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Hazard Ratio Bias in Cohort Studies

Lin, Jia-Chun; Lee, Wen-Chung

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doi: 10.1097/EDE.0b013e31829f65a7

To the Editor:

Hernán1 has pointed out that the hazard (rate) ratio has a built-in selection bias in the follow-up of a randomized experiment. Although comparability between the treatment and the control groups is ensured at the study outset, with a follow-up over time, the two arms of a randomized experiment may diverge because of differential depletion of susceptible persons, thus creating bias. We report here that a cohort study that follows a homogeneous population is still prone to selection bias, even though unmeasured confounding and differential susceptibility is not an issue.

Assume that the natural history of disease is divided into a healthy state (H), a preclinical state (P), and disease diagnosis (D) and that the study population is stable, with prevalence (of P and D above) and incidence rates (I1 and I2 below) that do not change with time.2 Let I1 denote the transition rate from H to P, and I2, the rate from P to D. eAppendix 1 ( shows that (Number of H subjects)/ (Number of P subjects)

. Therefore, a cohort study (sample size = N, after excluding the diseased subjects, D) in the population is expected to have the following numbers of subjects at cohort inception (t = 0):


, respectively. Afterward, a researcher will observe a total person-time from t to (

) as

, a total number of new events as

, and a disease incidence at time t as

Assume all above apply to


of a time-invariant dichotomous exposure. The exposure can affect I1 (

, a time-invariant parameter of interest) but not


). At time t, the observed incidence rate ratio is

, and the relative bias,

. (The nature of bias here differs from infected cohorts.3,4) An R code is developed to perform simulations (eAppendix 2,, taking into account that a researcher may try to reduce selection bias by excluding


) proportion of P subjects at t = 0, and/or performing active follow-up so as to raise I2 to





; unit: 1/year),


(Figure A), the bias is negative if

and positive if

, that is, toward the null (eAppendix 3, When

, the initial bias is

. It takes 9 years of follow-up for the bias to approach zero. If

, the initial bias is

with 6(3) years for it to approach zero. If ratios of average incidence rates

are calculated, the time course of the bias can be even more protracted (eAppendix 4,

Selection bias when
(red lines:
blue lines:
solid lines:
; dash lines:
; dotted lines: 
). A, k = 0, r = 1; (B)k = 0.5, r = 1; (C) k = 1, r = 1; (D) k = 0, r = 2; (E) k = 0.5, r = 2; (F) k = 1, r = 2; (G) k = 0,r = 5; (H) k = 0.5, r = 5; (I) k = 1, r = 5; (J) k = 0, r = 10; (K) k = 0.5, r = 10; (L) k = 1, r = 10.

When k = 0.5 (B), the bias is smaller but still requires long-term follow-up to approach zero. When k = 1, the initial bias is significantly reduced (C). However, after a period of follow-up, bias increases instead and then decreases. Under active follow-up (r = 2), the follow-up period it takes to exclude the bias is reduced (D vs. A). If the follow-up is even more active, it takes an even shorter time to exclude the bias (

in G;

in J). Similar findings can be found for

(B, E, H, K) and

(C, F, I, L). For smaller

(eFigures,, the initial bias is smaller but the time course of the bias is the same as in the Figure.

The initial bias is dependent on

. For total cancer,5

and the bias is merely

. Prevalent diseases have higher values of


for hypertension; ∼

for diabetes)6 and the magnitude of the biases cannot be taken lightly (

for hypertension;

for diabetes). The follow-up period required for the exclusion of bias depends on


). This means 1.5 years for a study on hepatitis B infection (

year)7 and 24 years for a study on diabetes (

years).8 Active follow-up can greatly shorten the period needed to exclude the bias.

Jia-Chun Lin

Wen-Chung Lee

Research Center for Genes, Environment and Human Health

Institute of Epidemiology and Preventive Medicine

College of Public Health

National Taiwan University

Taipei, Taiwan

[email protected]


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2. Rothman KJ, Greenland SRothman KJ, Greenland S, Lash TL. Measures of occurrence. Modern Epidemiology. 20083rd ed Philadelphia, PA Lippincott Williams and Wilkins:32–50
3. Törner A, Duberg AS, Dickman P, Svensson A. A proposed method to adjust for selection bias in cohort studies. Am J Epidemiol. 2010;171:602–608
4. Törner A, Dickman P, Duberg AS, et al. A method to visualize and adjust for selection bias in prevalent cohort studies. Am J Epidemiol. 2011;174:969–976
5. Health Statistics, Bureau of Health Promotion, Taiwan. Available at, Accessed November 1, 2012
6. Health Statistics, Bureau of Health Promotion, Taiwan. Available at: Accessed 1 November 2012
7. Sorrell MF, Belongia EA, Costa J, et al. National Institutes of Health consensus development conference statement: management of hepatitis B. Hepatology. 2009;49(5 suppl):S4–S12
8. Kuo HS, Chang HJ, Chou P, Teng L, Chen TH. A Markov chain model to assess the efficacy of screening for non-insulin dependent diabetes mellitus (NIDDM). Int J Epidemiol. 1999;28:233–240

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