Deviance Residuals Research Articles

Cross-validatory Z-Residual for Diagnosing Shared Frailty Models

Accurate model performance assessment in survival analysis is imperative for robust predictions and informed decision-making. Traditional residual diagnostic tools like martingale and deviance residuals lack a well-characterized reference distribution for censored regression, making numerical statistical tests based on these residuals challenging. Recently, the introduction of Z-residuals for diagnosing survival models addresses this limitation. However, concerns arise from conventional methods that utilize the entire dataset for both model parameter estimation and residual assessment, which may cause optimistic biases. This paper introduces cross-validatory Z-residuals as an innovative approach to address these limitations. Employing a cross-validation (CV) framework, the method systematically partitions the dataset into training and testing sets to reduce the optimistic bias. Our simulation studies demonstrate that, for goodness-of-fit tests and outlier detection, cross-validatory Z-residuals are significantly more powerful (e.g. power increased from 0.2 to 0.6). and more discriminative (e.g. AUC increased from 0.58 to 0.85) than Z-residuals without CV. We also compare the performance of Z-residuals with and without CV in identifying outliers in a real application that models the recurrence time of kidney infection patients. Our findings suggest that cross-validatory Z-residuals can identify outliers, which Z-residuals without CV fail to identify. The CV Z-residual is a more powerful tool than the No-CV Z-residual for checking survival models, particularly in goodness-of-fit tests and outlier detection. We have published a generic function, which is collected in an R package called Zresidual, for computing CV Z-residual for the output of the widely used survival R package.

Diagnostics for categorical response models based on quantile residuals and distance measures

Polytomous categorical data are frequent in studies, that can be obtained with an individual or grouped structure. In both structures, the generalized logit model is commonly used to relate the covariates on the response variable. After fitting a model, one of the challenges is the definition of an appropriate residual and choosing diagnostic techniques. Since the polytomous variable is multivariate, raw, Pearson, or deviance residuals are vectors and their asymptotic distribution is generally unknown, which leads to difficulties in graphical visualization and interpretation. Therefore, the definition of appropriate residuals and the choice of the correct analysis in diagnostic tools is important, especially for nominal data, where a restriction of methods is observed. This paper proposes the use of randomized quantile residuals associated with individual and grouped nominal data, as well as Euclidean and Mahalanobis distance measures, as an alternative to reduce the dimension of the residuals. We developed simulation studies with both data structures associated. The half-normal plots with simulation envelopes were used to assess model performance. These studies demonstrated a good performance of the quantile residuals, and the distance measurements allowed a better interpretation of the graphical techniques. We illustrate the proposed procedures with two applications to real data.

Z-residual diagnostic tool for assessing covariate functional form in shared frailty models

ABSTRACT Survival analysis often involves modeling hazard functions while considering frailty to account for unobserved cluster-level factors in clustered survival data. Shared frailty models have gained popularity for this purpose, but assessing covariate functional form in these models presents unique challenges. Martingale and deviance residuals are commonly used for visually assessing covariate functional form against continuous covariates. Nevertheless, their subjective nature and lack of a reference distribution make it challenging to derive numerical statistical tests from these residuals. To address these limitations, we propose ‘Z-residuals’, a novel diagnostic tool designed for shared frailty models, leveraging the concept of randomized survival probability and introducing both graphical and numerical tests. To implement this approach, we develop an R package to compute Z-residuals for shared frailty models. Through extensive simulation studies, we demonstrate the high power of our derived numerical test for assessing the functional form of covariates. To validate the effectiveness of our method, we apply it to a real data application concerning the modelling of survival time for acute myeloid leukemia patients. Our Z-residual diagnosis results reveal the inadequacy of log-transformation of the covariate, highlighting the limitations of other diagnostic methods for effectively assessing covariate functional form in shared frailty models.

Double Probability Integral Transform Residuals for Regression Models with Discrete Outcomes

The assessment of regression models with discrete outcomes is challenging and has many fundamental issues. With discrete outcomes, standard regression model assessment tools such as Pearson and deviance residuals do not follow the conventional reference distribution (normal) under the true model, calling into question the legitimacy of model assessment based on these tools. To fill this gap, we construct a new type of residuals for regression models with general discrete outcomes, including ordinal and count outcomes. The proposed residuals are based on two layers of probability integral transformation. When at least one continuous covariate is available, the proposed residuals closely follow a uniform distribution (or a normal distribution after transformation) under the correctly specified model. One can construct visualizations such as QQ plots to check the overall fit of a model straightforwardly, and the shape of QQ plots can further help identify possible causes of misspecification such as overdispersion. We provide theoretical justification for the proposed residuals by establishing their asymptotic properties. Moreover, in order to assess the mean structure and identify potential covariates, we develop an ordered curve as a supplementary tool, which is based on the comparison between the partial sum of outcomes and of fitted means. Through simulation, we demonstrate empirically that the proposed tools outperform commonly used residuals for various model assessment tasks. We also illustrate the workflow of model assessment using the proposed tools in data analysis. Supplementary materials for this article are available online.

Shewhart ridge profiling for the Gamma response model

ABSTRACT When product quality follows the Gamma distribution and is related to one or more covariate(s), then Gamma regression model (GRM) profiling will be used. The Gamma profiling is generally based on a maximum likelihood estimator (MLE). In GRM profiling, when two or more covariates are linearly related, the MLE-based GRM profiling is unsuitable. In this situation, an alternative to MLE-based profiling is required. So, this study develops the Gamma ridge regression profiling based on Pearson and deviance residuals. The performance of the proposed profiling is evaluated with the help of a simulation study under different conditions, where average run length is considered as the performance evaluation criterion. The simulation findings show that the Pearson residual based profiling with ridge estimator is better than the deviance residuals-based profiling with MLE as well as ridge estimator. Furthermore, we consider an application to evaluate the performance of the proposed methods.

Patient reported outcomes following whole brain radiotherapy in patients with brain metastases in NSIA-LUTH Cancer Center

BackgroundBrain metastases (BM) are a common complication in advanced cancer patients, and extremely challenging to treat. Consequently, whole brain radiotherapy (WBRT) remains the standard palliative intervention for patients with BM. The present study set to evaluate the clinical benefits of WBRT by assessing the quality of life (QoL) in WBRT-treated patients with BM, in Nigeria.MethodsThis was a prospective, longitudinal, hospital-based single-centre study. Consecutive sampling methodology was used to recruit 52 patients with BM undergoing WBRT. Patients were followed up on days 7, 30, 90 and 180 after WBRT. The EORTC QLQ-C15-PAL and EORTC QLQ-BN20 were employed to report patients’ responses. The likert scale responses were linearly converted into 0 – 100 scores, and the descriptive analysis was conducted using IBM SPSS Statistics 29.0, at 95% confidence interval, using the two-tailed t-test for continuous variables or the chi-square test for categorical values. The overall survival was calculated with the Kaplan Maier method and the difference tested with Log-rank method, considering the interval from the baseline until death or end of the study.ResultsThe study cohort was predominantly females (82.7%), and accordingly, 65.4% of the respondents had a breast primary tumor. A goodness-of-fit test yielded non-significant Chi square Pearson (p = 0.325) and Deviance (p = 1.000) residuals, indicating the best fit. The median overall survival was 180 days (~ 6 months). A total of 20 patients (38%) that survived up to 180 days reported alleviated symptoms and better functioning. A significant improvement in physical functioning (p < 0.001) and emotional functioning (p = 0.031) was reported at 180 days post WBRT, compared to baseline.ConclusionsWBRT is an effective palliative intervention in patients with BM, resulting in improved QoL. More than 50% of patients that survived ~ 3 months reported alleviation of pain, and 38% of patients that survived for ~ 6 months reported a significantly improved functioning. This demonstrated the clinical benefits of WBRT in palliative care and will add to the body of data on the use of WBRT, from Africa.

Analyzing County-Level COVID-19 Vaccination Rates in Texas: A New Lindley Regression Model

This work aims to study the factors that explain the COVID-19 vaccination rate through a generalized odd log-logistic Lindley regression model with a shape systematic component. To accomplish this, a dataset of the vaccination rate of 254 counties in the state of Texas, US, was used, and simulations were performed to investigate the accuracy of the maximum likelihood estimators in the proposed regression model. The mathematical properties investigated provide important information about the characteristics of the distribution. Diagnostic analysis and deviance residuals are addressed to examine the fit of the model. The proposed model shows effectiveness in identifying the key variables of COVID-19 vaccination rates at the county level, which can contribute to improving vaccination campaigns. Moreover, the findings corroborate with prior studies, and the new distribution is a suitable alternative model for future works on different datasets.

Patient Reported Outcomes Following Palliative Whole Brain Radiation Therapy in Patients with Brain Metastasis

Brain metastases (BM) are a common occurrence in patients with advanced cancers, and extremely challenging to treat, resulting in short survival periods. Consequently, whole brain radiation therapy (WBRT) remains the standard palliative intervention for patients with BM. The present study set to evaluate the efficacy of WBRT by assessing the quality of life (QoL) and survival outcomes in WBRT-treated patients with BM. It was hypothesized that the WBRT would enhance the QoL through alleviation of symptoms and functionality, and prolong patient survival. This was a prospective, longitudinal, hospital-based single-center study. Consecutive sampling methodology was used to recruit 52 patients with BM undergoing WBRT. Patients were followed up on days 7, 30, 90 and 180 after WBRT. The European Organization for Research and Treatment of Cancer Quality of Life Questionnaire-Core 15-Palliative care (EORTC QLQ-C15-PAL) was employed to report patient responses in a Likert scale. A descriptive analysis and multi-trait scaling correlation was computed using IBM SPSS Statistics 29.0, 95% confidence interval. The overall survival analysis (Kaplan-Meier) was also performed. The study cohort was predominantly females (82.7%), and accordingly, 65.4% of the respondents had a breast primary tumor. A Shapiro-Wilk test of normality revealed that the data was not normally distributed (sig. < 0.05), however, statistic values (W) closer to 1 suggested a good fit. A goodness-of-fit test ascertained the assumption, yielding non-significant Chi square Pearson (p = 0.325) and Deviance (p = 1.000) residuals. There was a significant correlation (p < 0.001) between physical functioning and emotional functioning. Median overall survival was 180 days (∼6 months). A total of 20 patients (38%) that survived up to 180 days reported alleviated symptoms and better functioning. A significant improvement in physical functioning (p < 0.001) and emotional functioning (p = 0.031) was reported at 180 days post WBRT, compared to baseline. Similarly, a significant improvement in visual disorders (p = 0.002), motor dysfunction (p = 0.031), and communication deficit (p = 0.001) was also reported. A significant alleviation of pain was reported at day 90 (p = 0.042) by 53% of the patients that survived. WBRT is an effective palliative intervention in patients with BM, resulting in improved QoL and prolonged survival. More than 50% of patients that survived 3 months reported a significant alleviation of pain, and 38% of patients that survived for 6 months reported a significantly improved functioning. This demonstrated the effectiveness of WBRT in palliative care and will add to the body of data on the efficacy of radiotherapy.

Deviance residual-based Shewhart control chart for monitoring Conway-Maxwell-Poisson profile under the r-k class estimator

ABSTRACT In this study, we first propose the r-k class estimator in COM-Poisson regression, and then we introduce the Shewhart control chart using the deviance residuals obtained from the r-k class estimator. The performance analysis of the new control chart over maximum likelihood, principal components regression, and ridge-based Shewhart control charts is evaluated via a numerical example and a simulation study by using average run length, the standard deviation of run length, the average alarm rate and percentile of run length criteria when different types of shifts are present in data.

Interval-Censored Regression with Non-Proportional Hazards with Applications

Proportional hazards models and, in some situations, accelerated failure time models, are not suitable for analyzing data when the failure ratio between two individuals is not constant. We present a Weibull accelerated failure time model with covariables on the location and scale parameters. By considering the effects of covariables not only on the location parameter, but also on the scale, a regression should be able to adequately describe the difference between treatments. In addition, the deviance residuals adapted for data with the interval censored and the exact time of failure proved to be satisfactory to verify the fit of the model. This information favors the Weibull regression as an alternative to the proportional hazards models without masking the effect of the explanatory variables.

Deviance and Pearson Residuals-Based Control Charts with Different Link Functions for Monitoring Logistic Regression Profiles: An Application to COVID-19 Data

In statistical process control, the control charts are an effective tool to monitor the process. When the process is examined based on an exponential family distributed response variable along with a single explanatory variable, the generalized linear model (GLM) provides better estimates and GLM-based charts are preferred. This study is designed to propose GLM-based control charts using different link functions (i.e., logit, probit, c-log-log, and cauchit) with the binary response variable. The Pearson residuals (PR)- and deviance residuals (DR)-based control charts for logistic regression are proposed under different link functions. For evaluation purposes, a simulation study is designed to evaluate the performance of the proposed control charts. The results are compared based on the average run length (ARL). Moreover, the proposed charts are implemented on a real application for COVID-19 death monitoring. The Monte Carlo simulation study and real applications show that the performance of the model-based control charts with the c-log-log link function gives a better performance as compared to model-based control charts with other link functions.

Influence Diagnostic in Log-Exponential-Inverse- Exponential {Weibull} Regression Failure Model

An exponential-inverse-exponential {Weibull} regression failure model is introduced. Some of its properties like density function, survival function, and hazard function are derived. Maximum likelihood estimates of the parameters of the new model from censored data are obtained. To assess the local influence diagnostic(s) on the parameter estimates, the appropriate matrices are derived. Also, global influence and local influence are used to detect influential observations. Martingale and Deviance residuals are obtained and used to detect outliers and evaluate the model assumptions. A real data is analyzed under Log-Exponential-Inverse-Exponential {Weibull} regression model to show the usefulness of the model. A simulation study is performed to investigate the behavior of the estimates for different sample sizes and censoring percentages.

Improving efficiency of fitting Cox proportional hazards models for time-to-event outcomes in genome-wide association studies (GWAS).

Technologies identifying single nucleotide polymorphisms (SNPs) in DNA sequencing yield an avalanche of data requiring analysis and interpretation. Standard methods may require many weeks of processing time. The use of statistical methods requiring data sorting, matrix inversions of a high-dimension and replication in subsets of the data on multiple outcomes exacerbate these times.A method which reduces the computational time in problems with time-to-event outcomes and hundreds of thousands/millions of SNPs using Cox-Snell residuals after fitting the Cox proportional hazards model (PH) to a fixed set of concomitant variables is proposed. This yields coefficients for SNP effect from a Cox-Snell adjusted Poisson model and shows a high concordance to the adjusted PH model.The method is illustrated with a sample of 10000 SNPs from a genome-wide association study in a diabetic population. The gain in processing efficiency using the proposed method based on Poisson modelling can be as high as 62%. This could result in saving of over three weeks processing time if 5 million SNPs require analysis. The method involves only a single predictor variable (SNP), offering a simpler, computationally more stable approach to examining and identifying SNP patterns associated with the outcome(s) allowing for a faster development of genetic signatures. Use of deviance residuals from the PH model to screen SNPs demonstrates a large discordance rate at a 0.2% threshold of concordance. This rate is 15 times larger than that based on the Cox-Snell residuals from the Cox-Snell adjusted Poisson model. The method is simple to implement as the procedures are available in most statistical packges. The approach involves obtaining Cox-Snell residuals from a PH model, to a binary time-to-event outcome, for factors which need to be common when assessing each SNP. Each SNP is then fitted as a predictor to the outcome of interest using a Poisson model with the Cox-Snell as the exposure variable.

Yaşam Süresi Verilerinin Çok Değişkenli Uyumlu Regresyon Analizinde Kullanımı: Simülasyon Çalışması

Objective: The multivariate adaptive regression splines (MARS) is very effective in order to model linear or nonlinear relationships. The Cox regression residuals-based MARS model, which integrates Cox regression and MARS approaches, was created to assess the relationships between efficient risk factors on the survival. The purpose of this study is to introduce the Survival-MARS (SM) model which uses the Cox-Snell, Martingale, and deviance residuals. Also,our aim is to compare the performance of the models created with residuals at different sample sizes and correlation levels with the simulation study in order to determine the most effective residual type that can be used in the SM model. Material and Methods: Performances of SM models that use CoxSnell,Martingale, and deviance residual types were compared at different sample sizes (n = 30, 100, 150, 250, 500, 1.000), with both no correlation (r = 0.00) and medium (r = 0.50) and high (r = 0.90) correlations between predictors. SM model performances were compared via minimum generalized cross-validation and the sum of mean squared error values.. Results: In all scenarios, SM models with Cox-Snell residuals have the best performance compared to other models established with other residuals. Martingale and deviance residuals were affected by high correlation and low sample sizes. Conclusion: In case of linear relationship between risk factors, SM models with Cox-Snell residuals are quite successful in explaining these relationship structures and enable the effects on the dependent variable to easily interpret.

The effect of climate on melioidosis incidence in Townsville, Australia: a dry tropical region.

Townsville is in the dry tropics in Northern Australia and an endemic region for melioidosis. Melioidosis is an infectious disease caused by Burkholderia pseudomallei, a soil dwelling organism. The incidence of melioidosis is associated with high levels of rainfall and has been linked to multiple weather variables in other melioidosis endemic regions such as in Darwin. In contrast to Townsville, Darwin is in the wet-dry tropics in Northern Australia and receives 40% more rainfall. We assessed the relationship between melioidosis incidence and weather conditions in Townsville and compared the patterns to the findings in Darwin and other melioidosis endemic regions. Performing a time series analysis from 1996 to 2020, we applied a negative binomial regression model to evaluate the link between the incidence of melioidosis in Townsville and various weather variables. Akaike's information criterion was used to assess the most parsimonious model with best predictive performance. Fourier terms and lagged deviance residuals were included to control long term seasonal trends and temporal autocorrelation. Humidity is the strongest predictor for melioidosis incidence in Townsville. Furthermore, the incidence of melioidosis showed a three-times rise in the Townsville region when >200 mm of rain fell within the fortnight. Prolonged rainfall had more impact than a heavy downpour on the overall melioidosis incident rate. There was no statistically significant increase in incidence with cloud cover in the multivariable model. Consistent with other reports, melioidosis incidence can be attributed to humidity and rainfall in Townsville. In contrast to Darwin, there was no strong link between melioidosis cases and cloud cover and nor single large rainfall events.

Profile monitoring of residuals control charts under gamma regression model

The statistical control chart is considered one of the superlative tools in quality control. Currently, control charts are being widely used in various areas, one of them being manufacturing processes. They are essential instruments that can impart crucial insights to quality controllers for maintaining productivity. The quality of a product or process can be characterized by a relationship between two or more variables, which is typically referred to as a profile. Also, public health surveillance is considered another important area that widely used control charts. In this regard, they are very useful and reliable tools for detecting outbreaks of infectious diseases. On the other hand, the gamma regression model (GRM) is a popular model considered in medical and other fields. It is applied when the response variable is continuous and positively skewed and well fitted to the gamma distribution. This paper presents a scheme for monitoring the profile. Based upon the generalized linear model (GLM) in the case of two link functions: identity and log link function. Exponentially weighted moving average control charts (EWMA) are proposed using deviance residuals and Pearson residuals for detecting any disturbance in the control variable of the gamma regression model. A detailed simulation study is designed to scrutinize and evaluate the performance of the control charts in phase I analysis and in phase II under parametric maximum likelihood estimation (MLE) using the average run length (ARL) measure. It turns out that using deviance residuals under the identity link function seems more suitable than Pearson residuals. Also, with increasing the sample size, the percentages of out-of-control (OC) samples increased which is theoretically acceptable

Influence diagnostics for the Cox proportional hazards regression model: method, simulation and applications

This article investigates the performance of several residuals for the Cox proportional hazards regression model to diagnose the influential observations. The standardized and adjusted forms of residuals are proposed for Cox proportional hazards regression model. In addition, Cook’s distance is proposed for both standardized and adjusted residuals. The assessment of different residuals for the identification of influential observations is made through the Monte Carlo simulation. A real dataset of bone marrow transplant Leukaemia is analyzed to show the benefit of the proposed methods. Simulation and application results show that the standardized and adjusted residuals based on the Cox–Snell method perform best for the detection of influential points. Furthermore, the standardized, and adjusted Martingale and deviance residuals work better when the sample size is large.

An Extended Weibull Regression for Censored Data: Application for COVID-19 in Campinas, Brazil

This work aims to study the factors that increase the risk of death of hospitalized patients diagnosed with COVID-19 through the odd log-logistic regression model for censored data with two systematic components, as well as provide new mathematical properties of this distribution. To achieve this, a dataset of individuals residing in the city of Campinas (Brazil) was used and simulations were performed to investigate the accuracy of the maximum likelihood estimators in the proposed regression model. The provided properties, such as stochastic representation, identifiability, and moments, among others, can help future research since they provide important information about the distribution structure. The simulation results revealed the consistency of the estimates for different censoring percentages and show that the empirical distribution of the modified deviance residuals converge to the standard normal distribution. The proposed model proved to be efficient in identifying the determinant variables for the survival of the individuals in this study, which can help to find more opportune treatments and medical interventions. Therefore, the new model can be considered an interesting alternative for future works that evaluate censored lifetimes.

Testing microbiome associations with survival times at both the community and individual taxon levels.

Finding microbiome associations with possibly censored survival times is an important problem, especially as specific taxa could serve as biomarkers for disease prognosis or as targets for therapeutic interventions. The two existing methods for survival outcomes, MiRKAT-S and OMiSA, are restricted to testing associations at the community level and do not provide results at the individual taxon level. An ad hoc approach testing each taxon with a survival outcome using the Cox proportional hazard model may not perform well in the microbiome setting with sparse count data and small sample sizes. We have previously developed the linear decomposition model (LDM) for testing continuous or discrete outcomes that unifies community-level and taxon-level tests into one framework. Here we extend the LDM to test survival outcomes. We propose to use the Martingale residuals or the deviance residuals obtained from the Cox model as continuous covariates in the LDM. We further construct tests that combine the results of analyzing each set of residuals separately. Finally, we extend PERMANOVA, the most commonly used distance-based method for testing community-level hypotheses, to handle survival outcomes in a similar manner. Using simulated data, we showed that the LDM-based tests preserved the false discovery rate for testing individual taxa and had good sensitivity. The LDM-based community-level tests and PERMANOVA-based tests had comparable or better power than MiRKAT-S and OMiSA. An analysis of data on the association of the gut microbiome and the time to acute graft-versus-host disease revealed several dozen associated taxa that would not have been achievable by any community-level test, as well as improved community-level tests by the LDM and PERMANOVA over those obtained using MiRKAT-S and OMiSA. Unlike existing methods, our new methods are capable of discovering individual taxa that are associated with survival times, which could be of important use in clinical settings.

Development and Modeling of Decision Tree for Survival Data with Multiple Events Using Deviance and Cox-Snell Residuals within Node Homogeneity Technique

It is very common in medical studies for a patient to experience more than one event rather than one of interest. This led to exposing an individual to multiple risks and medical practitioners need to account for these risks concerning some prognostic factors. There are many methods of dealing with multiple events in survival data classically, however, these methods break down when considering the top-down effect of the prognostic factors concurrently and when the risks of events are correlated (competing risks). This study aimed to develop a decision tree using a within-node homogeneity procedure in survival analysis with multiple events to classify individual risks for the competing risks. Since the CART methodology involves recursive portioning of covariates into different subgroups, this study considers the use of Deviance and Modified Cox-Snell residuals as a measure of impurity in the Classification Regression Tree (CART) during the process of partitioning. The flexibility and predictive accuracy of our learning algorithm would then be compared with other existing methods through simulation and the freely available online real-life data. The results of the simulation revealed that: using Deviance and Cox-Snell residuals as a response within the node homogeneity classification tree performs better than using other residuals irrespective of performance indices. Results from empirical studies of the two real-life data that the proposed model with Cox-Snell residual (Deviance=16.6498) performs better than both the Martingale residual (deviance=160.3592) and Deviance residual (Deviance=556.8822). Conclusively, using Cox-Snell residual (Mean Square Error (MSE)=0.01783563) as a measure of impurity in CART revealed improved performance than using any other residual methods (MSE=0.1853148, 0.8043366). This implies that the proposed methods have the capability of accounting for individual effects based on the prognostic biomarkers.