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.     

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.     

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.     

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.     

Bias and inference from misspecified mixed‐effect models in stepped wedge trial analysis

Many stepped wedge trials (SWTs) are analysed by using a mixed‐effect model with a random intercept and fixed effects for the intervention and time periods (referred to here as the standard model). However, it is not known whether this model is robust to misspecification. We simulated SWTs with three groups of clusters and two time periods; one group received the intervention during the first period and two groups in the second period. We simulated period and intervention effects that were either common‐to‐all or varied‐between clusters. Data were analysed with the standard model or with additional random effects for period effect or intervention effect. In a second simulation study, we explored the weight given to within‐cluster comparisons by simulating a larger intervention effect in the group of the trial that experienced both the control and intervention conditions and applying the three analysis models described previously. Across 500 simulations, we computed bias and confidence interval coverage of the estimated intervention effect. We found up to 50% bias in intervention effect estimates when period or intervention effects varied between clusters and were treated as fixed effects in the analysis. All misspecified models showed undercoverage of 95% confidence intervals, particularly the standard model. A large weight was given to within‐cluster comparisons in the standard model. In the SWTs simulated here, mixed‐effect models were highly sensitive to departures from the model assumptions, which can be explained by the high dependence on within‐cluster comparisons. Trialists should consider including a random effect for time period in their SWT analysis model. © 2017 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

The impossibility of separating age,period and cohort effects

This commentary discusses the age–period–cohort identification problem. It shows that, despite a plethora of proposed solutions in the literature, no model is able to solve the identification problem because the identification problem is inherent to the real-world processes being modelled. As such, we cast doubt on the conclusions of a number of papers, including one presented here (Page, Milner, Morrell, & Taylor, 2013). We conclude with some recommendations for those wanting to model age, period and cohort in a compelling way.     

Identifying implausible gestational ages in preterm babies with Bayesian mixture models

Infant birth weight and gestational age are two important variables in obstetric research. The primary measure of gestational age used in US birth data is based on a mother's recall of her last menstrual period, which has been shown to introduce random or systematic errors. To mitigate some of those errors, Oja et al., Platt et al., and Tentoni et al. estimated the probabilities of gestational ages being misreported under the assumption that the distribution of infant birth weights for a true gestational age is approximately Gaussian. From this assumption, Oja et al. fitted a three‐component mixture model, and Tentoni et al. and Platt et al. fitted two‐component mixture models. We build on their methods and develop a Bayesian mixture model. We then extend our methods using reversible jump Markov chain Monte Carlo to incorporate the uncertainty in the number of components in the model. We conduct simulation studies and apply our methods to singleton births with reported gestational ages of 23–32 weeks using 2001–2008 US birth data. Results show that a three‐component mixture model fits the birth data better for gestational ages reported as 25 weeks or less; and a two‐component mixture model fits better for the higher gestational ages. Under the assumption that our Bayesian mixture models are appropriate for US birth data, our research provides useful statistical tools to identify records with implausible gestational ages, and the techniques can be used in part of a multiple‐imputation procedure for missing and implausible gestational ages. Published 2012. This article is a US Government work and is in the public domain in the USA.     

Generalized survival models for correlated time‐to‐event data

Our aim is to develop a rich and coherent framework for modeling correlated time‐to‐event data, including (1) survival regression models with different links and (2) flexible modeling for time‐dependent and nonlinear effects with rich postestimation. We extend the class of generalized survival models, which expresses a transformed survival in terms of a linear predictor, by incorporating a shared frailty or random effects for correlated survival data. The proposed approach can include parametric or penalized smooth functions for time, time‐dependent effects, nonlinear effects, and their interactions. The maximum (penalized) marginal likelihood method is used to estimate the regression coefficients and the variance for the frailty or random effects. The optimal smoothing parameters for the penalized marginal likelihood estimation can be automatically selected by a likelihood‐based cross‐validation criterion. For models with normal random effects, Gauss‐Hermite quadrature can be used to obtain the cluster‐level marginal likelihoods. The Akaike Information Criterion can be used to compare models and select the link function. We have implemented these methods in the R package rstpm2. Simulating for both small and larger clusters, we find that this approach performs well. Through 2 applications, we demonstrate (1) a comparison of proportional hazards and proportional odds models with random effects for clustered survival data and (2) the estimation of time‐varying effects on the log‐time scale, age‐varying effects for a specific treatment, and two‐dimensional splines for time and age.     

Skew‐normal Bayesian nonlinear mixed‐effects models with application to AIDS studies

Studies of HIV dynamics in AIDS research are very important in understanding the pathogenesis of HIV‐1 infection and also in assessing the effectiveness of antiviral therapies. Nonlinear mixed‐effects (NLME) models have been used for modeling between‐subject and within‐subject variations in viral load measurements. Mostly, normality of both within‐subject random error and random‐effects is a routine assumption for NLME models, but it may be unrealistic, obscuring important features of between‐subject and within‐subject variations, particularly, if the data exhibit skewness. In this paper, we develop a Bayesian approach to NLME models and relax the normality assumption by considering both model random errors and random‐effects to have a multivariate skew‐normal distribution. The proposed model provides flexibility in capturing a broad range of non‐normal behavior and includes normality as a special case. We use a real data set from an AIDS study to illustrate the proposed approach by comparing various candidate models. We find that the model with skew‐normality provides better fit to the observed data and the corresponding estimates of parameters are significantly different from those based on the model with normality when skewness is present in the data. These findings suggest that it is very important to assume a model with skew‐normal distribution in order to achieve robust and reliable results, in particular, when the data exhibit skewness. Copyright © 2010 John Wiley & Sons, Ltd.     

Meta‐Analysis of Sequencing Studies With Heterogeneous Genetic Associations

Recent advances in sequencing technologies have made it possible to explore the influence of rare variants on complex diseases and traits. Meta‐analysis is essential to this exploration because large sample sizes are required to detect rare variants. Several methods are available to conduct meta‐analysis for rare variants under fixed‐effects models, which assume that the genetic effects are the same across all studies. In practice, genetic associations are likely to be heterogeneous among studies because of differences in population composition, environmental factors, phenotype and genotype measurements, or analysis method. We propose random‐effects models which allow the genetic effects to vary among studies and develop the corresponding meta‐analysis methods for gene‐level association tests. Our methods take score statistics, rather than individual participant data, as input and thus can accommodate any study designs and any phenotypes. We produce the random‐effects versions of all commonly used gene‐level association tests, including burden, variable threshold, and variance‐component tests. We demonstrate through extensive simulation studies that our random‐effects tests are substantially more powerful than the fixed‐effects tests in the presence of moderate and high between‐study heterogeneity and achieve similar power to the latter when the heterogeneity is low. The usefulness of the proposed methods is further illustrated with data from National Heart, Lung, and Blood Institute Exome Sequencing Project (NHLBI ESP). The relevant software is freely available.     

A comparison of seven random‐effects models for meta‐analyses that estimate the summary odds ratio

Comparative trials that report binary outcome data are commonly pooled in systematic reviews and meta‐analyses. This type of data can be presented as a series of 2‐by‐2 tables. The pooled odds ratio is often presented as the outcome of primary interest in the resulting meta‐analysis. We examine the use of 7 models for random‐effects meta‐analyses that have been proposed for this purpose. The first of these models is the conventional one that uses normal within‐study approximations and a 2‐stage approach. The other models are generalised linear mixed models that perform the analysis in 1 stage and have the potential to provide more accurate inference. We explore the implications of using these 7 models in the context of a Cochrane Review, and we also perform a simulation study. We conclude that generalised linear mixed models can result in better statistical inference than the conventional 2‐stage approach but also that this type of model presents issues and difficulties. These challenges include more demanding numerical methods and determining the best way to model study specific baseline risks. One possible approach for analysts is to specify a primary model prior to performing the systematic review but also to present the results using other models in a sensitivity analysis. Only one of the models that we investigate is found to perform poorly so that any of the other models could be considered for either the primary or the sensitivity analysis.     

Mortality cohort effects from mid-19th to mid-20th century Britain: did they exist?

PurposeIdentification is a central problem with age–period–cohort analysis. Because age + cohort = period, there is no unique solution to the linear effect using generalized linear modeling, but cohort effects have caused greater controversy than age and period effects. To illustrate the magnitude of cohort effects given the presence of collinearity, we reanalyze data from the seminal study by Kermack et al, with an update.MethodsRelative mortality data in England and Wales between year 1845 and 1995 were analyzed using partial least squares regression. There were seven age groups ranging from 5 to 74 years old and 16 periods with 22 cohorts.ResultsOur reanalysis seemed to support the existence of cohort effects in the mortality trends. Period and cohort effects were generally consistent with changes in the social, economic, and environmental factors taking place in the last two centuries. Our analysis also showed a declining trend in period effects up to 1950s.ConclusionsPartial least squares and related methods provide intuitive pointers toward the separation of linear age, period, and cohort effects. Because statistical algorithms cannot distinguish between relative and actual mortality rates, cohort effects may be underestimated because of contamination by negative age effects.     

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.     

Time‐varying effect modeling with longitudinal data truncated by death: conditional models,interpretations, and inference

Recent studies found that infection‐related hospitalization was associated with increased risk of cardiovascular (CV) events, such as myocardial infarction and stroke in the dialysis population. In this work, we develop time‐varying effects modeling tools in order to examine the CV outcome risk trajectories during the time periods before and after an initial infection‐related hospitalization. For this, we propose partly conditional and fully conditional partially linear generalized varying coefficient models (PL‐GVCMs) for modeling time‐varying effects in longitudinal data with substantial follow‐up truncation by death. Unconditional models that implicitly target an immortal population is not a relevant target of inference in applications involving a population with high mortality, like the dialysis population. A partly conditional model characterizes the outcome trajectory for the dynamic cohort of survivors, where each point in the longitudinal trajectory represents a snapshot of the population relationships among subjects who are alive at that time point. In contrast, a fully conditional approach models the time‐varying effects of the population stratified by the actual time of death, where the mean response characterizes individual trends in each cohort stratum. We compare and contrast partly and fully conditional PL‐GVCMs in our aforementioned application using hospitalization data from the United States Renal Data System. For inference, we develop generalized likelihood ratio tests. Simulation studies examine the efficacy of estimation and inference procedures. Copyright © 2015 John Wiley & Sons, Ltd.     

Data‐generating models of dichotomous outcomes: Heterogeneity in simulation studies for a random‐effects meta‐analysis

Simulation studies to evaluate performance of statistical methods require a well‐specified data‐generating model. Details of these models are essential to interpret the results and arrive at proper conclusions. A case in point is random‐effects meta‐analysis of dichotomous outcomes. We reviewed a number of simulation studies that evaluated approximate normal models for meta‐analysis of dichotomous outcomes, and we assessed the data‐generating models that were used to generate events for a series of (heterogeneous) trials. We demonstrate that the performance of the statistical methods, as assessed by simulation, differs between these 3 alternative data‐generating models, with larger differences apparent in the small population setting. Our findings are relevant to multilevel binomial models in general.     

What is a cohort effect? Comparison of three statistical methods for modeling cohort effects in obesity prevalence in the United States, 1971–2006

Analysts often use different conceptual definitions of a cohort effect, and therefore different statistical methods, which lead to differing empirical results. A definition often used in sociology assumes that cohorts have unique characteristics confounded by age and period effects, whereas epidemiologists often conceive that period and age effects interact to produce cohort effects. The present study aims to illustrate these differences by estimating age, period, and cohort (APC) effects on obesity prevalence in the U.S. from 1971 to 2006 using both conceptual approaches. Data were drawn from seven cross-sectional waves of the National Health and Nutrition Examination Survey. Obesity was defined as BMI ≥ 30 for adults and ≥95th percentile for children under the age of 20. APC effects were estimated using the classic constraint-based method (first-order effects estimated and interpreted), the Holford method (first-order effects estimated but second-order effects interpreted), and median polish method (second-order effects are estimated and interpreted). Results indicated that all methods report significant age and period effects, with lower obesity prevalence in early life as well as increasing prevalence in successive surveys. Positive cohort effects for more recently born cohorts emerged based on the constraint-based model; when cohort effects were considered second-order estimates, no significant effects emerged. First-order estimates of age–period–cohort effects are often criticized because of their reliance on arbitrary constraints, but may be conceptually meaningful for sociological research questions. Second-order estimates are statistically estimable and produce conceptually meaningful results for epidemiological research questions. Age–period–cohort analysts should explicitly state the definition of a cohort effect under consideration. Our analyses suggest that the prevalence of obesity in the U.S. in the latter part of the 20th century rose across all birth cohorts, in the manner expected based on estimated age and period effects. As such, the absence or presence of cohort effects depends on the conceptual definition and therefore statistical method used.     

A comparison of change point models with application to longitudinal lung function measurements in children with cystic fibrosis

Cystic fibrosis (CF) is a hereditary lung disease characterized by loss of lung function over time. Lung function in CF is believed to decline at a higher rate during the adolescence period. It has been also hypothesized that there is a subgroup of individuals for whom lung disease remains relatively stable with only a slight decline over their lifetime. Using data from the University of Colorado CF Children's Registry, we investigate four change point models to model the decline of lung function in children and adolescents: (i) a two‐component mixture random change point model, (ii) a two‐component mixture‐fixed change point model, (iii) a random change point model, and (iv) a fixed change point model. The models are investigated through posterior predictive simulation at the individual and population levels, and a simulation study examining the effects of model misspecification. The data support the mixed random change point model as the preferred model, with roughly 30% of adolescents experiencing a steady decline of 0.5 %FEV₁ per year and 70% experiencing an increase in decline of 4.4 %FEV₁ per year beginning on average at 14.6 years of age. Copyright © 2016 John Wiley & Sons, Ltd.     

Secular trends and sociodemographic regularities of coital debut age in Norway

Gender and birth cohort differences and the influence of social background variables on the coital debut age were investigated in the general population of Norway. The data derive from a 1987 questionnaire on sexual behavior mailed to a random sample of 10,000 Norwegians of both sexes, ages 18 through 60; 63% responded to the questionnaire. 94.5% reported that they had experienced intercourse. The median coital debut age was 18.2 years. There were both cohort and gender differences. Younger cohorts have lower coital debut ages, and women younger than 35 years experienced their first intercourse at an earlier age than men in the same age group. When each independent variable was analyzed separately, there were substantial differences between educational levels and social classes with respect to age of coital debut. Multivariate analysis of six separate cohorts revealed independent effects of gender in the two youngest cohorts, whereas educational level had significant independent effects in all but the oldest cohort. Social class did not reveal any independent effect on coital debut age. Population density of the place of residence of the respondents was not substantially related to age of coital debut. Seen together, the independent variables explain about 13% of the variance in coital debut age (by multiple regression).This project was supported by grants from the Norwegian Research Council for Science and the Humanities and by the Ministry of Health and Social Affairs.     

Model‐based standardization to adjust for unmeasured cluster‐level confounders with complex survey data

Model‐based standardization uses a statistical model to estimate a standardized, or unconfounded, population‐averaged effect. With it, one can compare groups had the distribution of confounders been identical in both groups to that of the standard population. We develop two methods for model‐based standardization with complex survey data that accommodate a categorical confounder that clusters the individual observations into a very large number of subgroups. The first method combines a random‐intercept generalized linear mixed model with a conditional pseudo‐likelihood estimator of the fixed effects. The second method combines a between–within generalized linear mixed model with census data on the cluster‐level means of the individual‐level covariates. We conduct simulation studies to compare the two approaches. We apply the two methods to the 2008 Florida Behavioral Risk Factor Surveillance System survey data to estimate standardized proportions of people who drink alcohol, within age groups, adjusting for measured individual‐level and unmeasured cluster‐level confounders. Copyright © 2015 John Wiley & Sons, Ltd.