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Estimation of the number needed to treat, the number needed to be exposed, and the exposure impact number with instrumental variables

  • Valentin Vancak EMAIL logo and Arvid Sjölander
Published/Copyright: August 21, 2024
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Epidemiologic Methods
From the journal Epidemiologic Methods Volume 13 Issue s2

Abstract

Objectives

The Number Needed to Treat (NNT) is an efficacy index defined as the average number of patients needed to treat to attain one additional treatment benefit. In observational studies, specifically in epidemiology, the adequacy of the populationwise NNT is questionable since the exposed group characteristics may substantially differ from the unexposed. To address this issue, groupwise efficacy indices were defined: the Exposure Impact Number (EIN) for the exposed group and the Number Needed to be Exposed (NNE) for the unexposed. Each defined index answers a unique research question since it targets a unique sub-population. In observational studies, the group allocation is typically affected by confounders that might be unmeasured. The available estimation methods that rely either on randomization or the sufficiency of the measured covariates for confounding control result in statistically inconsistent estimators of the true EIN, NNE, and NNT. This study presents a theoretical framework for statistically consistent point and interval estimation of the NNE, EIN and NNE in observational studies with unmeasured confounders.

Methods

Using Rubin’s potential outcomes framework, this study explicitly defines the NNT and its derived indices, EIN and NNE, as causal measures. Then, we use instrumental variables to introduce a novel method to estimate the three aforementioned indices in observational studies where the omission of unmeasured confounders cannot be ruled out. To illustrate the novel methods, we present two analytical examples – double logit and double probit models. Next, a corresponding simulation study and a real-world data example are presented.

Results

This study provides an explicit causal formulation of the EIN, NNE, and NNT indices and a comprehensive theoretical framework for their point and interval estimation using the G-estimators in observational studies with unmeasured confounders. The analytical proofs and the corresponding simulation study illustrate the improved performance of the new estimation method compared to the available methods in terms of consistency and the confidence intervals empirical coverage rates.

Conclusions

In observational studies, traditional estimation methods to estimate the EIN, NNE, or NNT result in statistically inconsistent estimators. We introduce a novel estimation method that overcomes this pitfall. The novel method produces consistent estimators and reliable CIs for the true EIN, NNE, and NNT. Such a method may facilitate more accurate clinical decision-making and the development of efficient public health policies.

Keywords: NNT; NNE; EIN; instrumental variables; G-estimation
Corresponding author: Valentin Vancak, Department of Data Science, Holon Institute of Technology, Holon, Israel; and Department of Medical Epidemiology and Biostatistics, Karolinska Institute, Stockholm, Sweden, E-mail:

Funding source: Vetenskapsrådet

Award Identifier / Grant number: 2020-01188

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Competing interests: The authors state no conflct of interests.

  5. Research funding: Vetenskapsrådet, Grant/Award Number: 2020-01188.

  6. Data availability: The generated and analyzed datasets and the corresponding source code of the simulation section of the current study are available in the first author’s GitHub repository, https://github.com/vancak/nne_iv. The dataset analyzed in the real-world data example section of the current study is available in the ivtools R-package CRAN repository, https://cran.r-project.org/web/packages/ivtools/index.html.

Proof of unbiasedness of the G-estimators estimating equations

Without the loss of generality, we present the proof for the exposed group, a=1. The proof for the unexposed group can be readily obtained using analogical steps for a=0. Assume that ξ is a link function for the structural mean model (6) and let η be the link function for the association model η([E|Z, L, A; β I ]). Let the estimating equation for the G-estimator of ψ 1 be as defined in eq. (21). Therefore,

(31) E [ D 1 ( Z , L ; π Z ) h ( 1 ; Z , L , 1 , β I , ψ 1 ) ] = E D 1 ( Z , L ; π Z ) ξ − 1 η ( E [ I | Z , L , A = 1 ; β I ] ) − m 1 T ( L ) ψ 1

(32) = E D 1 ( Z , L ; π Z ) ξ − 1 η ( E [ I 1 | Z , L , A = 1 ; β I ] ) − m 1 T ( L ) ψ 1

(33) = E [ D 1 ( Z , L ; π Z ) E [ I 0 | Z , L , A = 1 ] ]

(34) = E E [ D 1 ( Z , L ; π Z ) I 0 | Z , L , A = 1 ]

(35) = E [ D 1 ( Z , L ; π Z ) I 0 ]

= 0 .

Equation (31) is a direct consequence of the assumed GSMM structure in eq. (6). Equation (32) stems from the consistency assumption [30]. Equation (33) is obtained by plugging-in eq. (8) for a=1. Equation (34) holds since the potential outcome I 0 is independent of the observed allocation A. The last step in eq. (35) stems from the validity of the IV. Namely, the estimating equations are unbiased as long as the IV Z satisfies conditions of a valid IV, i.e., I 0 ⫫ Z | L .

The double probit model example

Assume a binary outcome I ∈ {0, 1}, a binary exposure A ∈ {0, 1}, and a binary instrument Z ∈ {0, 1}. Assume there are no measured confounders, i.e., L=∅ and m(L)=1. For the probit link function ξ, we define the GSMM (6) as

(36) Φ − 1 E [ I 1 = 1 | Z , A = a ] − Φ − 1 E [ I 0 = 1 | Z , A = a ] = ψ a , a ∈ { 0,1 } ,

where Φ−1(x) is the inverse of the standard normal random variable’s cumulative distribution function Φ ( x ) = ∫ − ∞ x ( 2 π ) − 1 / 2 ⁡ exp { − s 2 / 2 } d s . Additionally, assume the following saturated probit model for the observed outcome

(37) Φ − 1 E [ I | Z , A ; β I ] = β 0 + β 1 A + β 2 Z + β 3 Z A .

Using the GSMM and the probit association model, we can express explicitly the general conditional exposure benefit (14) as a function of β I and ψ

(38) p b ( Z , A , β I , ψ ) = Φ β 0 + β 1 A + β 2 Z + β 3 Z A + ψ 0 ( 1 − A ) ) − Φ β 0 + β 1 A + β 2 Z + β 3 Z A − ψ 1 A .

To compute the populationwise NNT, we apply the function g to the marginal exposure benefit p b =E[p b (Z, A, β I , ψ)] (11), i.e., NNT=g(p b ). If we set A to 0 in eq. (38) we obtain the conditional exposure benefit (12) for the unexposed a=0 as a function of β I and ψ 0

(39) p b ( 0 ; Z , β I , ψ 0 ) = Φ ( β 0 + β 2 Z + ψ 0 ) − Φ ( β 0 + β 2 Z ) ,

To compute the NNE, we apply the function g to the marginal exposure benefit for the unexposed p b (0)=E[p b (0; Z, β I , ψ 0)|A=0] (13), i.e., NNE=g(p b (0)). Analogically, for the conditional exposure benefit (12) for the exposed a=1, we set A to one in eq. (17)

(40) p b ( 1 ; Z , β I , ψ 1 ) = Φ ( β 0 + β 1 + β 2 Z + β 3 Z ) − Φ ( β 0 + β 1 + β 2 Z + β 3 Z − ψ 1 ) .

To compute the EIN, we apply the function g to the marginal exposure benefit for the exposed p b (1)=E[p b (1; Z, β I , ψ 1)|A=1] (13), i.e., EIN=g(p b (1)).

Explicit form of valid IV conditions

The double logit model

Let the GSMM be defined as in eq. (15), and the marginal distribution of the IV Z and the conditional distribution of the exposure A are as specified in (28). For a double logit model, the explicit form of eq. (30) for the potential outcome I 0 is

(41) expit ( β 0 + β 2 ) ( 1 − expit ( γ 0 + γ 1 ) ) + expit ( β 0 + β 1 + β 2 + β 3 − ψ 1 ) expit ( γ 0 + γ 1 ) = expit ( β 0 ) ( 1 − expit ( γ 0 ) ) + expit ( β 0 + β 1 − ψ 1 ) expit ( γ 0 ) ,

and for the potential outcome I 1 is

(42) expit ( β 0 + β 2 + ψ 0 ) ( 1 − expit ( γ 0 + γ 1 ) ) + expit ( β 0 + β 1 + β 2 + β 3 ) expit ( γ 0 + γ 1 ) = expit ( β 0 + ψ 0 ) ( 1 − expit ( γ 0 ) ) + expit ( β 0 + β 1 ) expit ( γ 0 ) .

The double probit model

Let the GSMM be defined as in eq. (36), and the function ξ be the probit Φ−1 function. The explicit forms of the exposure benefits are given in eqs. (39) and (40). The explicit forms of eq. (30) for the potential outcomes I 0 and I 1 are obtained by replacing the expit function with the inverse probit function Φ in eqs. (41) and (42), respectively.

Simulation study, Step 2: estimation

This subsection presents the explicit form of the vector-valued function Q(Z, A; θ) components. The vector of unbiased estimating functions (24) of β I and π Z consists of the score functions of the association model (29) and the binary instrument model (28), namely,

S ( Z , A ; β I , π Z ) = ( I − ξ − 1 ( β 0 + β 1 A + β 2 Z + β 3 A Z ) ) ( I − ξ − 1 ( β 0 + β 1 A + β 2 Z + β 3 A Z ) ) A ( I − ξ − 1 ( β 0 + β 1 A + β 2 Z + β 3 A Z ) ) Z ( I − ξ − 1 ( β 0 + β 1 A + β 2 Z + β 3 A Z ) ) A Z ( Z − π Z ) .

The vector-valued function Dh(Z, A; β I , π Z , ψ) as in eq. (21) of the two estimating functions for the causal parameters ψ is

D h ( Z , A ; β I , π Z , ψ ) = ( Z − π Z ) ξ − 1 β 0 + β 1 A + β 2 Z + β 3 A Z + ψ 0 ( 1 − A ) ( Z − π Z ) ξ − 1 β 0 + β 1 A + β 2 Z + β 3 A Z − ψ 1 A .

The vector-valued function p Z , L , A ; β I , ψ , p b ( 0 ) , p b ( 1 ) , p b as in eq. (25) consists of the three estimating equations for the exposure benefits

p Z , L , A ; β I , ψ , p b ( 0 ) , p b ( 1 ) , p b = ξ − 1 β 0 + β 2 Z + ψ 0 − ξ − 1 β 0 + β 2 Z − p ( 0 ) ( 1 − A ) ξ − 1 β 0 + β 1 + β 2 Z + β 3 Z − ξ − 1 β 0 + β 1 + β 2 Z + β 3 Z − ψ 1 − p ( 1 ) A ξ − 1 β 0 + β 1 A + β 2 Z + β 3 A Z + ψ 0 ( 1 − A ) − ξ − 1 β 0 + β 1 A + β 2 Z + β 3 A Z − ψ 1 A − p b .

In all components of the estimating function Q(Z, A; θ), ξ −1 is the expit and the inverse probit Φ functions for the double logit and double probit models, respectively. Notably, the vector valued function g p b ( 0 ) , p b ( 1 ) , p b , NNE, EIN, NNT is independent of the data, and thus its form remains the same as in its definition in (26).

Simulation study – setting II

This simulation setting is analogical to the simulations in Simulation study, except for smaller values of the causal parameters ψ and higher consequent values of the corresponding EIN, NNE and NNT measures. Namely, we use the double logit and double probit models described in the simulation setup in Subsection 6.1 with the same sample sizes. This setting can serve as a preliminary sensitivity analysis of the methodology for combining small sample sizes with small causal effects.

Figure 6: Logit and probit models examples: binary outcome with logit and probit causal models, respectively. The true causal parameters for both models are ψ=(0.5, 1) T . The true populationwise NNTs are 8.00, and 5.30, respectively. The red boxplots denote the IV-based estimators of the NNT, which are based on G-estimators of ψ. The unadjusted estimators of the NNT were infinitely large (the estimated benefits were negative before the application of the function g (3)); therefore, they were omitted from the graph. The red dashed line denotes the true NNT value for each model. The calculations were repeated m=1,000 times for four different sample sizes: n=500, 1,000, 2000, 4,000. For both models, the marginal P(A=1)=0.6, π Z =0.5, γ=(−0.83, 3) T and the marginal probability of the outcome is P(I=1)=0.3. (a) Double logit model. True NNT=8.00. (b) Double probit model. True NNT=5.30.
Figure 6:

Logit and probit models examples: binary outcome with logit and probit causal models, respectively. The true causal parameters for both models are ψ=(0.5, 1) T . The true populationwise NNTs are 8.00, and 5.30, respectively. The red boxplots denote the IV-based estimators of the NNT, which are based on G-estimators of ψ. The unadjusted estimators of the NNT were infinitely large (the estimated benefits were negative before the application of the function g (3)); therefore, they were omitted from the graph. The red dashed line denotes the true NNT value for each model. The calculations were repeated m=1,000 times for four different sample sizes: n=500, 1,000, 2000, 4,000. For both models, the marginal P(A=1)=0.6, π Z =0.5, γ=(−0.83, 3) T and the marginal probability of the outcome is P(I=1)=0.3. (a) Double logit model. True NNT=8.00. (b) Double probit model. True NNT=5.30.

Figure 7: Double logit and probit models examples: binary outcome with logit and probit causal models, respectively. The causal parameters are ψ=(0.5, 1) T . The true EINs are 6.41 and 4.50, respectively. The red boxplots denote the IV-based estimators of the EIN, which are based on G-estimators of ψ. The unadjusted estimators of the EIN were infinitely large (the estimated benefits were negative before the application of the function g (3)); therefore, they were omitted from the graph. The red dashed line denotes the true EIN value in each model. The calculations were repeated m=1,000 times for four different sample sizes: n=500, 1,000, 2000, 4,000. For both models, the marginal P(A=1)=0.6, π Z =0.5, γ=(−0.83, 3) T and the marginal probability of the outcome is P(I=1)=0.3. (a) Double logit model. True EIN=6.41. (b) Double probit model. True EIN=4.50.
Figure 7:

Double logit and probit models examples: binary outcome with logit and probit causal models, respectively. The causal parameters are ψ=(0.5, 1) T . The true EINs are 6.41 and 4.50, respectively. The red boxplots denote the IV-based estimators of the EIN, which are based on G-estimators of ψ. The unadjusted estimators of the EIN were infinitely large (the estimated benefits were negative before the application of the function g (3)); therefore, they were omitted from the graph. The red dashed line denotes the true EIN value in each model. The calculations were repeated m=1,000 times for four different sample sizes: n=500, 1,000, 2000, 4,000. For both models, the marginal P(A=1)=0.6, π Z =0.5, γ=(−0.83, 3) T and the marginal probability of the outcome is P(I=1)=0.3. (a) Double logit model. True EIN=6.41. (b) Double probit model. True EIN=4.50.

Figure 8: Double logit and probit models examples: binary outcome with logit and probit causal models, respectively. The true causal parameters for both models are ψ=(0.5, 1) T . The true populationwise NNEs are 12.77, and 7.25, respectively. The red boxplots denote the IV-based estimators of the NNE, which are based on G-estimators of ψ. The blue boxplots denote the unadjusted estimators of the NNE. The red dashed line denotes the true NNE value for each model. The calculations were repeated m=1,000 times for four different sample sizes: n=500, 1,000, 2000, 4,000. For both models, the marginal P(A=1)=0.6, π Z =0.5, γ=(−0.83, 3) T and the marginal probability of the outcome is P(I=1)=0.3. (a) Double logit model. True NNE=12.77. (b) Double probit model. True NNE=7.25.
Figure 8:

Double logit and probit models examples: binary outcome with logit and probit causal models, respectively. The true causal parameters for both models are ψ=(0.5, 1) T . The true populationwise NNEs are 12.77, and 7.25, respectively. The red boxplots denote the IV-based estimators of the NNE, which are based on G-estimators of ψ. The blue boxplots denote the unadjusted estimators of the NNE. The red dashed line denotes the true NNE value for each model. The calculations were repeated m=1,000 times for four different sample sizes: n=500, 1,000, 2000, 4,000. For both models, the marginal P(A=1)=0.6, π Z =0.5, γ=(−0.83, 3) T and the marginal probability of the outcome is P(I=1)=0.3. (a) Double logit model. True NNE=12.77. (b) Double probit model. True NNE=7.25.

Table 2:

Double logit and probit models examples: empirical coverage rates (coverage) of the sandwich-matrix-based 95 %-level CIs for the marginal EIN, NNE, and NNT, the estimators’ Monte Carlo standard errors (MCSE), average bias (Av. bias), and the percentage of non-informative extremely large (upper limit>1,000) CIs (% inf. CIs), as a function of sample size n=500, 1,000, 2000, 4,000. The number of iterations for each sample size is m=1,000. The strength of the IV was measured as the mean values of the Wald statistic of the IV regression coefficient in the exposure model, i.e., E[Z stat.] for γ 1 in (28). The estimated strength of the instrument was 11.90, 16.95, 24.00, and 33.94 for each sample size, respectively. “Bread” matrices with condition number of ≥ 1 0 12 were excluded from further analysis since they produce numerically singular covariance matrices and do not allow for the construction of the analytical CIs.

Model Logit Probit
n Measure EIN (6.41) NNE (12.77) NNT (8.00) EIN (4.50) NNE (7.25) NNT (5.30)
500 Coverage 0.957 0.937 0.941 0.994 0.948 0.979
MCSE 1.158 1.714 1.397 0.125 0.210 0.148
Av. bias 8.733 14.610 10.537 1.718 3.010 2.017
% inf. CIs 2.9 % 4.6 % 4.8 % 13.5 % 0.01 % 13.4 %
1,000 Coverage 0.953 0.946 0.948 0.989 0.955 0.977
MCSE 0.798 1.371 0.946 0.086 0.141 0.100
Av. bias 6.052 11.099 7.306 1.041 1.838 1.218
% inf. CIs 0.9 % 1.1 % 1.2 % 8.1 % 0.01 % 7.9 %
2000 Coverage 0.954 0.938 0.946 0.989 0.959 0.979
MCSE 0.466 0.879 0.571 0.026 0.046 0.031
Av. bias 3.173 5.970 3.867 0.619 1.076 0.719
% inf. CIs 0.5 % 0.5 % 0.6 % 2.6 % 0 % 2.5 %
4,000 Coverage 0.955 0.937 0.952 0.978 0.954 0.969
MCSE 0.063 0.121 0.077 0.018 0.030 0.020
Av. bias 1.381 2.604 1.679 0.438 0.742 0.507
% inf. CIs 0 % 0 % 0 % 0.05 % 0 % 0.04 %

In summary, a combination of small causal effects that lead to high EIN, NNE and NNT values may result in unstable estimators with inflated average bias. Graphical summary of the estimtors’ behaviour as a function of the sample size can be found in Figures 6–8. Table 2 presents the empirical coverage rates of the 95%-level CIs, the MCSE, the average bias, and percentage of extremely large CIs of the marginal EIN, NNE, and NNT. The proportion of non-informative extremely large CIs was non-neglectable for the EIN and NNT in the double-probit model for the sample size of 500. These caveats are mitigated only for moderate-size sample sizes of 2000 and higher. A possible direction for future research is comprehensive sensitivity analysis and the development of robust estimation methods for such scenarios.

References

1. 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.Search in Google Scholar PubMed

2. Lee, T-Y, Kuo, S, Yang, C-Y, Ou, H-T. 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.Search in Google Scholar PubMed PubMed Central

3. 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.Search in Google Scholar PubMed PubMed Central

4. Newcombe, RG. Confidence intervals for proportions and related measures of effect size. Boca Raton, Florida: CRC Press; 2012.10.1201/b12670Search in Google Scholar

5. 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.Search in Google Scholar PubMed PubMed Central

6. 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.Search in Google Scholar PubMed PubMed Central

7. 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.Search in Google Scholar PubMed

8. 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.Search in Google Scholar

9. 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.Search in Google Scholar PubMed

10. 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.Search in Google Scholar PubMed

11. 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.Search in Google Scholar PubMed

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

13. 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.Search in Google Scholar PubMed PubMed Central

14. 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.Search in Google Scholar PubMed PubMed Central

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

16. Pearl, J. Probabilities of causation: three counterfactual interpretations and their identification. In: Probabilistic and causal inference: the works of Judea Pearl; 2022:317–72 pp.10.1145/3501714.3501735Search in Google Scholar

17. Schulzer, M, Mancini, GBJ. ‘unqualified success’ and ‘unmitigated failure’number-needed-to-treat-related concepts for assessing treatment efficacy in the presence of treatment-induced adverse events. Int J Epidemiol 1996;25:704–12. https://doi.org/10.1093/ije/25.4.704.Search in Google Scholar PubMed

18. Laubender, RP, Bender, R. Estimating adjusted risk difference (RD) and number needed to treat (NNT) measures in the Cox regression model. Stat Med 2010;29:851–9. https://doi.org/10.1002/sim.3793.Search in Google Scholar PubMed

19. Walter, SD. Number needed to treat (NNT): estimation of a measure of clinical benefit. Stat Med 2001;20:3947–62. https://doi.org/10.1002/sim.1173.Search in Google Scholar PubMed

20. Angrist, JD, Imbens, GW. Two-stage least squares estimation of average causal effects in models with variable treatment intensity. J Am Stat Assoc 1995;90:431–42. https://doi.org/10.2307/2291054.Search in Google Scholar

21. Angrist, JD, Imbens, GW, Rubin, DB. Identification of causal effects using instrumental variables. J Am Stat Assoc 1996;91:444–55. https://doi.org/10.2307/2291629.Search in Google Scholar

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

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

24. 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.Search in Google Scholar PubMed

25. 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.Search in Google Scholar PubMed

26. 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.Search in Google Scholar PubMed

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

28. Fedorov, V, Mannino, F, Zhang, R. Consequences of dichotomization. Pharmaceut Stat 2009;8:50–61. https://doi.org/10.1002/pst.331.Search in Google Scholar PubMed

29. Senn, S. Disappointing dichotomies. Pharmaceut Stat 2003;2:239–40. https://doi.org/10.1002/pst.90.Search in Google Scholar

30. Didelez, V, Meng, S, Sheehan, NA. Assumptions of iv methods for observational epidemiology. Stat Sci 2010;25:22–40. https://doi.org/10.1214/09-sts316.Search in Google Scholar

31. Robins, JM. The analysis of randomized and non-randomized aids treatment trials using a new approach to causal inference in longitudinal studies. In: Health service research methodology: a focus on AIDS; 1989:113–59 pp.Search in Google Scholar

32. Robins, JM. Correcting for non-compliance in randomized trials using structural nested mean models. Commun Stat Theory Methods 1994;23:2379–412. https://doi.org/10.1080/03610929408831393.Search in Google Scholar

33. Vansteelandt, S, Goetghebeur, E. Causal inference with generalized structural mean models. J Roy Stat Soc B 2003;65:817–35. https://doi.org/10.1046/j.1369-7412.2003.00417.x.Search in Google Scholar

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

35. Vansteelandt, S, Bowden, J, Babanezhad, M, Goetghebeur, E. On instrumental variables estimation of causal odds ratios. Stat Sci 2011;26:403–22. https://doi.org/10.1214/11-STS360.Search in Google Scholar

36. Robins, J, Rotnitzky, A. Estimation of treatment effects in randomised trials with non-compliance and a dichotomous outcome using structural mean models. Biometrika 2004;91:763–83. https://doi.org/10.1093/biomet/91.4.763.Search in Google Scholar

37. Sjölander, A, Martinussen, T. Instrumental variable estimation with the R package ivtools. Epidemiol Methods 2019;8. https://doi.org/10.1515/em-2018-0024.Search in Google Scholar

38. Borchers, H. pracma: Practical Numerical Math Functions. R package version 2.4.2; 2022. Available from: https://CRAN.R-project.org/package=pracma.Search in Google Scholar

39. Burgess, S, Small, DS, Thompson, SG. A review of instrumental variable estimators for mendelian randomization. Stat Methods Med Res 2017;26:2333–55. https://doi.org/10.1177/0962280215597579.Search in Google Scholar PubMed PubMed Central

40. Skaaby, T, Husemoen, LLN, Pisinger, C, Jørgensen, T, Thuesen, BH, Fenger, M, et al.. Vitamin D status and incident cardiovascular disease and all-cause mortality: a general population study. Endocrine 2013;43:618–25. https://doi.org/10.1007/s12020-012-9805-x.Search in Google Scholar PubMed

41. Smith, GD, Ebrahim, S. Mendelian randomization’: can genetic epidemiology contribute to understanding environmental determinants of disease? Int J Epidemiol 2003;32:1–22. https://doi.org/10.1093/ije/dyg070.Search in Google Scholar PubMed

42. Smith, GD, Ebrahim, S. Mendelian randomization: prospects, potentials, and limitations. Int J Epidemiol 2004;33:30–42. https://doi.org/10.1093/ije/dyh132.Search in Google Scholar PubMed

43. Katan, MB. Apoupoprotein e isoforms, serum cholesterol, and cancer. Lancet 1986;327:507–8. https://doi.org/10.1016/s0140-6736(86)92972-7.Search in Google Scholar PubMed

44. Youngman, L, Keavney, B, Palmer, A, Parish, S, Clark, S, Danesh, J, et al.. Plasma fibrinogen and fibrinogen genotypes in 4685 cases of myocardial infarction and in 6002 controls: test of causality by “mendelian randomisation”. Circulation 2000;102:31–2.Search in Google Scholar

45. Martinussen, T, Sørensen, DN, Vansteelandt, S. Instrumental variables estimation under a structural cox model. Biostatistics 2019;20:65–79. https://doi.org/10.1093/biostatistics/kxx057.Search in Google Scholar PubMed

46. Heaney, RP, Holick, MF. Why the iom recommendations for vitamin d are deficient. J Bone Min Res 2011;26:455–7. https://doi.org/10.1002/jbmr.328.Search in Google Scholar PubMed

47. Staiger, D, JH Stock. Instrumental variables regression with weak instruments. Econometrica 1997;65:557–86. https://doi.org/10.2307/2171753.Search in Google Scholar

48. Menard, S. Six approaches to calculating standardized logistic regression coefficients. Am Stat 2004;58:218–23. https://doi.org/10.1198/000313004x946.Search in Google Scholar

49. Cameron, AC, Windmeijer, FAG. An r-squared measure of goodness of fit for some common nonlinear regression models. J Econom 1997;77:329–42. https://doi.org/10.1016/s0304-4076(96)01818-0.Search in Google Scholar

50. Anderson, TW, Rubin, H. Estimation of the parameters of a single equation in a complete system of stochastic equations. Ann Math Stat 1949;20:46–63. https://doi.org/10.1214/aoms/1177730090.Search in Google Scholar

51. Moreira, MJ. A conditional likelihood ratio test for structural models. Econometrica 2003;71:1027–48. https://doi.org/10.1111/1468-0262.00438.Search in Google Scholar

52. Vancak, V, Sjölander, A. Sensitivity analysis of G-estimators to invalid instrumental variables. Stat Med 2023;42:4257–81. https://doi.org/10.1002/sim.9859.Search in Google Scholar PubMed

Received: 2023-09-03
Accepted: 2024-08-03
Published Online: 2024-08-21

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Perspective
  2. Integrating Hill’s classical considerations with modern causal inference methods in observational studies: a ‘How-Questions’ framework
  3. Research Articles
  4. A compound representation of the multiple treatment propensity score with applications to marginal structural modeling
  5. Non-plug-in estimators could outperform plug-in estimators: a cautionary note and a diagnosis
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  7. Estimation of the number needed to treat, the number needed to be exposed, and the exposure impact number with instrumental variables
  8. Bounds for selection bias using outcome probabilities
https://doi.org/10.1515/em-2023-0034
Keywords for this article
NNT; NNE; EIN; instrumental variables; G-estimation

Articles in the same Issue

  1. Perspective
  2. Integrating Hill’s classical considerations with modern causal inference methods in observational studies: a ‘How-Questions’ framework
  3. Research Articles
  4. A compound representation of the multiple treatment propensity score with applications to marginal structural modeling
  5. Non-plug-in estimators could outperform plug-in estimators: a cautionary note and a diagnosis
  6. Item-level heterogeneous treatment effects of selective serotonin reuptake inhibitors (SSRIs) on depression: implications for inference, generalizability, and identification
  7. Estimation of the number needed to treat, the number needed to be exposed, and the exposure impact number with instrumental variables
  8. Bounds for selection bias using outcome probabilities
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