The Mizon-Richard encompassing test for the Cox and Aalen additive hazards models.

The Cox hazards model (Cox, 1972, Journal of the Royal Statistical Society, Series B34, 187-220) for survival data is routinely used in many applied fields, sometimes, however, with too little emphasis on the fit of the model. A useful alternative to the Cox model is the Aalen additive hazards model (Aalen, 1980, in Lecture Notes in Statistics-2, 1-25) that can easily accommodate time changing covariate effects. It is of interest to decide which of the two models that are most appropriate to apply in a given application. This is a nontrivial problem as these two classes of models are nonnested except only for special cases. In this article we explore the Mizon-Richard encompassing test for this particular problem. It turns out that it corresponds to fitting of the Aalen model to the martingale residuals obtained from the Cox regression analysis. We also consider a variant of this method, which relates to the proportional excess model (Martinussen and Scheike, 2002, Biometrika 89, 283-298). Large sample properties of the suggested methods under the two rival models are derived. The finite-sample properties of the proposed procedures are assessed through a simulation study. The methods are further applied to the well-known primary biliary cirrhosis data set.     

The Mizon–Richard Encompassing Test for the Cox and Aalen Additive Hazards Models

Summary .   The Cox hazards model ( Cox, 1972 , Journal of the Royal Statistical Society, Series B 34, 187–220) for survival data is routinely used in many applied fields, sometimes, however, with too little emphasis on the fit of the model. A useful alternative to the Cox model is the Aalen additive hazards model ( Aalen, 1980 , in Lecture Notes in Statistics-2 , 1–25) that can easily accommodate time changing covariate effects. It is of interest to decide which of the two models that are most appropriate to apply in a given application. This is a nontrivial problem as these two classes of models are nonnested except only for special cases. In this article we explore the Mizon–Richard encompassing test for this particular problem. It turns out that it corresponds to fitting of the Aalen model to the martingale residuals obtained from the Cox regression analysis. We also consider a variant of this method, which relates to the proportional excess model ( Martinussen and Scheike, 2002 , Biometrika 89, 283–298). Large sample properties of the suggested methods under the two rival models are derived. The finite-sample properties of the proposed procedures are assessed through a simulation study. The methods are further applied to the well-known primary biliary cirrhosis data set.     

A simple linear regression method for quantitative trait loci linkage analysis with censored observations

Standard quantitative trait loci (QTL) mapping techniques commonly assume that the trait is both fully observed and normally distributed. When considering survival or age-at-onset traits these assumptions are often incorrect. Methods have been developed to map QTL for survival traits; however, they are both computationally intensive and not available in standard genome analysis software packages. We propose a grouped linear regression method for the analysis of continuous survival data. Using simulation we compare this method to both the Cox and Weibull proportional hazards models and a standard linear regression method that ignores censoring. The grouped linear regression method is of equivalent power to both the Cox and Weibull proportional hazards methods and is significantly better than the standard linear regression method when censored observations are present. The method is also robust to the proportion of censored individuals and the underlying distribution of the trait. On the basis of linear regression methodology, the grouped linear regression model is computationally simple and fast and can be implemented readily in freely available statistical software.     

Origin and Morphogenetic Movements of the Pores of the Contractile Vacuoles in Paramecium qurelia

The position of the contractile vacuoles in non-dividing Paramecium aurelia, as judged by that of the pores, is relatively constant both laterally and longitudinally, as is the distance between them. Some variability of the distance between the pores persists from the recent fission among the smaller animals of normal shape. At the onset of fission two new vacuoles appear, one anterior to each of the old vacuoles; all the vacuoles must undergo relocation because their mean positions now are somewhat removed from that in the non-dividing animals. The pores of the front daughter are too close together: the anterior pore is too far back, and the posterior is too far forward; likewise the pores of the hind daughter are too close together: the anterior too far forward as well as the posterior. The relocation does not begin until shortly before separation of the daughters and is practically complete by the time the daughters have assumed their normal length/width ratio after separation. A neuroneme connects the pore to one of the basal granules adjacent to its place of origin, now far removed.
Occasionally more than one new pore appears at the usual time and place during very early fission; these are very close together and usually serve a single vacuole. However, most of them fail to survive the relocation following this and the next fission, so they are found in the posterior position very rarely. But, in some individuals they may survive and serve a single or separate vacuoles. In a certain clone of Paramecium aurelia multiple pores and vacuoles are rather frequent; these are, of course, the result of the tendency to produce multiple pores and vacuoles at the time of fission.     

Residuals for proportional hazards models with interval-censored survival data

We develop diagnostic tools for use with proportional hazards models for interval-censored survival data. We propose counterparts to the Cox-Snell, Lagakos (or martingale), deviance, and Schoenfeld residuals. Many of the properties of these residuals carry over to the interval-censored case. In particular, the interval-censored versions of the Lagakos and Schoenfeld residuals may be derived as components of suitable score statistics. The Lagakos residuals may be used to check regression relationships, while the Schoenfeld residuals can help to detect nonproportional hazards in semiparametric models. The methods apply to parametric models and to the semiparametric model with discrete observation times.     

Birnbaum–Saunders frailty regression models: Diagnostics and application to medical data

In survival models, some covariates affecting the lifetime could not be observed or measured. These covariates may correspond to environmental or genetic factors and be considered as a random effect related to a frailty of the individuals explaining their survival times. We propose a methodology based on a Birnbaum–Saunders frailty regression model, which can be applied to censored or uncensored data. Maximum‐likelihood methods are used to estimate the model parameters and to derive local influence techniques. Diagnostic tools are important in regression to detect anomalies, as departures from error assumptions and presence of outliers and influential cases. Normal curvatures for local influence under different perturbations are computed and two types of residuals are introduced. Two examples with uncensored and censored real‐world data illustrate the proposed methodology. Comparison with classical frailty models is carried out in these examples, which shows the superiority of the proposed model.     

Functional form diagnostics for Cox's proportional hazards model

We propose a new type of residual and an easily computed functional form test for the Cox proportional hazards model. The proposed test is a modification of the omnibus test for testing the overall fit of a parametric regression model, developed by Stute, González Manteiga, and Presedo Quindimil (1998, Journal of the American Statistical Association93, 141-149), and is based on what we call censoring consistent residuals. In addition, we develop residual plots that can be used to identify the correct functional forms of covariates. We compare our test with the functional form test of Lin, Wei, and Ying (1993, Biometrika80, 557-572) in a simulation study. The practical application of the proposed residuals and functional form test is illustrated using both a simulated data set and a real data set.     

Comparison of methods for analysis of selective genotyping survival data

Survival traits and selective genotyping datasets are typically not normally distributed, thus common models used to identify QTL may not be statistically appropriate for their analysis. The objective of the present study was to compare models for identification of QTL associated with survival traits, in particular when combined with selective genotyping. Data were simulated to model the survival distribution of a population of chickens challenged with Marek disease virus. Cox proportional hazards (CPH), linear regression (LR), and Weibull models were compared for their appropriateness to analyze the data, ability to identify associations of marker alleles with survival, and estimation of effects when all individuals were genotyped (full genotyping) and when selective genotyping was used. Little difference in power was found between the CPH and the LR model for low censoring cases for both full and selective genotyping. The simulated data were not transformed to follow a Weibull distribution and, as a result, the Weibull model generally resulted in less power than the other two models and overestimated effects. Effect estimates from LR and CPH were unbiased when all individuals were genotyped, but overestimated when selective genotyping was used. Thus, LR is preferred for analyzing survival data when the amount of censoring is low because of ease of implementation and interpretation. Including phenotypic data of non-genotyped individuals in selective genotyping analysis increased power, but resulted in LR having an inflated false positive rate, and therefore the CPH model is preferred for this scenario, although transformation of the data may also make the Weibull model appropriate for this case. The results from the research presented herein are directly applicable to interval mapping analyses.     

Statistical analysis of survival data from radiation countermeasure experiments

We present an introduction to, and examples of, Cox proportional hazards regression in the context of animal lethality studies of potential radioprotective agents. This established method is seldom used to analyze survival data collected in such studies, but is appropriate in many instances. Presenting a hypothetical radiation study that examines the efficacy of a potential radioprotectant both in the absence and presence of a potential modifier, we detail how to implement and interpret results from a Cox proportional hazards regression analysis used to analyze the survival data, and we provide relevant SAS? code. Cox proportional hazards regression analysis of survival data from lethal radiation experiments (1) considers the whole distribution of survival times rather than simply the commonly used proportions of animals that survived, (2) provides a unified analysis when multiple factors are present, and (3) can increase statistical power by combining information across different levels of a factor. Cox proportional hazards regression should be considered as a potential statistical method in the toolbox of radiation researchers.     

The impact of urbanization on body size of Barn Swallows Hirundo rustica gutturalis

Urbanization implies a dramatic impact on ecosystems, which may lead to drastic phenotypic differences between urban and nonurban individuals. For instance, urbanization is associated with increased metabolic costs, which may constrain body size, but urbanization also leads to habitat fragmentation, which may favor increases in body mass when for instance it correlates with dispersal capacity. However, this apparent contradiction has rarely been studied. This is particularly evident in China where the urbanization process is currently occurring at an unprecedented scale. Moreover, no study has addressed this issue across large geographical areas encompassing locations in different climates. In this regard, Barn Swallows (Hirundo rustica) are a suitable model to study the impact of urbanization on wild animals because they are a widely distributed species tightly associated with humans. Here, we collected body mass and wing length data for 359 breeding individuals of Barn Swallow (H. r. gutturalis) from 128 sites showing different levels of urbanization around the whole China. Using a set of linear mixed‐effects models, we assessed how urbanization and geography influenced body size measured using body mass, wing length, and their regression residuals. Interestingly, we found that the impact of urbanization was sex‐dependent, negatively affecting males’ body mass, its regression residuals, and females’ wing length. We also found that northern and western individuals were larger, regarding both body mass and wing length, than southern and eastern individuals. Females were heavier than males, yet males had slightly longer wings than females. Overall, our results showed that body mass of males was particularly sensitive trait to urbanization, latitude, and longitude, while it only showed a weak response to latitude in females. Conversely, while wing length showed a similar geographical pattern, it was only affected by urbanization in the case of females. Further research is needed to determine whether these phenotypic differences are associated with negative effects of urbanization or potential selective advantages.     

A New High-Throughput Approach to Genotype Ancient Human Gastrointestinal Parasites

Human gastrointestinal parasites are good indicators for hygienic conditions and health status of past and present individuals and communities. While microscopic analysis of eggs in sediments of archeological sites often allows their taxonomic identification, this method is rarely effective at the species level, and requires both the survival of intact eggs and their proper identification. Genotyping via PCR-based approaches has the potential to achieve a precise species-level taxonomic determination. However, so far it has mostly been applied to individual eggs isolated from archeological samples. To increase the throughput and taxonomic accuracy, as well as reduce costs of genotyping methods, we adapted a PCR-based approach coupled with next-generation sequencing to perform precise taxonomic identification of parasitic helminths directly from archeological sediments. Our study of twenty-five 100 to 7,200 year-old archeological samples proved this to be a powerful, reliable and efficient approach for species determination even in the absence of preserved eggs, either as a stand-alone method or as a complement to microscopic studies.     

Spatial Autocorrelation Approaches to Testing Residuals from Least Squares Regression

In geo-statistics, the Durbin-Watson test is frequently employed to detect the presence of residual serial correlation from least squares regression analyses. However, the Durbin-Watson statistic is only suitable for ordered time or spatial series. If the variables comprise cross-sectional data coming from spatial random sampling, the test will be ineffectual because the value of Durbin-Watson’s statistic depends on the sequence of data points. This paper develops two new statistics for testing serial correlation of residuals from least squares regression based on spatial samples. By analogy with the new form of Moran’s index, an autocorrelation coefficient is defined with a standardized residual vector and a normalized spatial weight matrix. Then by analogy with the Durbin-Watson statistic, two types of new serial correlation indices are constructed. As a case study, the two newly presented statistics are applied to a spatial sample of 29 China’s regions. These results show that the new spatial autocorrelation models can be used to test the serial correlation of residuals from regression analysis. In practice, the new statistics can make up for the deficiencies of the Durbin-Watson test.     

Tests for Hazard Transformation

The semiparametric Cox proportional hazards model is routinely adopted to model time-to-event data. Proportionality is a strong assumption, especially when follow-up time, or study duration, is long. Zeng and Lin (J. R. Stat. Soc., Ser. B, 69:1–30, 2007) proposed a useful generalisation through a family of transformation models which allow hazard ratios to vary over time. In this paper we explore a variety of tests for the need for transformation, arguing that the Cox model is so ubiquitous that it should be considered as the default model, to be discarded only if there is good evidence against the model assumptions. Since fitting an alternative transformation model is more complicated than fitting the Cox model, especially as procedures are not yet incorporated in standard software, we focus mainly on tests which require a Cox fit only. A score test is derived, and we also consider performance of omnibus goodness-of-fit tests based on Schoenfeld residuals. These tests can be extended to compare different transformation models. In addition we explore the consequences of fitting a misspecified Cox model to data generated under a true transformation model. Data on survival of 1043 leukaemia patients are used for illustration.     

Survival of Patients with Primary Brain Tumors: Comparison of Two Statistical Approaches

Purpose

We reviewed the survival time for patients with primary brain tumors undergoing treatment with stereotactic radiation methods at the Masaryk Memorial Cancer Institute Brno. We also identified risk factors and characteristics, and described their influence on survival time.

Methods

In summarizing survival data, there are two functions of principal interest, namely, the survival function and the hazard function. In practice, both of them can depend on some characteristics. We focused on nonparametric methods, propose a method based on kernel smoothing, and compared our estimates with the results of the Cox regression model. The hazard function is conditional to age and gross tumor volume and visualized as a color-coded surface. A multivariate Cox model was also designed.

Results

There were 88 patients with primary brain cancer, treated with stereotactic radiation. The median survival of our patient cohort was 47.8 months. The estimate of the hazard function has two peaks (about 10 months and about 40 months). The survival time of patients was significantly different for various diagnoses (p≪0.001), KI (p = 0.047) and stereotactic methods (p = 0.033). Patients with a greater GTV had higher risk of death. The suitable threshold for GTV is 20 cm³. Younger patients with a survival time of about 50 months had a higher risk of death. In the multivariate Cox regression model, the selected variables were age, GTV, sex, diagnosis, KI, location, and some of their interactions.

Conclusion

Kernel methods give us the possibility to evaluate continuous risk variables and based on the results offer risk-prone patients a different treatment, and can be useful for verifying assumptions of the Cox model or for finding thresholds of continuous variables.     

Mutation-tolerant protein identification by mass spectrometry.

Database search in tandem mass spectrometry is a powerful tool for protein identification. High-throughput spectral acquisition raises the problem of dealing with genetic variation and peptide modifications within a population of related proteins. A method that cross-correlates and clusters related spectra in large collections of uncharacterized spectra (i.e., from normal and diseased individuals) would be very valuable in functional proteomics. This problem is far from being simple since very similar peptides may have very different spectra. We introduce a new notion of spectral similarity that allows one to identify related spectra even if the corresponding peptides have multiple modifications/mutations. Based on this notion, we developed a new algorithm for mutation-tolerant database search as well as a method for cross-correlating related uncharacterized spectra.     

Region-Based Association Analysis of Human Quantitative Traits in Related Individuals

Regional-based association analysis instead of individual testing of each SNP was introduced in genome-wide association studies to increase the power of gene mapping, especially for rare genetic variants. For regional association tests, the kernel machine-based regression approach was recently proposed as a more powerful alternative to collapsing-based methods. However, the vast majority of existing algorithms and software for the kernel machine-based regression are applicable only to unrelated samples. In this paper, we present a new method for the kernel machine-based regression association analysis of quantitative traits in samples of related individuals. The method is based on the GRAMMAR+ transformation of phenotypes of related individuals, followed by use of existing kernel machine-based regression software for unrelated samples. We compared the performance of kernel-based association analysis on the material of the Genetic Analysis Workshop 17 family sample and real human data by using our transformation, the original untransformed trait, and environmental residuals. We demonstrated that only the GRAMMAR+ transformation produced type I errors close to the nominal value and that this method had the highest empirical power. The new method can be applied to analysis of related samples by using existing software for kernel-based association analysis developed for unrelated samples.     

Evidence for a pleiotropic QTL on chromosome 5q13 influencing both time to asthma onset and asthma score in French EGEA families

Although many genome screens have been conducted for asthma as a binary trait, there is limited information regarding the genetic factors underlying variation of asthma expression. Phenotypes related to variable disease expression include time to asthma onset and variation in clinical expression as measured by an asthma score built from EGEA data. A recent genome scan conducted for this score led to detection of a new region (18p11) not revealed by analysis of dichotomous asthma. Our goal was to characterize chromosomal regions harboring genes underlying time to asthma onset and to search for pleiotropic QTL influencing both time to asthma onset and the asthma score. We conducted a genome-wide linkage screen for time to asthma onset, modeled by martingale residuals from Cox survival model, in EGEA families with at least two asthmatic sibs. This was followed by a bivariate linkage scan of these residuals and asthma score. Univariate linkage analysis was performed using the Maximum Likelihood Binomial method that we extended to bivariate analysis. This screen revealed two regions potentially linked to time to asthma onset, 1p31 (LOD = 1.70, P = 0.003) and 5q13 (LOD = 1.87, P = 0.002). Bivariate linkage analysis led to a substantial improvement of the linkage signal on 5q13 (P = 0.00007), providing evidence for a pleiotropic QTL influencing both variation of time to asthma onset and of clinical expression. Use of quantitative phenotypes of variable disease expression and suitable statistical methodology can improve the power to detect new regions harboring genes which may play an important role in onset and course of disease.     

Development of a risk scoring system for patients with papillary thyroid cancer

As the importance of personalized therapeutics in aggressive papillary thyroid cancer (PTC) increases, accurate risk stratification is required. To develop a novel prognostic scoring system for patients with PTC (n = 455), we used mRNA expression and clinical data from The Cancer Genome Atlas. We performed variable selection using Network‐Regularized high‐dimensional Cox‐regression with gene network from pathway databases. The risk score was calculated using a linear combination of regression coefficients and mRNA expressions. The risk score and clinical variables were assessed by several survival analyses. The risk score showed high discriminatory power for the prediction of event‐free survival as well as the presence of metastasis. In multivariate analysis, the risk score and presence of metastasis were significant risk factors among the clinical variables that were examined together. In the current study, we developed a risk scoring system that will help to identify suitable therapeutic options for PTC.     

Combining gene signatures improves prediction of breast cancer survival

Background

Several gene sets for prediction of breast cancer survival have been derived from whole-genome mRNA expression profiles. Here, we develop a statistical framework to explore whether combination of the information from such sets may improve prediction of recurrence and breast cancer specific death in early-stage breast cancers. Microarray data from two clinically similar cohorts of breast cancer patients are used as training (n = 123) and test set (n = 81), respectively. Gene sets from eleven previously published gene signatures are included in the study.

Principal Findings

To investigate the relationship between breast cancer survival and gene expression on a particular gene set, a Cox proportional hazards model is applied using partial likelihood regression with an L2 penalty to avoid overfitting and using cross-validation to determine the penalty weight. The fitted models are applied to an independent test set to obtain a predicted risk for each individual and each gene set. Hierarchical clustering of the test individuals on the basis of the vector of predicted risks results in two clusters with distinct clinical characteristics in terms of the distribution of molecular subtypes, ER, PR status, TP53 mutation status and histological grade category, and associated with significantly different survival probabilities (recurrence: p = 0.005; breast cancer death: p = 0.014). Finally, principal components analysis of the gene signatures is used to derive combined predictors used to fit a new Cox model. This model classifies test individuals into two risk groups with distinct survival characteristics (recurrence: p = 0.003; breast cancer death: p = 0.001). The latter classifier outperforms all the individual gene signatures, as well as Cox models based on traditional clinical parameters and the Adjuvant! Online for survival prediction.

Conclusion

Combining the predictive strength of multiple gene signatures improves prediction of breast cancer survival. The presented methodology is broadly applicable to breast cancer risk assessment using any new identified gene set.     

IDUA,NDST1, SAP30L,CRYBA4, and SI as novel prognostic signatures clear cell renal cell carcinoma

Carcinoma of the kidney is one of the most prevalent carcinoma worldwide. The majority types of carcinoma are clear cell renal cell carcinoma (CCRCC), which consist more than 80% of the cases. As a genetically diverse disease, identification of prognosis-related genes has utmost importance in the early diagnosis and prognosis of the CCRCC. In this study, we performed gene expression profiling to identify prognosis-related genes for CCRCC. In addition, we developed and validated a gene signature-based risk score to comprehensively assess the prognostic function of differentially expressed genes. Furthermore, we performed a ROC analysis to identify the optimal cut-off point for classification risk level of the patients. Univariate Cox regression models were used to assess the association between differentially expressed genes in relation to the prognosis of patients with different stages of CCRCC. Five genes were identified significantly differentially expressed in CCRCC and associated with their survival time, namely: IDUA, NDST1, SAP30L, CRYBA4, and SI. A 5-gene signature-based risk score was developed based on the Cox coefficient of the individual genes. The prognostic value of this risk score was validated in an internal testing data set. In summary, a gene-based risk score was identified and validated, which can predict CCRCC patient survival. The potential functions of this gene expression signature and individual differentially expressed gene as prognostic targets of CCRCC were revealed by this study. Furthermore, these findings may have important implications in the understanding of the potential therapeutic method for the CCRCC patients.