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.     

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.     

A flexible parametric approach to examining spatial variation in relative survival

Most of the few published models used to obtain small‐area estimates of relative survival are based on a generalized linear model with piecewise constant hazards under a Bayesian formulation. Limitations of these models include the need to artificially split the time scale, restricted ability to include continuous covariates, and limited predictive capacity. Here, an alternative Bayesian approach is proposed: a spatial flexible parametric relative survival model. This overcomes previous limitations by combining the benefits of flexible parametric models: the smooth, well‐fitting baseline hazard functions and predictive ability, with the Bayesian benefits of robust and reliable small‐area estimates. Both spatially structured and unstructured frailty components are included. Spatial smoothing is conducted using the intrinsic conditional autoregressive prior. The model was applied to breast, colorectal, and lung cancer data from the Queensland Cancer Registry across 478 geographical areas. Advantages of this approach include the ease of including more realistic complexity, the feasibility of using individual‐level input data, and the capacity to conduct overall, cause‐specific, and relative survival analysis within the same framework. Spatial flexible parametric survival models have great potential for exploring small‐area survival inequalities, and we hope to stimulate further use of these models within wider contexts. Copyright © 2016 John Wiley & Sons, Ltd.     

Evidence synthesis from aggregate recurrent event data for clinical trial design and analysis

Information from historical trials is important for the design, interim monitoring, analysis, and interpretation of clinical trials. Meta‐analytic models can be used to synthesize the evidence from historical data, which are often only available in aggregate form. We consider evidence synthesis methods for trials with recurrent event endpoints, which are common in many therapeutic areas. Such endpoints are typically analyzed by negative binomial regression. However, the individual patient data necessary to fit such a model are usually unavailable for historical trials reported in the medical literature. We describe approaches for back‐calculating model parameter estimates and their standard errors from available summary statistics with various techniques, including approximate Bayesian computation. We propose to use a quadratic approximation to the log‐likelihood for each historical trial based on 2 independent terms for the log mean rate and the log of the dispersion parameter. A Bayesian hierarchical meta‐analysis model then provides the posterior predictive distribution for these parameters. Simulations show this approach with back‐calculated parameter estimates results in very similar inference as using parameter estimates from individual patient data as an input. We illustrate how to design and analyze a new randomized placebo‐controlled exacerbation trial in severe eosinophilic asthma using data from 11 historical trials.     

Multilevel mixed effects parametric survival models using adaptive Gauss–Hermite quadrature with application to recurrent events and individual participant data meta‐analysis

Multilevel mixed effects survival models are used in the analysis of clustered survival data, such as repeated events, multicenter clinical trials, and individual participant data (IPD) meta‐analyses, to investigate heterogeneity in baseline risk and covariate effects. In this paper, we extend parametric frailty models including the exponential, Weibull and Gompertz proportional hazards (PH) models and the log logistic, log normal, and generalized gamma accelerated failure time models to allow any number of normally distributed random effects. Furthermore, we extend the flexible parametric survival model of Royston and Parmar, modeled on the log‐cumulative hazard scale using restricted cubic splines, to include random effects while also allowing for non‐PH (time‐dependent effects). Maximum likelihood is used to estimate the models utilizing adaptive or nonadaptive Gauss–Hermite quadrature. The methods are evaluated through simulation studies representing clinically plausible scenarios of a multicenter trial and IPD meta‐analysis, showing good performance of the estimation method. The flexible parametric mixed effects model is illustrated using a dataset of patients with kidney disease and repeated times to infection and an IPD meta‐analysis of prognostic factor studies in patients with breast cancer. User‐friendly Stata software is provided to implement the methods. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimating cancer incidence using a Bayesian back‐calculation approach

We propose a Bayesian hierarchical model for the calculation of incidence counts from mortality data by a convolution equation that expresses mortality through its relationship with incidence and the survival probability density. The basic idea is to use mortality data together with an estimate of the survival distribution from cancer incidence to cancer mortality to reconstruct the numbers of individuals who constitute previously incident cases that give rise to the observed pattern of cancer mortality. This model is novel because it takes into account the uncertainty from the survival distribution; thus, a Bayesian‐mixture cure model for survival is introduced. Furthermore, projections are obtained starting from a Bayesian age‐period‐cohort model. The main advantage of the proposed approach is its consideration of the three components of the model: the convolution equation, the survival mixture cure model and the age‐period‐cohort projection within a directed acyclic graph model. Furthermore, the estimation are obtained through the Gibbs sampler. We applied the model to cases of women with stomach cancer using six age classes [15–45], [45–55], [55–65], [65–75], [75–85] and [85–95] and validated it by using data from the Tuscany Cancer Registry. The model proposed and the program implemented are convenient because they allow different cancer disease to be analysed because the survival time is modelled by flexible distributions that are able to describe different trends. Copyright © 2014 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.     

A model for space–time cluster detection using spatial clusters with flexible temporal risk patterns

Maps of estimated disease rates over multiple time periods are useful tools for gaining etiologic insights regarding potential exposures associated with specific locations and times. In this paper, we describe an extension of the Gangnon–Clayton model for spatial clustering to spatio‐temporal data. As in the purely spatial model, a large set of circular regions of varying radii centered at observed locations are considered as potential clusters, e.g. subregions with a different pattern of risk than the remainder of the study region. Within the spatio‐temporal model, no specific parametric form is imposed on the temporal pattern of risk within each cluster. In addition to the clusters, the proposed model incorporates spatial and spatio‐temporal heterogeneity effects and can readily accommodate regional covariates. Inference is performed in a Bayesian framework using MCMC. Although formal inferences about the number of clusters could be obtained using a reversible jump MCMC algorithm, we use local Bayes factors from models with a fixed, but overly large, number of clusters to draw inferences about both the number and the locations of the clusters. We illustrate the approach with two applications of the model to data on female breast cancer mortality in Japan and evaluate its operating characteristics in a simulation study. Copyright © 2010 John Wiley & Sons, Ltd.     

Semiparametric Bayesian inference on skew–normal joint modeling of multivariate longitudinal and survival data

We propose a semiparametric multivariate skew–normal joint model for multivariate longitudinal and multivariate survival data. One main feature of the posited model is that we relax the commonly used normality assumption for random effects and within‐subject error by using a centered Dirichlet process prior to specify the random effects distribution and using a multivariate skew–normal distribution to specify the within‐subject error distribution and model trajectory functions of longitudinal responses semiparametrically. A Bayesian approach is proposed to simultaneously obtain Bayesian estimates of unknown parameters, random effects and nonparametric functions by combining the Gibbs sampler and the Metropolis–Hastings algorithm. Particularly, a Bayesian local influence approach is developed to assess the effect of minor perturbations to within‐subject measurement error and random effects. Several simulation studies and an example are presented to illustrate the proposed methodologies. Copyright © 2014 John Wiley & Sons, Ltd.     

Estimation of death rates in US states with small subpopulations

In US states with small subpopulations, the observed mortality rates are often zero, particularly among young ages. Because in life tables, death rates are reported mostly on a log scale, zero mortality rates are problematic. To overcome the observed zero death rates problem, appropriate probability models are used. Using these models, observed zero mortality rates are replaced by the corresponding expected values. This enables logarithmic transformations and, in some cases, the fitting of the eight‐parameter Heligman–Pollard model to produce mortality estimates for ages 0–130 years, a procedure illustrated in terms of mortality data from several states. Copyright © 2014 John Wiley & Sons, Ltd.     

Bayesian semiparametric model with spatially–temporally varying coefficients selection

In spatiotemporal analysis, the effect of a covariate on the outcome usually varies across areas and time. The spatial configuration of the areas may potentially depend on not only the structured random intercept but also spatially varying coefficients of covariates. In addition, the normality assumption of the distribution of spatially varying coefficients could lead to potential biases of estimations. In this article, we proposed a Bayesian semiparametric space–time model where the spatially–temporally varying coefficient is decomposed as fixed, spatially varying, and temporally varying coefficients. We nonparametrically modeled the spatially varying coefficients of space–time covariates by using the area‐specific Dirichlet process prior with weights transformed via a generalized transformation. We modeled the temporally varying coefficients of covariates through the dynamic model. We also took into account the uncertainty of inclusion of the spatially–temporally varying coefficients by variable selection procedure through determining the probabilities of different effects for each covariate. The proposed semiparametric approach shows its improvement compared with the Bayesian spatial–temporal models with normality assumption on spatial random effects and the Bayesian model with the Dirichlet process prior on the random intercept. We presented a simulation example to evaluate the performance of the proposed approach with the competing models. We used an application to low birth weight data in South Carolina as an illustration. Copyright © 2013 John Wiley & Sons, Ltd.     

中国居民1987-2014年肺癌死亡趋势分析

目的 分析中国居民1987-2014年肺癌死亡的时间变化趋势。方法 汇总中国居民1987-2014年肺癌死亡率数据,利用Joinpoint模型估算各年龄组人群肺癌死亡率的时间变化趋势,负二项回归模型探索肺癌死亡在人群水平上的危险因素。结果 肺癌死亡的风险比值,城市居民是农村居民的1.43倍（95%CI=1.35~1.50,P<0.01）,男性是女性的2.28倍（95%CI=2.17~2.41,P<0.01）,每增加5岁,肺癌死亡风险平均增大62%（OR=1.62,95%CI=1.60~1.63,P<0.01）,每过一年平均增大1%（OR=1.01,95%CI=1.00~1.01,P<0.01）;农村居民肺癌中标死亡率上升明显[男性死亡率年度变化百分比（APC）=2.58%,女性APC=2.54%：P<0.01],城市女性略有下降（APC=-0.74%,P<0.01）,城市男性无明显趋势（APC=-0.23%,P=0.11）;城市居民在20~74岁肺癌死亡率逐年下降,≥75岁逐年上升;农村低龄组居民无明显下降趋势,农村男性≥40岁、农村女性≥50岁呈明显上升趋势。结论 中国肺癌死亡率的变化趋势有年龄差异;农村地区肺癌死亡率逐年增加。     

Bayesian projections: what are the effects of excluding data from younger age groups?

Bayesian age-period-cohort models are used increasingly to project cancer incidence and mortality rates. Data for younger age groups for which rates are low are often discarded from the analysis. The authors explored the effect of excluding these data, in terms of the precision and accuracy of projections, for selected cancer mortality data sets. Projections were made by using a generalized Bayesian age-period-cohort model. Smoothing was applied to each time scale to reduce random variation between adjacent parameter estimates. The sum of squared standardized residuals was used to assess the accuracy of projections, and 90% credible intervals were calculated to assess precision. For the data sets considered, inclusion of all age groups in the analysis provided more precise age-standardized and age-specific projections as well as more accurate age-specific projections for younger age groups. An overall improvement in the accuracy of age-standardized rates was demonstrated for males but not females, which may suggest that analysis of the full data set is beneficial when projecting cancer rates with strong cohort effects.     

2008—2018年佳木斯市肺癌死亡趋势时空分析

目的 分析 2008—2018 年佳木斯市肺癌死亡率的时空分布特征。方法 肺癌死亡数据来自于佳木斯市疾病预防控制中心死因登记系统,计算肺癌中国人口标化死亡率和世界人口标化死亡率。应用 Joinpoint 回归模型计算肺癌死亡率的年百分比变化（the annual percentage change,APC）和年均百分比变化（the average annual percentage change,AAPC）用以评估时间变化趋势,应用 Geoda 1.14.0 软件进行肺癌死亡率的空间自相关分析。结果 2008—2018 年佳木斯市肺癌死亡率最低为 27.80/10 万,最高为 41.43/10 万,2014 年前肺癌死亡率呈上升趋势,2014 年后,女性肺癌死亡率呈下降趋势,男性变化趋势不显著。2009—2011 年,佳木斯市女性肺癌死亡率存在空间聚集性（Moran’s I = 0.244, P＜0.05）,但随后聚集现象消失。结论 佳木斯市肺癌死亡率较高,应开展肺癌危险因素和干预研究,以期降低该地区肺癌死亡率。     

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.     

Multivariate non‐parametric methods for Mann–Whitney statistics to analyse cross‐over studies with two treatment sequences

A non‐parametric strategy for the analysis of ordinal data from cross‐over studies with two treatment sequences and d(⩾2) periods is examined through Mann–Whitney rank measures of association. For each period, these statistics estimate the probability of larger response for a randomly selected patient in one group relative to a randomly selected patient in the other group. Such estimates are as well formed for comparisons between groups for u pairs of periods with the same treatment. Methods for U‐statistics are used to produce a consistent estimate of the covariance matrix for the (d+u) Mann–Whitney estimates. The effects of periods and treatments on the respective Mann–Whitney estimates are evaluated through linear (or log‐linear) models. For estimation of the parameters in these models, a modified weighted least squares method is applied through a (2d−1)⩽(d+u) dimensional basis which effectively addresses potentially near singularities in the estimated covariance matrix of the Mann–Whitney estimates. The proposed methods are applicable to response variables with an interval or an ordered categorical scale. Their scope additionally has capabilities for controlling strata in the design of a cross‐over study or concomitant variables for which covariance adjustment is of interest for reduction of variance. Applications of the methods are illustrated through three cross‐over studies with different specifications for the two sequences of two treatments during two to four periods. Copyright © 1999 John Wiley & Sons, Ltd.     

Objective Bayesian model selection for Cox regression

There is now a large literature on objective Bayesian model selection in the linear model based on the g‐prior. The methodology has been recently extended to generalized linear models using test‐based Bayes factors. In this paper, we show that test‐based Bayes factors can also be applied to the Cox proportional hazards model. If the goal is to select a single model, then both the maximum a posteriori and the median probability model can be calculated. For clinical prediction of survival, we shrink the model‐specific log hazard ratio estimates with subsequent calculation of the Breslow estimate of the cumulative baseline hazard function. A Bayesian model average can also be employed. We illustrate the proposed methodology with the analysis of survival data on primary biliary cirrhosis patients and the development of a clinical prediction model for future cardiovascular events based on data from the Second Manifestations of ARTerial disease (SMART) cohort study. Cross‐validation is applied to compare the predictive performance with alternative model selection approaches based on Harrell's c‐Index, the calibration slope and the integrated Brier score. Finally, a novel application of Bayesian variable selection to optimal conditional prediction via landmarking is described. Copyright © 2016 John Wiley & Sons, Ltd.     

Avoiding zero between‐study variance estimates in random‐effects meta‐analysis

Fixed‐effects meta‐analysis has been criticized because the assumption of homogeneity is often unrealistic and can result in underestimation of parameter uncertainty. Random‐effects meta‐analysis and meta‐regression are therefore typically used to accommodate explained and unexplained between‐study variability. However, it is not unusual to obtain a boundary estimate of zero for the (residual) between‐study standard deviation, resulting in fixed‐effects estimates of the other parameters and their standard errors. To avoid such boundary estimates, we suggest using Bayes modal (BM) estimation with a gamma prior on the between‐study standard deviation. When no prior information is available regarding the magnitude of the between‐study standard deviation, a weakly informative default prior can be used (with shape parameter 2 and rate parameter close to 0) that produces positive estimates but does not overrule the data, leading to only a small decrease in the log likelihood from its maximum. We review the most commonly used estimation methods for meta‐analysis and meta‐regression including classical and Bayesian methods and apply these methods, as well as our BM estimator, to real datasets. We then perform simulations to compare BM estimation with the other methods and find that BM estimation performs well by (i) avoiding boundary estimates; (ii) having smaller root mean squared error for the between‐study standard deviation; and (iii) better coverage for the overall effects than the other methods when the true model has at least a small or moderate amount of unexplained heterogeneity. Copyright © 2013 John Wiley & Sons, Ltd.     

AUTOREGRESSIVE AGE–PERIOD–COHORT MODELS

Age–period–cohort analysis of vital data has received much attention recently, and it is already well known that the exact linear relation of the three time factors creates a non-identifiability problem. Previous studies have shown that the curvature terms of these factors are estimable but the linear trends are not. However, little attention has been paid to the possibility that the effects due to cohort and/or period might change through time stochastically rather than deterministically and hence display a stochastic trend. In this paper, we model the cohort effects as an AR(1) process and use lung cancer mortality data from 1966 to 1990 for males in Taiwan as an example. The parameters are identifiable in the proposed model and the estimates are found to be stable. However, the assumption made in the model should be carefully considered before using our methodology.