concept does add value: Not only does the concept of natural effects provide a definition of direct effects in the presence of interaction, they also lead to a definition of indirect effects. A natural indirect effect can be defined as the change in outcome when the exposure is fixed and the distribution of the mediator is changed. The reader is referred elsewhere for a comprehensive review of natural effects PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5QZWFybDwvQXV0aG9yPjxZZWFyPjIwMDU8L1llYXI+PFJl
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ADDIN EN.CITE.DATA (7, 27, 33). Interactions and moderationThe assumption of homogeneity. An additionalAn assumption that all regression model make is that the effect of a given variable is homogeneous across all levels of all other variables, irrespective of whether those variables are measured and included in the model or not. For example, if we estimate a model in which depression is a predictor of cardiac disease, we implicitly make the assumption that the association between depression and disease is the same (within sampling error) for men and women, the old and the young, across ethnicities, genotypes, etc. Even if these other variables were measured and included in the model as adjustment covariables, such a model would not yield any information about this possible heterogeneity. There are several ways we might tackle this question. One intuitively appealing method would be to divide the sample into subgroups and evaluate the regression coefficient within each of the groups. For example, we might divide our sample based on gender, and estimate the relation between depression and cardiac disease separately for me and women. Subgroup tests, however, are highly controversial and generally discouraged by statisticians for a number of reasons ADDIN EN.CITE <EndNote><Cite><Author>Assmann</Author><Year>2000</Year><RecNum>76</RecNum><record><rec-number>76</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">76</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Assmann, Susan F.</author><author>Pocock, Stuart J.</author><author>Enos, Laura E.</author><author>Kasten, Linda E.</author></authors></contributors><titles><title>Subgroup analysis and other (mis)uses of baseline data in clinical trials</title><secondary-title>Lancet</secondary-title></titles><periodical><full-title>Lancet</full-title><abbr-1>Lancet</abbr-1><abbr-2>Lancet</abbr-2></periodical><pages>1064-1069</pages><volume>355</volume><number>9209</number><dates><year>2000</year></dates><isbn>0140-6736</isbn><urls><related-urls><url>;(34).? Among these objections to subgroup testing, the two most important are the inflated error rate, and the differential power of the tests, and the increased imprecision of the parameter estimates due to the smaller sample sizes.? Conducting many tests of any kind inflates the Type I error rate.? In the case of subgroup tests, of course, all the parameters in a given model are re-estimated within each subgroup, creating a whole host of new opportunities for capitalizing on the idiosyncrasies of sample, with the added disadvantage of conducting those tests on fewer data points! Correction for multiple testing in these cases can be of some help, but unless the study was designed specifically for the subgroup test, the power can and usually will be quite different for different subgroups.? Hence, some subgroup tests will have more power than others, making it virtually impossible to manage the error rate coherently.? If subgroup tests are of interest, the sampling plan must take them into account before the study is carried out to ensure adequate and consistent power across them.? The inferences from pre-planned subgroup analyses are, of course, more robust than those which arose from post hoc analyses.? If the design did not take these tests into account, subgroup analyses should either not be conducted at all, or should be interpreted as highly preliminary.? Finally, if we are interested in studying heterogeneity of associations, the preferred approach is to test the corresponding interaction term rather to examine subgroups separately PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5BbHRtYW48L0F1dGhvcj48WWVhcj4yMDAzPC9ZZWFyPjxS
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ADDIN EN.CITE.DATA (37, 38)]).? For example, if one is interested in whether a treatment is more effective in one ethnic group than another, the proper test is a treatment group by ethnicity interaction term.? In a multivariable model setting, when more than one interaction term is of interest, the error rate can be minimized by entering all the interaction terms of interest in the model as a block simultaneously and testing the change in model fit associated with the block ADDIN EN.CITE <EndNote><Cite><Author>Cohen</Author><Year>2002</Year><RecNum>92</RecNum><record><rec-number>92</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">92</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Cohen, J.</author><author>West, S.G.</author><author>Aiken, L.</author><author>Cohen, P.</author></authors></contributors><titles><title>Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences</title></titles><edition>3rd</edition><dates><year>2002</year></dates><pub-location>London</pub-location><publisher>Taylor and Francis</publisher><urls></urls></record></Cite><Cite><Author>Harrell</Author><Year>2001</Year><RecNum>10</RecNum><record><rec-number>10</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">10</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Harrell, F. E.</author></authors></contributors><titles><title>Regression Modeling Strategies: With applications to linear modeling, logistic regression, and survival analysis</title></titles><keywords><keyword>analysis</keyword><keyword>regression</keyword><keyword>survival</keyword><keyword>Survival Analysis</keyword></keywords><dates><year>2001</year></dates><pub-location>New York</pub-location><publisher>Springer</publisher><accession-num>151</accession-num><urls></urls></record></Cite></EndNote>(39, 40).? If the test of the entire block is not significant, then the individual interaction terms are interpreted as inconclusive or noise. ?I add a reminder here that in most statistical models nowadays all lower order component terms must be included in the model with a higher order terms such as interaction.? For example, if we are testing a treatment group by ethnicity interaction, we also must include the treatment group and ethnicity main effects—otherwise the interaction term is not really interpretable as an interaction in the conventional sense of the concept.Preserve measurement information wherever possible. On a final note, when testing interactions, one might be tempted to create dichotomies or groups out of continuously measured variables. Researchers also make artificial categories for other reasons, such as ease of interpretation, evaluating nonlinearity, to parallel clinical cutpoints, or even in the belief that the grouping somehow improves measurement precision.? Indeed, creating groups out of continuous variables has a long history in psychology, medicine, and epidemiology.? What many modern researchers fail to realize, however, is that this tradition arose strictly out of necessity.? In the early days of modern statistical practice, it was apparently well understood that the practice of grouping was less than ideal, but there was little choice given the lack of computational power. With the availability of ample computational power, modern authorities in methodology have repeatedly discouraged researchers from adopting this practice PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5NYWNDYWxsdW08L0F1dGhvcj48WWVhcj4yMDAyPC9ZZWFy
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ADDIN EN.CITE.DATA (41-44). Compared to the categorized version of a variable, using the continuous form yields substantially greater statistical power ADDIN EN.CITE <EndNote><Cite><Author>Cohen</Author><Year>1983</Year><RecNum>9</RecNum><record><rec-number>9</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">9</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Cohen, J.</author></authors></contributors><titles><title>The cost of dichotomization</title><secondary-title>Applied Psychological Measurement</secondary-title></titles><periodical><full-title>Applied Psychological Measurement</full-title><abbr-1>Appl Psychol Meas</abbr-1><abbr-2>Appl Psychol Meas</abbr-2></periodical><pages>249-253</pages><volume>7</volume><number>3</number><keywords><keyword>Psycinfo</keyword></keywords><dates><year>1983</year></dates><accession-num>2697</accession-num><urls></urls></record></Cite></EndNote>(43), is less likely to produce spurious significance ADDIN EN.CITE <EndNote><Cite><Author>Maxwell</Author><Year>1993</Year><RecNum>37</RecNum><record><rec-number>37</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">37</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Maxwell, S.E.</author><author>Delaney, H. D.</author></authors></contributors><titles><title>Bivariate Median Splits and Spurious Statistical Significance</title><secondary-title>Psychological Bulletin</secondary-title></titles><periodical><full-title>Psychological Bulletin</full-title><abbr-1>Psychol. Bull.</abbr-1><abbr-2>Psychol Bull</abbr-2></periodical><pages>20</pages><volume>113</volume><section>181</section><dates><year>1993</year></dates><urls></urls></record></Cite></EndNote>(45), and, from a measurement perspective, is a more reliable instantiation of the variable under study ADDIN EN.CITE <EndNote><Cite><Author>Harrell</Author><Year>2008</Year><RecNum>35</RecNum><record><rec-number>35</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">35</key></foreign-keys><ref-type name="Web Page">12</ref-type><contributors><authors><author>Harrell, F.E.</author></authors></contributors><titles><title>Problems Caused by Categorizing Continuous Variables</title></titles><dates><year>2008</year></dates><urls><related-urls><url>;(44). The much preferred alternative to categorizing is to model the continuous variable as measured. If nonlinearity is a concern, techniques such as splines ADDIN EN.CITE <EndNote><Cite><Author>Harrell</Author><Year>2001</Year><RecNum>10</RecNum><record><rec-number>10</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">10</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Harrell, F. E.</author></authors></contributors><titles><title>Regression Modeling Strategies: With applications to linear modeling, logistic regression, and survival analysis</title></titles><keywords><keyword>analysis</keyword><keyword>regression</keyword><keyword>survival</keyword><keyword>Survival Analysis</keyword></keywords><dates><year>2001</year></dates><pub-location>New York</pub-location><publisher>Springer</publisher><accession-num>151</accession-num><urls></urls></record></Cite></EndNote>(40) or fractional polynomials ADDIN EN.CITE <EndNote><Cite><Author>Royston</Author><Year>1994</Year><RecNum>735</RecNum><record><rec-number>735</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">735</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Royston, Patrick</author><author>Altman, Douglas G.</author></authors></contributors><titles><title>Regression Using Fractional Polynomials of Continuous Covariates: Parsimonious Parametric Modelling</title><secondary-title>Journal of the Royal Statistical Society. Series C (Applied Statistics)</secondary-title></titles><periodical><abbr-1>Journal of the Royal Statistical Society. Series C (Applied Statistics)</abbr-1></periodical><pages>429-467</pages><volume>43</volume><number>3</number><dates><year>1994</year></dates><publisher>Blackwell Publishing for the Royal Statistical Society</publisher><isbn>00359254</isbn><urls><related-urls><url>;(46) will allow for a nonlinear association without discarding information or making arbitrary cutpoints. Despite the overwhelming evidence of the inadequacy of the categorization approach, a quick glance at many scientific journals suggests that the force of tradition is apparently quite strong. We once again appeal to readers to avoid this fundamental error in data analysis. Some more additional considerations on regression modelsSample size in multivariable models. We now turn to a last few concepts that bear directly on the above material in terms of producing replicable models. Earlier, we alluded to the idea that although it is a good idea to include potential confounders and additional predictors of the response in a model, the number we can include in a model and still obtain reproducible results is determined by the sample size we have to work with.? Before the advent of simulation studies, statisticians often offered rules of thumb based on their experience. One well-known rule of thumb for linear regression models is that there should be at least 10, preferably 15, cases for every degree of freedom used in estimating the equation.? Typically, each predictor uses one degree of freedom.? For example, is we want to study 10 predictors with no interactions or curvilinear terms, we should have at least 100 observations?in our sample.? Perhaps it is not a surprise, but modern simulation studies have tended to support this rule of thumb, demonstrating empirically that following this guideline will result in a regression model that is more likely to replicate in new samples.? There are also rules of thumb that have been empirically tested for logistic regression models and also survival models such as Cox regression.? ?The rules of thumb for logistic and time to event models are similar to that for linear regression, about 10-15 observations per predictor.? However, there is an important difference in how the number of observations is counted in the logistic and time to event models.? In these models, the number of observations is based on something called the effective sample size.? The effective sample size for a time to event regression model is simply the number of events.? So, if there are 1000 participants in a study, and only 10 of them sustain the event being study, the effective sample size is 10.? For logistic regression models, in which the outcome is a binary variable, the effective sample size is the count of events or nonevents, whichever is the smaller number of the two.? For example, if there are 200 individuals in the sample, and 20 had an event, the effective sample size is 20, not 200, and at best 2 variables can be studied with reasonable confidence.? If there were 180 events rather than 20, the effective sample size would still be 20.? In more technical parlance, the number of cases in a logistic regression model with a binary response is min(q, n-q), where min represents “the minimum of the following quantities”, q is the number of events, and n is the total sample size.? Finally, for ordinal logistic regression models, that is, models with more than two ordered category as the response, the effective sample size is given by n-1n2i=1kni3 where n is the sample size and k is the number of response categories ADDIN EN.CITE <EndNote><Cite><Author>Harrell</Author><Year>2001</Year><RecNum>10</RecNum><record><rec-number>10</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">10</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Harrell, F. E.</author></authors></contributors><titles><title>Regression Modeling Strategies: With applications to linear modeling, logistic regression, and survival analysis</title></titles><keywords><keyword>analysis</keyword><keyword>regression</keyword><keyword>survival</keyword><keyword>Survival Analysis</keyword></keywords><dates><year>2001</year></dates><pub-location>New York</pub-location><publisher>Springer</publisher><accession-num>151</accession-num><urls></urls></record></Cite></EndNote>(40).What are the consequences of studying more variables than the guidelines suggest? ? Perhaps the most serious consequence of trying to squeeze too many variables in a model is overfitting.? Overfitting is a condition in which the idiosyncrasies of the sample lead to an overly optimistic overall fit of the model.? Intuitively, we might say that there is simply not enough information (in terms of observations) to distinguish noise from true signal.? The fewer observations per degree of freedom in a model, the more likely the model will be overfit.? Overfitting is discussed in greater detail in Babyak ADDIN EN.CITE <EndNote><Cite><Author>Babyak</Author><Year>2004</Year><RecNum>5</RecNum><record><rec-number>5</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">5</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Babyak, M. A.</author></authors></contributors><titles><title>What you see may not be what you get: a brief, nontechnical introduction to overfitting in regression-type models</title><secondary-title>Psychosomatic Medicine</secondary-title></titles><periodical><full-title>Psychosomatic Medicine</full-title><abbr-1>Psychosom. Med.</abbr-1><abbr-2>Psychosom Med</abbr-2></periodical><pages>411-421</pages><volume>66</volume><keywords><keyword>models</keyword></keywords><dates><year>2004</year></dates><accession-num>158</accession-num><urls></urls></record></Cite></EndNote>(47) and Steyerberg ADDIN EN.CITE <EndNote><Cite><Author>Steyerberg</Author><Year>2009</Year><RecNum>47</RecNum><record><rec-number>47</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">47</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Steyerberg, E.W.</author></authors></contributors><titles><title>Clinical Prediction Models</title></titles><dates><year>2009</year></dates><pub-location>New York</pub-location><publisher>Springer</publisher><urls></urls></record></Cite></EndNote>(8).? Figure 1 displays the results of a series of simulations carried out by Babyak ADDIN EN.CITE <EndNote><Cite><Author>Babyak</Author><Year>2004</Year><RecNum>5</RecNum><record><rec-number>5</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">5</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Babyak, M. A.</author></authors></contributors><titles><title>What you see may not be what you get: a brief, nontechnical introduction to overfitting in regression-type models</title><secondary-title>Psychosomatic Medicine</secondary-title></titles><periodical><full-title>Psychosomatic Medicine</full-title><abbr-1>Psychosom. Med.</abbr-1><abbr-2>Psychosom Med</abbr-2></periodical><pages>411-421</pages><volume>66</volume><keywords><keyword>models</keyword></keywords><dates><year>2004</year></dates><accession-num>158</accession-num><urls></urls></record></Cite></EndNote>(47).? The plot shows the distribution of model r-square values for various levels of predictors/observations for a model with 10 predictors whose values are merely randomly generated, i.e. are pure noise.? Because the predictor values are randomly generated, the 'true' model should have an r-square value of zero, with any non-zero r-square arising simply due to random sampling fluctuation.? The plot demonstrates that when there are relatively many observations per predictor, the vast majority of r-square values are zero or very close to zero.? However, as the predictor/observation ratio becomes smaller, the typical r-square values become larger and more varied, with some even reflecting a fairly large amount of variance explained.? In addition to generating overly optimistic model fit, having too few observations per predictor also results in bias in the estimates for the individual parameters.? Peduzzi et al. ADDIN EN.CITE <EndNote><Cite><Author>Peduzzi</Author><Year>1996</Year><RecNum>79</RecNum><record><rec-number>79</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">79</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Peduzzi, Peter</author><author>Concato, John</author><author>Kemper, Elizabeth</author><author>Holford, Theodore R.</author><author>Feinstein, Alvan R.</author></authors></contributors><titles><title>A simulation study of the number of events per variable in logistic regression analysis</title><secondary-title>Journal of Clinical Epidemiology</secondary-title></titles><periodical><full-title>Journal of Clinical Epidemiology</full-title><abbr-1>J. Clin. Epidemiol.</abbr-1><abbr-2>J Clin Epidemiol</abbr-2></periodical><pages>1373-1379</pages><volume>49</volume><number>12</number><keywords><keyword>Monte Carlo</keyword><keyword>bias</keyword><keyword>precision</keyword><keyword>significance testing</keyword></keywords><dates><year>1996</year></dates><isbn>0895-4356</isbn><urls><related-urls><url>;(48) showed in a series of simulations that an inadequate predictors/observations ratio also leads to serious bias in the estimates of the regression coefficients in logistic regression and time-to-event models. Some have argued that in the case of models in which we are interested in a single predictor and are merely concerned about ruling out confounding, fewer variables per predictor may be required.? Vittinghoff et al. ADDIN EN.CITE <EndNote><Cite><Author>Vittinghoff</Author><Year>2007</Year><RecNum>81</RecNum><record><rec-number>81</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">81</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Vittinghoff, Eric</author><author>McCulloch, Charles E.</author></authors></contributors><titles><title>Relaxing the Rule of Ten Events per Variable in Logistic and Cox Regression</title><secondary-title>American Journal of Epidemiology</secondary-title></titles><periodical><full-title>American Journal of Epidemiology</full-title><abbr-1>Am. J. Epidemiol.</abbr-1><abbr-2>Am J Epidemiol</abbr-2></periodical><pages>710-718</pages><volume>165</volume><dates><year>2007</year></dates><urls><related-urls><url>;(49) has argued that in this circumstance, perhaps as few as 5 events/case per predictor may be sufficient, but the authors also show that under some circumstances even more than 15 per predictor may not be enough.? Perhaps the most prudent advice is that more is always better when it comes to sample size and that when there are relatively fewer cases than the guidelines suggest, interpreting such results with great caution.Reducing the degrees of freedom in a model. If you are confronted with a situation in which you wish to study more variables than the sample size allows, what are the alternatives?? A popular approach in the past has been to use automated 'stepwise' methods.? There are actually a variety of these techniques, but they are typically characterized by sequentially entering and removing variables based on the correlations and partial correlations between the predictors and response variable until some arbitrary criterion is met.? For example, in forward stepwise selection, the algorithm scans the correlations between the predictors and response variable and selects the predictor with the largest correlation with the response.? In the next step, the correlations between the remaining candidate predictors and the response are partialled for the effect of the first variable that was chosen, and the algorithm selects the largest of these partialled correlations.? The process continues until some predetermined measure of fit is achieved.? Unfortunately, these algorithms have been subsequently shown to be significantly flawed in terms of inference.? They do generate models that will fit the sample data well, but when used in the way that most of us have used them, they are almost certain to not produce a replicable model.? That is, when we compare the fit of the model and the parameter estimates from the stepwise model to a model based on a new sample, not much will be the same.? Intuitively, the overly optimistic fit can be understood as a function of the fact that we have tested many variables, and that by chance alone (i.e. random sampling fluctuation), we are bound to find at least a few, and sometimes even many, predictor variables that display a non-trivial association with the response variable. On the other side of the same coin, the automated algorithm will also miss potentially important variables, again due to sampling error, yielding a model with parameter estimates that may not be appropriately adjusted, i.e., a misspecified model. Moreover,Further problems arise with automated algorithms when there are correlations among the candidate predictor variables. In these instances, the choice to select one or the other by the algorithm can be quite arbitrary.? Not surprisingly, in recent years, the use of automated stepwise methods has been almost uniformly discouraged by statisticians.? Several journals, in fact, will not accept papers that are based on conventional stepwise analyses ADDIN EN.CITE <EndNote><Cite><Author>Freedland</Author><Year>2005</Year><RecNum>91</RecNum><record><rec-number>91</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">91</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Freedland, K.E.</author><author>Babyak, M.A.</author><author>McMahon, R.J.</author><author>Jennings, J.R.</author><author>Golden, R.N.</author><author>Sheps, D.S.</author></authors></contributors><titles><title>Statistical Guidelines for Psychosomatic Medicine</title><secondary-title>Psychosomatic Medicine</secondary-title></titles><periodical><full-title>Psychosomatic Medicine</full-title><abbr-1>Psychosom. Med.</abbr-1><abbr-2>Psychosom Med</abbr-2></periodical><pages>167</pages><volume>67</volume><dates><year>2005</year></dates><urls></urls></record></Cite><Cite><Author>Thompson</Author><Year>1995</Year><RecNum>732</RecNum><record><rec-number>732</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">732</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Thompson, B.</author></authors></contributors><titles><title>Stepwise regression and stepwise discriminant analysis need not apply here: A guidelines editorial</title><secondary-title>Educational and Psychological Measurement</secondary-title></titles><periodical><full-title>Educational and Psychological Measurement</full-title><abbr-1>Ed Psychol Meas</abbr-1><abbr-2>Ed Psychol Meas</abbr-2></periodical><pages>525-534</pages><volume>55</volume><number>4</number><dates><year>1995</year><pub-dates><date>August 1, 1995</date></pub-dates></dates><urls><related-urls><url>;(50, 51).??A commonly used alternative to stepwise selection is univariate prescreening of variables.? In this approach, the researcher evaluates the univariate relation between each predictor and the response variable and selects those which are statistically significant for entry in a final regression model.? Unfortunately, this technique suffers from essentially the same, and at times worse shortcomings, though perhaps not quite as dire, asthan those seen in the automated stepwise algorithms. The fit is again biased toward being too good, because we are selecting predictors whose parameters of the largest magnitude without accounting for the possibility that the magnitude of the predictor is also influenced by random sampling error.? Steyerberg ADDIN EN.CITE <EndNote><Cite><Author>Steyerberg</Author><Year>2009</Year><RecNum>47</RecNum><record><rec-number>47</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">47</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Steyerberg, E.W.</author></authors></contributors><titles><title>Clinical Prediction Models</title></titles><dates><year>2009</year></dates><pub-location>New York</pub-location><publisher>Springer</publisher><urls></urls></record></Cite></EndNote>(8) calls selection based on p-values “testimation bias” As a more general principle, using the sample data to determine what to include in a model will produce fit that may be too good and parameters that are too large.? A further difficulty with univariate prescreening is that variables behave differently in univariate setting compared to a multivariate model.? It is entirely possible, for example, for a potential predictor to look quite uninteresting in a univariate setting and then come to life when partialled for other variables.Arguably the best alternative to automated techniques and prescreening is to specify the model in its entirety before even collecting the data. A prespecified model is preferable for a number of reasons.? First and foremost, it requires a thoughtful consideration of the phenomenon under study before collecting the data.? Second, it is transparent.? There is no doubt as to whether other variables were considered but just not reported.? Finally, the p-values for the fit of the model and for the parameters will be 'honest.'? In other words, once predictors are tested either during pretesting or some other selection process and discarded, the tests of the model with the remaining variables, as well as the test of model fit will be too optimistic (for a simulation study demonstrating this principle, see ADDIN EN.CITE <EndNote><Cite><Author>Budtz-J?rgensen</Author><Year>2007</Year><RecNum>736</RecNum><record><rec-number>736</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">736</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Budtz-J?rgensen, Esben</author><author>Keiding, Niels</author><author>Grandjean, Philippe</author><author>Weihe, Pal</author></authors></contributors><titles><title>Confounder Selection in Environmental Epidemiology: Assessment of Health Effects of Prenatal Mercury Exposure</title><secondary-title>Annals of Epidemiology</secondary-title></titles><periodical><full-title>Annals of Epidemiology</full-title><abbr-1>Ann. Epidemiol.</abbr-1><abbr-2>Ann Epidemiol</abbr-2></periodical><pages>27-35</pages><volume>17</volume><number>1</number><keywords><keyword>Confounding Factors (Epidemiology)</keyword><keyword>Regression Analysis</keyword><keyword>Statistical Models</keyword></keywords><dates><year>2007</year></dates><isbn>1047-2797</isbn><urls><related-urls><url>;(52)).Sometimes, of course, it is not possible or even desirable to have a single prespecified model.? We simply may not know quite enough about the entire system of variables we are studying, or perhaps collecting some of the data is expensive and we want to cull as many of the non-important variables out of the equation.? There are a variety of approaches that will either allow us to include more variables that the rules of thumb suggest, or that will remove extraneous variables with the correct adjustment. The simplest technique for reducing degrees of freedom is to combine predictors in some rational way.? Combining is useful when there are variables that are acting solely as nuisance or adjustment variables for which we are not particularly interested in their individual regression coefficients, but still want the information they provide to be included in the model.? We can simply create a composite score from two or more variables, by summing their ranks or converting the variables to standardized scores and summing them.? Alternatively, we can use a clustering technique such as principle components or common factor analysis to develop a composite that captures the information in the variables.? The resulting composite that we create is then used instead of the individual variables in the model.? More details on these approaches are available in Harrell ADDIN EN.CITE <EndNote><Cite><Author>Harrell</Author><Year>2001</Year><RecNum>10</RecNum><record><rec-number>10</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">10</key></foreign-keys><ref-type name="Book">6</ref-type><contributors><authors><author>Harrell, F. E.</author></authors></contributors><titles><title>Regression Modeling Strategies: With applications to linear modeling, logistic regression, and survival analysis</title></titles><keywords><keyword>analysis</keyword><keyword>regression</keyword><keyword>survival</keyword><keyword>Survival Analysis</keyword></keywords><dates><year>2001</year></dates><pub-location>New York</pub-location><publisher>Springer</publisher><accession-num>151</accession-num><urls></urls></record></Cite></EndNote>(40).More sophisticated methods for automated model selection have been developed recently and are now becoming more widely available in popular software packages. The techniques include the lasso and least angle regression approaches developed by Tibshirani ADDIN EN.CITE <EndNote><Cite><Author>Tibshirani</Author><Year>1997</Year><RecNum>108</RecNum><record><rec-number>108</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">108</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Tibshirani, R.</author></authors></contributors><auth-address>Department of Preventive Medicine and Biostatistics, University of Toronto, Ontario, Canada.</auth-address><titles><title>The lasso method for variable selection in the Cox model</title><secondary-title>Statistics in Medicine</secondary-title><alt-title>Stat Med</alt-title></titles><periodical><full-title>Statistics in Medicine</full-title><abbr-1>Stat. Med.</abbr-1><abbr-2>Stat Med</abbr-2></periodical><alt-periodical><full-title>Statistics in Medicine</full-title><abbr-1>Stat. Med.</abbr-1><abbr-2>Stat Med</abbr-2></alt-periodical><pages>385-95</pages><volume>16</volume><number>4</number><keywords><keyword>Humans</keyword><keyword>Karnofsky Performance Status</keyword><keyword>Likelihood Functions</keyword><keyword>Liver Cirrhosis/et [Etiology]</keyword><keyword>Liver Cirrhosis/mo [Mortality]</keyword><keyword>Lung Neoplasms/mo [Mortality]</keyword><keyword>*Proportional Hazards Models</keyword><keyword>Randomized Controlled Trials as Topic/mt [Methods]</keyword><keyword>*Survival Analysis</keyword></keywords><dates><year>1997</year><pub-dates><date>Feb 28</date></pub-dates></dates><accession-num>9044528</accession-num><urls></urls><language>English</language></record></Cite></EndNote>(53), Bayesian model averaging ADDIN EN.CITE <EndNote><Cite><Author>Hoeting</Author><Year>1999</Year><RecNum>733</RecNum><record><rec-number>733</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">733</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Hoeting, J.A.</author><author>Madigan, D.</author><author>Raftery, A.E.</author><author>Volinsky, C.T.</author></authors></contributors><titles><title>Bayesian model averaging: a tutorial</title><secondary-title>Statistical Science</secondary-title></titles><periodical><abbr-1>Statistical Science</abbr-1></periodical><pages>382-417</pages><volume>14</volume><dates><year>1999</year></dates><urls></urls></record></Cite></EndNote>(54), and the use of penalization ADDIN EN.CITE <EndNote><Cite><Author>Moons</Author><Year>2004</Year><RecNum>59</RecNum><record><rec-number>59</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">59</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Moons, K. G. M.</author><author>Donders, A. Rogier T.</author><author>Steyerberg, E. W.</author><author>Harrell, F. E.</author></authors></contributors><titles><title>Penalized maximum likelihood estimation to directly adjust diagnostic and prognostic prediction models for overoptimism: a clinical example</title><secondary-title>Journal of Clinical Epidemiology</secondary-title></titles><periodical><full-title>Journal of Clinical Epidemiology</full-title><abbr-1>J. Clin. Epidemiol.</abbr-1><abbr-2>J Clin Epidemiol</abbr-2></periodical><pages>1262-1270</pages><volume>57</volume><number>12</number><keywords><keyword>Prediction research</keyword><keyword>Overoptimism</keyword><keyword>Overfitting</keyword><keyword>Penalization</keyword><keyword>Bootstrapping</keyword><keyword>Shrinkage</keyword></keywords><dates><year>2004</year></dates><isbn>0895-4356</isbn><urls><related-urls><url>;(55) or random effects ADDIN EN.CITE <EndNote><Cite><Author>Greenland</Author><Year>2000</Year><RecNum>109</RecNum><record><rec-number>109</rec-number><foreign-keys><key app="EN" db-id="tpazepzt7f9f59ez0ar52xdqe0fxte9d205w">109</key></foreign-keys><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Greenland, S.</author></authors></contributors><auth-address>Department of Epidemiology, UCLA School of Public Health 90095-1772, USA.</auth-address><titles><title>When should epidemiologic regressions use random coefficients?</title><secondary-title>Biometrics</secondary-title><alt-title>Biometrics</alt-title></titles><periodical><full-title>Biometrics</full-title><abbr-1>Biometrics</abbr-1><abbr-2>Biometrics</abbr-2></periodical><alt-periodical><full-title>Biometrics</full-title><abbr-1>Biometrics</abbr-1><abbr-2>Biometrics</abbr-2></alt-periodical><pages>915-21</pages><volume>56</volume><number>3</number><keywords><keyword>Breast Neoplasms/ep [Epidemiology]</keyword><keyword>Breast Neoplasms/et [Etiology]</keyword><keyword>Carcinogens</keyword><keyword>Case-Control Studies</keyword><keyword>Diet</keyword><keyword>*Epidemiologic Methods</keyword><keyword>Female</keyword><keyword>Humans</keyword><keyword>*Models, Statistical</keyword><keyword>Odds Ratio</keyword><keyword>Random Allocation</keyword><keyword>*Regression Analysis</keyword><keyword>Reproducibility of Results</keyword></keywords><dates><year>2000</year><pub-dates><date>Sep</date></pub-dates></dates><accession-num>10985237</accession-num><urls></urls><language>English</language></record></Cite></EndNote>(56). The details of these techniques are far beyond the scope of this chapter, but they do show some promise in terms of allowing an algorithm to make reasonable selections of variables while accounting for uncertainty. Because these approaches properly correct for capitalizing on the idiosyncrasies of the sample, however, many researcher may be quite displeased with the failure to find ‘significant’ results. Nevertheless, these approaches generate far more realistic appraisals of the extent to which our results will replicate in a new sample.?Summary. This paper has reviewed some of the issues involved in the estimation of regression models in terms of variable selection and underlying causal models. Specifically, regression models that attempt to illuminate causal understanding are most useful when we try to account for potential confounders, include additional variables that enhance precision, and test for mediators. For mediation, SEMs are currently the best choice for the applied researcher because they are linear, provide consistent decomposition of the total effect into direct and indirect contributions and allow the investigator to take measurement error into account. If interactions among two or more variables are suspected, care must be taken to design the study in such a way that these potential interactions can be adequately studies. When testing mediation, if there are strong interactions between the exposure and the outcome, methods beyond simple SEMs are needed. Finally, in order to increase the likelihood that our models will replicate, and hence be generalizable, attention should be paid to the number of parameters we seek to estimate in the context of sample size. Figure CaptionResults of simulation automated stepwise regression with 15 candidate predictor variables. In the true model, predictors were randomly generated and therefore unrelated to the response variable, meaning that the true r-square was zero. The ratio of predictors to sample size was then manipulated by altering the sample size. The frequency of falsely high r-squares increases as the sample size to predictors ratio decreases.References ADDIN EN.REFLIST 1.McCullagh P, Nelder J. Generalized Linear Models. London: Chapman and Hall; 1989.2.Cox DR, Oakes D. Analysis of survival data. London: Chapman & Hall; 1984.3.Muthen LK, Muthen B. Mplus User's Guide. 3rd ed. Los Angeles, CA: Muthen and Muthen; 2004.4.Glare PGW. Oxford Latin dictionary. Oxford University Press; 1982.5.Rubin DB. Estimating causal effects from large data sets using propensity scores. Ann Intern Med 1997;127:757-63.6.Glymour MM, Greenland S, Rothman KJ, Lash TL. Causal diagrams. Modern Epidemiology, vol. 3rd. Philadelphia: Lippincott Williams & Wilkins; 2008, p. 183-212.7.VanderWeele TJ, Vansteelandt S. 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When should epidemiologic regressions use random coefficients? Biometrics 2000;56:915-21. ................
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