Mediation and Interaction Analysis

Transcription

1 Mediation and Interaction Analysis Andrea Bellavia May 17, 2017 Andrea Bellavia Mediation and Interaction May 17, / 43

2 Epidemiology, public health, and clinical research are largely about assessing and explaining exposure-outcome associations X Y Andrea Bellavia Mediation and Interaction May 17, / 43

3 Epidemiology, public health, and clinical research are largely about assessing and explaining exposure-outcome associations X Y However, observing a statistical association often only represents the starting point Addressing additional questions is required before implementing public health intervention and recommendations Andrea Bellavia Mediation and Interaction May 17, / 43

4 In this lecture we will focus on two specific questions: How is the association generated? Does the association only exist for specific subgroups? Andrea Bellavia Mediation and Interaction May 17, / 43

5 In this lecture we will focus on two specific questions: How is the association generated? Does the association only exist for specific subgroups? Statistically/epidemiologically speaking, we are investigating the contribution of other covariates in the X-Y association G M X Y Andrea Bellavia Mediation and Interaction May 17, / 43

6 This afternoon we will see another common situation: X2 X3 X1 Y Andrea Bellavia Mediation and Interaction May 17, / 43

7 Motivating example - HIV infection at birth and adolescence health HIV-infected children born to women living with HIV experience delays in pubertal onset and sexual maturity, with higher subsequent risk of the related social and medical outcomes. Differences in growth (i.e lower height and higher BMI) have been reported among children living with HIV. Growth is one of the main predictors of pubertal timing. Andrea Bellavia Mediation and Interaction May 17, / 43

8 Motivating example - HIV infection at birth and adolescence health HIV-infected children born to women living with HIV experience delays in pubertal onset and sexual maturity, with higher subsequent risk of the related social and medical outcomes. Differences in growth (i.e lower height and higher BMI) have been reported among children living with HIV. Growth is one of the main predictors of pubertal timing. Is the delay in puberty observed among HIV+ children due to differences in growth? To what extent? (Bellavia et al. 2017) Interventions aimed at improving growth during childhood and adolescence might prevent/reduce the difference. Andrea Bellavia Mediation and Interaction May 17, / 43

9 Potential fields of application Andrea Bellavia Mediation and Interaction May 17, / 43

10 Potential fields of application Lifestyle and social factors Andrea Bellavia Mediation and Interaction May 17, / 43

11 Potential fields of application Lifestyle and social factors Psychology (a lot) Andrea Bellavia Mediation and Interaction May 17, / 43

12 Potential fields of application Lifestyle and social factors Psychology (a lot) Interaction between genetic and lifestyle factors. Andrea Bellavia Mediation and Interaction May 17, / 43

13 Potential fields of application Lifestyle and social factors Psychology (a lot) Interaction between genetic and lifestyle factors. Medical predecessors of common diseases (e.g. diabetes, hypertension, depression) Andrea Bellavia Mediation and Interaction May 17, / 43

14 Potential fields of application Lifestyle and social factors Psychology (a lot) Interaction between genetic and lifestyle factors. Medical predecessors of common diseases (e.g. diabetes, hypertension, depression) Epigenetics (and genes-environment interactions). Andrea Bellavia Mediation and Interaction May 17, / 43

15 Potential fields of application Lifestyle and social factors Psychology (a lot) Interaction between genetic and lifestyle factors. Medical predecessors of common diseases (e.g. diabetes, hypertension, depression) Epigenetics (and genes-environment interactions). Personalized treatments Andrea Bellavia Mediation and Interaction May 17, / 43

16 Potential fields of application Lifestyle and social factors Psychology (a lot) Interaction between genetic and lifestyle factors. Medical predecessors of common diseases (e.g. diabetes, hypertension, depression) Epigenetics (and genes-environment interactions). Personalized treatments... many other Andrea Bellavia Mediation and Interaction May 17, / 43

17 Potential fields of application Lifestyle and social factors Psychology (a lot) Interaction between genetic and lifestyle factors. Medical predecessors of common diseases (e.g. diabetes, hypertension, depression) Epigenetics (and genes-environment interactions). Personalized treatments... many other Investigating the contribution of third variables in an X-Y association generally requires merging multiple research fields ->multidisciplinary approach. Andrea Bellavia Mediation and Interaction May 17, / 43

18 Mediation and Interaction Interaction and mediation analysis are the statistical tools that address our two questions of interest: Mediation: how an effect occurs Interaction: for whom an effect occurs Andrea Bellavia Mediation and Interaction May 17, / 43

19 The biological question comes before the statistical one Andrea Bellavia Mediation and Interaction May 17, / 43

20 The biological question comes before the statistical one Statistical methods to assess interaction and mediation (or even confounding), are generally simple (and often similar) Andrea Bellavia Mediation and Interaction May 17, / 43

21 The biological question comes before the statistical one Statistical methods to assess interaction and mediation (or even confounding), are generally simple (and often similar) To correctly identify the potential role of each variable involved in the association is crucial. Methods to conceptualise the causal pathway are available and increasingly recommended (e.g. DAGs) Andrea Bellavia Mediation and Interaction May 17, / 43

22 The biological question comes before the statistical one Statistical methods to assess interaction and mediation (or even confounding), are generally simple (and often similar) To correctly identify the potential role of each variable involved in the association is crucial. Methods to conceptualise the causal pathway are available and increasingly recommended (e.g. DAGs) Failing to identify the correct DAG may lead to severe bias once we move to statistical analysis (i.e. the correct method is used to address a wrong question) Andrea Bellavia Mediation and Interaction May 17, / 43

23 Concept of Mediation A mediator is a covariate that mediates the association between X and Y M X Y Part of the effect of X on Y is due to the fact that X causes M, which in turn causes Y Andrea Bellavia Mediation and Interaction May 17, / 43

24 Concept of Mediation We aim to disentangle the total effect of X on Y into a direct effect that goes through all possible pathways but M... M X Y Andrea Bellavia Mediation and Interaction May 17, / 43

25 Concept of Mediation... and an indirect effect that goes through M M X Y Andrea Bellavia Mediation and Interaction May 17, / 43

26 Assessing Mediation - continuous outcomes The classical approach to Mediation Analysis (Baron & Kenny, 1986) E[Y x] = α 0 + α 1 x E[Y x, m] = β 0 + β 1 x + β 2 m E[M x] = γ 0 + γ 1 x Andrea Bellavia Mediation and Interaction May 17, / 43

27 Assessing Mediation - continuous outcomes The classical approach to Mediation Analysis (Baron & Kenny, 1986) E[Y x] = α 0 + α 1 x E[Y x, m] = β 0 + β 1 x + β 2 m E[M x] = γ 0 + γ 1 x Once the three statistical models are correctly identified we can estimate direct and indirect effects: Direct Effect = β 1 Indirect Effect = α 1 β 1 = β 2 γ 1 Andrea Bellavia Mediation and Interaction May 17, / 43

28 Adjusting a model for a mediator gives an estimate of the direct effect rather than the total effect. Crucial to distinguish confounders and mediators! Andrea Bellavia Mediation and Interaction May 17, / 43

29 Adjusting a model for a mediator gives an estimate of the direct effect rather than the total effect. Crucial to distinguish confounders and mediators! All models can be adjusted for potential confounders Andrea Bellavia Mediation and Interaction May 17, / 43

30 Adjusting a model for a mediator gives an estimate of the direct effect rather than the total effect. Crucial to distinguish confounders and mediators! All models can be adjusted for potential confounders The product=difference statement is not valid under some scenarios (binary and survival outcomes, exposure-mediator interaction, missing values on mediators) A common measure to summarise results from a mediation model is the proportion mediated (only when effects are all in the same direction) PM = IndirectEffect TotalEffect Andrea Bellavia Mediation and Interaction May 17, / 43

31 Assessing Mediation - binary outcomes logit[y x] = α 0 + α 1 x logit[y x, m] = β 0 + β 1 x + β 2 m E[M x] = γ 0 + γ 1 x Andrea Bellavia Mediation and Interaction May 17, / 43

32 Assessing Mediation - binary outcomes logit[y x] = α 0 + α 1 x logit[y x, m] = β 0 + β 1 x + β 2 m E[M x] = γ 0 + γ 1 x Once the three statistical models are correctly identified we can estimate direct and indirect effects: Direct Effect = exp(β 1 ) Indirect Effect = exp(β 2 γ 1 ) The product method is recommended as the equivalence with the difference method only yields when the outcome is rare. Andrea Bellavia Mediation and Interaction May 17, / 43

33 Assessing Mediation - survival outcomes Accelerated failure time models (AFT) log[t x] = α 0 + α 1 x log[t x, m] = β 0 + β 1 x + β 2 m E[M x] = γ 0 + γ 1 x Andrea Bellavia Mediation and Interaction May 17, / 43

34 Assessing Mediation - survival outcomes Accelerated failure time models (AFT) Proportional hazard model log[t x] = α 0 + α 1 x log[t x, m] = β 0 + β 1 x + β 2 m E[M x] = γ 0 + γ 1 x h(t x) = h 0 exp(α 1 x) h(t x, m) = h 0 exp(β 1 x + β 2 m) E[M x] = γ 0 + γ 1 x Formulas for direct and indirect effects are the same as for binary outcomes Andrea Bellavia Mediation and Interaction May 17, / 43

35 Mediation with Multiple Mediators M 1 X Y M 2 Andrea Bellavia Mediation and Interaction May 17, / 43

36 Assessing Mediation - Multiple mediators E[Y x] = α 0 + α 1 x E[Y x, m 1, m 2 ] = β 0 + β 1 x + β 2 m 1 + β 3 m 2 + β 4 m 1 m 2 E[M 1 x] = γ 0 + γ 1 x E[M 2 x] = δ 0 + δ 1 x E[Z x] = ω 0 + ω 1 x where Z = M 1 M 2. (Required to assure that the sum of the mediated proportions be equal to the joint mediated proportion) Andrea Bellavia Mediation and Interaction May 17, / 43

37 Assessing Mediation - Multiple mediators E[Y x] = α 0 + α 1 x E[Y x, m 1, m 2 ] = β 0 + β 1 x + β 2 m 1 + β 3 m 2 + β 4 m 1 m 2 E[M 1 x] = γ 0 + γ 1 x E[M 2 x] = δ 0 + δ 1 x E[Z x] = ω 0 + ω 1 x where Z = M 1 M 2. (Required to assure that the sum of the mediated proportions be equal to the joint mediated proportion) Direct and indirect effects are estimated by: Direct Effect = β 1 Indirect Effect through M 1 only= β 2 γ 1 Indirect Effect through M 2 only= β 3 δ 1 Indirect Effect through M 1 and M 2 = β 4 ω 1 Andrea Bellavia Mediation and Interaction May 17, / 43

38 The basic approach to multiple mediators only requires including an interaction between the two mediators and specifying two additional statistical models (Vanderweele & Vansteelandt 2013) Binary and continuous outcomes can be evaluated Andrea Bellavia Mediation and Interaction May 17, / 43

39 The basic approach to multiple mediators only requires including an interaction between the two mediators and specifying two additional statistical models (Vanderweele & Vansteelandt 2013) Binary and continuous outcomes can be evaluated When multiple mediators are of interest it may be difficult to specify correct statistical models for all of them and for their interactions. A weighting approach can be used in those situations Andrea Bellavia Mediation and Interaction May 17, / 43

40 Illustrative example Study population: 2539 children born to HIV-infected mothers. Exposure: HIV status. 453 perinatally HIV-exposed uninfected children (PHEU); 2086 perinatally HIV-infected children (PHIV) Mediators: age- and sex-adjusted Z-scores for height and BMI, collected at multiple visits Outcome: Pubertal staging assessed by visual inspection of 1) breast development; 2) pubic hair; (girls) 3) genitalia 4) pubic hair (boys). Various confounders were also included Andrea Bellavia Mediation and Interaction May 17, / 43

41 Height/BMI HIV infection Age at sexual maturity Andrea Bellavia Mediation and Interaction May 17, / 43

42 Illustrative example - 2 Accelerated Failure Time (AFT) models with normal distribution for interval-censored outcomes (R package survreg) T = β 0 + β 1 HIV + β T 2 C + ɛ Andrea Bellavia Mediation and Interaction May 17, / 43

43 Illustrative example - 2 Accelerated Failure Time (AFT) models with normal distribution for interval-censored outcomes (R package survreg) T = β 0 + β 1 HIV + β T 2 C + ɛ Linear regression models for height and BMI Z-scores, at different time points (16 statistical models) E[Height] = α 0 + α 1 HIV + α T 2 C E[BMI ] = θ 0 + θ 1 HIV + θ T 2 C Andrea Bellavia Mediation and Interaction May 17, / 43

44 Illustrative example - multiple mediators E[T ] = β 0 + β 1 HIV + β T 2 C E[T ] = γ 0 + γ 1 HIV + γ 2 Height + γ 3 BMI + γ 4 Height BMI + γ T 5 C E[Height] = α 0 + α 1 HIV + α T 2 C E[BMI ] = θ 0 + θ 1 HIV + θ T 2 C E[Height BMI ] = φ 0 + φ 1 HIV + φ T 2 C Andrea Bellavia Mediation and Interaction May 17, / 43

45 Illustrative example - multiple mediators E[T ] = β 0 + β 1 HIV + β T 2 C E[T ] = γ 0 + γ 1 HIV + γ 2 Height + γ 3 BMI + γ 4 Height BMI + γ T 5 C E[Height] = α 0 + α 1 HIV + α T 2 C E[BMI ] = θ 0 + θ 1 HIV + θ T 2 C E[Height BMI ] = φ 0 + φ 1 HIV + φ T 2 C Total effect = β 1 Direct effect = γ 1 Indirect effect = γ 2 α 1 + γ 3 θ 1 + γ 4 φ 1 Andrea Bellavia Mediation and Interaction May 17, / 43

46 Results - Total effects Girls Breast Pubic Hair PHEU 14.9 ( ) 15.0 ( ) PHIV 15.5 ( )** 15.5 ( )* Boys Genitalia Pubic Hair PHEU 15.3 ( ) 15.3 ( ) PHIV 15.9 ( )* 15.8 ( )* *p-value<0.1; **p-value<0.05 Andrea Bellavia Mediation and Interaction May 17, / 43

47 Results - Mediation analysis Andrea Bellavia Mediation and Interaction May 17, / 43

48 Concept of Interaction The effect of a given exposure on the outcome of interest may depend on the presence or absence of another exposure Andrea Bellavia Mediation and Interaction May 17, / 43

49 Concept of Interaction The effect of a given exposure on the outcome of interest may depend on the presence or absence of another exposure Interaction and effect modification (EM) have a similar biological meaning. There is no agreement on their use and definition ->confusion Vanderweele 2009: On the distinction between Int. and EM. Andrea Bellavia Mediation and Interaction May 17, / 43

50 Concept of Interaction The effect of a given exposure on the outcome of interest may depend on the presence or absence of another exposure Interaction and effect modification (EM) have a similar biological meaning. There is no agreement on their use and definition ->confusion Vanderweele 2009: On the distinction between Int. and EM. Interaction can be defined on the additive or multiplicative scale Andrea Bellavia Mediation and Interaction May 17, / 43

51 Dichotomous exposures Suppose we have a dichotomous outcome Y and two dichotomous exposures G and E We are interested in P(Y = 1 G = g, E = e), the probability of Y given the two exposures Andrea Bellavia Mediation and Interaction May 17, / 43

52 Additive Interaction Definition of additive interaction p 11 = p 10 + p 01 p = 0.45 Andrea Bellavia Mediation and Interaction May 17, / 43

53 Multiplicative Interaction Definition of multiplicative interaction p 11 = (p 10 p 01 )/p 00 ( )/0.20 = Andrea Bellavia Mediation and Interaction May 17, / 43

54 Additive vs Multiplicative Interaction The sign of the interaction depends on the chosen scale. A common situation is to have negative interaction on the multiplicative scale, but positive interaction on the additive scale. In general, if both exposures have an effect on the outcome there should be an interaction on some scale. Andrea Bellavia Mediation and Interaction May 17, / 43

55 Additive vs Multiplicative Interaction The sign of the interaction depends on the chosen scale. A common situation is to have negative interaction on the multiplicative scale, but positive interaction on the additive scale. In general, if both exposures have an effect on the outcome there should be an interaction on some scale. Additive interaction is the more relevant public health measure. It can be used to identify a particular subgroup that should be treated. For example, presence of additive interaction in the previous situation implies that the public health consequence of an intervention on E would be larger in the G = 1 group Andrea Bellavia Mediation and Interaction May 17, / 43

56 Statistical interaction Statistical interaction is commonly assessed by including a product term in the regression model When dealing with dichotomous exposures, this method yields the same results of a simple stratification Andrea Bellavia Mediation and Interaction May 17, / 43

57 Statistical interaction Statistical interaction is commonly assessed by including a product term in the regression model When dealing with dichotomous exposures, this method yields the same results of a simple stratification Whether additive or multiplicative interaction is evaluated depends on the chosen statistical model Andrea Bellavia Mediation and Interaction May 17, / 43

58 Statistical interaction Statistical interaction is commonly assessed by including a product term in the regression model When dealing with dichotomous exposures, this method yields the same results of a simple stratification Whether additive or multiplicative interaction is evaluated depends on the chosen statistical model A statistical model on the linear scale can be used to evaluate statistical interaction on the additive scale P(Y = 1 G = g, E = e) = α 0 + α 1 g + α 2 e + α 3 ge Andrea Bellavia Mediation and Interaction May 17, / 43

59 Statistical interaction Statistical interaction is commonly assessed by including a product term in the regression model When dealing with dichotomous exposures, this method yields the same results of a simple stratification Whether additive or multiplicative interaction is evaluated depends on the chosen statistical model A statistical model on the linear scale can be used to evaluate statistical interaction on the additive scale P(Y = 1 G = g, E = e) = α 0 + α 1 g + α 2 e + α 3 ge A statistical model on the log-linear scale can be used to evaluate statistical interaction on the multiplicative scale log[p(y = 1 G = g, E = e)] = β 0 + β 1 g + β 2 e + β 3 ge Andrea Bellavia Mediation and Interaction May 17, / 43

60 Interaction: Extensions and limitations To derive additive interaction when the statistical model is log-linear, measures such as the RERI can be used (Li & Chambless, 2007). In survival analysis, additive models can also be used (Rod et al., 2012; Bellavia et al., 2016) Andrea Bellavia Mediation and Interaction May 17, / 43

61 Interaction: Extensions and limitations To derive additive interaction when the statistical model is log-linear, measures such as the RERI can be used (Li & Chambless, 2007). In survival analysis, additive models can also be used (Rod et al., 2012; Bellavia et al., 2016) Interpreting a measure of interaction is simple and intuitive when dealing with dichotomous exposures, complicated with categorical, and extremely challenging with continuous Andrea Bellavia Mediation and Interaction May 17, / 43

62 Interaction: Extensions and limitations To derive additive interaction when the statistical model is log-linear, measures such as the RERI can be used (Li & Chambless, 2007). In survival analysis, additive models can also be used (Rod et al., 2012; Bellavia et al., 2016) Interpreting a measure of interaction is simple and intuitive when dealing with dichotomous exposures, complicated with categorical, and extremely challenging with continuous There is no agreement on the definition of interaction, neither on the distinction between interaction and effect modification. One can also use the term combined effect Andrea Bellavia Mediation and Interaction May 17, / 43

63 Challenges Mediation and interaction can occur simultaneously. The classical approach for mediation analysis does not allow taking into account exposure-mediator interaction. Andrea Bellavia Mediation and Interaction May 17, / 43

64 Challenges Mediation and interaction can occur simultaneously. The classical approach for mediation analysis does not allow taking into account exposure-mediator interaction. Causal interpretation is not straightforward. Confounding operate at different levels, including the mediator-outcome association. Andrea Bellavia Mediation and Interaction May 17, / 43

65 Exposure-mediator interaction E[Y x] = α 0 + α 1 x E[Y x, m] = β 0 + β 1 x + β 2 m + β 3 x m E[M x] = γ 0 + γ 1 x In this situation the formulas for direct and indirect effects previously presented do not yield valid estimates. Andrea Bellavia Mediation and Interaction May 17, / 43

66 Mediator-outcome confounding If control is not made for the mediator-outcome confounders then results from the standard approach to mediation can be highly biased M C X Y Andrea Bellavia Mediation and Interaction May 17, / 43

67 Mediator-outcome confounding If control is not made for the mediator-outcome confounders then results from the standard approach to mediation can be highly biased M C X Y A famous example, worth reading, is the birth-weight paradox (Hernandez-Diaz et al., 2006) Andrea Bellavia Mediation and Interaction May 17, / 43

68 Mediator-outcome confounding If control is not made for the mediator-outcome confounders then results from the standard approach to mediation can be highly biased M C X Y A famous example, worth reading, is the birth-weight paradox (Hernandez-Diaz et al., 2006) Randomized clinical trials are also affected! Andrea Bellavia Mediation and Interaction May 17, / 43

69 The counterfactual approach to mediation analysis The counterfactual approach to mediation analysis was introduced to overcome these important limitations (recently summarized in Vanderweele 2015). Andrea Bellavia Mediation and Interaction May 17, / 43

70 The counterfactual approach to mediation analysis The counterfactual approach to mediation analysis was introduced to overcome these important limitations (recently summarized in Vanderweele 2015). This approach defines direct and indirect effects in terms of the counterfactual intervention [i.e. fixing exposure and mediator to a predefined value (controlled), or fixing the exposure to a predefined value and the mediator to the value that naturally follows (natural)]. Andrea Bellavia Mediation and Interaction May 17, / 43

71 The counterfactual approach to mediation analysis The counterfactual approach to mediation analysis was introduced to overcome these important limitations (recently summarized in Vanderweele 2015). This approach defines direct and indirect effects in terms of the counterfactual intervention [i.e. fixing exposure and mediator to a predefined value (controlled), or fixing the exposure to a predefined value and the mediator to the value that naturally follows (natural)]. Vanderweele (2014) also showed that by using the counterfactual approach the total effect can be decomposed into four different components: a direct effect (controlled), a proportion due to mediation alone (natural indirect effect), a proportion due to interaction alone (reference interaction), and a proportion due to both mediation and interaction (mediated interaction). Andrea Bellavia Mediation and Interaction May 17, / 43

72 Component CDE INTref INTmed PNIE Interpretation Treatment effect neither due to mediation nor interaction Treatment effect only due to interaction Treatment effect due to both mediation and interaction Treatment effect only due to mediation Andrea Bellavia Mediation and Interaction May 17, / 43

73 Mediation: other relevant extensions Causal Mediation Analysis (Robins & Greenland 1992, Pearl 2001) Non-parametric approach (Imai et al. 2010) Time-varying Exposures and Mediators (van der Laan & Petersen 2008, Vanderweele 2009) Longitudinal analysis (Bind et al. 2017) Andrea Bellavia Mediation and Interaction May 17, / 43

74 Mediation: software Stata: paramed, med4way R: mediation SAS: macro from Valeri & Vanderweele 2013 Andrea Bellavia Mediation and Interaction May 17, / 43

75 Summary Mediation and interaction are about explaining mechanisms that underline exposure-outcome associations. Andrea Bellavia Mediation and Interaction May 17, / 43

76 Summary Mediation and interaction are about explaining mechanisms that underline exposure-outcome associations. Interaction provides information on for whom a given effect occurs Mediation provides insight on how an effect occurs Andrea Bellavia Mediation and Interaction May 17, / 43

77 Summary Mediation and interaction are about explaining mechanisms that underline exposure-outcome associations. Interaction provides information on for whom a given effect occurs Mediation provides insight on how an effect occurs Recent methodological developments allow evaluating interaction and mediation in a large variety of settings These methods can be applied with different data structures and to address different research questions, particularly those dealing when non-modifiable exposures. Andrea Bellavia Mediation and Interaction May 17, / 43

78 Causality - assumptions and conditions A temporal sequence of exposure, mediator, and outcome, is generally recommended. Andrea Bellavia Mediation and Interaction May 17, / 43

79 Causality - assumptions and conditions A temporal sequence of exposure, mediator, and outcome, is generally recommended. Exposure-mediator, exposure-outcome, mediator-outcome, and mediator-mediator confounding must be all taken into account and are dealt in different ways. Studies are often designed without thinking of mediator-outcome confounders. Andrea Bellavia Mediation and Interaction May 17, / 43

80 Causality - assumptions and conditions A temporal sequence of exposure, mediator, and outcome, is generally recommended. Exposure-mediator, exposure-outcome, mediator-outcome, and mediator-mediator confounding must be all taken into account and are dealt in different ways. Studies are often designed without thinking of mediator-outcome confounders. Two assumptions of unmeasured confounding are required to causally interpret controlled direct effects. Two additional assumptions are required for the other effects. Andrea Bellavia Mediation and Interaction May 17, / 43

81 Referecens Baron, R. M., & Kenny, D. A. (1986). The moderator mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of personality and social psychology, 51(6), Bellavia, A., Bottai, M., Orsini, N. (2016). Evaluating Additive Interaction Using Survival Percentiles. Epidemiology, 27(3), 360. Bellavia A, Williams PL, Dimeglio LA, Hazra R, Abzug MJ, Patel K, Jacobson DL, Van Dyke RB, Geffner ME. (2017) Delay in sexual maturation in perinatally HIV-infected youth is mediated by poor growth. AIDS. Bind, M. A., Vanderweele, T. J., Coull, B. A., & Schwartz, J. D. (2016). Causal mediation analysis for longitudinal data with exogenous exposure. Biostatistics, 17(1), Fulchner, E., Tchetgen Tchetgen, E., and Williams, P. (2017). Mediation Analysis for Censored Survival Data under an Accelerated Failure Time Model. Epidemiology Gamborg, M., Jensen, G. B., Srensen, T. I., & Andersen, P. K. (2011). Dynamic path analysis in life-course epidemiology. American journal of epidemiology, 173(10), Hernandez-Diaz S, Schisterman EF, Hernn MA. (2006) The birth weight paradox uncovered?. American journal of epidemiology. 1;164(11): Imai, K., Keele, L., & Tingley, D. (2010). A general approach to causal mediation analysis. Psychological methods, 15(4), 309. Imai, K., Keele, L., Tingley, D., Yamamoto, T. (2010). Causal mediation analysis using R. In: H.D. Vinod (ed.), Advances in Social Science Research Using R. New York: Springer (Lecture Notes in Statistics), p Andrea Bellavia Mediation and Interaction May 17, / 43

82 Lange, T., & Hansen, J. V. (2011). Direct and indirect effects in a survival context. Epidemiology, 22(4), Li, R., & Chambless, L. (2007). Test for additive interaction in proportional hazards models. Annals of epidemiology, 17(3), Pearl, J. (2001). Direct and indirect effects. In Proceedings of the seventeenth conference on uncertainty in artificial intelligence (pp ). Morgan Kaufmann Publishers Inc.. Robins, J. M., & Greenland, S. (1992). Identifiability and exchangeability for direct and indirect effects. Epidemiology, Rod, N. H., Lange, T., Andersen, I., Marott, J. L., & Diderichsen, F. (2012). Additive interaction in survival analysis: use of the additive hazards model. Epidemiology, 23(5), Rothman KJ. (1976). Causes. Am J of Epidemiol 104: Strohmaier, S., Rysland, K., Hoff, R., Borgan,., Pedersen, T. R., & Aalen, O. O. (2015). Dynamic path analysisa useful tool to investigate mediation processes in clinical survival trials. Statistics in medicine, 34(29), Tein, J. Y., & MacKinnon, D. P. (2003). Estimating mediated effects with survival data. In New developments in psychometrics (pp ). Springer Japan. VanderWeele, T. J. (2011). Causal mediation analysis with survival data. Epidemiology (Cambridge, Mass.), 22(4), 582. VanderWeele, T.J. (2015). Explanation in causal inference: methods for mediation and interaction. Oxford University Press. VanderWeele, T. J. (2009). On the distinction between interaction and effect modification. Epidemiology, 20(6), Andrea Bellavia Mediation and Interaction May 17, / 43

83 VanderWeele, T.J., & Vansteelandt, S. (2013). Mediation analysis with multiple mediators. Epidemiologic methods, 2(1), VanderWeele, T.J. and Vansteelandt, S. (2010). Odds ratios for mediation analysis with a dichotomous outcome. American Journal of Epidemiology, 172: VanderWeele, T. J. (2009). Marginal structural models for the estimation of direct and indirect effects. Epidemiology, 20(1), VanderWeele, T. J. (2014). A unification of mediation and interaction: a 4-way decomposition. Epidemiology, 25(5), VanderWeele, T.J. and Knol, M.J. (2014). A tutorial on interaction. Epidemiologic Methods. Valeri, L., & VanderWeele, T. J. (2013). Mediation analysis allowing for exposuremediator interactions and causal interpretation: Theoretical assumptions and implementation with SAS and SPSS macros. Psychological methods, 18(2), 137. Andrea Bellavia Mediation and Interaction May 17, / 43