American Journal of Epidemiology Advance Access originally published online on August 20, 2007
American Journal of Epidemiology 2007 166(7):861-862; doi:10.1093/aje/kwm229
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LETTERS TO THE EDITOR |
RE: "CONFIDENCE INTERVALS FOR BIOMARKER-BASED HUMAN IMMUNODEFICIENCY VIRUS INCIDENCE ESTIMATES AND DIFFERENCES USING PREVALENT DATA"
Department of Epidemiology, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD
(e-mail: scole{at}jhsph.edu)
We recently reported on the estimation of confidence intervals for biomarker-based human immunodeficiency virus (HIV) incidence estimates and differences using prevalent data (1). There is a growing interest in the use of Bayesian methods in epidemiology (2). Our proposed Monte Carlo method (1) can be viewed as a semi-Bayes approach, with prior information on the duration T between positive highly sensitive and less sensitive enzyme immunoassays but no prior information on the incidence of HIV. Here, we describe how our proposed Monte Carlo method can be extended easily to create fully Bayesian credible intervals in place of a confidence interval. A fully Bayesian approach would allow one to utilize prior information regarding the incidence, as well as T.
Extensions of our method to prevalent infection and missing data on the less sensitive enzyme immunoassay are presented elsewhere (3).
Let N be the number of subjects tested, A be the number with recent HIV infection, and B be the number with established HIV infection, as previously described (1). We assume that A follows a Poisson distribution with parameter 
x PPT, where is the incidence and PPT is the pseudo-person-time defined as (N – B)(T/365.25). Similar to Berry et al. (4), if we assume a conjugate Gamma(
, ß) prior distribution for the biomarker-based incidence, given by 
, then the posterior distribution of is distributed as Gamma( + A, ß + PPT).
For example, in 1997, before publication of the example data presented in our paper (1), we found two published reports of the incidence of HIV seroconversion among blood donors in developed countries (5, 6). Pooling these data yields 20 seroconversions in 585,524 person-years or an incidence of 3.42 seroconversions per 105 person-years (95 percent confidence interval: 2.20, 5.29 per 105 person-years). Using this data summary as a conjugate Gamma prior for the biomarker-based incidence with shape parameter = 20 and scale parameter ß = 585,524, we can extend the Monte Carlo approach to be fully Bayesian using the following procedure.
First, draw Tj from a normal distribution with mean T = 129 and a standard error of 10.2 and calculate the pseudo-person-time as PPTj = (N – B)(Tj/365.25). When appropriate, the log-normal, gamma, or chi-squared distributions could be used in place of the normal distribution as the prior for T. Second, draw the posterior incidence j from a gamma distribution with shape parameter 20 + A and scale parameter 585,524 + PPTj. Repeat these steps a larger number of times, say J = 105, and take either the posterior median or the mean of the resulting j as a Bayesian point estimate of incidence, and the 2.5 and 97.5 percentiles of j as the lower and upper limits of a 95 percent Bayesian credible interval, respectively.
For example, using the 1993 American Red Cross data regarding 15 recent infections in 460,385 donors (7), we previously found an incidence of 9.22 cases per 105 person-years with a 95 percent confidence interval of 4.77, 14.62 (1). Applying the above Bayesian method, we find an incidence estimate of 4.63 cases per 105 person-years and a 95 percent Bayesian credible interval of 3.26, 6.37 cases. As expected (2), the resulting Bayesian estimate is an information-weighted average of the prior incidence of 3.42 cases per 105 person-years and the observed incidence of 9.22 cases per 105 person-years.
On a related note, the SAS macro code (SAS Institute, Inc., Cary, North Carolina) given in appendix 2 of our paper (1) has two mistakes. An incorrect variable name "d" should be replaced with the correct variable name "t"; specifically, the relevant two lines should read as follows: pt = nmb*(t/365.25) and pt0 = nmb0*(t/365.25).
In conclusion, if one wishes to incorporate prior information into the point and interval estimation of biomarker-based incidence, then a Bayesian approach is ideal. Otherwise, the Monte Carlo approach previously described (1) provides an exact (within chosen simulation error) 1 – confidence interval.
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ACKNOWLEDGMENTS |
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Conflict of interest: none declared.
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References |
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 TOP References |
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- Cole SR, Chu H, Brookmeyer R. Confidence intervals for biomarker-based human immunodeficiency virus incidence estimates and differences using prevalent data. Am J Epidemiol (2007) 165:94–100.
[Abstract/Free Full Text] - Greenland S. Bayesian perspectives for epidemiological research: I. Foundations and basic methods. Int J Epidemiol (2006) 35:765–75.
[Abstract/Free Full Text] - Chu H, Cole SR. Estimating biomarker-based HIV incidence using prevalent data in high risk groups with missing outcomes. Biom J (2006) 48:772–9.[CrossRef][Web of Science][Medline]
- Berry DA, Wolff MC, Sack D. Decision making during a phase III randomized controlled trial. Control Clin Trials (1994) 15:360–78.[CrossRef][Web of Science][Medline]
- Remis RS, Delage G, Palmer RW. Risk of HIV infection from blood transfusion in Montreal. CMAJ (1997) 157:375–82.[Abstract]
- Whyte GS, Savoia HF. The risk of transmitting HCV, HBV or HIV by blood transfusion in Victoria. Med J Aust (1997) 166:584–6.[Web of Science][Medline]
- Janssen RS, Satten GA, Stramer SL, et al. New testing strategy to detect early HIV-1 infection for use in incidence estimates and for clinical and prevention purposes. JAMA (1998) 280:42–8.
[Abstract/Free Full Text]
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