Definition, identification, and estimation of the direct and indirect number needed to treat

Valentin Vancak; Arvid Sjölander

doi:10.1515/em-2025-0018

Artikel

Definition, identification, and estimation of the direct and indirect number needed to treat

Valentin Vancak und Arvid Sjölander

Veröffentlicht/Copyright: 1. Januar 2026

Veröffentlicht von

Veröffentlichen auch Sie bei De Gruyter Brill

Manuskript einreichen Informationen für Autor*innen Erkunden Sie dieses Fachgebiet

Aus der Zeitschrift Epidemiologic Methods Band 14 Heft 1

Abstract

Objectives

The number needed to treat (NNT) is a widely used efficacy and effect-size measure in epidemiology and meta-analysis, originally defined as the average number of patients who need be treated to obtain one additional beneficial outcome. In this study, we introduce novel direct and indirect formulations of the NNT, the number needed to be exposed (NNE) and the exposure impact number (EIN) - quantifying the average number needed to treat (expose) to achieve benefit via the treatment's direct and indirect effects in the respective treatment group.

Methods

Using nested potential outcomes, we formally define the direct effect NNT, NNE, and EIN (DNNT, DNNE, and DEIN, respectively) and the indirect effect NNT, NNE, and EIN (INNT, INNE, and IEIN, respectively). We then derive their identification conditions in observational studies. We introduce an estimation method and illustrate it with two analytical examples.

Results

The identification results provide explicit conditions under which the novel direct and indirect indices are estimable in observational studies. Simulation studies demonstrate that the proposed estimators are consistent and that the associated analytic confidence intervals attain their nominal coverage rates. Two analytical examples clarify implementation and interpretation.

Conclusions

We formalize novel path-dependent efficacy measures - DNNT, INNT, DNNE, INNE, DEIN, and IEIN - and derive their identification conditions for observational studies. We also introduce an estimation method and demonstrate its efficiency and accuracy using theoretical results and a simulation study. The widespread use of NNT and NNE, together with the need for path-dependent reporting, supports the utility of the proposed indices. These indices can help disentangle direct and indirect benefits and guide the choice among intervention strategies in public health, clinical practice, and other policy decision-making contexts.

Keywords: NNT; NNE; indirect effects; identification; meditation; effect size

Corresponding author: Valentin Vancak, Department of Data Science, Holon Institute of Technology, Holon, Israel, E-mail: valentin.vancak@gmail.com

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: LLM was used to refine phrasing and to help refactor R code for the boxplots.
Conflict of interest: The authors state no conflict of interest.
Research funding: Not applicable.
Data availability: All results are based on simulated data. The complete simulation source code, together with scripts to reproduce the datasets, figures, and tables in the manuscript, is openly available at the first author’s GitHub repository: https://github.com/vancak/indirect_nnt.

Appendix A

A.1 Twin causal networks

A twin network is a graphical method that presents two networks together – one for the hypothetical world and the other for the factual world, alternatively two networks for two distinct hypothetical worlds [52]. Such networks provide a graphical way to check and test independence between factual and counterfactual quantities. The DAGs in Figures 5 and 6 illustrate the first part of the sequential ignorability assumption, i.e., the conditional independence of I a ′ , m , M a and A, given L; I a ′ , m , M a ⊥ ⊥ A | L . The DAG in Figure 7 illustrates the second part of the sequential ignorability, i.e., the conditional independence of I _a′,m and M _a, given L and A=a; I a ′ , m ⊥ ⊥ M a | L , A = a . For further information on Twin networks, please refer to Chapter 7 in Ref. [53].

$Figure 5: DAG of a twin network causal model with a mediator that illustrates the first part of the sequential ignorability (17), namely, the conditional independence of I a′,m and A, given L; I a ′ , m ⊥ ⊥ A | L ${I}_{{a}^{\prime },m}\perp \perp A\vert L$ . The left-hand side of the DAG represents the observed world, while the right-hand side represents the hypothetical potential world. On the left-hand side, M is the mediator, A is the exposure, and I is the outcome. On the right-hand side, the exposure A is set to a, the mediator M is set to m, and I a′,m is the potential outcome where the exposure is set to a′, and the mediator to m. For both DAGs, L is the set of measured confounders, ϵ M and ϵ I represent all the unmeasured exogenous factors that determine the values of M and I, respectively.$

Figure 5:

DAG of a twin network causal model with a mediator that illustrates the first part of the sequential ignorability (17), namely, the conditional independence of I _a′,m and A, given L; I a ′ , m ⊥ ⊥ A | L . The left-hand side of the DAG represents the observed world, while the right-hand side represents the hypothetical potential world. On the left-hand side, M is the mediator, A is the exposure, and I is the outcome. On the right-hand side, the exposure A is set to a, the mediator M is set to m, and I _a′,m is the potential outcome where the exposure is set to a′, and the mediator to m. For both DAGs, L is the set of measured confounders, ϵ _M and ϵ _I represent all the unmeasured exogenous factors that determine the values of M and I, respectively.

$Figure 6: DAG of a twin network causal model with a mediator that illustrates the first part of the sequential ignorability (17), namely, the conditional independence of M a and A, given L; M a ⊥ ⊥ A | L ${M}_{a}\perp \perp A\vert L$ . The left-hand side of the DAG represents the observed world, while the right-hand side represents the hypothetical potential world. On the left-hand side, M is the mediator, A is the exposure, and I is the outcome. On the right-hand side, A=a is the specified value of the exposure, M a is the potential value of the mediator where A is set to a, and I a , M a ${I}_{a,{M}_{a}}$ is the potential outcome where the exposure A is set to a, and the mediator attains the value it would have attained for A=a. For both DAGs, L represents the set of measured confounders, where ϵ M and ϵ I represent all the unmeasured exogenous factors that determine the values of M and I, respectively.$

Figure 6:

DAG of a twin network causal model with a mediator that illustrates the first part of the sequential ignorability (17), namely, the conditional independence of M _a and A, given L; M a ⊥ ⊥ A | L . The left-hand side of the DAG represents the observed world, while the right-hand side represents the hypothetical potential world. On the left-hand side, M is the mediator, A is the exposure, and I is the outcome. On the right-hand side, A=a is the specified value of the exposure, M _a is the potential value of the mediator where A is set to a, and I a , M a is the potential outcome where the exposure A is set to a, and the mediator attains the value it would have attained for A=a. For both DAGs, L represents the set of measured confounders, where ϵ _M and ϵ _I represent all the unmeasured exogenous factors that determine the values of M and I, respectively.

$Figure 7: DAG of a twin network causal model with a mediator that illustrates the second part of the sequential ignorability (17), namely, the conditional independence of I a′,m and M a , given L and A=a; I a ′ , m ⊥ ⊥ M a | L , A = a ${I}_{{a}^{\prime },m}\perp \perp {M}_{a}\vert L,A=a$ . The left-hand side of the DAG represents the hypothetical world where only the exposure is set to a, while the right-hand side represents the hypothetical world where both the exposure and the mediator are set to a′ and m, respectively. Specifically, on the left-hand side, M a is the potential mediator where A is set to a, and I a , M a ${I}_{a,{M}_{a}}$ is the nested potential outcome for a and M a . On the right-hand side, the exposure A is set to a', and mediator M is set to m, thus I a′,m is the potential outcome for a′≠a, and m. For both DAGs, L represented the set of all measured confounders, while ϵ M and ϵ I represent all the unmeasured exogenous factors that determine the values of M and I, respectively.$

Figure 7:

DAG of a twin network causal model with a mediator that illustrates the second part of the sequential ignorability (17), namely, the conditional independence of I _a′,m and M _a, given L and A=a; I a ′ , m ⊥ ⊥ M a | L , A = a . The left-hand side of the DAG represents the hypothetical world where only the exposure is set to a, while the right-hand side represents the hypothetical world where both the exposure and the mediator are set to a^′ and m, respectively. Specifically, on the left-hand side, M _a is the potential mediator where A is set to a, and I a , M a is the nested potential outcome for a and M _a. On the right-hand side, the exposure A is set to a', and mediator M is set to m, thus I _a′,m is the potential outcome for a′≠a, and m. For both DAGs, L represented the set of all measured confounders, while ϵ _M and ϵ _I represent all the unmeasured exogenous factors that determine the values of M and I, respectively.

A.2 Controlled direct effect NNT

Definition 4.

The marginal controlled direct effect (CDE) is defined as

(38) p c ( m ) = E [ I 1 , m − I 0 , m ] , m ∈ M ,

where M is the support set of the mediator M. Notably, setting M to a fixed value m removes the effect represented by the path A → M in Figure 2, thereby isolating the direct effect of A on I through the path A → I, conditional on M=m. To identify p _c(m) from observed data, we apply the law of total expectation:

p c ( m ) = E E [ I 1 , m − I 0 , m ∣ M ] .

However, since m is fixed, the inner conditional expectation E [ I a , m ∣ M ] is no longer a function of the random variable M, and the outer expectation becomes redundant. That is,

p c ( m ) = E [ I 1 , m ] − E [ I 0 , m ] .

To express this difference in terms of observed data, we use the consistency assumption: I=I _a,m when A=a and M=m, and the conditional ignorability assumption, i.e., I a ⊥ ⊥ A ∣ M (or I a ⊥ ⊥ A ∣ M , L if covariates L are present, as illustrated in Figure 5). These yield the following identification formulas

E [ I 1 , m ] = E [ I ∣ A = 1 , M = m ] , E [ I 0 , m ] = E [ I ∣ A = 0 , M = m ] ,

therefore,

p c ( m ) = E [ I ∣ A = 1 , M = m ] − E [ I ∣ A = 0 , M = m ] .

These quantities can be estimated using standard regression techniques. Applying the function g to p _c(m) yields the controlled direct effect on the NNT scale, also interpretable as the NNT conditional on M=m. An analogous definition holds for the group-specific controlled direct effect:

p c ( m ; a ) = E [ I 1 , m − I 0 , m ∣ A = a ] , a = 0,1 , m ∈ M .

For additional details and applications of this framework, see Refs. [17], 54]. As a final note, unlike other effect types, there is no corresponding “controlled indirect effect,” so we do not pursue this direction further.

A.3 Proof of Proposition 1

Proof.

The NIE is defined as E [ I a , M 1 − I a , M 0 ] , for a certain value a of the exposure A. Without loss of generality, we assume a=0 for consistency with the INNT (INNE, IEIN) defined in Definition 3. Assume a binary outcome I, and a binary mediator M, we have

E [ I 0 , M 1 − I 0 , M 0 ] = E E [ I 0 , M 1 − I 0 , M 0 | M 0 , M 1 ] = E [ I 0,1 − I 0,0 | M 0 = 0 , M 1 = 1 ] P ( M 0 = 0 , M 1 = 1 ) + E [ I 0,0 − I 0,1 | M 0 = 1 , M 1 = 0 ] P ( M 0 = 1 , M 1 = 0 ) = E [ I 0,1 − I 0,0 ] P ( M 0 = 0 , M 1 = 1 ) − P ( M 0 = 1 , M 1 = 0 ) = E [ I 0,1 − I 0,0 ] E [ M 1 − M 0 ] .

The first equality applies the law of total expectation. The second equality uses the fact that for M ₁=M ₀, the term E [ I 0 , M 1 − I 0 , M 0 ] vanishes. The third equality stems from effect homogeneity over principal strata, i.e., E I 0,1 − I 0,0 ∣ M 0 , M 1 = E I 0,1 − I 0,0 . The final equality follows from the following identity

E [ M 1 − M 0 ] = P ( M 0 = 0 , M 1 = 1 ) − P ( M 0 = 1 , M 1 = 0 ) + 0 × P ( M 0 = M 1 ) = P ( M 0 = 0 , M 1 = 1 ) − P ( M 0 = 1 , M 1 = 0 ) ,

which concludes the proof. □

A.4 Proof of Proposition 2

Proof.

The NDE is defined as E [ I 1 , M a − I 0 , M a ] , where the exposure A is set to a for the potential mediator M _a. Without loss of generality, we assume a=1 for consistency with the DNNT (DNNE, DEIN) as defined in Definition 2. Assume a binary outcome I, and a binary mediator M. Therefore, the NDE in the ath group can be written as follows

where the last equality assumes E [ I 1 , m − I 0 , m ∣ A = a , M a = m ] = E [ I 1 , m − I 0 , m ∣ A = a ] , for m=0, 1. □

A.5 Identification of the indirect effect in the ath group

Although the illustration below focuses on the unexposed group A=0, the derivation for the exposed group A=1 is entirely analogous, with all expectations conditioned on A=1 instead. For the INNE, which is defined as g(p _i(0)), we show the identification procedure for the first multiplicative term of eq. (12). Therefore, we need to express E [ M 1 − M 0 | A = 0 ] as a function of the observed data and the fitted model for the unexposed group A=0. Assume the exposure-mediator model as in eq. (24), thus

E [ M 1 − M 0 | A = 0 ] = E [ M 1 | A = 0 ] − E [ M 0 | A = 0 ] = E [ E [ M | A = 1 , L ] | A = 0 ] − E [ E [ M | A = 0 , L ] | A = 0 ] = E η 1 , L ; γ | A = 0 − E η ( 0 , L ; γ ) | A = 0 = E η 1 , L ; γ − η 0 , L ; γ | A = 0 .

The derivation for the second multiplicative term in eq. (12) follows the same steps for the conditional outcome model. Namely, assuming a conditional outcome model as in eq. (25), E [ I 0,1 − I 0,0 | A = 0 ] is identified and computed as

E ξ 0,1 , L ; β − ξ 0,0 , L ; β | A = 0 .

Therefore, the indirect effect for the unexposed group p _i(0; θ) is identified by

(39) p i ( 0 ; θ ) = E η 1 , L ; γ − η 0 , L ; γ | A = 0 E ξ 0,1 , L ; β − ξ 0,0 , L ; β | A = 0 ,

where θ denotes the set of all unknown parameters that the indirect effect is dependent on. Finally, the INNE is identified by g(p _i(0; θ)). Replacing the conditioning set with A=1, that is, evaluating the expectations with respect to the exposed group, yields the identification formula for the indirect effect among the exposed group p _i(1; θ), and consequently for the IEIN, defined as g(p _i(1; θ)).

References

1. Nguyen, TQ, Schmid, I, Stuart, EA. Clarifying causal mediation analysis for the applied researcher: defining effects based on what we want to learn. Psychol Methods 2021;26:255. https://doi.org/10.1037/met0000299.Suche in Google Scholar

2. Rijnhart, JJM, Lamp, SJ, Valente, MJ, MacKinnon, DP, Twisk, JWR, Heymans, MW. Mediation analysis methods used in observational research: a scoping review and recommendations. BMC Med Res Methodol 2021;21:1–7. https://doi.org/10.1186/s12874-021-01426-3.Suche in Google Scholar

3. MacKinnon, DP, Lockwood, CM, Hoffman, JM, West, SG, Sheets, V. A comparison of methods to test mediation and other intervening variable effects. Psychol Methods 2002;7:83. https://doi.org/10.1037//1082-989x.7.1.83.Suche in Google Scholar

4. Preacher, KJ, Kelley, K. Effect size measures for mediation models: quantitative strategies for communicating indirect effects. Psychol Methods 2011;16:93. https://doi.org/10.1037/a0022658.Suche in Google Scholar

5. Lange, T, Starkopf, L. Commentary: mediation analyses in the real world. Epidemiology 2016;27:677–81. https://doi.org/10.1097/ede.0000000000000518.Suche in Google Scholar

6. Ditlevsen, S, Christensen, U, Lynch, J, Damsgaard, MT, Keiding, N. The mediation proportion: a structural equation approach for estimating the proportion of exposure effect on outcome explained by an intermediate variable. Epidemiology 2005:114–20. https://doi.org/10.1097/01.ede.0000147107.76079.07.Suche in Google Scholar

7. Newcombe, RG. Confidence intervals for proportions and related measures of effect size. Boca Raton, Florida: CRC Press; 2012.10.1201/b12670Suche in Google Scholar

8. Vancak, V, Goldberg, Y, Levine, SZ. Guidelines to understand and compute the number needed to treat. BMJ Ment Health 2021;24:131–6. https://doi.org/10.1136/ebmental-2020-300232.Suche in Google Scholar

9. Lee, TY, Kuo, S, Yang, CY, Ou, HT. Cost-effectiveness of long-acting insulin analogues vs intermediate/long-acting human insulin for type 1 diabetes: a population-based cohort followed over 10 years. Br J Clin Pharmacol 2020;86:852–60. https://doi.org/10.1111/bcp.14188.Suche in Google Scholar

10. Verbeek, JGE, Atema, V, Mewes, JC, van Leeuwen, M, Oldenburg, HSA, van Beurden, M, et al.. Cost-utility, cost-effectiveness, and budget impact of internet-based cognitive behavioral therapy for breast cancer survivors with treatment-induced menopausal symptoms. Breast Cancer Res Treat 2019;178:573–85. https://doi.org/10.1007/s10549-019-05410-w.Suche in Google Scholar

11. da Costa, BR, Rutjes, AWS, Johnston, BC, Reichenbach, S, Nüesch, E, Tonia, T, et al.. Methods to convert continuous outcomes into odds ratios of treatment response and numbers needed to treat: meta-epidemiological study. Int J Epidemiol 2012;41:1445–59. https://doi.org/10.1093/ije/dys124.Suche in Google Scholar

12. Mendes, D, Alves, C, Batel-Marques, F. Number needed to treat (nnt) in clinical literature: an appraisal. BMC Med 2017;15:1–13. https://doi.org/10.1186/s12916-017-0875-8.Suche in Google Scholar

13. Bender, R, Blettner, M. Calculating the “number needed to be exposed” with adjustment for confounding variables in epidemiological studies. J Clin Epidemiol 2002;55:525–30. https://doi.org/10.1016/s0895-4356(01)00510-8.Suche in Google Scholar

14. Laupacis, A, Sackett, DL, Roberts, RS. An assessment of clinically useful measures of the consequences of treatment. N Engl J Med 1988;318:1728–33. https://doi.org/10.1056/nejm198806303182605.Suche in Google Scholar

15. Kristiansen, IS, Gyrd-Hansen, D, Nexøe, J, Nielsen, JB. Number needed to treat: easily understood and intuitively meaningful? Theoretical considerations and a randomized trial. J Clin Epidemiol 2002;55:888–92. https://doi.org/10.1016/s0895-4356(02)00432-8.Suche in Google Scholar

16. Mueller, S, Pearl, J. Personalized decision making–a conceptual introduction. J Causal Inference 2023;11:20220050. https://doi.org/10.1515/jci-2022-0050.Suche in Google Scholar

17. Vancak, V, Sjölander, A. Estimation of the number needed to treat, the number needed to be exposed, and the exposure impact number with instrumental variables. Epidemiol Methods 2024;13:20230034. https://doi.org/10.1515/em-2023-0034.Suche in Google Scholar

18. Vancak, V, Goldberg, Y, Levine, SZ. Systematic analysis of the number needed to treat. Stat Methods Med Res 2020;29:2393–410. https://doi.org/10.1177/0962280219890635.Suche in Google Scholar

19. Kraemer, HC, Kupfer, DJ. Size of treatment effects and their importance to clinical research and practice. Biol Psychiatry 2006;59:990–6. https://doi.org/10.1016/j.biopsych.2005.09.014.Suche in Google Scholar

20. Grieve, AP. The number needed to treat: a useful clinical measure or a case of the emperor’s new clothes? Pharm Stat J Appl Stat Pharm Ind 2003;2:87–102. https://doi.org/10.1002/pst.33.Suche in Google Scholar

21. Hutton, JL. Number needed to treat: properties and problems. J Roy Stat Soc Ser A (Stat Soc) 2000;163:381–402. https://doi.org/10.1111/1467-985x.00175.Suche in Google Scholar

22. Snapinn, S, Jiang, Q. On the clinical meaningfulness of a treatment’s effect on a time-to-event variable. Stat Med 2011;30:2341–8. https://doi.org/10.1002/sim.4256.Suche in Google Scholar

23. Kristiansen, IS, Gyrd-Hansen, D. Cost-effectiveness analysis based on the number-needed-to-treat: common sense or non-sense? Health Econ 2004;13:9–19. https://doi.org/10.1002/hec.797.Suche in Google Scholar

24. Breitborde, NJK, Srihari, VH, Pollard, JM, Addington, DN, Woods, SW. Mediators and moderators in early intervention research. Early Interv Psychiatry 2010;4:143–52. https://doi.org/10.1111/j.1751-7893.2010.00177.x.Suche in Google Scholar

25. Tsai, CL, Camargo, CAJr. Methodological considerations, such as directed acyclic graphs, for studying “acute on chronic” disease epidemiology: chronic obstructive pulmonary disease example. J Clin Epidemiol 2009;62:982–90. https://doi.org/10.1016/j.jclinepi.2008.10.005.Suche in Google Scholar

26. Du Toit, G, Foong, RXM, Lack, G. Prevention of food allergy-early dietary interventions. Allergol Int 2016;65:370–7. https://doi.org/10.1016/j.alit.2016.08.001.Suche in Google Scholar

27. Carr, A. Cardiovascular risk factors in hiv-infected patients. JAIDS J Acquir Immune Defic Syndr 2003;34:S73–8. https://doi.org/10.1097/00126334-200309011-00011.Suche in Google Scholar

28. Pellegrini, V, De Cristofaro, V, Salvati, M, Giacomantonio, M, Leone, L. Social exclusion and anti-immigration attitudes in Europe: the mediating role of interpersonal trust. Soc Indic Res 2021;155:697–724. https://doi.org/10.1007/s11205-021-02618-6.Suche in Google Scholar

29. Rubin, DB. Causal inference using potential outcomes: design, modeling, decisions. J Am Stat Assoc 2005;100:322–31. https://doi.org/10.1198/016214504000001880.Suche in Google Scholar

30. Pearl, J. The causal mediation formula – a guide to the assessment of pathways and mechanisms. Prev Sci 2012;13:426–36. https://doi.org/10.1007/s11121-011-0270-1.Suche in Google Scholar

31. Forastiere, L, Mattei, A, Ding, P. Principal ignorability in mediation analysis: through and beyond sequential ignorability. Biometrika 2018;105:979–86. https://doi.org/10.1093/biomet/asy053.Suche in Google Scholar

32. Robins, JM, Greenland, S. Identifiability and exchangeability for direct and indirect effects. Epidemiology 1992;3:143–55. https://doi.org/10.1097/00001648-199203000-00013.Suche in Google Scholar

33. Docquier, F, Rapoport, H. Globalization, brain drain, and development. J Econ Lit 2012;50:681–730. https://doi.org/10.1257/jel.50.3.681.Suche in Google Scholar

34. Beine, M, Docquier, F, Rapoport, H. Brain drain and economic growth: theory and evidence. J Dev Econ 2001;64:275–89. https://doi.org/10.1016/s0304-3878(00)00133-4.Suche in Google Scholar

35. Beine, M, Docquier, F, Rapoport, H. Brain drain and human capital formation in developing countries: winners and losers. Econ J 2008;118:631–52. https://doi.org/10.1111/j.1468-0297.2008.02135.x.Suche in Google Scholar

36. Holland, PW. Statistics and causal inference. J Am Stat Assoc 1986;81:945–60. https://doi.org/10.2307/2289064.Suche in Google Scholar

37. Rubin, DB. Estimating causal effects of treatments in randomized and nonrandomized studies. J Educ Psychol 1974;66:688. https://doi.org/10.1037/h0037350.Suche in Google Scholar

38. Pearl, J. On the consistency rule in causal inference: axiom, definition, assumption, or theorem? Epidemiology 2010;21:872–5. https://doi.org/10.1097/ede.0b013e3181f5d3fd.Suche in Google Scholar

39. Petersen, ML, Porter, KE, Gruber, S, Wang, Y, Van Der Laan, MJ. Diagnosing and responding to violations in the positivity assumption. Stat Methods Med Res 2012;21:31–54. https://doi.org/10.1177/0962280210386207.Suche in Google Scholar

40. Imai, K, Keele L, Yamamoto T. Identification, inference and sensitivity analysis for causal mediation effects. Statistical Science 2010;25:51–71. https://doi.org/10.1214/10-sts321.Suche in Google Scholar

41. Imai, K, Keele, L, Tingley, D. A general approach to causal mediation analysis. Psychol Methods 2010;15:309. https://doi.org/10.1037/a0020761.Suche in Google Scholar

42. Frangakis, CE, Rubin, DB. Principal stratification in causal inference. Biometrics 2002;58:21–9. https://doi.org/10.1111/j.0006-341x.2002.00021.x.Suche in Google Scholar

43. Ding, P, Lu, J. Principal stratification analysis using principal scores. J Roy Stat Soc B Stat Methodol 2017;79:757–77. https://doi.org/10.1111/rssb.12191.Suche in Google Scholar

44. Cole, SR, Frangakis, CE. The consistency statement in causal inference: a definition or an assumption? Epidemiology 2009;20:3–5. https://doi.org/10.1097/ede.0b013e31818ef366.Suche in Google Scholar

45. Sjölander, A. Estimation of causal effect measures with the r-package stdreg. Eur J Epidemiol 2018;33:847–58. https://doi.org/10.1007/s10654-018-0375-y.Suche in Google Scholar

46. Bender, R, Kuss, O, Hildebrandt, M, Gehrmann, U. Estimating adjusted NNT measures in logistic regression analysis. Stat Med 2007;26:5586–95. https://doi.org/10.1002/sim.3061.Suche in Google Scholar

47. Nevo, D, Liao, X, Spiegelman, D. Estimation and inference for the mediation proportion. Int J Biostat 2017;13:20170006. https://doi.org/10.1515/ijb-2017-0006.Suche in Google Scholar

48. Bender, R, Vervölgyi, V. Estimating adjusted nnts in randomised controlled trials with binary outcomes: a simulation study. Contemp Clin Trials 2010;31:498–505. https://doi.org/10.1016/j.cct.2010.07.005.Suche in Google Scholar

49. Stefanski, LA, Boos, DD. The calculus of m-estimation. Am Stat 2002;56:29–38. https://doi.org/10.1198/000313002753631330.Suche in Google Scholar

50. VanderWeele, TJ. Bias formulas for sensitivity analysis for direct and indirect effects. Epidemiology 2010;21:540–51. https://doi.org/10.1097/ede.0b013e3181df191c.Suche in Google Scholar

51. Valeri, L, VanderWeele, TJ. Mediation analysis allowing for exposure–mediator interactions and causal interpretation: theoretical assumptions and implementation with sas and spss macros. Psychol Methods 2013;18:137. https://doi.org/10.1037/a0031034.Suche in Google Scholar

52. Balke, A, Pearl J. Probabilistic evaluation of counterfactual queries. In: Probabilistic and causal inference: the works of Judea Pearl. New York, NY, USA: Association for Computing Machinery; 2022:237–54 pp.10.1145/3501714.3501733Suche in Google Scholar

53. Pearl, J. Causality. Cambridge, UK: Cambridge University Press; 2009.Suche in Google Scholar

54. Vancak, V, Goldberg, Y, Levine, SZ. The number needed to treat adjusted for explanatory variables in regression and survival analysis: theory and application. Stat Med 2022;41:3299–320. https://doi.org/10.1002/sim.9418.Suche in Google Scholar

55. Pearl, J. Direct and indirect effects. In: Probabilistic and causal inference: the works of Judea Pearl. New York, NY, USA: Association for Computing Machinery; 2022:373–392 pp.10.1145/3501714.3501736Suche in Google Scholar

Received: 2025-04-30

Accepted: 2025-10-29

Published Online: 2026-01-01

Sie haben derzeit keinen Zugang zu diesem Inhalt.

Artikel in diesem Heft

https://doi.org/10.1515/em-2025-0018

Schlagwörter für diesen Artikel

NNT; NNE; indirect effects; identification; meditation; effect size