Different conventional and causal approaches have been proposed for mediation analysis to better understand the mechanism of a treatment. Count and zero-inflated count data occur in biomedicine, economics, and social sciences. This paper considers mediation analysis for count and zero-inflated count data under the potential outcome framework with nonlinear models. When there are post-treatment confounders which are independent of, or affected by, the treatment, we first define the direct, indirect, and total effects of our interest and then discuss various conditions under which the effects of interest can be identified. Proofs are provided for the sensitivity analysis proposed in the paper. Simulation studies show that the methods work well. We apply the methods to the Detroit Dental Health Project’s Motivational Interviewing DVD trial for the direct and indirect effects of motivational interviewing on count and zero-inflated count dental caries outcomes.
MacKinnonDPLuecenLJ. Statistical analysis for identifying mediating variables in public health dentistry interventions. J Pub Health Dentist2011; 71(Suppl 1): S37–S46.
2.
IsmailAIOndersmaSWillem JedeleJMet al.Evaluation of a brief tailored motivational intervention to prevent early childhood caries. Commun Dentist Oral Epidemiol2011; 39: 433–448.
3.
BaronRMKennyDA. The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. J Personal Soc Psychol1986; 51: 1173–1182.
4.
ColeDAMaxwellSE. Testing mediation models with longitudinal data: Questions and tips in the use of structural equation modeling. J Abnormal Psychol2003; 112: 558–577.
5.
MacKinnonDP.Introduction to statistical mediation analysis, New York: Erlbaum, 2008.
6.
RobinsJMGreenlandS. Identifiability and exchangeability for direct and indirect effects. Epidemiology1992; 3: 143–155.
7.
PearlJDirect and indirect effects. In: BreeseJKollerD (eds). Proceedings of the 17th conference on uncertainty in artificial intelligence, San Francisco, CA: Morgan Kaufmann Publishers Inc, 2001, pp. 411–420.
8.
RubinD. Direct and indirect causal effects via potential outcomes. Scand J Stat2004; 31: 161–170.
9.
Ten HaveTRJoffeMLynchKet al.Causal mediation analyses with rank preserving models. Biometrics2007; 63: 926–934.
10.
AlbertJM. Mediation Analysis via potential outcomes models. Stat Med2008; 27: 1282–1304.
11.
van der LaanMPetersenM. Direct effect models. Int J Biostat2008; 4: Article 23.
12.
SobelME. Identification of causal parameters in randomized studies with mediating variables. J Educ Behav Stat2008; 33: 230–251.
13.
GoetgelukSVansteelandtSGoetghebeurE. Estimation of controlled direct effects. J R Stat Soc Ser B2009; 70: 1049–1066.
14.
VanderWeeleTJVansteelandtS. Conceptual issues concerning mediation, interventions and composition. Stat Interf2009; 2: 457–468.
15.
VanderWeeleTJVansteelandtS. Odds ratios for mediation analysis for a dichotomous outcome. Am J Epidemiol2010; 172: 1339–1348.
16.
ElliottMRRaghunathanTELiY. Bayesian inference for causal mediation effects using principal stratification with dichotomous mediators and outcomes. Biostatistics2010; 11: 353–372.
17.
ImaiKKeeleLYamamotoT. Identification, inference and sensitivity analysis for causal mediation effects. Stat Sci2010; 25: 51–71.
18.
VansteelandtS. Estimation of controlled direct effects on a dichotomous outcome using logistic structural direct effect models. Biometrika2010; 97: 921–934.
19.
JoBStuartEAMacKinnonDPet al.The use of propensity scores in mediation analysis. Multi Behav Res2011; 46: 425–452.
20.
Small D. Mediation analysis without sequential ignorability: Using baseline covariates interacted with random assignment as instrumental variables. arXiv:1109.1070v1, http://arxiv.org/abs/1109.1070 (2011 accessed 14 December 2016).
21.
DanielsMJRoyJKimCet al.Bayesian Inference for the Causal Effect of Mediation. Biometrics2012; 68: 1028–1036.
22.
SteyerRMayerAFiegeCCausal inference on total, direct, and indirect effects. In: MichalosAC (ed). Encyclopedia of quality of life and well-being research, Dordrecht, The Netherlands: Springer, 2014, pp. 606–631.
23.
NeymanJ. On the application of probability theory to agricultural experiments. Essay on principles (with discussion). Section 9 (translated). Stat Sci1923; 5: 465–480.
24.
RubinDB. Estimating causal effects of treatments in randomized and nonrandomized studies. J Educ Psychol1974; 66: 688–701.
AlbertJM. Mediation analysis for nonlinear models with confounding. Epidemiology2012; 23: 879–888.
29.
ValeriLVanderWeeleTJ. Mediation analysis allowing for exposure-mediator interactions and causal interpretation: Theoretical assumptions and implementation with SAS and SPSS macros. Psychol Meth2013; 18: 137–150.
30.
WangWAlbertJM. Estimation of mediation effects for zero-inflated regression models. Stat Med2012; 31: 3118–3132.
31.
RobinsJMSemantics of causal DAG models and the identification of direct and indirect effects. In: GreenPJHjortNLRichardsonS (eds). Highly structured stochastic system, New York, NY: Oxford University Press, 2003, pp. 70–81.
32.
Ten HaveTRJoffeM. A review of causal estimation of effects in mediation analyses. Stat Meth Med Res2012; 21: 77–107.
33.
MinYAgrestiA. Modeling nonnegative data with clumping at zero: A survey. J Iran Stat Soci2002; 1: 7–33.
34.
LambertD. Zero-inflated poisson regression, with an application to defects in manufacturing. Technometrics1992; 34: 1–14.
35.
LongJS.Regression models for categorical and limited dependent variables, Thousand Oaks, CA: Sage, 1997.
36.
FollmannDFayMPProschanM. Chop-Lump tests for vaccine trials. Biometrics2009; 65: 885–893.
37.
KingGTomzMWittenbergJ. Making the most of statistical analyses: Improving interpretation and presentation. Am J Polit Sci2000; 44: 341–355.
38.
VansteelandtS. Estimating direct effects in cohort and case-control studies. Epidemiology2009; 20: 851–860.
39.
JoffeMGreeneT. Related causal frameworks for surrogate outcomes. Biometrics2009; 65: 530–538.
40.
SjlanderA. Bounds on natural direct effects in the presence of confounded intermediate variables. Stat Med2009; 28: 558571–558571.
41.
KaufmanSKaufmanJSMacLehoseRF. Analytic bounds on causal risk differences in directed acyclic graphs involving three observed binary variables. J Stat Plan Inferen2009; 139: 3473–3487.
42.
RobinsJMRichardsonTSAlternative graphical causal models and the identification of direct effects. In: ShroutP (ed). Causality and psychopathology: Finding the determinants of disorders and their cures, Oxford University Press, 2011, pp. 103–158.
43.
Tchetgen TchetgenEJShpitserI. Semiparametric theory for causal mediation analysis: Efficiency bounds, multiple robustness, and sensitivity analysis. Ann Stat2012; 40: 1816–1845.
44.
VanderWeeleTJChibaY. Sensitivity analysis for direct and indirect effects in the presence of exposure-induced mediator-outcome confounders. Epidemiol Biostat Pub Health2014; 11: e9027–e9027.
45.
Tchetgen TchetgenEJVanderWeeleTJ. Identification of natural direct effects when a confounder of the mediator is directly affected by exposure. Epidemiology2014; 25: 282–291.
46.
TaguriMChibaY. A principal stratification approach for evaluating natural direct and indirect effects in the presence of treatment-induced intermediate confounding. Stat Med2015; 34: 131–144.
ImaiKYamamotoT. Identification and sensitivity analysis for multiple causal mechanisms: Revisiting evidence from framing experiments. Polit Anal2013; 21: 141–171.
49.
VanderWeeleTJVansteelandtS. Mediation analysis with multiple mediators. Epidemiol Meth2013; 2: 95–115.
50.
ChengJChaffeeBChengFNet al.Understanding treatment effect mechanisms of the CAMBRA randomized trial in reducing caries increment. J Dental Res2015; 94: 44–51.
51.
BelloniAChenDChernozhukovVet al.Sparse models and methods for optimal instruments with an application to eminent domain. Econometrica2012; 80: 2369–2429.
52.
ChernozhukovVHansenCSpindlerM. Post-selection and post-regularization inference in linear models with many controls and instruments. Am Econ Rev2015; 105: 486–490.
53.
VuongQH. Likelihood ratio tests for model seletion and non-nested hypothese. Econometrica1989; 57: 307–333.
