American Journal of Epidemiology Advance Access originally published online on June 19, 2006
American Journal of Epidemiology 2006 164(3):292-293; doi:10.1093/aje/kwj221
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Letter to the Editor |
RE: "BAYESIAN PROJECTIONS: WHAT ARE THE EFFECTS OF EXCLUDING DATA FROM YOUNGER AGE GROUPS?"
1 National Centre for Epidemiology and Population Health, The Australian National University, Canberra, ACT 0200, Australia
2 Institute for Statistical and Epidemiological Cancer Research, Finnish Cancer Registry, FI-00170 Helsinki, Finland
3 Biostatistics Modeling and Methods, Fred Hutchinson Cancer Research Center, Seattle, WA 98109
(e-mail: mark.clements{at}anu.edu.au)
Baker and Bray (1
) continue Bray's thoughtful development of a particular Bayesian age-period-cohort model for cancer rate projections. This model has an elegant formulation and has brought to the forefront a Bayesian interpretation of cancer projections. However, we feel that there are more appropriate models for cancer rate projections. Why? The Bayesian age-period-cohort model suffers from very wide credible intervals for predictions outside observed data (2), which can be attributed to the choice of model priors. Although the second-order autoregressive priors for the age, period, and cohort effects are flexible, alternative priors allow for realistic projections with considerably improved precision.
Predictive performance can be measured by splitting observed data into a training data set and a test data set (3, 4). The model is fitted with the training data set, and then the deviance or another loss criterion is calculated for predictions on the test data. Taking a Bayesian perspective, we can calculate 1) the plug-in deviance, which is the deviance of the mean predicted values, and 2) the predictive deviance, which is the mean deviance over the posterior distribution. Importantly, the plug-in deviance is a measure of bias, while the predictive deviance is a measure of both bias and precision, providing an average loss criterion on the test data.
Clements et al. (3) found that generalized additive models, predicated on a smoothness assumption, with two-dimensional smoothing splines had considerably improved performance for predicting lung cancer mortality rates over the Bayesian age-period-cohort model. Even for the single data set when the Bayesian age-period-cohort model had a smaller plug-in deviance, the predictive deviance was considerably larger for the Bayesian age-period-cohort model. Moreover, a generalized additive model based on univariate smoothing splines for age, period, and cohort had similar plug-in deviance to that of the Bayesian age-period-cohort model, while the generalized additive model had a considerably smaller predictive deviance. Other smooth priors that deserve attention include the one- and two-dimensional penalized splines, which can be fitted as Bayesian generalized linear mixed models (5). The smoothness assumption is likely to be valid for most cancers, aside from sites where there has been a rapid introduction of either screening or treatment therapies.
Alternatively, probably the best-validated, age-period-cohort models are those developed by the Cancer Registry of Norway (6). For an approximate Bayesian interpretation of these models, resampling from the posterior distribution can proceed using the bootstrap.
Bray (7) has provided some validation using a small subset of alternative models, although she did not use the predictive deviance or any related measure incorporating precision. The development of a benchmark for rate projections methods would allow for a more comprehensive comparison between methods, to assist in selecting a model for performing cancer rate projections.
ACKNOWLEDGMENTS
Conflict of interest: none declared.
References
- Baker A, Bray I. Bayesian projections: what are the effects of excluding data from younger age groups? Am J Epidemiol 2005;162:798–805.
[Abstract/Free Full Text] - Bashir SA, Estève J. Projecting cancer incidence and mortality using Bayesian age-period-cohort models. J Epidemiol Biostat 2001;6:287–96.[CrossRef][Medline]
- Clements MS, Armstrong BK, Moolgavkar SH. Lung cancer rate predictions using generalized additive models. Biostatistics 2005;6:576–89.
[Abstract/Free Full Text] - Hastie T, Tibshirani R, Friedman J. The elements of statistical learning: data mining, inference, and prediction. New York, NY: Springer, 2001.
- Ruppert D, Wand MP, Carroll RJ. Semiparametric regression. Cambridge, United Kingdom: Cambridge University Press, 2003.
- Møller B, Fekjaer H, Hakulinen T, et al. Prediction of cancer incidence in the Nordic countries: empirical comparison of different approaches. Stat Med 2003;22:2751–66.[CrossRef][Web of Science][Medline]
- Bray I. Application of Markov chain Monte Carlo methods to projecting cancer incidence and mortality. Appl Stat 2002;51:151–64.
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