# Revised estimates of influenza-associated excess mortality, United States, 1995 through 2005

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## Abstract

### Background

Excess mortality due to seasonal influenza is thought to be substantial. However, influenza may often not be recognized as cause of death. Imputation methods are therefore required to assess the public health impact of influenza. The purpose of this study was to obtain estimates of monthly excess mortality due to influenza that are based on an epidemiologically meaningful model.

### Methods and Results

U.S. monthly all-cause mortality, 1995 through 2005, was hierarchically modeled as Poisson variable with a mean that linearly depends both on seasonal covariates and on influenza-certified mortality. It also allowed for overdispersion to account for extra variation that is not captured by the Poisson error. The coefficient associated with influenza-certified mortality was interpreted as ratio of total influenza mortality to influenza-certified mortality. Separate models were fitted for four age categories (<18, 18–49, 50–64, 65+). Bayesian parameter estimation was performed using Markov Chain Monte Carlo methods. For the eleven year study period, a total of 260,814 (95% CI: 201,011–290,556) deaths was attributed to influenza, corresponding to an annual average of 23,710, or 0.91% of all deaths.

### Conclusion

Annual estimates for influenza mortality were highly variable from year to year, but they were systematically lower than previously published estimates. The excellent fit of our model with the data suggest validity of our estimates.

### Keywords

Influenza Respiratory Syncytial Virus Excess Mortality Deviance Information Criterion Excess Death## Background

To estimate excess mortality due to influenza, two fundamental approaches have previously been used. The most popular one is based on Serfling's seasonal regression method [18] and has resulted in numerous estimates of excess mortality due influenza [3, 4, 5, 6, 7, 8, 12, 14, 15, 19]. This periodical regression approach is based on parametric estimation of a sinusoidal "baseline" function that represents mortality in absence of influenza. The difference between the baseline function and the observed numbers of deaths is then interpreted as the number of excess deaths due to influenza. Typically, the baseline mortality function is fitted to weekly or monthly mortality rates or numbers during non-influenza months, using two or more Fourier terms [18]. This approach is intuitively appealing as it captures the strong seasonal periodicity of mortality. However, the particular choice of a parametric baseline function lacks epidemiological justification: Why should the baseline function be sinusoidal rather than of any other periodic form? Depending on the shape of the "true" baseline function, under- or overestimation of excess mortality due to influenza might result. If, for example, the true baseline function is "higher" (i.e. the definite integral of the true function is larger) than the assumed sinusoidal function, then overestimation would result and *vice versa*. Another, potentially more important shortcoming of the periodical regression approach lies in the fact that seasonally correlated causes of mortality, including influenza, are not controlled for, which might lead to confounded estimates of excess mortality.

To avoid this difficulty, one could gauge all-cause mortality with some independent measure of influenza transmission (or mortality). Following this rationale, Thompson et al. [11] estimated excess mortality due to both influenza and respiratory syncytial virus (RSV). They used a generalized linear model (GLM) with a Poisson distribution and a logarithmic link function to model the weekly number of deaths. They also used two Fourier terms in their model, but, in addition, used indicators of influenza and RSV transmission. These indicator variables were defined by the proportions of specimens testing positive for influenza A(H1N1), influenza A(H3N2), influenza B and RSV. Several potential shortcomings of this methods are, however, apparent. First, this model also makes *a priori* assumptions about the baseline mortality function–in this case an exponentiated sinusoidal function. Although this might conceivably be true, there is little empirical evidence to support this assumption. Second, the multiplicative form of the model implies that excess mortality, given a certain amount of influenza activity, depends on the current level of all-cause mortality. Again, this does not appear to be a well-founded assumption. Finally, the proportion of test positive specimens is likely to be a poor measure of excess mortality. While a high proportion of test positive specimens is compatible with high levels of influenza transmission (and excess mortality), this is not necessarily true. The model, however, implies that five hundred influenza positives, obtained from a thousand tests, are associated with less excess mortality than two influenza positives, obtained from three tests. This appears to be an unrealistic assumption. The seasonally changing frequency of influenza testing [20] is, at least partly, due to the seasonally changing incidence of other agents causing influenza-like illness (ILL).

Alternatively, one could postulate that mortality directly attributed to influenza (influenza-certified mortality) represents a certain proportion of all mortality attributable to influenza. This assumption implies that the coefficient associated with influenza-certified mortality represents the ratio of total influenza mortality to influenza certified mortality [17, 21]. Here we use a method for the estimation of influenza excess mortality which is similar to the one recently presented by Schanzer and colleagues [17]: we adopt the proportionality assumption and avoid specific parametric assumptions about the baseline function. In addition, and deviating from the Schanzer model, we allow for random variability of influenza-certified mortality by adopting a hierarchical modeling approach. We present the resulting estimates of U.S. excess mortality due to influenza for the years 1995 to 2005. We compare these to estimates obtained from a Thompson-like model [11], as well as to previously published estimates of influenza-associated excess mortality.

## Methods

### Data

We used Multiple Cause-of-Death Data for the years 1995 to 2005 (Multiple Cause-of-Death Microdata, 1995–2005, National Center for Health Statistics, Hyattsville, Maryland). This dataset is in the public domain and can be electronically downloaded from the web site of the National Bureau of Economic Research http://www.nber.org/data/vital-statistics-mortality-data-multiple-cause-of-death.html. We defined deaths as influenza-certified if influenza was given as underlying cause of death. The corresponding diagnostic code for ICD-9 (1995 to 1998) was 487 and and for ICD-10 (1999 onwards) the code range was J10–J12. Influenza years were defined as lasting from July 1 of one year to June 30 of the following year. We defined four age categories: < 18, 18–49, 50–64 and 65+. Observations with missing age (N = 4,490) were not included in this analysis.

### Statistical model

*Y*

_{ i }is the observed all-cause mortality count during index months

*i*= 1, ..., 132. The variable

*Y*which represents a number and not a rate, is assumed to follow a Poisson distribution with a mean parameter

*θ*

_{ i }. The Poisson mean parameter

*θ*

_{ i }has an identity link and is distributed as Normal with mean

*μ*

_{ i }and variance

*τ*. This parametrization for the Poisson mean

*θ*

_{ i }allows for overdispersion. In the implementation,

*θ*

_{ i }is restricted to positive values to ensure the positivity of the generated samples. The model for

*μ*

_{ i }has two parts. The first part concerns mortality due to non-influenza related causes (baseline mortality) which includes a random intercept ${\lambda}_{{m}_{i}}$ for calendar month

*m*

_{ i }(

*m*

_{ i }= 1, ..., 12) that models the seasonal background mortality, and also includes linear, quadratic and cubic effects for temporal changes due to health, demographic or socioeconomic factors. The variable

*t*

_{ i }(

*t*

_{ i }= 0, ..., 10) indicates the calendar year;

*t*

_{ i }= 0 corresponds to the year 1995. The regression coefficients

*β*

_{1},

*β*

_{2}and

*β*

_{3}measure these changes. The second part of the model for

*μ*

_{ i }concerns mortality due to influenza. The symbol

*γ*

_{ i }is the Poisson parameter from the second level of hierarchy for the observed influenza-certified mortality,

*X*

_{ i }. The parameter

*ϕ*measures the effect of influenza-certified mortality on all cause mortality assuming that all other effects are fixed. This is the parameter of interest. It can also be interpreted as the ratio of total influenza mortality to influenza-certified mortality. Thus, the total excess influenza mortality for index month

*i*, ${X}_{i}^{*}$, is given by

To estimate excess mortality due to influenza, ${\widehat{X}}_{i}^{*}$ is calculated using expression 5, with posterior estimates of *γ*_{ i }and *ϕ*. As total influenza mortality cannot be lower than influenza-certified mortality, the minimum value for the range in the prior distribution for *ϕ* was set to one (see additional file 1).

*β*

_{0}is an intercept,

*β*

_{1}and

*β*

_{2}are defined as above,

*α*

_{1}and

*α*

_{2}represent the parameters associated with the Fourier terms and

*λ*is the natural logarithm of the rate ratio associated with influenza-certified mortality. In contrast to Thompson et al. we used monthly, rather than weekly data and used observed influenza-certified mortality, rather than proportion of positive influenza tests, as indicator for total influenza mortality. For this Thompson-like (TL) model, because of its multiplicative nature, total excess mortality due to influenza, ${X}_{i}^{*}$, given by the expression

To calculate estimated excess mortality due to influenza, all parameters in 8 are replaced by their posterior estimates

### Statistical analysis

The parameters for this hierarchical model were estimated using a Markov chain Monte Carlo (MCMC) algorithm implemented in WinBUGS, version 1.4.1 (Imperial College and Medical Research Council, UK) [22]. Uninformative prior distributions were used (additional file 1). To ensure positivity of all *θ*_{ i }, the normal priors of this parameter were truncated at non-positive values (additional file 1). The empirical posterior distributions of the parameters were obtained from MCMC samples of 30,000, resulting from three chains with 200,000 burn-in iterations and 10,000 samples each. Posterior means and 95% credible intervals (CIs) were calculated for all parameters of interest after ensuring convergence of all model parameters.

The parameters of the TL model could easily be estimated using a GLM procedure in any standard statistical software package. However, to allow for direct comparison of the model fit we used the same estimation procedure as for the current model. The fit of the two age-specific models was compared using the deviance information criterion (DIC) [23]. DIC penalizes the model goodness-of-fit for additional complexity. The complexity is measured by the effective number of parameters.

where *E*(·) and *V*(·) are the operators for the posterior mean and empirical variance, respectively and *e*_{ i }= *Y*_{ i }- *μ*_{ i }. The empirical variance of *e* is computed for each iteration.

## Results

Age category-specific estimates for the detection ratio *ϕ*.

Age Category | Posterior Mean (95% CI) |
---|---|

< 18 | 3.47 (1.61–5.40) |

18–49 | 9.60 (1.152–26.95) |

50–64 | 22.96(15.66–30.67) |

65+ | 21.16 (18.06–24.32) |

Estimated Numbers of Deaths Attributable to Influenza, United States, 1995–2005, according the the current model.

influenza Year | Posterior Mean (95% CI) |
---|---|

1995/96 | 12,067 (8,594–13,898) |

1996/97 | 19,373 (14,750–21,895) |

1997/98 | 36,778 (29,368–41,555) |

1998/99 | 26,666 (20,813–30,381) |

1999/00 | 43,339 (33,886–46,708) |

2000/01 | 5,479 (3,540–6,434) |

2001/02 | 14,995 (11,256–17,405) |

2002/03 | 5,371 (3,523–6,625) |

2003/04 | 49,925 (39,181–52,919) |

2004/05 | 36,726 (28,914–40,881) |

Comparison of the age-specific fit (DIC) of the current with the TL model.

Age Category | Current Model | TL Model |
---|---|---|

< 18 | 2,279.30 | 2,021.39 |

18–49 | 2,235.22 | 6,123.61 |

50–64 | 2,298.53 | 5,656.66 |

65+ | 2,912.66 | 27,298.70 |

## Discussion

Our estimates of excess mortality due to influenza are substantial, especially for the influenza years 1997/98, 1999/2000 and 2003/04, during which influenza A(H3N2) predominated. The lowest estimates were obtained for the years 2000/01 and 2002/03, when influenza A(H1N1) and B viruses predominated. Nevertheless, our estimates are markedly lower than previous estimates. For the year 1995/96, for example, we attributed 12,067 excess deaths to influenza. For the same period, Simonsen et al. [14] estimated that 25,071 deaths were attributable to influenza in ages 65+ alone. Thompson et al. [11] estimated the number of excess deaths during that influenza year at 36,280–more than three times our estimate. The obvious question arises, which of these estimates are closest to the true excess mortality? As pointed out above, the method of Simonsen et al. [14] is problematic for two reasons. First, it does not account for temporal correlation between baseline mortality and influenza excess mortality. The resulting estimates of influenza excess mortality may therefore be confounded. Second, their model makes *a priori* assumptions about the parametric shape of the baseline function; these assumptions may or may not be true. They should, in any event, be validated. The Thompson model [11], which superficially resembles a hybrid between the Simonsen model and the model proposed by Schanzer et al. [17] (or the current model), addresses the issue of temporal confounding by controlling for the proportion of influenza test positives. As pointed out in the Background section, the use of that specific variable to control for influenza mortality may not be appropriate. We compared estimates from the TL model with estimates from the current model. The TL model is based on the Thompson model, but influenza-certified mortality is substituted for proportion positives. Although the resulting estimates were about a sixth higher than our estimates, the seasonal pattern was highly consistent with the pattern seen with the current model. This consistency implies relative robustness of excess deaths estimates to the choice of a specific baseline function. The vast difference between our and Thompson's estimates [11] can therefore not be explained by differences in model structure, nor in the way the baseline function is modeled. They may rather be due to the use of proportion of specimens testing positive to control for influenza mortality.

Schanzer et al. [17], like us, used a Poisson model with linear (rather than logarithmic) link function, to analyze weekly mortality data from Canada. Modeling weekly mortality has the advantage of giving higher temporal resolution to the analysis. On the other hand, deaths associated with, but not attributed to influenza may occur with some delay and may thus be partially decoupled from influenza-certified mortality. However, Schanzer and colleagues did not find an obvious lag between weekly influenza-certified mortality and mortality due to other causes. Future studies will be needed to determine what level of temporal aggregation results in the best estimates.

To take into account random variability in influenza-certified mortality, we used a hierarchical model. While the point estimates for *ϕ* (corresponding to *β*_{3} in [17]) obtained from a GLM are very similar to the ones obtained from the hierarchical model (21.35 and 21.16, respectively, for 65+), the confidence limits are much wider for the latter (95% credible interval 18.06, 24.32 vs. Wald 95% confidence interval 20.91, 21.80). This may even be more pronounced for weekly data, where numbers of influenza-certified deaths are often quite small. To the extent that our hierarchical model takes into account random variability of influenza-certified deaths and thus leads to wider confidence limits around the resulting excess mortality estimates, it is more conservative than non-hierarchical GLM models.

## Conclusion

Previous estimates of excess mortality due to influenza may be biased and inflated. We propose a two-level hierarchical Poisson model where the baseline mortality varies with time. The goodness-of-fit statistic indicates that this model fits the data very well, explaining well above 90% of the observed variation of all-cause mortality during the eleven years study period. The resulting estimates are therefore likely of high validity. Future attempts to quantify the public health burden of influenza should also explore demographic approaches that take into account life expectancy.

## Notes

### Acknowledgements

MMH was partly supported by the National Institutes of Health (NIH grant # 1 R03 CA125828-01). We thank anonymous reviewers whose criticism helped to substantially improve this paper. We would also like to thank Drs. Eric Brenner, MD, and Robert T. Ball, MD, MPH for their insight into the accuracy of the death certificate diagnosis "influenza".

## Supplementary material

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