Startseite A comparison of approaches for estimating combined population attributable risks (PARs) for multiple risk factors
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A comparison of approaches for estimating combined population attributable risks (PARs) for multiple risk factors

  • Yibing Ruan , Stephen D. Walter , Christine M. Friedenreich , Darren R. Brenner ORCID logo EMAIL logo und on behalf of the ComPARe Study Team
Veröffentlicht/Copyright: 9. Oktober 2020
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Epidemiologic Methods
Aus der Zeitschrift Epidemiologic Methods Band 9 Heft 1

Abstract

Objectives

The methods to estimate the population attributable risk (PAR) of a single risk factor or the combined PAR of multiple risk factors have been extensively studied and well developed. Ideally, the estimation of combined PAR of multiple risk factors should be based on large cohort studies, which account for both the joint distributions of risk exposures and for their interactions. However, because such individual-level data are often lacking, many studies estimate the combined PAR using a comparative risk assessment framework. It involves estimating PAR of each risk factor based on its prevalence and relative risk, and then combining the individual PARs using an approach that relies on two key assumptions: that the distributions of exposures to the risk factors are independent and that the relative risks are multiplicative. While such assumptions rarely hold true in practice, no studies have investigated the magnitude of bias incurred if the assumptions are violated.

Methods

Using simulation-based models, we compared the combined PARs obtained with this approach to the more accurate estimates of PARs that are available when the joint distributions of exposures and risks can be established.

Results

We show that the assumptions of exposure independence and risk multiplicativity are sufficient but not necessary for the combined PAR to be unbiased. In the simplest situation of two risk factors, the bias of this approach is a function of the strength of association and the magnitude of risk interaction, for any values of exposure prevalence and their associated risks. In some cases, the combined PAR can be strongly under- or over-estimated, even if the two assumptions are only slightly violated.

Conclusions

We encourage researchers to quantify likely biases in their use of the M–S method, and here, we provided level plots and R code to assist.

Keywords: bias analysis; population attributable risk; risk interaction
Corresponding author: Darren R. Brenner, Department of Cancer Epidemiology and Prevention Research, CancerControl Alberta, Alberta Health Services, Holy Cross Centre – Room 513C, Box ACB, 2210-2nd, St. SW, Calgary, AB, Canada, T2S 3C3; and Departments of Oncology and Community Health Sciences, Cumming School of Medicine, University of Calgary, Calgary, AB, Canada, Phone: +1 403 698 8178, E-mail:
Additional Members of the ComPARe Study Team: Eduardo Franco, Gerald Bronfman Department of Oncology, Division of Cancer Epidemiology, McGill University, Montréal, Québec, Canada; Will King, Department of Public Health Sciences, Queen’s University, Kingston, Ontario, Canada; Paul Demers, Occupational Cancer Research Centre, Toronto, Ontario, Canada; Paul Villeneuve, Department of Health Sciences, Carleton University, Ottawa, Ontario, Canada; Prithwish De, Cancer Care Ontario, Toronto, Ontario, Canada; Leah Smith, Canadian Cancer Society, Toronto, Ontario, Canada; Abbey Poirier, Department of Cancer Epidemiology and Prevention Research, CancerControl Alberta, Alberta Health Services, Calgary, Alberta, Canada; Elizabeth Holmes, Canadian Cancer Society, Toronto, Ontario, Canada; Dylan O’Sullivan, Department of Public Health Sciences, Queen’s University, Kingston, Ontario, Canada; Karena Volesky, Gerald Bronfman Department of Oncology, Division of Cancer Epidemiology, McGill University, Montréal, Québec, Canada; Zeinab El-Masri, Cancer Care Ontario, Toronto, Ontario, Canada; Robert Nuttall, Health Quality Ontario, Toronto, Ontario, Canada; Mariam El-Zein, Gerald Bronfman Department of Oncology, Division of Cancer Epidemiology, McGill University, Montréal, Québec, Canada; Tasha Narain, Department of Public Health Sciences, Queen’s University, Kingston, Ontario, Canada; Priyanka Gogna, Department of Public Health Sciences, Queen’s University, Kingston, Ontario, Canada.

Funding source: Canadian Cancer Society

Award Identifier / Grant number: 703106

  1. Research funding: This research is supported by the Canadian Cancer Society Partner Prevention Research Grant (grant #703106).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission. All authors contributed equally to this work.

  3. Competing interests: Authors state no conflict of interest.

Appendix

Proof 1:

exposure independence and risk multiplicativity are sufficient conditions for Miettinen–Steenland approach to be unbiased.

Assume a disease associated with K risk factors is present in a population of N individuals. The disease incidence among non-exposed (i.e., not exposed to any of the K risk factors) is I0. If all K risk factors are dichotomous, then the N individuals can be placed in 2K mutually exclusive strata formed by cross-classifying the risk factors. For risk factor i (i=1, 2,…, K), the marginal prevalence is pi and the unconfounded relative risk is ri (i.e., ri=IiI0 , in which Ii is the disease incidence among those exposed and only exposed to risk factor i). Depending on the presence of risk factor i in stratum j, the prevalence and relative risk for factor i can be defined as

Pij={pi, if  factor i is present in stratum j1−pi, if  factor  i is not present in stratum j.

and

RRij={ri,  if  factor i is present in stratum j1,  if  factor i is not  present  in  stratum  j.

Given the independent exposure assumption, the proportion of population in stratum j is the product of all Pij, and the number of individuals in stratum j is Nj=N∏i=1KPij.

Based on the multiplicative risk assumption, the disease incidence in stratum j is the product of I0 and the relative risks of exposures present in the stratum: Ij=I0∏i=1KRRij.

The true combined PAR is the proportion of the excess number of cases from all strata compared to all cases of disease:

PARcombined=∑(IjNj−I0Nj)∑IjNj=1−I0∑Nj∑IjNj=1−I0N∑IjNj=1−1∑j=12K∏i=1KPij∏i=1KRRij

Considering any exposure i (for example, let us consider exposure 1, i.e., i=1), the 2K strata can be split into two 2K−1 strata with respect to the presence of exposure 1. The two 2K−1 strata have identical combinations of all other exposures except exposure 1. In the 2K−1 strata that exposure 1 is present, all strata have a common factor of p1r1. In the 2K−1 strata that exposure 1 is absent, all strata have a common factor of 1 × (1 − p1). Therefore,

PARcombined=1−1∑j=12K∏i=1KPij∏i=1KRRij=1−1p1r1∑∏i=2KPij∏i=2KRRij+(1−p1)∑∏i=2KPij∏i=2KRRij=1−1(p1r1+1−p1)∑∏i=2KPij∏i=2KRRij=1−11+p1(r1−1)⋅1∑∏i=2KPij∏i=2KRRij

Using the same approach, each of K the risk factors can be extracted, which results in

PARcombined=1−∏i=1K11+pi(ri−1)

The independent exposure and multiplicative risk assumptions also indicate that there are no confounding or effect measure modifications among the K risk factors, in which case the adjusted RR would be equal to unconfounded RR. Therefore,

11+pi(ri−1)=11+pi(riadj−1)=1−pi(riadj−1)1+pi(riadj−1)=1−PARi

and therefore

PARcombined=1−∏i=1K(1−PARi)=PARM−S

Proof 2:

f(RRa, RRb, Pa, Pb, Y, Z |ra, rb, pa, pb)=0 is a quadratic equation ofY

Because

f(RRa,RRb,Pa,Pb,Y,Z)=1(1−Pa−Pb+PaPbZ)+RRa(Pa−PaPbZ)+RRb(Pb−PaPbZ)+RRaRRbPaPbYZ−1(1+Pa(RRA−1))×(1+Pb(RRB−1))

For f(RRa, RRb, Pa, Pb, Y, Z|ra, rb, pa, pb)=0, it is equivalent that

(1+pa(rrA−1))×(1+pb(rrB−1))−(1−pa−pb+papbZ)+ra(pa−papbZ)+rb(pb−papbZ)+rarbpapbYZ=0

Define the following terms:

Za=pa−papbZ;Zb=pb−papbZ;Z0=1−pa−pb+papbZ;p¯a=1−pa;p¯b=1−pb.

The Mantel–Haenszel adjusted RR can be rearranged as:

rrA=raZaZ0+rarbpap¯bZbZYZaZ0+rbpap¯bZbZ

And

1+pa(rrA−1)=(para+p¯a)ZaZ0+rbpap¯ap¯bZbZ+rarbpapap¯bZbZYZaZ0+rbpap¯bZbZ

Similarly

1+pb(rrB−1)=(pbrb+p¯b)ZbZ0+rapbp¯ap¯bZaZ+rarbpbpbp¯aZaZYZbZ0+rapbp¯aZaZ

Therefore, the equation can be rearranged as:

[(para+p¯a)ZaZ0+rbpap¯ap¯bZbZ+rarbpapap¯bZbZY]×[(pbrb+p¯b)ZbZ0+rapbp¯ap¯bZaZ+rarbpbpbp¯aZaZY]−(ZaZ0+rbpap¯bZbZ)×(ZbZ0+rapbp¯aZaZ)×(Z0+raZa+rbZb+rarbpapbZY)=0

This equation is a quadratic equation of Y in the form of A⋅Y2 + B⋅Y + C=0, in which

A=pa2ra2pb2rb2p¯ap¯bZaZbZ2B=rarb[pb2p¯a(para+p¯a−paraZ)Za2Z0Z+pa2p¯b(pbrb+p¯b−pbrbZ)Zb2Z0Z+pap¯apbp¯b(pap¯bra+pbp¯arb−rarbpapbZ)Z2ZaZb−papbZ⋅Z02ZaZb]C=pbrap¯a[p¯b(para+p¯a)−(Z0+raZa+rbZb)]Za2Z0Z+parbp¯b[p¯a(pbrb+p¯b)−(Z0+raZa+rbZb)]Zb2Z0Z+rarbpapbp¯ap¯b[p¯ap¯b−(Z0+raZa+rbZb)]Z2ZaZb+[(para+p¯a)(pbrb+p¯b)−(Z0+raZa+rbZb)]Z02ZaZb

Potentially, Y has two solutions (Y=−B±B2−4AC2A), representing two functions of Z, which are restricted on the plausible range of Z.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/em-2019-0021).

Received: 2019-11-28
Accepted: 2020-05-14
Published Online: 2020-10-09

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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  1. Editorial
  2. The mean prevalence
  3. Research Articles
  4. Heterogeneous indirect effects for multiple mediators using interventional effect models
  5. Sleep habits and their association with daytime sleepiness among medical students of Tanta University, Egypt
  6. Population attributable fractions for continuously distributed exposures
  7. A real-time search strategy for finding urban disease vector infestations
  8. Disease mapping models for data with weak spatial dependence or spatial discontinuities
  9. A comparison of cause-specific and competing risk models to assess risk factors for dementia
  10. A simple index of prediction accuracy in multiple regression analysis
  11. A comparison of approaches for estimating combined population attributable risks (PARs) for multiple risk factors
  12. Posterior predictive treatment assignment methods for causal inference in the context of time-varying treatments
  13. Random effects tumour growth models for identifying image markers of mammography screening sensitivity
  14. Extrapolating sparse gold standard cause of death designations to characterize broader catchment areas
  15. Extending balance assessment for the generalized propensity score under multiple imputation
  16. Regression analysis of unmeasured confounding
  17. The Use of Logic Regression in Epidemiologic Studies to Investigate Multiple Binary Exposures: An Example of Occupation History and Amyotrophic Lateral Sclerosis
  18. Meeting the Assumptions of Inverse-Intensity Weighting for Longitudinal Data Subject to Irregular Follow-Up: Suggestions for the Design and Analysis of Clinic-Based Cohort Studies
https://doi.org/10.1515/em-2019-0021
Schlagwörter für diesen Artikel
bias analysis; population attributable risk; risk interaction

Artikel in diesem Heft

  1. Editorial
  2. The mean prevalence
  3. Research Articles
  4. Heterogeneous indirect effects for multiple mediators using interventional effect models
  5. Sleep habits and their association with daytime sleepiness among medical students of Tanta University, Egypt
  6. Population attributable fractions for continuously distributed exposures
  7. A real-time search strategy for finding urban disease vector infestations
  8. Disease mapping models for data with weak spatial dependence or spatial discontinuities
  9. A comparison of cause-specific and competing risk models to assess risk factors for dementia
  10. A simple index of prediction accuracy in multiple regression analysis
  11. A comparison of approaches for estimating combined population attributable risks (PARs) for multiple risk factors
  12. Posterior predictive treatment assignment methods for causal inference in the context of time-varying treatments
  13. Random effects tumour growth models for identifying image markers of mammography screening sensitivity
  14. Extrapolating sparse gold standard cause of death designations to characterize broader catchment areas
  15. Extending balance assessment for the generalized propensity score under multiple imputation
  16. Regression analysis of unmeasured confounding
  17. The Use of Logic Regression in Epidemiologic Studies to Investigate Multiple Binary Exposures: An Example of Occupation History and Amyotrophic Lateral Sclerosis
  18. Meeting the Assumptions of Inverse-Intensity Weighting for Longitudinal Data Subject to Irregular Follow-Up: Suggestions for the Design and Analysis of Clinic-Based Cohort Studies
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