Mixed models,linear dependency,and identification in age‐period‐cohort models

This paper examines the identification problem in age‐period‐cohort models that use either linear or categorically coded ages, periods, and cohorts or combinations of these parameterizations. These models are not identified using the traditional fixed effect regression model approach because of a linear dependency between the ages, periods, and cohorts. However, these models can be identified if the researcher introduces a single just identifying constraint on the model coefficients. The problem with such constraints is that the results can differ substantially depending on the constraint chosen. Somewhat surprisingly, age‐period‐cohort models that specify one or more of ages and/or periods and/or cohorts as random effects are identified. This is the case without introducing an additional constraint. I label this identification as statistical model identification and show how statistical model identification comes about in mixed models and why which effects are treated as fixed and which are treated as random can substantially change the estimates of the age, period, and cohort effects. Copyright © 2017 John Wiley & Sons, Ltd.     

Proportional hazards models and age–period–cohort analysis of cancer rates

Age–period–cohort (APC) analysis is widely used in cancer epidemiology to model trends in cancer rates. We develop methods for comparative APC analysis of two independent cause‐specific hazard rates assuming that an APC model holds for each one. We construct linear hypothesis tests to determine whether the two hazards are absolutely proportional or proportional after stratification by cohort, period, or age. When a given proportional hazards model appears adequate, we derive simple expressions for the relative hazards using identifiable APC parameters. To demonstrate the utility of these new methods, we analyze cancer incidence rates in the United States in blacks versus whites for selected cancers, using data from the National Cancer Institute's Surveillance, Epidemiology, and End Results Program. The examples illustrate that each type of proportionality may be encountered in practice. Published in 2010 by John Wiley & Sons, Ltd.     

Bayesian approach to cancer‐trend analysis using age‐stratified Poisson regression models

Annual Percentage Change (APC) summarizes trends in age‐adjusted cancer rates over short time‐intervals. This measure implicitly assumes linearity of the log‐rates over the intervals in question, which may not be valid, especially for relatively longer time‐intervals. An alternative is the Average Annual Percentage Change (AAPC), which computes a weighted average of APC values over intervals where log‐rates are piece‐wise linear. In this article, we propose a Bayesian approach to calculating APC and AAPC values from age‐adjusted cancer rate data. The procedure involves modeling the corresponding counts using age‐specific Poisson regression models with a log‐link function that contains unknown joinpoints. The slope‐changes at the joinpoints are assumed to have a mixture distribution with point mass at zero and the joinpoints are assumed to be uniformly distributed subject to order‐restrictions. Additionally, the age‐specific intercept parameters are modeled nonparametrically using a Dirichlet process prior. The proposed method can be used to construct Bayesian credible intervals for AAPC using age‐adjusted mortality rates. This provides a significant improvement over the currently available frequentist method, where variance calculations are done conditional on the joinpoint locations. Simulation studies are used to demonstrate the success of the method in capturing trend‐changes. Finally, the proposed method is illustrated using data on prostate cancer incidence. Copyright © 2010 John Wiley & Sons, Ltd.     

Age‐period‐cohort models using smoothing splines: a generalized additive model approach

Age‐period‐cohort (APC) models are used to analyze temporal trends in disease or mortality rates, dealing with linear dependency among associated effects of age, period, and cohort. However, the nature of sparseness in such data has severely limited the use of APC models. To deal with these practical limitations and issues, we advocate cubic smoothing splines. We show that the methods of estimable functions proposed in the framework of generalized linear models can still be considered to solve the non‐identifiability problem when the model fitting is within the framework of generalized additive models with cubic smoothing splines. Through simulation studies, we evaluate the performance of the cubic smoothing splines in terms of the mean squared errors of estimable functions. Our results support the use of cubic smoothing splines for APC modeling with sparse but unaggregated data from a Lexis diagram. Copyright © 2013 John Wiley & Sons, Ltd.     

Age‐Period‐Cohort approaches to back‐calculation of cancer incidence rate

A compartment model for cancer incidence and mortality is developed in which healthy subjects may develop cancer and subsequently die of cancer or another cause. In order to adequately represent the experience of a defined population, it is also necessary to allow for subjects who are diagnosed at death, as well as subjects who migrate and are subsequently lost to follow‐up. Expressions are derived for the number of cancer deaths as a function of the number of incidence cases and vice versa, which allows for the use of mortality statistics to obtain estimates of incidence using survival information. In addition, the model can be used to obtain estimates of cancer prevalence, which is useful for health care planning. The method is illustrated using data on lung cancer among males in Connecticut. Copyright © 2015 John Wiley & Sons, Ltd.     

A Bayesian generalized age–period–cohort power model for cancer projections

Age–period–cohort (APC) models are the state of art in cancer projections, assessing past and recent trends and extrapolating mortality or incidence data into the future. Nordpred is a well‐established software, assuming a Poisson distribution for the counts and a log‐link or power‐link function with fixed power; however, its predictive performance is poor for sparse data. Bayesian models with log‐link function have been applied, but they can lead to extreme estimates. In this paper, we address criticisms of the aforementioned models by providing Bayesian formulations based on a power‐link and develop a generalized APC power‐link model, which assumes a random rather than fixed power parameter. In addition, a power model with a fixed power parameter of five was formulated in the Bayesian framework. The predictive performance of the new models was evaluated on Swiss lung cancer mortality data using model‐based estimates of observed periods. Results indicated that the generalized APC power‐link model provides best estimates for male and female lung cancer mortality. The gender‐specific models were further applied to project lung cancer mortality in Switzerland during the periods 2009–2013 and 2014–2018. Copyright © 2014 John Wiley & Sons, Ltd.     

Approaches to fitting age-period-cohort models with unequal intervals

Age-period-cohort models have provided useful insights into the analysis of time trends for disease rates, in spite of the well known identifiability problem. Unique parameter estimates that avoid arbitrary constraints are provided by estimable functions of the parameter estimates. For data that are generated using equal interval widths for age and period, the identifiability issue may be expressed in terms of the age, period and cohort slopes. However, when the interval widths are not the same for age and period, additional identifiability problems arise. These may be represented in terms of macro-trends, which have the identical identifiability problem seen in the equal interval case, and micro-trends, which are the source of the additional problems. A framework for testing estimability is presented, and a variety of potentially interesting functions of the parameters considered. Unlike the equal interval case, drift is not estimable for unequal intervals, but local drift may be. In addition, the available functions for forecasting are much more restrictive in the latter case. This estimability problem induces cyclical patterns in the estimates of trend as is demonstrated using data on leukaemia in Connecticut males, but this can be avoided through the use of smoothing splines. These methods of are illustrated for three-year period and five-year age intervals using data on lung cancer mortality in Californian women.     

Bayesian age–period–cohort models with versatile interactions and long‐term predictions: mortality and population in Finland 1878–2050

Age–period–cohort (APC) models are widely used for studying time trends of disease incidence or mortality. Model identifiability has become less of a problem with Bayesian APC models. These models are usually based on random walk (RW1, RW2) smoothing priors. For long and complex time series and for long predicted periods, these models as such may not be adequate. We present two extensions for the APC models. First, we introduce flexible interactions between the age, period and cohort effects based on a two‐dimensional conditional autoregressive smoothing prior on the age/period plane. Our second extension uses autoregressive integrated (ARI) models to provide reasonable long‐term predictions. To illustrate the utility of our model framework, we provide stochastic predictions for the Finnish male and female population, in 2010–2050. For that, we first study and forecast all‐cause male and female mortality in Finland, 1878–2050, showing that using an interaction term is needed for fitting and interpreting the observed data. We then provide population predictions using a cohort component model, which also requires predictions for fertility and migration. As our main conclusion, ARI models have better properties for predictions than the simple RW models do, but mixing these prediction models with RW1 or RW2 smoothing priors for observed periods leads to a model that is not fully consistent. Further research with our model framework will concentrate on using a more consistent model for smoothing and prediction, such as autoregressive integrated moving average models with state‐space methods or Gaussian process priors. Copyright © 2013 John Wiley & Sons, Ltd.     

Predicting county‐level cancer incidence rates and counts in the USA

Many countries, including the USA, publish predicted numbers of cancer incidence and death in current and future years for the whole country. These predictions provide important information on the cancer burden for cancer control planners, policymakers and the general public. Based on evidence from several empirical studies, the joinpoint (segmented‐line linear regression) model (JPM) has been adopted by the American Cancer Society to estimate the number of new cancer cases in the USA and in individual states since 2007. Recently, cancer incidence in smaller geographic regions such as counties, and local policy makers are increasingly interested with Federal Information Processing Standard code regions. The natural extension is to directly apply the JPM to county‐level cancer incidence data. The direct application has several drawbacks and its performance has not been evaluated. To address the concerns, we developed a spatial random‐effects JPM for county‐level cancer incidence data. The proposed model was used to predict both cancer incidence rates and counts at the county level. The standard JPM and the proposed method were compared through a validation study. The proposed method outperformed the standard JPM for almost all cancer sites, especially for moderate or rare cancer sites and for counties with small population sizes. As an application, we predicted county‐level prostate cancer incidence rates and counts for the year 2011 in Connecticut. Published 2013. This article is a US Government work and is in the public domain in the USA.     

A case‐cohort approach for multi‐state models in hospital epidemiology

Analysing the determinants and consequences of hospital‐acquired infections involves the evaluation of large cohorts. Infected patients in the cohort are often rare for specific pathogens, because most of the patients admitted to the hospital are discharged or die without such an infection. Death and discharge are competing events to acquiring an infection, because these individuals are no longer at risk of getting a hospital‐acquired infection. Therefore, the data is best analysed with an extended survival model – the extended illness‐death model. A common problem in cohort studies is the costly collection of covariate values. In order to provide efficient use of data from infected as well as uninfected patients, we propose a tailored case‐cohort approach for the extended illness‐death model. The basic idea of the case‐cohort design is to only use a random sample of the full cohort, referred to as subcohort, and all cases, namely the infected patients. Thus, covariate values are only obtained for a small part of the full cohort. The method is based on existing and established methods and is used to perform regression analysis in adapted Cox proportional hazards models. We propose estimation of all cause‐specific cumulative hazards and transition probabilities in an extended illness‐death model based on case‐cohort sampling. As an example, we apply the methodology to infection with a specific pathogen using a large cohort from Spanish hospital data. The obtained results of the case‐cohort design are compared with the results in the full cohort to investigate the performance of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Incidence estimation using a single cross‐sectional age‐specific prevalence survey with differential mortality

Here, we present a method for incidence estimation of a curable, non‐recurring disease when data from a single cross‐sectional survey are used together with population‐level mortality rates and an assumption of differential mortality of diseased versus non‐diseased individuals. The motivating example is cataract, and the VISION2020 goal to eliminate avoidable blindness globally by 2020. Reliable estimates of current and future cataract disease burden are required to predict how many surgeries would need to be performed to meet the VISION2020 goals. However, incidence estimates, needed to derive future burden, are not as easily available, due to the cost of conducting cohort studies. Disease is defined at the person‐level in accordance with the WHO person‐level definition of blindness. An extension of the standard time homogeneous illness–death model to a four‐state model is described, which allows the disease to be cured, whereby surgery is performed on at least one diseased eye. Incidence is estimated, and the four‐state model is used to predict disease burden assuming different surgical strategies whilst accounting for the competing risk of death. The method is applied to data from approximately 10 000 people from a survey of visual impairment in Nigeria. Copyright © 2013 John Wiley & Sons, Ltd.     

Prediction of cancer incidence in the Nordic countries: empirical comparison of different approaches

Prediction of the future number of cancer cases is of great interest to society. The classical approach is to use the age-period-cohort model for making cancer incidence predictions. We made an empirical comparison of different versions of this model, using data from cancer registries in the Nordic countries for the period 1958-1997. We have applied 15 different methods to 20 sites for each sex in Denmark, Finland, Norway and Sweden. Median absolute value of the relative difference between observed and predicted numbers of cases for these 160 combinations of site, sex and country was calculated. The medians varied between 10.4 per cent and 15.3 per cent in predictions 10 years ahead, and between 15.1 per cent and 32.0 per cent for 20 year predictions. We have four main conclusions: (i) projecting current trends worked better than assuming that future rates are equal to present rates; (ii) the method based on the multiplicative APC model often overestimated the number of cancer cases due to its exponential growth over time, but using a power function to level off this growth improved the predictions; (iii) projecting only half of the trend after the first 10 years also gave better long-term predictions; (iv) methods that emphasize trends in the last decade seem to perform better than those that include earlier time trends.     

Relative risk and excess rate models in exposure-response analysis

The cumulative dose of an environmental exposure and age are usually correlated. It is shown that if an excess rate model is the true relationship, a relative risk model based on standardized mortality ratios (SMR) may show a biased and even contradictory dose-response relationship if the incidence rate in the reference population increases rapidly by age. The choice of model should be based on biological plausibility rather than on available software, and in many cases it may be reasonable to analyze the data both according to a relative and to an excess risk model as the prior knowledge rarely favors one of these models. In a review of a volume of this journal, there were eight cohort studies analyzing dose-response relationship by comparing SMRs without reporting or analyzing excess rates. Thus, there seems to be some lack of awareness of the possible bias in analyzing exposure-disease associations by SMRs. Am. J. Ind. Med. 31:399–402, 1997. © 1997 Wiley-Liss, Inc.     

Meta‐analysis of pairs of survival curves under heterogeneity: A Poisson correlated gamma‐frailty approach

We address the problem of meta‐analysis of pairs of survival curves under heterogeneity. Starting point for the meta‐analysis is a set of studies, each comparing the same two treatments, containing information about multiple survival outcomes. Under heterogeneity, we model the number of events using an extension of the Poisson correlated gamma‐frailty model with serial within‐arm and positive between‐arm correlations. The parameters of the models are estimated following a two‐stage estimation procedure. In the first stage the underlying hazards and between‐study variance are estimated using the marginals, while a second stage is used to estimate both within‐arm and between‐arm correlations. The methodology is illustrated with an observational study on breast cancer. Copyright © 2009 John Wiley & Sons, Ltd.     

Bayesian spatial modeling of HIV mortality via zero‐inflated Poisson models

In this paper, we investigate the effects of poverty and inequality on the number of HIV‐related deaths in 62 New York counties via Bayesian zero‐inflated Poisson models that exhibit spatial dependence. We quantify inequality via the Theil index and poverty via the ratios of two Census 2000 variables, the number of people under the poverty line and the number of people for whom poverty status is determined, in each Zip Code Tabulation Area. The purpose of this study was to investigate the effects of inequality and poverty in addition to spatial dependence between neighboring regions on HIV mortality rate, which can lead to improved health resource allocation decisions. In modeling county‐specific HIV counts, we propose Bayesian zero‐inflated Poisson models whose rates are functions of both covariate and spatial/random effects. To show how the proposed models work, we used three different publicly available data sets: TIGER Shapefiles, Census 2000, and mortality index files. In addition, we introduce parameter estimation issues of Bayesian zero‐inflated Poisson models and discuss MCMC method implications. Copyright © 2012 John Wiley & Sons, Ltd.     

Bayesian approach to predicting cancer incidence for an area without cancer registration by using cancer incidence data from nearby areas

This paper compares three different methods for performing cancer incidence prediction in an area without a cancer registry under a Bayesian framework, using linear and log-linear age-period models with either age-specific slopes or a common slope across age groups. The three methods assume that a nearby area with a cancer registration has similar incidence and mortality patterns as the area of interest without a cancer registry where the cancer incidence prediction is carried out. The three methods differ in modeling strategies: (i) modeling the incidence rate directly; (ii) modeling the ratio of the number of incident cases to that of mortality cases; and (iii) modeling the difference between the incidence rate and the mortality rate. Strategy (iii) is a new approach in this type of projection. Empirical assessment is made using real data from the cancer registry of Tarragona, Spain, to predict cancer incidence in Girona, Spain, and vice versa. Predictions of short-term (3-4 years) incidence were made for 2001 in Tarragona using observed cancer incidence and mortality data for 1994-1998 from Girona. Short-term predictions were made for 2002 in Girona using Tarragona's 1994-1998 data. Additionally, long-term (10 years) incidence rate predictions were made for 2002 in Girona using data from Tarragona for the period 1985-1992. Our results suggest that extrapolating time-trends of incidence rates minus mortality rates may have the best predictive performance overall. These methods of population-level disease-incidence prediction are highly relevant to health care planning and policy decisions.     

Benefits Gained,Benefits Lost: Comparing Baby Boomers to Other Generations in a Longitudinal Cohort Study of Self‐Rated Health

Policy Points

• Despite beliefs that baby boomers are healthier than previous generations, we found no evidence that the health of baby boomers is substantially different from that of the previous or succeeding cohorts.
• The effects of increased education, higher income, and lower smoking rates on improving self-rated health were nearly counterbalanced by the adverse effect of increasing body mass index (BMI).
• Assumptions that baby boomers will require less health care as they age because of better education, more prosperity, and less propensity to smoke may not be realized because of increases in obesity.

Context

Baby boomers are commonly believed to be healthier than the previous generation. Using self-rated health (SRH) as an indicator of health status, this study examines the effects of age, period, and birth cohort on the trajectory of health across 4 generations: World War II (born between 1935 and 1944), older baby boomers (born between 1945 and 1954), younger baby boomers (born between 1955 and 1964), and Generation X (born between 1965 and 1974).

Methods

We analyzed Canada’s longitudinal National Population Health Survey 1994-2010 (n = 8,570 at baseline), using multilevel growth models to estimate the age trajectory of SRH by cohort, accounting for period and incorporating the influence of changes in education, household income, smoking status, and body mass index (BMI) on SRH over time.

Findings

SRH worsened with increasing age in all cohorts. Cohort differences in SRH were modest (p = 0.034), but there was a significant period effect (p = 0.002). We found marked cohort effects for increasing education, income, and BMI, and decreasing smoking from the youngest to the oldest cohorts, which were much reduced (education and smoking) or removed (income and BMI) once period was taken into account. At the population level, multivariable analysis showed the benefits of increasing education and income and declines in smoking on the trajectory of improving SRH were almost counterbalanced by the effects of increasing BMI (obesity).

Conclusions

We found no evidence to support the expectation that baby boomers will age more or less healthily than previous cohorts did. We also found that increasing BMI has likely undermined improvements in health that might have otherwise occurred, with possible implications for the need for health care. Period effects had a more profound effect than birth cohort effects. This suggests that interventions to improve health, such as reducing obesity, can be targeted to the entire, or a major portion of the, population and need not single out particular birth cohorts.     

Confidence intervals for ranks of age‐adjusted rates across states or counties

Health indices provide information to the general public on the health condition of the community. They can also be used to inform the government's policy making, to evaluate the effect of a current policy or healthcare program, or for program planning and priority setting. It is a common practice that the health indices across different geographic units are ranked and the ranks are reported as fixed values. We argue that the ranks should be viewed as random and hence should be accompanied by an indication of precision (i.e., the confidence intervals). A technical difficulty in doing so is how to account for the dependence among the ranks in the construction of confidence intervals. In this paper, we propose a novel Monte Carlo method for constructing the individual and simultaneous confidence intervals of ranks for age‐adjusted rates. The proposed method uses as input age‐specific counts (of cases of disease or deaths) and their associated populations. We have further extended it to the case in which only the age‐adjusted rates and confidence intervals are available. Finally, we demonstrate the proposed method to analyze US age‐adjusted cancer incidence rates and mortality rates for cancer and other diseases by states and counties within a state using a website that will be publicly available. The results show that for rare or relatively rare disease (especially at the county level), ranks are essentially meaningless because of their large variability, while for more common disease in larger geographic units, ranks can be effectively utilized. Copyright © 2014 John Wiley & Sons, Ltd.