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hello everybody and welcome to module 10. we're just about halfway through the
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multi institute and i hope that you're becoming more comfortable with the knowledge and you're able to apply it to
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your capstone projects if you haven't already please respond to
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the email requesting your preferred capstone presentation date we'd like to
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honor your first preference if at all possible today we have the third lecture in our
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quantitative methods series we are bringing great britain to multi
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today and kicking us off is dr richard emsley from king's college london
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dr ensley has a teaching style like no one else with his ability to take the
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complicated topics of moderation and mediation and apply the concepts to
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multi-level models hello everyone my name is richard ensley
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i'm from king's college london in the uk and i'm a professor of medical statistics and clinical trials
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methodology i'm delighted to be joining you today to take you through module 10 on the mlti
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course looking at quantitative analytic techniques
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our learning objectives for this session are to understand what is meant by mediator and moderator analysis
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to understand the aspects of study designs allowing such analyses to be applied and to identify suitable
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quantitative methods for addressing these questions in multi-level interventions
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this session will be split into two parts in the first we'll look at some explanatory questions in clinical
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effectiveness and implementation settings and the issue of moderation and in the second video we'll look at
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mediation of effects via post-treatment variables and some extensions to multi-level models
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let's look at some explanatory questions in clinical effectiveness trials
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so one question might be what's a moderation of a treatment effect or prediction of the treatment response we
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want to know for whom does the treatment work and typically this will be on some variable that we refer to as a moderator
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which is measured at baseline we might be interested in the mediation of a treatment effect by a particular
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treatment target this this answers the question how does the treatment work
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fire which clinical targets does the treatment bring about to change in the clinical outcome
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typically those mediators are measured post baseline we may be also interested in some
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therapeutic process evaluation asking is the treatment effect stronger
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depending on some therapeutic process and again typically we'd measure the therapeutic process post baseline
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and we also need strong baseline predictors of that process status itself
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how do those questions translate into implementation effectiveness trials
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well for moderation we could be looking at the moderation of the effect of an implementation strategy
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for whom is the implementation strategy effective for example for which type of therapist
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or clinic was the implementation strategy effective for mediation we may be interested in
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mediation of a treatment event via the implementation aspect so how much of the treatment effect is transmitted via some
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implementation aspects and for post randomization modification
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this would be implementation strategy process evaluation so for example is the
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treatment effect stronger dependent on some therapeutic or organizational
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processes which can only be measured after the start of the study
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one of the key differences between these two is what we refer to as the unit of measurement so for clinical
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effectiveness these measures are typically at the level of the individual participant in the study
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whereas for implementation effectiveness these these measures could be at the level of either the individual
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participant the trainer or the therapist who's delivering an intervention or change
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or the organizational level for example clinics hospital systems or workplaces
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we're now going to consider the topic of moderation by baseline variables
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to do this we'll introduce some notation that we'll use throughout this session
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this slide is here as a reference slide for you to come back to and see what each of the letters correspond to
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we know that if there is a significant correlation between two variables say an exposure variable r and some
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outcome y then there has to be one of four possible reasons for that
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significant association either r is a direct cause of y
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as shown in the first diagram either y causes r
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as shown from the arrow going from the y box to r
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either r and y share a common cause so that we see an association between
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our two variables of interest but this is because there's it's common cause c which is a cause of
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both r and y indicated by these red arrows here
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all r and y are conditioned on a common descendant in either the data design or
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the data analysis so if there is a common variable say s which is a cause of both r and y
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and we condition on that variable either by including it in our statistical models or through selection
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in the study design we can induce an association between r and y
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and if we see an association it has to be because of one of these four explanations
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when we talk about moderation of treatment effects the key point here is that the
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association between r and y is not the same at different values or
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levels of some covariate x in other words there's a modifier
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which is a variable which alters our relationship between this independent variable of interest r and our dependent
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variable y and this is defined in a paper by byron
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and kenny as treatment moderators specify for whom and under what conditions the treatment works that's
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clearly a very very pertinent and relevant question to ask in many contexts
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so it informs clinicians perhaps which of their patients might be most responsive to the treatment and for
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which patients other perhaps more appropriate treatments might be solved and this is
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part of the underlying concept of personalized or stratified medicine
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moderators can also be used to identify subpopulations with possibly different causal mechanisms or course of illness
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and this may be helpful for example in restructuring diagnostic classification based on underlying etiology rather than
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presentation of symptoms the key distinction to make is that
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between prognosis and prediction so a prognostic biomarker is a
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biological measurement made before treatment which indicates the long-term outcome for patients whether they're
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treated or untreated and so they basically predict which
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participants will have an improved outcome regardless of the treatment that they receive
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this contrasts with a predictive or moderation effect where this is a
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measurement made before treatment to identify which participant is likely or unlikely to benefit from a particular
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treatment so we're talking here about things that predict differential response to
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treatment as an example within the oncology field boys talks about over expression of the
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her2 gene in patients with early breast cancer this is an example of a biomarker that has birth the prognostic value because
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patients with her2 overexpression typically have a worse prognosis but also has a predictive value for the
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treatment herceptin because patients with a her2 over expression are more likely to be
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deriving a benefit from that particular treatment
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this slide shows the difference between prognostic and a predictive marker in the graph on the left hand side we
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see age represented as a prognostic marker so on the x-axis we have
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our age and on the y-axis we have improvement our outcome and let's assume
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that a higher score indicates more improvement the red line represents
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the treatment arm and the black line represents the control arm and what we can see is that as age
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increases so the value of the outcome also increases
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but that this increase is the same in both the treatment and the control arms so the treatment effect which is the
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difference between the treatment and the control arms is the same across the
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levels of age on the right hand side we see a representation of age as a moderator or
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predictive marker so again age is on the x-axis our red line represents the treatment
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arm and our black line represents the control arm and what we can see is that whilst age
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is also prognostic so the value of the outcome varies according to age it
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differentially changes between the treatment and the control arm and in fact there is an inflection point
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where to the left of where the lines cross we would recommend giving the control
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because the black line is higher than the red line and to the right of that cross point we would recommend giving
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treatment because the red line is higher than the black line and so this is what's known as a
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qualitative interaction
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we can represent the difference between prognosis and moderating effects using these kind of structural diagrams
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that correspond to the graphs we saw on the previous slide so on the left we see a diagram of a
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variable which is prognostic of treatment response we have an effective treatment
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influencing outcome and we have an additional predictor such as age also predicting outcome
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on the right we see a moderator of treatment response so treatment is a predictor of outcome
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the predictor itself is associated with the outcome and the treatment by predictor interaction term
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the cross product between those is also associated with the outcome and this
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interaction is what answers the question for whom does the treatment work and is the essence of personalized or precision
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medicine how does one assess affect modification
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well in a regression model approach we would include baseline moderators of the treatment outcome variables
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so we would include in the set of predictors or independent variables in the model both the treatment time
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variable the baseline moderator variable and the interaction between the
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treatment arm and the baseline variable so in this context of regression
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analysis when we're assessing effect modification or moderation this is the same as assessing for an interaction
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effect between the two variables and this will be true in both the single level and in the multi-level modeling
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setting to assess the significance of effect
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modification we'd follow three steps the first step is that we would need a
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new variable the cross product or the interaction between our exposure r and
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our modifier x and so we denote this by r by x
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in step two we consider the linear regression to establish the relationships between our outcome y our
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exposure on the covariate x and if we had a linear regression model
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where y is a function of the exposure and the covariance we would add this new
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interaction or cross product term to the regression model here
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so we have that our outcome is a function of an intercept plus an effect
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of the randomization plus the effect of the covariates plus the interaction effect of the covariance
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in step three our primary focus for effect modification would be a significance test of the coefficient
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beta3 so we would have a null hypothesis that beta3 is equal to zero an alternative
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hypothesis that is not equal to zero and if our p-value for that two-tailed
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hypothesis test was less than 0.05 then we may conclude that there is significant effect modification
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that the effect of our exposure or treatment on the outcome
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depends upon the level of this covariate x
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having established our model with the interaction term let's consider the interpretation of the regression
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coefficients so beta1 is interpreted as the effect of r on y when x is equal to zero
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similarly beta2 represents the effect of x on y when r is equal to zero
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now in effect b to one and b to two are no longer that useful unless the zero values of those respective predictors
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are of particular interest these are what we refer to as the main effects here
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beta 3 is interpreted as the difference of the effect of r on y depending on the
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levels of the x variable if our hypothesis test for beta3 concludes that it does significantly
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differ from zero that means that typically both r and x are associated with y
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that the effect of r and y will depend on x and vice versa
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and so the r by x interaction is interpreted as the difference of the effect of r as x goes from zero to one
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or for any one unit increase in our covariate our moderator variable
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let's apply that to the example we saw earlier with the her2 gene expression in
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this example our outcome might be some continuous symptom score our treatments are is herceptin or
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placebo and our moderator variable is the presence of the her2g yes or no
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so we may fit the linear regression model that we specified earlier and the interpretation of the coefficients is
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that beta 1 is the effect of herceptin on the outcome in the absence of the her2 gene ie when x is equal to zero
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beta2 is the effect of the her2g non-outcome in the placebo group i.e
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when r is equal to zero and beta3 is the difference in the effect of percepting on outcome in the
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subgroup with the her2 gene present and the subgroup without the her2 gene present
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to summarize what we've learned about effect modification or moderation if the treatment effect is thought to
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vary with subgroups or some moderators then this should be specified in advance
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in order to avoid accusations of cherry picking or alternatively incorporated into the design of the study itself
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the analysis should typically proceed in the following order so we would add the interaction term between the moderator
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and the treatment to the model and then we may look at explanatory analysis within those relevant subgroups
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if we find a significant interaction effect if one is doing this then we should report the results of all the subgroups
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and the interactions which have been carried out so that we're not just presenting those which have got significant or
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interesting findings also know that the statistical power will be lowered to detect an interaction
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effect than for the main effect and so wouldn't conclude that absence of evidence is not evidence of absence
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that a non-significant beta 3 does not mean there's no effect modification it may be that the difference is small and
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we simply didn't have a large enough sample size and therefore enough statistical power to detect that small
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effect
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welcome back to the mlti module 10. in this session we're going to look at
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quantitative analytic techniques for mediation and components
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let's first consider the mediation of effects via post treatment variables
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so what do we mean by mediation hyman in a paper in 1955 described this
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as when the analyst interprets a relationship he determines the process through which the assumed cause is
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related to what we take to be its effect how did the result come about what are
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the links between the two variables described in formal terms the interpretation of a statistical
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relationship between two variables involves the introduction of further variables and an examination of the
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resulting interrelationships between all the factors so this is similar to the diagrams we
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saw in the video in part one where we linked r and y our two variables and said what
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are the underlying causes or explanations for this association
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david kenny on his website refers to this as one reason for testing mediation is trying
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to understand the mechanism through which the causal variable affects the outcome so in other words by mediation
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we're thinking about a mechanisms evaluation
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in their 1986 paper bound and kenny defined mediation as the generative
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mechanism through which the focal independent variable is able to influence the dependent variable of interest
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this gives rise to a definition of a mediator this is a variable which occurs in the causal pathway between an
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exposure and an outcome variable it causes variation in the outcome and
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itself is caused to vary by the exposure variable so note that this implies the kind of
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temporal relationship that r occurs before m and the m occurs before
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y so the exposure changes the mediator and the mediator changes the outcome
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if we were to represent this mediated effect pictorially then we may have the diagram here which represents complete
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mediation so in this diagram we have an effective r on m and an effective m on y
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and note here that m is the only mechanism by which r can bring about a
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change in y
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that might be an unrealistic assumption there may be other effects through which
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r can change why they don't act through a particular mediator and that's what leads us to what's often referred to as
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the mediation triangle so we have partial mediation at the effect of r on y to a mediator variable
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m and we're interested in estimating all three of the pathways represented by the
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arrows between the boxes in this picture
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so why would we want to assess mediation in any of our studies the first reason might be to develop or
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confirm a mechanistic theory of how a treatment is working how it benefits an
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outcome the second might be to explain why a study or trial has produced a negative
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finding is it because the treatment failed to change the hypothesized mediator or is
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it that the mediator failed to influence the outcome in the way that theory had suggested
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it could be that the effects actually counter balance other harmful effects and so we see that there's no
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overall effect third reason might be because we want to improve the treatment by identifying
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better target variables in earlier phase trials we're generally
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concerned with a more exploratory analysis to maybe identify which of the
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putative mediators are present from a larger set of pool of potential variables and that this might
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be considered more hypothesis generating and that any hypothesis would then be confirmed in a new trial or a new
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independent sample so later phase trials what we might consider phase three trials are
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generally then concerned with confirming and or testing the existence of a
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mediation hypothesis let's give you a couple of more concrete
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examples of what we mean by target mechanisms or target intermediate variables so if a treatment targets a
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particular intermediate variable in order to bring about a change in a clinical outcome such as symptoms for
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example in cognitive behavior therapy or talking therapies
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often the target is a person's thinking or their beliefs in order to bring about
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a change in their symptoms a more classic example is that beta blockers
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help to control blood pressure and particularly variation in blood pressure and that brings about a reduction in
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stroke risk and the idea is that an explanatory analysis of a trial or study would seek
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to establish that this is actually the mechanism through which that effect is happening so to assess the mediated
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pathway in order to do this we need to consider
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the statistical models that we might consider what's represented here is the single mediator model so in the top
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diagram we have the effect of r on y which is our total effect and we
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represent this by some coefficient c in the second diagram we have our
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mediation triangle so we have the effect of r on m represented by a
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we have the effect of m on y represented by b and we have what's called the direct
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effect the effect of r on y which doesn't go through m which is represented by here the c prime
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and in the box we have a reminder of what each of these variables represents
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u in this picture represents some measured confounders and i'll talk more about the role that these play a little
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bit later so we refer to this top effect here the c as being the total
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effect we refer to c prime the effect of r1y in the presence of the mediator as being a
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direct effect and we refer to the effect of r1y that
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goes through m as being the indirect or the mediated effect
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so the aim here is to decompose these effects into direct and indirect effects
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we want to partition the total effect of an exposure on an outcome into that bit of the effect which
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operates via this putative mediator and that's what we refer to as the indirect
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effect and the bit that doesn't go through that mediator and that's what referred to as the direct effect
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and it's important to know that the definition of a direct effect includes any effects via other mediating
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variables which are not included in that model so the interpretation of this direct effect is always relative to the
00:25:17.000 --> 00:25:23.000
variable whose mediating effect is being modeled so a more effective term might be the
00:25:23.000 --> 00:25:29.000
residual direct effect typically the way that we would estimate
00:25:29.000 --> 00:25:35.000
these is by using what's known as the positive coefficient method so if we have a continuous mediator and a
00:25:35.000 --> 00:25:40.000
continuous outcome we may fit three linear regression models
00:25:40.000 --> 00:25:47.000
the first is a model for the total effect so we fit a model for our outcome
00:25:47.000 --> 00:25:53.000
y where we include as a covariate our randomization or exposure
00:25:53.000 --> 00:25:59.000
variable r and we estimate our parameter c that gives us our estimate of the
00:25:59.000 --> 00:26:06.000
total effect in a randomized trial context this may be equivalent to an intention to treat
00:26:06.000 --> 00:26:12.000
analysis the second model is a model where the mediator variable is the dependent
00:26:12.000 --> 00:26:19.000
variable and we include the randomization variable as a predictor and the coefficient of the randomization
00:26:19.000 --> 00:26:25.000
variable is a and that corresponds to this arrow between r and m this a
00:26:25.000 --> 00:26:32.000
pathway here the third model is a model for the outcome again where we include both the
00:26:32.000 --> 00:26:38.000
randomization or exposure indicator and the mediating variable in the same model
00:26:38.000 --> 00:26:44.000
and this gives us estimates of both the b path the effect of m on y
00:26:44.000 --> 00:26:51.000
and the c prime path the effect of r on y in the presence of m
00:26:51.000 --> 00:26:56.000
and from these we can get our estimates of a b c prime and c
00:26:56.000 --> 00:27:01.000
and so we can estimate the indirect or mediated effect as the product of the
00:27:01.000 --> 00:27:08.000
coefficients of the relevant pathways that is the effect of r1y through m is a
00:27:08.000 --> 00:27:13.000
times b now we know that our decomposition means
00:27:13.000 --> 00:27:19.000
that the total effect c is equal to a times b plus c prime and
00:27:19.000 --> 00:27:25.000
that means that we can rearrange this equation and get another estimate of our indirect effect as
00:27:25.000 --> 00:27:31.000
a times b is equal to c minus c prime that is our total effect minus our
00:27:31.000 --> 00:27:36.000
estimate of the direct effect
00:27:36.000 --> 00:27:42.000
so the top of this slide just picks up on that point and this is sometimes what's referred to as the difference in
00:27:42.000 --> 00:27:48.000
coefficients approach because we're no longer calculating the indirect vectors the product of a times
00:27:48.000 --> 00:27:54.000
b but instead is the difference in these coefficients c and c prime
00:27:54.000 --> 00:27:59.000
one advantage of this is that it only actually requires fitting two of those three models it requires fitting a model
00:27:59.000 --> 00:28:06.000
for the outcome for the total effect and a model for the outcome for the direct effect and the difference between
00:28:06.000 --> 00:28:12.000
these two coefficients is taken as representing the indirect effect this is often an approach used in
00:28:12.000 --> 00:28:18.000
epidemiology because it involves just fitting models for the outcome rather than for the mediator which may be more
00:28:18.000 --> 00:28:24.000
familiar in fact for the rest of this session i'm just going to proceed with estimating a
00:28:24.000 --> 00:28:31.000
times b which is perhaps more common in the randomized trial context but all of the thinking in the methods apply
00:28:31.000 --> 00:28:37.000
similarly regardless of which of the methods you use now that we've defined the relevant
00:28:37.000 --> 00:28:42.000
coefficients let's go back to consider the difference between complete and partial mediation so we know that a
00:28:42.000 --> 00:28:48.000
mediator can account for either all or part of this total effect
00:28:48.000 --> 00:28:53.000
complete mediation would occur if the mediator accounts for all of that total effect
00:28:53.000 --> 00:29:00.000
and you would see that this is the case if the direct effect the path c prime drops to an estimate of almost zero and
00:29:00.000 --> 00:29:07.000
is statistically insignificant after controlling for the mediator variable in the model for the outcome
00:29:07.000 --> 00:29:15.000
partial mediation would be said to occur if the mediating variable accounts for some but not all of that total effect so
00:29:15.000 --> 00:29:20.000
in this case the path c prime will be smaller in magnitude than the total
00:29:20.000 --> 00:29:25.000
effect or weaker as i've labeled it here but it's not fully eliminated after
00:29:25.000 --> 00:29:31.000
controlling for the mediator variable so it may still be statistically significant but it's certainly its
00:29:31.000 --> 00:29:37.000
magnitude would be somewhat smaller note that it's also generally
00:29:37.000 --> 00:29:44.000
sensible to consider mediation when both the indirect effects and the direct effect have the same sign
00:29:44.000 --> 00:29:49.000
i.e both the positive or both negative if they act in different directions
00:29:49.000 --> 00:29:54.000
then each of those two pathways may be statistically significant but that some of those
00:29:54.000 --> 00:30:01.000
may contribute to a total effect c which is not significant and this is what's referred to as inconsistent mediation or
00:30:01.000 --> 00:30:08.000
suppression within the literature as i stated earlier the differencing
00:30:08.000 --> 00:30:14.000
coefficient method is popular in epidemiology because it only involves fitting models for this clinical outcome
00:30:14.000 --> 00:30:20.000
however in trials the reason we may be more interested in the product of coefficients approach is because we
00:30:20.000 --> 00:30:27.000
actually want to see what the treatment effect on that intermediate variable m is so we want to know the size and the
00:30:27.000 --> 00:30:32.000
significance of the a pathway and in our work we refer to this a
00:30:32.000 --> 00:30:38.000
pathway as being the target effect it's the effect on our target intermediate mechanism
00:30:38.000 --> 00:30:44.000
regardless of your estimation approach it's somewhat essential to show that the treatment has shifted the mediator for
00:30:44.000 --> 00:30:50.000
mediation through this variable to occur so that means that if there is no significance
00:30:50.000 --> 00:30:57.000
a pathway that there's no difference between your exposure levels on that mediating variable then that is not a
00:30:57.000 --> 00:31:04.000
mediator of the relationship between your exposure and the outcome
00:31:04.000 --> 00:31:10.000
what we've considered so far is how to estimate the size of the indirect effect
00:31:10.000 --> 00:31:17.000
multiplying the a and the b paths together in linear models but to make inference we need to know
00:31:17.000 --> 00:31:23.000
the standard error of that product so so will divide a formula for estimating this
00:31:23.000 --> 00:31:29.000
asymptotic standard error where it only requires both the estimates of a and b and the standard errors of a and b
00:31:29.000 --> 00:31:37.000
themselves and symmetric confidence intervals using this rely on asymptotic normality but
00:31:37.000 --> 00:31:44.000
because we're multiplying two coefficients together that asymptotic normalities are likely to hold and so
00:31:44.000 --> 00:31:53.000
these days we would prefer to use bootstrapping as a way of estimating our standard error
00:31:53.000 --> 00:31:59.000
so to get a bootstrapped confidence interval for the indirect effect we often use bootstrapping
00:31:59.000 --> 00:32:05.000
this requires taking a large number of samples with replacement from the original data set
00:32:05.000 --> 00:32:12.000
in each of those individual samples we then estimate our indirect effect a times b
00:32:12.000 --> 00:32:19.000
we then plot the distribution of those um estimates from each of the samples
00:32:19.000 --> 00:32:27.000
and this gives us a non-parametric distribution of the indirect effect and for a given level of significance we
00:32:27.000 --> 00:32:33.000
can simply calculate the percentile bootstrap limits by finding
00:32:33.000 --> 00:32:40.000
the 2.5 and the 97.5 limits of that non-parametric
00:32:40.000 --> 00:32:46.000
distribution across the bootstrap samples and importantly to note if one's doing
00:32:46.000 --> 00:32:52.000
this is that because this is a random sampling procedure from the original data set one needs to set the same seed
00:32:52.000 --> 00:32:58.000
value so that you can replicate the results if you need to go back to them
00:32:58.000 --> 00:33:04.000
now i'd like to consider one of the major challenges in estimating the mediation effects we're illustrating
00:33:04.000 --> 00:33:11.000
this using a randomized trial so on the left hand side we have random allocation we have our mediator and our outcome
00:33:11.000 --> 00:33:18.000
they represent the mediation triangle and we're trying to estimate the pathways between these three variables
00:33:18.000 --> 00:33:24.000
now in a trial we as investigators have control over the random allocation we're
00:33:24.000 --> 00:33:32.000
assigning individuals to different randomized groups however our mediator has to occur after
00:33:32.000 --> 00:33:37.000
randomization and prior to the outcome and that means that this is the mediator
00:33:37.000 --> 00:33:44.000
is a post-randomization variable in of itself what that means is that it's not under
00:33:44.000 --> 00:33:50.000
the direct control of the investigators typically an observation that we make on the individuals
00:33:50.000 --> 00:33:56.000
and that leads to a challenge that both the mediator and the outcome are post-randomization measures which
00:33:56.000 --> 00:34:02.000
themselves may be affected by other factors not under the control of the investigators in this case they're
00:34:02.000 --> 00:34:07.000
represented by this you these unmeasured confounding terms
00:34:07.000 --> 00:34:13.000
if we know there are factors which might influence for example both blood pressure and stroke risk or someone's
00:34:13.000 --> 00:34:20.000
thinking style and their symptoms then we can measure those and we can account for them as covariates in our
00:34:20.000 --> 00:34:26.000
linear models that we saw previously the challenge comes that we can never rule out that there are unmeasured
00:34:26.000 --> 00:34:32.000
confounders between our mediator and our outcome that may be accounting for
00:34:32.000 --> 00:34:40.000
any association that we see you'll recognize this picture is one of the ones that we saw when i first introduced
00:34:40.000 --> 00:34:46.000
one of the explanations for why there may be an association between two variables
00:34:46.000 --> 00:34:52.000
so what we're wanting to do is estimate unbiasedly the pathway from the mediator to the outcome and the problem is that
00:34:52.000 --> 00:34:58.000
in the presence of these unmeasured confounders we can't get an unbiased estimate of this pathway
00:34:58.000 --> 00:35:03.000
and in fact because this opens up a an association between
00:35:03.000 --> 00:35:09.000
the mediator and the outcome which is not explained as a direct path
00:35:09.000 --> 00:35:16.000
in fact we can also not get a unbiased estimate of the effect of random allocation on outcome that is this
00:35:16.000 --> 00:35:22.000
pathway because we estimate this in the same model for the outcome and in fact what we're doing by
00:35:22.000 --> 00:35:29.000
conditioning on this variable here in the literature this is known as conditioning on a collider we open up an
00:35:29.000 --> 00:35:36.000
association between random allocation and outcome that acts through this unmeasured confounder
00:35:36.000 --> 00:35:41.000
and so we don't we fail to get a an accurate unbiased estimate of this
00:35:41.000 --> 00:35:48.000
pathway and a number assessment of this pathway and the whole thing falls down
00:35:48.000 --> 00:35:54.000
i'm just going to show you this in action using a very simple example using simulated data so i've made up some data
00:35:54.000 --> 00:36:00.000
where we have a continuous mediation outcome all the paths in this diagram are represented by
00:36:00.000 --> 00:36:07.000
a value of 0.25 and we have a covariate here which we know is influencing both m
00:36:07.000 --> 00:36:13.000
and y and the data that i've made up and i'm going to fit models that include and exclude that variable to see the impact
00:36:13.000 --> 00:36:18.000
of that so here's just some standard output from
00:36:18.000 --> 00:36:25.000
a package we have 2 000 observations in our data set this is our first linear model so we fit
00:36:25.000 --> 00:36:31.000
a linear regression of the outcome on r exactly as i described previously and we
00:36:31.000 --> 00:36:36.000
get an estimate of the total effect of 0.313 which is what we expect in our
00:36:36.000 --> 00:36:41.000
data if we figure the second linear model
00:36:41.000 --> 00:36:48.000
which is a model for the outcome conditional on r so now we do a linear regression of m
00:36:48.000 --> 00:36:54.000
on r then we get an estimate of the coefficient of 0.25
00:36:54.000 --> 00:37:01.000
and again that's what we set this to be in the data so we're obtaining an unbiased estimate of the a pathway the
00:37:01.000 --> 00:37:08.000
effect of r on m
00:37:08.000 --> 00:37:15.000
now if we fit a model for the outcome conditional on the mediator and the exposure so we do a linear
00:37:15.000 --> 00:37:20.000
regression of y on m and r we know that there is an unmeasured
00:37:20.000 --> 00:37:25.000
confounder that we're not accounting for in this analysis model that is actually present
00:37:25.000 --> 00:37:31.000
in the data and what you can see is that we get biased estimates of both the m to y
00:37:31.000 --> 00:37:38.000
relationship and the r to y relationship exactly as i described a couple of slides ago
00:37:38.000 --> 00:37:45.000
so we overestimate the effect of m on y and underestimate the effect of r on y
00:37:45.000 --> 00:37:51.000
both of these coefficients should be approximately 0.25 this would mean that we would
00:37:51.000 --> 00:37:56.000
overestimate the indirect effect because we're saying that this b pathway is far
00:37:56.000 --> 00:38:02.000
stronger than it is in practice and so we'd over over estimate the size of the
00:38:02.000 --> 00:38:13.000
indirect effect and conclude that there's a mechanism that is stronger than what might actually be present
00:38:13.000 --> 00:38:18.000
and in fact the only way we can gain unbiased estimates and recover the true
00:38:18.000 --> 00:38:25.000
values that we know were there when we generated the data is by including the covariate that we
00:38:25.000 --> 00:38:30.000
use in the generating model so now we recover the correct estimates of 0.25
00:38:30.000 --> 00:38:38.000
for all of the coefficients and of course the challenge in practice in real data is we never know when we've
00:38:38.000 --> 00:38:43.000
collected all of those covariates and confound us or not so we can never rule out
00:38:43.000 --> 00:38:56.000
that there's an unmeasured confounder which may be accounting for an association that we see in our data
00:38:56.000 --> 00:39:03.000
here's just an example of how one can go further and gain some influence for the indirect effects themselves so i talked
00:39:03.000 --> 00:39:08.000
previously about how sobel had introduced an asymptotic standard error
00:39:08.000 --> 00:39:14.000
for the indirect effect and here in this output we can see our estimate for the a
00:39:14.000 --> 00:39:19.000
coefficient the b coefficient an estimate of the indirect effect the
00:39:19.000 --> 00:39:25.000
direct effect and the total effect so these are all terms and concepts that should now be familiar and map onto that
00:39:25.000 --> 00:39:30.000
mediation triangle that we've been describing and
00:39:30.000 --> 00:39:37.000
the proportion mediated is often reported this is the ratio of the indirect effect of the total effect so
00:39:37.000 --> 00:39:44.000
it is expressed as how much of the total effect is acting through the particular mediator investigated in this case that
00:39:44.000 --> 00:39:49.000
estimate is about 19
00:39:49.000 --> 00:39:55.000
this example just shows how one can get inference for the indirect effects using the bootstrap approach which is the
00:39:55.000 --> 00:40:03.000
recommended approach that i highlighted earlier so if we run a bootstrap replication with 2000 applications
00:40:03.000 --> 00:40:11.000
and these are the results within status statistical software package but all packages can run bootstrap commands we
00:40:11.000 --> 00:40:17.000
see that our estimates of the indirect effects and the direct effect are reported here and the coefficients
00:40:17.000 --> 00:40:24.000
themselves are exactly the same as we saw on the previous slide we now get an estimated bootstrap standard error and
00:40:24.000 --> 00:40:31.000
from this we would probably report the percentile confidence intervals as a measure of the direct and indirect
00:40:31.000 --> 00:40:37.000
effect and of course what one is looking for is that this indirect effect is
00:40:37.000 --> 00:40:44.000
um statistically significant or the confidence interval doesn't cross the zero threshold in this case because of
00:40:44.000 --> 00:40:50.000
the way we generated the data we know that there is an indirect effect corresponding to an effect size of about
00:40:50.000 --> 00:40:57.000
0.06 and that's exactly what we've recovered in this example
00:40:57.000 --> 00:41:02.000
so we've illustrated how to look at mediation in a single mediator context
00:41:02.000 --> 00:41:08.000
an obvious extension that people want to do is to look at potentially two mediators in the same model and again
00:41:08.000 --> 00:41:13.000
that can be done straightforwardly with an extension to what we've done before so this is a diagram we've seen
00:41:13.000 --> 00:41:19.000
previously but now we've added an additional mediator m2 here and so now
00:41:19.000 --> 00:41:24.000
there are two indirect effects there's an indirect effect that acts 2 m1 and
00:41:24.000 --> 00:41:32.000
there's an indirect effect that acts through m2 and we estimate the effect of the exposure of randomization on our second
00:41:32.000 --> 00:41:39.000
mediator as being this path a2 and the effect of the mediator on the outcome has been this path b2
00:41:39.000 --> 00:41:45.000
and so our indirect effect through the second mediator is simply a2 times b2
00:41:45.000 --> 00:41:51.000
using the product of coefficients approach in this picture this decomposition of
00:41:51.000 --> 00:41:56.000
the total effect into the indirect and direct effects still holds as you'll see
00:41:56.000 --> 00:42:03.000
here it's the sum of each of the separate indirect effects plus the direct effect
00:42:03.000 --> 00:42:10.000
and the reason that this holds is that we've assumed that there's no relationship between these two mediators and often
00:42:10.000 --> 00:42:16.000
this is an assumption used to simplify the analysis and one needs to be aware of whether this is biologically
00:42:16.000 --> 00:42:22.000
plausible within your own example and data
00:42:22.000 --> 00:42:27.000
so with this artificial data i've shown how it can be fairly straightforward to estimate
00:42:27.000 --> 00:42:33.000
mediation effects using effectively three linear regression models
00:42:33.000 --> 00:42:38.000
however embedded within that are a number of assumptions and so these pose as
00:42:38.000 --> 00:42:44.000
challenges for establishing mediation the first is that we've highlighted the possible confounding effect
00:42:44.000 --> 00:42:50.000
of mediator on outcome if there are unmeasured confounders in that
00:42:50.000 --> 00:42:56.000
data that are acting to drive that association that we're not taking into account then we will potentially
00:42:56.000 --> 00:43:02.000
get biased estimates of the indirect effect actually we see the same thing happens
00:43:02.000 --> 00:43:09.000
if our mediator variable is measured with error so if we're trying to collect for example blood pressure as our mediator
00:43:09.000 --> 00:43:14.000
or some measure of tumor activity then we have to know that we're collecting
00:43:14.000 --> 00:43:20.000
reliable and valid measures of that because any measurement error in the mediator when it's controlled for the
00:43:20.000 --> 00:43:27.000
covariate in the model for the outcome will then induce bias in the coefficients and it has the same impact
00:43:27.000 --> 00:43:32.000
that confounding does we've seen that the possibility of multiple mediators that are either
00:43:32.000 --> 00:43:37.000
working with power there or sequentially will massively complicate the definition of in
00:43:37.000 --> 00:43:44.000
the indirect effects because there are multiple indirect effects acting through each of the possible pathways or arrows
00:43:44.000 --> 00:43:50.000
and in general we need to be aware of the data structure so we may have repeated or serial assessments of both
00:43:50.000 --> 00:43:56.000
the mediator and the outcome or we may have in in the case of oncology see with assessments of the mediator and a
00:43:56.000 --> 00:44:02.000
survival time for the outcome and that becomes much more challenging than the simple linear regression models
00:44:02.000 --> 00:44:09.000
that i've illustrated here and i'd point towards the further reading and some of the references for further information
00:44:09.000 --> 00:44:15.000
about these in part one of this session we talked about moderation and here i'm showing
00:44:15.000 --> 00:44:22.000
how we can combine our thinking of mediation and moderation together in the first diagram we have what's called
00:44:22.000 --> 00:44:28.000
mediated moderation so if we find an interaction between r and x that
00:44:28.000 --> 00:44:34.000
influences y we may assume that the entire effect of that interaction is through our mediator m
00:44:34.000 --> 00:44:39.000
so that our r by x interaction represented by this red line here is
00:44:39.000 --> 00:44:44.000
acting through our mediator on the outcome
00:44:44.000 --> 00:44:50.000
contrast this with moderated mediation and this is where the effect of a mediator
00:44:50.000 --> 00:44:56.000
on the outcome varies depending on the level of our moderator x so here the red
00:44:56.000 --> 00:45:01.000
arrow is now pointing at the m to y pathway indicating that it's the effect
00:45:01.000 --> 00:45:08.000
of m on y which varies across the level of x
00:45:08.000 --> 00:45:13.000
throughout your course you have been introduced to logic models and encouraged to produce these for how your
00:45:13.000 --> 00:45:20.000
interventions may influence outcomes and what i'm going to illustrate here is an example of a logic model and how this
00:45:20.000 --> 00:45:25.000
maps onto the thinking around mechanisms and mediation that we've discussed
00:45:25.000 --> 00:45:31.000
so we have some inputs and activities leading to our short-term
00:45:31.000 --> 00:45:36.000
outcomes medium-term and long-term outcomes
00:45:36.000 --> 00:45:42.000
and if we put this in the context of the variables we've defined we see that our
00:45:42.000 --> 00:45:49.000
activities and outputs are what we might consider the exposures that we've been talking about our short-term and
00:45:49.000 --> 00:45:56.000
medium-term outcomes might be what we consider our mediators and our long-term outcome is the clinical outcome that we
00:45:56.000 --> 00:46:01.000
may express so these three variables would form what we've considered as our
00:46:01.000 --> 00:46:08.000
mediation mechanisms at the bottom we have our contextual variables in this example and they're
00:46:08.000 --> 00:46:15.000
what we might refer to as moderators so that each stage of our logic model is influenced by these different contexts
00:46:15.000 --> 00:46:22.000
and that's an example of moderation that's saying that the context may influence the way that each of these
00:46:22.000 --> 00:46:29.000
pathways work and that might lead us to either mediated moderation or moderated
00:46:29.000 --> 00:46:34.000
mediation so if one builds a logic model and is able to measure each of the components
00:46:34.000 --> 00:46:43.000
of that you can then fit those measures into our statistical models that we've been discussing today
00:46:43.000 --> 00:46:50.000
i focused on using regression models for mediation analysis and here is a summary of all of the key assumptions which are
00:46:50.000 --> 00:46:55.000
needed in order to validate the statistical methods and ensure that you
00:46:55.000 --> 00:47:01.000
get unbiased estimates by using those regression models and i would recommend you to read
00:47:01.000 --> 00:47:06.000
through these in your time what i'd like to finish with is an
00:47:06.000 --> 00:47:14.000
extension to multi-level models for mediation analysis so when we have multi-level data the
00:47:14.000 --> 00:47:19.000
variables themselves can come from different levels of the model generally in a multi-level modeling
00:47:19.000 --> 00:47:26.000
approach the outcome is always measured at level one that is at the lowest level in the data but depending on the data
00:47:26.000 --> 00:47:31.000
the independent variable or the mediator could be either level one or level two
00:47:31.000 --> 00:47:37.000
variables and as a general rule of thumb an independent variable can be mediated by
00:47:37.000 --> 00:47:43.000
a variable at the same level or by a variable a level lower and so a level two
00:47:43.000 --> 00:47:48.000
exposure can be mediated by a level two or a level one variable but a level one
00:47:48.000 --> 00:47:55.000
exposure can only be mediated by another level one variable logically a level one predictor can't
00:47:55.000 --> 00:48:02.000
influence a level two mediator that is something measured on the individual can't then drive something measured at
00:48:02.000 --> 00:48:09.000
level two the healthcare professional or the or the cluster
00:48:09.000 --> 00:48:16.000
so here's an example of multi-level mediation where the data is split into two levels level one the lowest level of
00:48:16.000 --> 00:48:22.000
observation and level two the clustered level and within the data and we may
00:48:22.000 --> 00:48:28.000
have two what's this diagram refers to as antecedent variables or exposure variables as we've termed them
00:48:28.000 --> 00:48:35.000
and at the lower level this is referred to as xij our mediator variable is also measured
00:48:35.000 --> 00:48:41.000
at level one this is denoted m i j and our dependent variable y i j
00:48:41.000 --> 00:48:49.000
and what is referred to as the one one one model illustrates the levels at which each of the exposure mediator and
00:48:49.000 --> 00:48:56.000
outcomes are measured so when they're all at level one we can estimate our a pathway and our b pathway on the level
00:48:56.000 --> 00:49:03.000
one data and our c prime as the direct effect as we defined previously
00:49:03.000 --> 00:49:10.000
if our exposure variable is something occurring at level 2 then this would be denoted just by xj so it doesn't have
00:49:10.000 --> 00:49:17.000
the the i subscript and this is referred to as a 2 1 one model and again we estimate our a
00:49:17.000 --> 00:49:24.000
pathway in our scene c prime pathway as the effects of the exposure on the mediator and the outcome respectively
00:49:24.000 --> 00:49:30.000
from this level two to the level one outcomes
00:49:30.000 --> 00:49:36.000
we can refer to the models as being either one one one two one one or two two one depending on the various levels
00:49:36.000 --> 00:49:42.000
of the exposure the mediating the outcome which are being analyzed an example of fitting a one-on-one model
00:49:42.000 --> 00:49:49.000
in multi-level data would be that each of our outcomes is measured at level one so y i
00:49:49.000 --> 00:49:55.000
j and mij and we introduce random effects to capture the clustering which is
00:49:55.000 --> 00:50:02.000
present in the outcomes at level two so we may have data on individuals which is captured within treatment centers or
00:50:02.000 --> 00:50:09.000
some hierarchy within the data and we need to count for that clustering by introducing random intercepts at level
00:50:09.000 --> 00:50:17.000
two and respectively these are represented by uj c u j m and u j y and this is a
00:50:17.000 --> 00:50:24.000
standard multi-level or random effects modeling framework but the interpretation of our coefficients in
00:50:24.000 --> 00:50:29.000
terms of our total effect c our indirect effect a and b and our
00:50:29.000 --> 00:50:36.000
direct effect c prime are unchanged we've just accounted for the lack of independence in our data due to the
00:50:36.000 --> 00:50:42.000
clustering an example of a 2-1-1 model in
00:50:42.000 --> 00:50:48.000
multi-level data and in this example our exposure is now occurring at level 2 so we see in the
00:50:48.000 --> 00:50:54.000
models that this is now indexed by rj rather than by r i j
00:50:54.000 --> 00:51:00.000
and this is now a level two or cluster level exposure
00:51:00.000 --> 00:51:07.000
the other thing to note here is that both of these approaches estimate a single b pathway the effect of m on y
00:51:07.000 --> 00:51:16.000
and that that doesn't account for both between and within cluster variation
00:51:16.000 --> 00:51:22.000
this highlights one of the limitations of using multi-level model framework for multi-level mediation
00:51:22.000 --> 00:51:28.000
firstly that the multi-level approach doesn't allow for outcome variables which occur at level two and so there
00:51:28.000 --> 00:51:34.000
couldn't be for example a one one two model or a one two two model
00:51:34.000 --> 00:51:40.000
in both of the models that we considered the 2-1-1 and the 1-1-1 model there's a conflation of the between and within
00:51:40.000 --> 00:51:46.000
effects of m on y so there's only one b-slope which is being estimated whereas in fact that might vary across the
00:51:46.000 --> 00:51:52.000
levels and so an extension to this would be to use a multi-level structural equation
00:51:52.000 --> 00:51:58.000
modeling framework for investigating these to overcome these issues and this is what's presented in this very nice
00:51:58.000 --> 00:52:04.000
paper by preacher and colleagues from psychological methods that i would encourage people to read if they want to
00:52:04.000 --> 00:52:10.000
know more about this topic so to summarize this session the effect
00:52:10.000 --> 00:52:16.000
of moderation or effect modification can be assessed by including interaction effects
00:52:16.000 --> 00:52:22.000
in a randomized trial if the treatment doesn't affect the mediator then there can't be mediation through that variable
00:52:22.000 -->  00:52:28.000
and it's possibly the only aspect which can be tested unbiasedly the models for mediation should be the
00:52:28.000 --> 00:52:35.000
same as used for modeling the outcomes and i think this is a particularly important point when you've got multi-level interventions that require
00:52:35.000 --> 00:52:40.000
multi-level modeling to account for those different levels and the clustering in the data the mediation
00:52:40.000 --> 00:52:46.000
model should match those and a final cautionary warning that mediation analysis contains a lot of
00:52:46.000 --> 00:52:52.000
implicit assumptions but the validity of those estimates depends exactly on those assumptions holding so one should be
00:52:52.000 --> 00:52:57.000
aware and consider those so thank you very much for the invite to
00:52:57.000 --> 00:53:09.000
present and for your attention in following these videos