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Rejoinder: The Authors Respond

Goldstein, Edwarda; Viboud, Cecileb; Lipsitch, Marca,c

doi: 10.1097/EDE.0b013e31826e12c1
Infectious diseases

From the aDepartment of Epidemiology, Center for Communicable Disease Dynamics, Harvard School of Public Health, Boston, MA; bDivision of International Epidemiology and Population Studies, Fogarty International Center, National Institutes of Health, Bethesda, MD; and cDepartment of Immunology and Infectious Diseases, Harvard School of Public Health, Boston, MA.

Editors’ Note: Related articles appear on pages 829 and 839.

Correspondence: Edward Goldstein, Harvard School of Public Health, Department of Epidemiology, 677 Huntington Ave, Kresge Room 506, Boston, MA 02115. E-mail:

We thank Thompson and colleagues1 for their commentary on our new approach to quantify influenza disease burden.2 We agree that there is no gold standard for determining a “best model,” but we believe that there is room for improvement in existing models, and that there are criteria for judging the mechanistic and statistical plausibility of a model.

Our article’s main contribution is to apply a better exposure model for influenza, borrowed from our earlier publication,3 which incorporates data on both the incidence of influenza-like illness in the community and the proportion of respiratory specimens collected from people with flu-like symptoms that test positive for each subtype. Those incidence proxies should, under certain assumptions,3 be linearly related to the incidence of symptomatic infection with the corresponding influenza subtypes, thus allowing one to formulate a linear model relating influenza incidence proxies to mortality.

Previous models, including those proposed by Thompson and colleagues,4,5 have used the proportions of samples from people with flu-like symptoms that test positive for each subtype as proxies for incidence. These proxies are expected to increase less than linearly with subtype-specific incidence of symptomatic infections.2,3 The use of an exponential link between these proxies and the mortality outcome helps counteract this effect to some extent. Nonetheless, the biological plausibility of this mathematical construct linking mortality with influenza proxies has been criticized.6

The eAppendix of our article in this issue of Epidemiology2 includes extensive sensitivity analyses comparing our estimate of flu-associated mortality with an estimate using the percent positive flu-incidence proxies. The comparison is done both for the linear estimation of mortality in terms of the percent positive incidence proxies (section S4 of the eAppendix), and the more traditionally used Poisson regression model with the exponential link between the percent positive incidence proxies and mortality (section S6 of the eAppendix). We have found that in all cases, our estimation method gives a better fit to the data for respiratory, circulatory, and all-cause mortality, which are typical outcomes in influenza disease burden models. Our method also produces positive estimates of influenza-associated mortality for several underlying causes not considered in previous work1,4,5,7: cancer, diabetes, central nervous system disease, renal disease, and chronic liver disease.

Thompson et al1 query the comparison of our results with those previously obtained using other methods and our use of different outcome measures. We have precisely addressed this point in our discussion2 and the eAppendix. Importantly, our estimates rely on mortality outcomes that are traditionally used to measure influenza burden (such as total mortality, or respiratory and cardiac mortality4,5,7). Further, we compared seasonal estimates of influenza-associated mortality rates by our method to those derived by the traditional Poisson regression model driven by virus percent positive (favored by Thompson and colleagues).4,5 The estimated influenza mortality burden during 1997–2007 was similar for the two approaches, with our model showing 11.9 deaths per 100,000 for all-cause mortality, 3.6 for respiratory mortality, and 4.6 for circulatory mortality, and the Thompson et al model showing 13.5, 3.6, and 5.1 for the same (eTable 6). This is encouraging in that it affirms broad agreement of previous models with the new approach proposed in our study,2 while providing a better biological rationale and statistical fit to mortality data than existing models. This broad agreement further suggests that reliable estimates of flu-associated mortality are possible in settings with limited influenza surveillance data, such as communities that test respiratory specimens for presence of influenza virus but lack sentinel surveillance of influenza-like illness.

The incidence proxies used in our article exhibit greater year-to-year variability than the percent positive incidence proxies, resulting in better fits to the mortality data and greater variability in annual estimates of flu-associated mortality. Thus, even if the average annual flu-associated mortality estimates are not very different for our method compared with the Poisson regression method (which reflects an inherent property of the regression process), we believe that yearly estimates obtained using Poisson regression with the percent positive proxies should be less reliable than ours.

We agree with Thompson et al1 that linear regression is not an ideal estimation framework for count data with autocorrelations in the noise. At the same time, other inference mechanisms (eg, through a maximum likelihood framework in a different model) would reflect a measure of belief about the accuracy of such a model, which may suffer from unknown biases (eg, a complex, nonstationary, autocorrelated structure in the noise). We stress that ordinary least squares regression analysis is used for inference because such models produce unbiased estimates. For comparison, we also sought to make inferences using a Poisson regression model with a linear link between influenza incidence and mortality counts. The resulting estimates are very similar to the estimates obtained through our main analysis for several underlying causes (eTable 5), though perhaps differences would be larger in other datasets with lower mortality counts. The autocorrelation in the noise suggests that confidence bounds for the ordinary least squares estimate are more narrow than the true ones. We have modeled the noise as a stationary AR(1) process attempting to give a better, albeit imperfect, presentation of the confidence bounds. Thompson et al’s criticism that we “include an autocorrelation term between subsequent observations” such that “the model absorbs some of the variance likely associated with influenza” perhaps reflects an erroneous impression that we treated the outcome as AR(1), whereas in fact we treated the noise as AR(1), which is not prone to this concern. We note also that a number of previous influenza burden models, including the traditional Poisson model,5 have not attempted to control for autocorrelated noise.

In summary, our article2 and its electronic supplement address the majority of Thompson et al’s concerns1 by explicitly validating influenza mortality estimates against those obtained using previous methods. Those previous methods include use of trigonometric rather than spline functions to model baseline seasonality, alternative influenza exposure proxy measures such as that in prior work by Thompson et al, and Poisson regression with linear and exponential links (eTables 4–6). We agree with their concern that the model is relatively complex given the limited data available for estimation. Although application to additional years’ data will be the best way to test the generality of the model, the consistency of spline baseline shapes across multiple causes of deaths (Fig. 4) argues against over-fitting of these models.

There is certainly room for improvement in all statistical approaches, including this one, and we hope to see finer analyses that incorporate some of the estimation structure we put forward, including the addition of environmental covariates that may independently affect mortality. We also hope that in addition to considerations of statistical correctness (which are vital), such advances will include attention to the mechanistic plausibility of the modeled relationship between influenza cases and outcomes. These should be linear rather than exponential unless there is a reason to believe otherwise. The mortality burden of influenza is still, unfortunately, the subject of debate.8 Refinements of existing models such as proposed in our study help to solidify these estimates,9 which in turn are essential to evaluate the benefits of interventions and inform vaccine recommendations.

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1. Thompson WW, Ridenhour BL, Barile JP, Shay DK. Time-series analyses of count data to estimate the burden of seasonal infectious diseases. Epidemiology. 2012;23:839–842
2. Goldstein E, Viboud C, Charu V, Lipsitch M. Improving the estimation of influenza associated mortality over a seasonal baseline. Epidemiology. 2012;23:829–838 Supplemental Digital Content available at:
3. Goldstein E, Cobey S, Takahashi S, Miller JC, Lipsitch M. Predicting the epidemic sizes of influenza A/H1N1, A/H3N2, and B: a statistical method. PLoS Med. 2011;8:e1001051
4. Thompson WW, Weintraub E, Dhankhar P, et al. Estimates of US influenza-associated deaths made using four different methods. Influenza Other Respi Viruses. 2009;3:37–49
5. Thompson WW, Shay DK, Weintraub E, et al. Mortality associated with influenza and respiratory syncytial virus in the United States. JAMA. 2003;289:179–186
6. Gay NJ, Andrews NJ, Trotter CL, Edmunds WJ. Estimating deaths due to influenza and respiratory syncytial virus. JAMA. 2003;289:2499; author reply 2500–2499; author reply 2502
7. . Estimates of deaths associated with seasonal influenza — United States, 1976–2007. MMWR Morb Mortal Wkly Rep. 2010;59:1057–1062
8. Doshi P. Trends in recorded influenza mortality: United States, 1900–2004. Am J Public Health. 2008;98:939–945
9. Simonsen L, Taylor R, Viboud C, Dushoff J, Miller M. US flu mortality estimates are based on solid science. BMJ. 2006;332:177–178

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