Simple parametric survival analysis with anonymized register data: A cohort study with truncated and interval censored event and censoring times

Background

To preserve patient anonymity, health register data may be provided as binned data only. Here we consider as example, how to estimate mean survival time after a diagnosis of metastatic colorectal cancer from Norwegian register data on time to death or censoring binned into 30 day intervals. All events occurring in the first three months (90 days) after diagnosis were removed to achieve comparability with a clinical trial. The aim of the paper is to develop and implement a simple, and yet flexible method for analyzing such interval censored and truncated data.

Methods

Considering interval censoring a missing data problem, we implement a simple multiple imputation strategy that allows flexible sensitivity analyses with respect to the shape of the censoring distribution. To allow identification of appropriate parametric models, a χ²-goodness-of-fit test--also imputation based--is derived and supplemented with diagnostic plots. Uncertainty estimates for mean survival times are obtained via a simulation strategy. The validity and statistical efficiency of the proposed method for varying interval lengths is investigated in a simulation study and compared with simpler alternatives.

Results

Mean survival times estimated from the register data ranged from 1.2 (SE = 0.09) to 3.2 (0.31) years depending on period of diagnosis and choice of parametric model. The shape of the censoring distribution within intervals did generally not influence results, whereas the choice of parametric model did, even when different models fit the data equally well. In simulation studies both simple midpoint imputation and multiple imputation yielded nearly unbiased analyses (relative biases of -0.6% to 9.4%) and confidence intervals with near-nominal coverage probabilities (93.4% to 95.7%) for censoring intervals shorter than six months. For 12 month censoring intervals, multiple imputation provided better protection against bias, and coverage probabilities closer to nominal values than simple midpoint imputation.

Conclusion

Binning of event and censoring times should be considered a viable strategy for anonymizing register data on survival times, as they may be readily analyzed with methods based on multiple imputation.

     

Can you trust the parametric standard errors in nonlinear least squares? Yes,with provisos

Background

Questions about the reliability of parametric standard errors (SEs) from nonlinear least squares (LS) algorithms have led to a general mistrust of these precision estimators that is often unwarranted.

A Review and Comparison of Methods for Recreating Individual Patient Data from Published Kaplan-Meier Survival Curves for Economic Evaluations: A Simulation Study

Background

In general, the individual patient-level data (IPD) collected in clinical trials are not available to independent researchers to conduct economic evaluations; researchers only have access to published survival curves and summary statistics. Thus, methods that use published survival curves and summary statistics to reproduce statistics for economic evaluations are essential. Four methods have been identified: two traditional methods 1) least squares method, 2) graphical method; and two recently proposed methods by 3) Hoyle and Henley, 4) Guyot et al. The four methods were first individually reviewed and subsequently assessed regarding their abilities to estimate mean survival through a simulation study.

Methods

A number of different scenarios were developed that comprised combinations of various sample sizes, censoring rates and parametric survival distributions. One thousand simulated survival datasets were generated for each scenario, and all methods were applied to actual IPD. The uncertainty in the estimate of mean survival time was also captured.

Results

All methods provided accurate estimates of the mean survival time when the sample size was 500 and a Weibull distribution was used. When the sample size was 100 and the Weibull distribution was used, the Guyot et al. method was almost as accurate as the Hoyle and Henley method; however, more biases were identified in the traditional methods. When a lognormal distribution was used, the Guyot et al. method generated noticeably less bias and a more accurate uncertainty compared with the Hoyle and Henley method.

Conclusions

The traditional methods should not be preferred because of their remarkable overestimation. When the Weibull distribution was used for a fitted model, the Guyot et al. method was almost as accurate as the Hoyle and Henley method. However, if the lognormal distribution was used, the Guyot et al. method was less biased compared with the Hoyle and Henley method.     

Association Tests for a Censored Quantitative Trait and Candidate Genes in Structured Populations with Multilevel Genetic Relatedness

Summary Several statistical methods for detecting associations between quantitative traits and candidate genes in structured populations have been developed for fully observed phenotypes. However, many experiments are concerned with failure‐time phenotypes, which are usually subject to censoring. In this article, we propose statistical methods for detecting associations between a censored quantitative trait and candidate genes in structured populations with complex multiple levels of genetic relatedness among sampled individuals. The proposed methods correct for continuous population stratification using both population structure variables as covariates and the frailty terms attributable to kinship. The relationship between the time‐at‐onset data and genotypic scores at a candidate marker is modeled via a parametric Weibull frailty accelerated failure time (AFT) model as well as a semiparametric frailty AFT model, where the baseline survival function is flexibly modeled as a mixture of Polya trees centered around a family of Weibull distributions. For both parametric and semiparametric models, the frailties are modeled via an intrinsic Gaussian conditional autoregressive prior distribution with the kinship matrix being the adjacency matrix connecting subjects. Simulation studies and applications to the Arabidopsis thaliana line flowering time data sets demonstrated the advantage of the new proposals over existing approaches.     

Waist-to-Height Ratio Is More Predictive of Years of Life Lost than Body Mass Index

Objective

Our aim was to compare the effect of central obesity (measured by waist-to-height ratio, WHtR) and total obesity (measured by body mass index, BMI) on life expectancy expressed as years of life lost (YLL), using data on British adults.

Methods

A Cox proportional hazards model was applied to data from the prospective Health and Lifestyle Survey (HALS) and the cross sectional Health Survey for England (HSE). The number of years of life lost (YLL) at three ages (30, 50, 70 years) was found by comparing the life expectancies of obese lives with those of lives at optimum levels of BMI and WHtR.

Results

Mortality risk associated with BMI in the British HALS survey was similar to that found in US studies. However, WHtR was a better predictor of mortality risk. For the first time, YLL have been quantified for different values of WHtR. This has been done for both sexes separately and for three representative ages.

Conclusion

This study supports the simple message “Keep your waist circumference to less than half your height”. The use of WHtR in public health screening, with appropriate action, could help add years to life.     

Placental weight mediates the effects of prenatal factors on fetal growth: the extent differs by preterm status

Objective:

Elevated pre‐pregnancy BMI, excessive gestational weight gain (GWG), and gestational diabetes mellitus (GDM) are known determinants of fetal growth. The role of placental weight is unclear. We aimed to examine the extent to which placental weight mediates the associations of pre‐pregnancy BMI, GWG, and GDM with birth weight‐for‐gestational age, and whether the relationships differ by preterm status.

Design and Methods:

We examined 1,035 mother‐infant pairs at birth from the Boston Birth Cohort. Data were collected by questionnaire and clinical measures. Placentas were weighed without membranes or umbilical cords. We performed sequential models excluding and including placental weight, stratified by preterm status.

Results:

We found that 21% of mothers were obese, 42% had excessive GWG, and 5% had GDM. Forty‐one percent were preterm. Among term births, after adjustment for sex, gestational age, maternal age, race, parity, education, smoking, and stress during pregnancy, birth weight‐for‐gestational age z‐score was 0.55 (0.30, 0.80) units higher for pre‐pregnancy obesity vs. normal weight. It was 0.34 (0.13, 0.55) higher for excessive vs. adequate GWG, 0.67 (0.24, 1.10) for GDM vs. no DM, with additional adjustment for pre‐pregnancy BMI. Adding placental weight to the models attenuated the estimates for pre‐pregnancy obesity by 20%, excessive GWG by 32%, and GDM by 21%. Among preterm infants, GDM was associated with 0.67 (0.34, 1.00) higher birth weight‐for‐gestational age z‐score, but pre‐pregnancy obesity and excessive GWG were not. Attenuation by placental weight was 36% for GDM.

Conclusions:

These results suggest that placental weight partially mediates the effects of pre‐pregnancy obesity, GDM, and excessive GWG on fetal growth among term infants.     

Statistical optimization of parametric accelerated failure time model for mapping survival trait loci

Most existing statistical methods for mapping quantitative trait loci (QTL) are not suitable for analyzing survival traits with a skewed distribution and censoring mechanism. As a result, researchers incorporate parametric and semi-parametric models of survival analysis into the framework of the interval mapping for QTL controlling survival traits. In survival analysis, accelerated failure time (AFT) model is considered as a de facto standard and fundamental model for data analysis. Based on AFT model, we propose a parametric approach for mapping survival traits using the EM algorithm to obtain the maximum likelihood estimates of the parameters. Also, with Bayesian information criterion (BIC) as a model selection criterion, an optimal mapping model is constructed by choosing specific error distributions with maximum likelihood and parsimonious parameters. Two real datasets were analyzed by our proposed method for illustration. The results show that among the five commonly used survival distributions, Weibull distribution is the optimal survival function for mapping of heading time in rice, while Log-logistic distribution is the optimal one for hyperoxic acute lung injury.     

Understanding dynamics using sensitivity analysis: caveat and solution

Background  

Parametric sensitivity analysis (PSA) has become one of the most commonly used tools in computational systems biology, in which the sensitivity coefficients are used to study the parametric dependence of biological models. As many of these models describe dynamical behaviour of biological systems, the PSA has subsequently been used to elucidate important cellular processes that regulate this dynamics. However, in this paper, we show that the PSA coefficients are not suitable in inferring the mechanisms by which dynamical behaviour arises and in fact it can even lead to incorrect conclusions.     

On the distribution of developmental errors: comparing the normal, gamma, and log-normal distribution

The amount of between‐individual variation in the unobservable developmental instability (DI) has been the subject of intense recent debates. The unexpectedly high estimates of between‐individual variation in DI based on distributional characteristics of observable asymmetry values (of on average bilaterally symmetric traits) rely on statistical models that assume an underlying normal distribution of developmental errors. This prompted doubts on the assumption of the Gaussian nature of developmental errors. However, when applying other candidate distributions [log‐normal and gamma (γ)], recent analyses of empirical datasets have indicated that estimates remain generally high. Yet, all estimates were based on bilaterally symmetric traits, which did not allow for a formal comparison of the alternative distributions. In the present study, we extend a recent statistical model to allow statistical comparison of the different distributions based on traits that developed repeatedly under the same conditions, such as flower traits and regrown feathers. We analyse simulated and empirical data and show that: (1) it is statistically difficult to differentiate among the three alternatives when variances are small relative to the mean, as is often the case with DI; (2) the normal distribution fits the log‐normal or γ relatively well under those circumstances; (3) the deviance information criterion (DIC) is able to pick up differences in model fit among the three alternative distributions, yet more strongly so when levels of DI were high; (4) empirical datasets show a better fit of the normal over the log‐normal and γ‐distributions as judged by the DIC; and (5) estimates of between‐individual variation in DI in the three empirical datasets were relatively high (> 50%) under each distributional assumption. In conclusion, and based on our three datasets, the normal approximation appears to be a reasonable choice for statistical models of DI and the remarkably high estimates of variation in DI cannot be attributed to non‐normal developmental noise. Nevertheless, our method should be applied to a broad range of traits and organisms to evaluate the generality of this result. We argue that there is an urgent need for studies that reveal the underlying mechanisms of developmental noise and stability, as well as the role of developmental selection, in order to be able to determine the biological importance of the highly skewed distributions of developmental instability often observed. © 2007 The Linnean Society of London, Biological Journal of the Linnean Society, 2007, 92 , 197–210.     

Bayesian regularization for flexible baseline hazard functions in Cox survival models

Fully Bayesian methods for Cox models specify a model for the baseline hazard function. Parametric approaches generally provide monotone estimations. Semi‐parametric choices allow for more flexible patterns but they can suffer from overfitting and instability. Regularization methods through prior distributions with correlated structures usually give reasonable answers to these types of situations. We discuss Bayesian regularization for Cox survival models defined via flexible baseline hazards specified by a mixture of piecewise constant functions and by a cubic B‐spline function. For those “semi‐parametric” proposals, different prior scenarios ranging from prior independence to particular correlated structures are discussed in a real study with microvirulence data and in an extensive simulation scenario that includes different data sample and time axis partition sizes in order to capture risk variations. The posterior distribution of the parameters was approximated using Markov chain Monte Carlo methods. Model selection was performed in accordance with the deviance information criteria and the log pseudo‐marginal likelihood. The results obtained reveal that, in general, Cox models present great robustness in covariate effects and survival estimates independent of the baseline hazard specification. In relation to the “semi‐parametric” baseline hazard specification, the B‐splines hazard function is less dependent on the regularization process than the piecewise specification because it demands a smaller time axis partition to estimate a similar behavior of the risk.     

Influence of calendar period on the association between BMI and coronary heart disease: A meta‐analysis of 31 cohorts

Objective:

The association between obesity and coronary heart disease (CHD) may have changed over time, for example due to improved pharmacological treatment of CHD risk factors. This meta‐analysis of 31 prospective cohort studies explores the influence of calendar period on CHD risk associated with body mass index (BMI).

Design and Methods:

The relative risks (RRs) of CHD for a five‐BMI‐unit increment and BMI categories were pooled by means of random effects models. Meta‐regression analysis was used to examine the influence of calendar period (>1985 v ≤1985) in univariate and multivariate analyses (including mean population age as a covariate).

Results:

The age, sex, and smoking adjusted RR (95% confidence intervals) of CHD for a five‐BMI‐unit increment was 1.28(1.22:1.34). For underweight, overweight and obesity, the RRs (compared to normal weight) were 1.11(0.91:1.36), 1.31(1.22:1.41), and 1.78(1.55:2.04), respectively. The univariate analysis indicated 31% (95%CI: ?56:0) lower RR of CHD associated with a five‐BMI‐unit increment and a 51% (95%CI: ?78: ?14)) lower RR associated with obesity in studies starting after 1985 (n = 15 and 10, respectively) compared to studies starting in or before 1985 (n = 16 and 10). However, in the multivariate analysis, only mean population age was independently associated with the RRs for a five‐BMI‐unit increment and obesity (?29(95%CI: ?55: ?5)) and ?31(95%CI: ?66:3), respectively) per 10‐year increment in mean age).

Conclusion:

This study provides no consistent evidence for a difference in the association between BMI and CHD by calendar period. The mean population age seems to be the most important factor that modifies the association between the risk of CHD and BMI, in which the RR decreases with increasing age.

     

A measure of explained variation for event history data

Summary There is no shortage of proposed measures of prognostic value of survival models in the statistical literature. They come under different names, including explained variation, correlation, explained randomness, and information gain, but their goal is common: to define something analogous to the coefficient of determination R² in linear regression. None however have been uniformly accepted, none have been extended to general event history data, including recurrent events, and many cannot incorporate time‐varying effects or covariates. We present here a measure specifically tailored for use with general dynamic event history regression models. The measure is applicable and interpretable in discrete or continuous time; with tied data or otherwise; with time‐varying, time‐fixed, or dynamic covariates; with time‐varying or time‐constant effects; with single or multiple event times; with parametric or semiparametric models; and under general independent censoring/observation. For single‐event survival data with neither censoring nor time dependency it reduces to the concordance index. We give expressions for its population value and the variance of the estimator and explore its use in simulations and applications. A web link to R software is provided.     

Disability prevalence among healthy weight,overweight, and obese adults

Objective:

Obesity is associated with adverse health outcomes in people with and without disabilities. However, little is known about disability prevalence among people who are obese. The purpose of this study is to determine the prevalence and type of disability among adults who are obese.

Design and Methods:

Pooled data from the 2003‐2009 National Health Interview Survey (NHIS) were analyzed to obtain national prevalence estimates of disability, disability type and obesity. The disability prevalence was stratified by body mass index (BMI): healthy weight (BMI 18.5‐<25.0), overweight (BMI 25.0‐<30.0), and obese (BMI ≥ 30.0).

Results:

In this pooled sample, among the 25.4% of US adults who were obese, 41.7% reported a disability. In contrast, 26.7% of those with a healthy weight and 28.5% of those who were overweight reported a disability. The most common disabilities among respondents with obesity were movement difficulty (32.5%) and work limitation (16.6%).

Conclusions:

This research contributes to the literature on obesity by including disability as a demographic in assessing the burden of obesity. Because of the high prevalence of disability among those who are obese, public health programs should consider the needs of those with disabilities when designing obesity prevention and treatment programs.     

Generalized Seasonal Autoregressive Integrated Moving Average Models for Count Data with Application to Malaria Time Series with Low Case Numbers

Introduction

With the renewed drive towards malaria elimination, there is a need for improved surveillance tools. While time series analysis is an important tool for surveillance, prediction and for measuring interventions’ impact, approximations by commonly used Gaussian methods are prone to inaccuracies when case counts are low. Therefore, statistical methods appropriate for count data are required, especially during “consolidation” and “pre-elimination” phases.

Methods

Generalized autoregressive moving average (GARMA) models were extended to generalized seasonal autoregressive integrated moving average (GSARIMA) models for parsimonious observation-driven modelling of non Gaussian, non stationary and/or seasonal time series of count data. The models were applied to monthly malaria case time series in a district in Sri Lanka, where malaria has decreased dramatically in recent years.

Results

The malaria series showed long-term changes in the mean, unstable variance and seasonality. After fitting negative-binomial Bayesian models, both a GSARIMA and a GARIMA deterministic seasonality model were selected based on different criteria. Posterior predictive distributions indicated that negative-binomial models provided better predictions than Gaussian models, especially when counts were low. The G(S)ARIMA models were able to capture the autocorrelation in the series.

Conclusions

G(S)ARIMA models may be particularly useful in the drive towards malaria elimination, since episode count series are often seasonal and non-stationary, especially when control is increased. Although building and fitting GSARIMA models is laborious, they may provide more realistic prediction distributions than do Gaussian methods and may be more suitable when counts are low.     

Midlife obesity and dementia: Meta‐analysis and adjusted forecast of dementia prevalence in the united states and china

Objective:

Obesity is a risk factor of dementia. Current forecasts of dementia prevalence fail to take the rising obesity prevalence into account.

Design and Methods:

Embase and Medline were searched for observational studies on the association between overweight (BMI 25‐30 kg/m²) or obesity (BMI > 30 kg/m²) and dementia and pooled the effect sizes by meta‐analysis. The population attributable risk (PAR) was calculated for different time points and adjusted them for confounders. Based on current prevalence rates of dementia and demographic forecasts, patient numbers were calculated and adjusted by the growth rates of PAR.

Results:

Compared to normal weight, midlife obesity increases the risk of dementia later in life (BMI 25‐30: RR = 1.34 [95% CI 1.08, 1.66], BMI > 30: RR = 1.91 [1.4, 2.62]). If obesity is included into forecast models, the prevalence of dementia is estimated to be 7.1 million (6.9, 7.3) and 11.3 million (10.9, 11.7) for the United States in 2030 and 2050, respectively. In China, the estimate is 13.1 million (12.8, 13.3) in 2030 and 26.2 million (25.1, 27.4) in 2050. These figures are 9% and 19% higher for the United States and China, respectively, than forecasts that rely solely on the demographic change.

Conclusion:

The past and ongoing increase in midlife obesity prevalence will contribute significantly to the future prevalence of dementia and public health measures to reduce midlife obesity are simultaneously primary prevention measures to reduce the risk of dementia.     

The Concordance Index C and the Mann–Whitney Parameter Pr(X>Y) with Randomly Censored Data

Harrell's c‐index or concordance C has been widely used as a measure of separation of two survival distributions. In the absence of censored data, the c‐index estimates the Mann–Whitney parameter Pr(X>Y), which has been repeatedly utilized in various statistical contexts. In the presence of randomly censored data, the c‐index no longer estimates Pr(X>Y); rather, a parameter that involves the underlying censoring distributions. This is in contrast to Efron's maximum likelihood estimator of the Mann–Whitney parameter, which is recommended in the setting of random censorship.     

The Predicted Impact of Ipilimumab Usage on Survival in Previously Treated Advanced or Metastatic Melanoma in the UK

Background

Evaluating long-term prognosis is important for physicians, patients and payers. This study reports the results of a model developed to predict long-term survival for UK patients receiving second-line ipilimumab.

Methods

MDX010-20 trial data were used to predict survival for ipilimumab versus UK best supportive care. Two aspects of this analysis required novel approaches: 1) The overall survival Kaplan–Meier data shape is unusual: an initial steep decline is observed before a ‘plateau’. 2) The need to extrapolate beyond the trial end (4.6 years). Based upon UK clinician advice, a three-part curve fit was used: from 0–1.5 years, Kaplan–Meier data from the trial; from 1.5–5 years, standard parametric curve fits; after 5 years, long-term data from the American Joint Committee on Cancer registry.

Results

This approach provided good internal validity: low mean absolute error and good match to median and mean trial data. Lifetime predicted means were 2.77 years for ipilimumab and 1.07 for best supportive care, driven by increased long-term survival with ipilimumab.

Conclusion

To understand the full benefit of treatment and to meet reimbursement requirements, accurate estimation of treatment benefit is key. Models, such as the one presented, can be used to extrapolate beyond trials.     

Obesity and the occurrence of bronchitis in adolescents

Objective:

Previous studies have shown that an elevated BMI was associated with higher risks of bronchitis among children. The magnitude of how increase in BMI influencing the risk of incident bronchitis remained unexplored. The objective of this study is to assess the association between BMI and the incidence of bronchitis in the Taiwan Children Health Study.

Design:

A school‐based prospective cohort study.

Methods:

We conducted a population‐based prospective cohort study among seventh‐grade school children in 14 Taiwanese communities. A total of 3,634 adolescents completed follow‐up questionnaire in 2009. Associations between BMI and incident bronchitis were analyzed by multiple Poisson regression models, taking overdispersion into account.

Results:

Among eligible cohort participants without bronchitis at study entry, the proportion of overweight and obesity were 32.1% and 17.9%. Overweight was 40.7% and obesity was 27.7% among those with incident bronchitis. The BMI percentile categories showed significant increasing trends for bronchitis in total eligible children and in girls (P for trend <0.001). Overweight and obesity were both associated with increased risks of incident bronchitis. This association was significant in girls only while stratified by gender.

Conclusions:

Our data showed that the BMI percentile and weight status were associated with higher risks of incident bronchitis in adolescents, especially in girls.     

Paradoxical protective effect of central obesity in patients with suspected stable coronary artery disease

Objective:

Increased body mass index (BMI) has been paradoxically inversely associated with the presence of angiographic coronary artery disease (CAD). Central obesity measures, considered to be more appropriate for assessing obesity‐related cardiovascular risk, have been little studied in relation to the presence of CAD. The aim was to investigate the association of central obesity with the presence of angiographic CAD as well as the prognostic significance of obesity measures in CAD prediction when added to other cardiovascular risk factors.

Design and Methods:

Patients with suspected stable CAD (n = 403, age 61 ± 10 years, 302 males) referred for diagnostic coronary angiography with documented anthropometric data were enrolled.

Results:

Significant angiographic CAD was found in 51% of patients. Both BMI (OR = 0.64 per 1 SD increase, P = 0.001) and waist circumference (WC) (OR = 0.54 per 1 SD increase, P < 0.001) were inversely associated with the presence of CAD even after adjustment for cardiovascular risk factors. In subgroup analysis, BMI and WC were significantly inversely associated with the presence of CAD in males, non diabetics, patients >60 years old and patients with Framingham risk score (FRS) >20% (P < 0.01 for all). The addition of BMI or WC in FRS‐based regression models improved prediction of CAD (P = 0.03 and P < 0.001 for BMI and WC respectively) without a significant difference between the two models (P = 0.08).

Conclusions:

Central and overall obesity were independently associated with a reduced prevalence of angiographic CAD, lending further credence to the existence of the ‘obesity paradox’. Obesity measures may further improve risk discrimination for the presence of CAD when added in an established risk score such as FRS.