Proportional Hazard Model Assumption Research Articles

Sample size calculation for mixture model based on geometric average hazard ratio and its applications to nonproportional hazard.

With the advent of cancer immunotherapy, some special features including delayed treatment effect, cure rate, diminishing treatment effect and crossing survival are often observed in survival analysis. They violate the proportional hazard model assumption and pose a unique challenge for the conventional trial design and analysis strategies. Many methods like cure rate model have been developed based on mixture model to incorporate some of these features. In this work, we extend the mixture model to deal with multiple non-proportional patterns and develop its geometric average hazard ratio (gAHR) to quantify the treatment effect. We further derive a sample size and power formula based on the non-centrality parameter of the log-rank test and conduct a thorough analysis of the impact of each parameter on performance. Simulation studies showed a clear advantage of our new method over the proportional hazard based calculation across different non-proportional hazard scenarios. Moreover, the mixture modeling of two real trials demonstrates how to use the prior information on the survival distribution among patients with different biomarker and early efficacy results in practice. By comparison with a simulation-based design, the new method provided a more efficient way to compute the power and sample size with high accuracy of estimation. Overall, both theoretical derivation and empirical studies demonstrate the promise of the proposed method in powering future innovative trial designs.

Communicating and understanding statistical measures when quantifying the between-group difference in competing risks.

Competing risks issues are common in clinical trials and epidemiological studies for patients in follow-up who may experience a variety of possible outcomes. Under such competing risks, two hazard-based statistical methods, cause-specific hazard (CSH) and subdistribution hazard (SDH), are frequently used to assess treatment effects among groups. However, the outcomes of the CSH-based and SDH-based methods have a close connection with the proportional hazards (CSH or SDH) assumption and may have an non-intuitive interpretation. Recently, restricted mean time lost (RMTL) has been used as an alternative summary measure for analysing competing risks, due to its clinical interpretability and robustness to the proportional hazards assumption. Considering the above approaches, we summarize the differences between hazard-based and RMTL-based methods from the aspects of practical interpretation, proportional hazards model assumption and the selection of restricted time points, and propose corresponding suggestions for the analysis of between-group differences under competing risks. Moreover, an R package 'cRMTL' and corresponding step-by-step guidance are available to help users for applying these approaches.

Borrowing Concurrent Information from Non-Concurrent Control to Enhance Statistical Efficiency in Platform Trials

In this paper, considering time-to-event endpoints under the proportional hazard model assumption, we introduce a new concept of NCC concurrent observation time (NCC COT), and propose to borrow NCC COT through left truncation. This assumes that the NCC COT and CC are comparable. If the protocol does not prohibit NCC patients to change the standard of care while on study, NCC COT and CC likely will share the same standard of care. A simulated example is provided to demonstrate the approach. Using exponential distributions, the simulated example assumes that NCC COT and CC have the same hazard, and the treatment group has a lower hazard. The estimated HR comparing treatment to the pooled control group is 0.744 (95% CI 0.575, 0.962), whereas the comparison to the CC group alone is 0.755 (95% CI 0.566, 1.008), with corresponding p-values of 0.024 versus 0.057, respectively. This suggests that borrowing NCC COT can improve statistical efficiency when the exchangeability assumption holds. This article proposes an innovative approach of borrowing NCC COT to enhance statistical inference in platform trials under appropriate scenarios.

Time to complementary feeding initiation and its predictors among children aged 9-23 months in Meket District, Northeast Ethiopia: a Cox Weibull regression.

.Globally and nationally, only 64⋅5 and 49⋅2 % of infants received solid or semi-solid foods, respectively. The available evidence indicates that the time to initiate complementary feeding practices is still poor and varies by region. The aim of the present study was to assess the time to initiation of complementary feeding and its predictors among children aged 9-23 months in Meket District, Amhara Region, Ethiopia. A community-based retrospective cohort study was conducted from June to July 2022 among 459 systematically selected mothers/caregivers with their children from 9 to 23 months of age. The result of descriptive statistics was reported by table, frequency, Kaplan-Meier curve and percent. The proportional hazard model assumption was checked, and a Weibull regression model was used to see the predictors of timely initiation of complementary feeding. An adjusted hazard ratio with a 95 % confidence interval and a P-value of 0⋅05 were used to declare the significant predictors. The median time of complementary feeding initiation was 6 months. Attending primary education (adjusted hazard ration (AHR) 1⋅8; 95 % CI 1⋅16, 2⋅78), occupation of the mother (AHR 1⋅43; 95 % CI 1⋅04, 1⋅95), home delivery (AHR 1⋅61; 95 % CI 1⋅09, 2⋅37) and birth preparedness (AHR 1⋅37; 95 % CI 1⋅03, 1⋅81) were the predictors of time to complementary feeding initiation. The median time to complementary feeding initiation was consistent with the WHO recommendation. Maternal education, maternal employment, place of delivery and birth preparedness were the predictors of time to initiation of complementary feeding. Therefore, working with the education sector, increasing the delivery rate in health facilities, strengthening counselling on birth preparation, increasing maternity leave until 6 months of age and initiating corner feeding should be part of the complementary feeding practices promotion agenda.

Incidence of anemia and predictors among Human Immunodeficiency Virus-infected children on antiretroviral therapy at public health facilities of Bahir Dar City, Northwest Ethiopia: multicenter retrospective follow up study

BackgroundAnemia is one of the common hematological problems among HIV-infected children. It impairs physical functioning, affects the quality of life, increases HIV progression, and decreases survival of HIV-infected children. In Ethiopia, limited studies were conducted on the incidence and predictors of anemia among HIV-infected children on antiretroviral therapy (ART). Therefore, this study aims to assess the incidence of anemia and predictors among HIV- infected children on ART at public health facilities of Bahir Dar City, Northwest Ethiopia.MethodsAn institution-based retrospective follow-up study was conducted among 403 HIV- infected children who have followed at ART clinics in public health facilities of Bahir Dar City from 2010 to 2020. A simple random sampling technique was employed to select the study units. Data was entered using Epi-data version 4.6 and analyzed using STATA 14.0. Cox proportional hazard model assumption was checked graphically and by scaled Schoenfeld residual test. Bivariable Cox-proportional hazards regression model was employed for each explanatory variable to check the association with the outcome variable. Variables with a p-value of < 0.2 in the bivariable analysis were candidates to the multivariable proportional hazard model. Cox proportional hazards model was used at a 5% level of significance to identify predictors of anemia.ResultsThe overall follow up time was 1587 person–years. The overall incidence density of anemia was 6.87 with 95% confidence interval (CI) = (5.60, 8.16) per 100 person-years. The independent predictors show an association were child age from 0.25 to 5 years adjusted hazard ratio (AHR) = (1.83; 95% CI = 1.22, 2.77), World health organization clinical stage III and IV (AHR = 1.80; 95% CI = 1.22, 2.67), being underweight (AHR = 1.5; 95% CI = 1.01, 2.26), having fair/poor adherence to anti-retroviral therapy (AHR = 1.75; 95% CI = 1.08, 2.85) and zidovidine based anti -retroviral therapy regimen (AHR = 1.72; 95% CI = 1.12, 2.64).ConclusionThe overall incidence rate of anemia was high compared to other country reports. Age, clinical, and ART-related variables provoked the incidence of anemia. Therefore, a need to emphasize the younger age group, prevent and manage opportunistic infections of WHO clinical stage III and IV, and select and monitor appropriate ART regimen types.

Time to recovery and predictors of severe community-acquired pneumonia among pediatric patients in Debre Markos referral hospital, North West Ethiopia: A retrospective follow-up study.

Globally, pneumonia is a major cause of morbidity and mortality among children which leads to over 156 million episodes and 14.9 million hospitalizations each year. Besides this fact, the recovery time and predictors of children's hospitalization related to severe community-acquired pneumonia is not well known. Therefore, the aim of this study was to estimate the median time to recovery and its predictors among severe community-acquired pneumonia patients admitted to the pediatric ward, Debre Markos referral hospital, North West Ethiopia. An institution-based retrospective follow-up study was employed among 352 records of children who were admitted starting from January 2016 to December 2018. Patients' charts were retrieved using a structured data extraction tool. Cox proportional hazard model assumption and model fitness was checked. Stratified Cox regression was fitted as a final model. Hazard ratio with its 95% confidence interval was used and P-value < 0.05 was considered as a statistically significant association. The overall median recovery time was 4 days IQR (3-7). Recovery rate from severe community acquired pneumonia was 16.25 (95% CI: 14.54-18.15) per 100 person day observation. Age (AHR; 0.94 95% CI (0.90-0.98)), being stunted (AHR; 0.62 95% CI (0.43-0.91)), presence of danger sign at admission (AHR; 0.61 95% CI (0.40-0.94)), late presentation to seek care(AHR; 0.64 95% CI (0.47-0.88)) and co-morbidity (AHR; 0.45 95% CI (0.35-0.58)) were significant predictors of recovery time. The median recovery time from severe community-acquired pneumonia was long so that measures to reduce recovery time should be strengthened.

Cancer immunotherapy trial design with cure rate and delayed treatment effect.

Cancer immunotherapy trials have two special features: a delayed treatment effect and a cure rate. Both features violate the proportional hazard model assumption and ignoring either one of the two features in an immunotherapy trial design will result in substantial loss of statistical power. To properly design immunotherapy trials, we proposed a piecewise proportional hazard cure rate model to incorporate both delayed treatment effect and cure rate into the trial design consideration. A sample size formula is derived for a weighted log-rank test under a fixed alternative hypothesis. The accuracy of sample size calculation using the new formula is assessed and compared with the existing methods via simulation studies. A real immunotherapy trial is used to illustrate the study design along with practical consideration of balance between sample size and follow-up time.

Marginal distribution test for checking proportional hazards model assumption

Let Z be the covariate vector and Y be the response variable with the joint cumulative distribution function FY,Z. Given a random sample from FY,Z, in order to analyze the data based on a certain proportional hazards (PH) model, say Θ0, one needs to test the null hypothesis H0: FY,Z∈Θ0 first. The existing tests to achieve this task make use of the residuals and are invalid in certain situations, such as when FY,Z is not from any PH model. To overcome this disadvantage, we propose a valid model checking test of H0. It is based on the weighted average of the difference between two estimators of the marginal distribution of the response variable: its non-parametric maximum likelihood estimator and its estimator under Θ0. This test is called the marginal distribution (MD) test. We give the theoretical justification of the MD test. The simulation study suggests that the MD test is always valid, whereas the existing tests may be invalid and they are often unlikely to reject the wrong PH model assumption when they are not valid.

A new test for single against competing risks models in duration analysis

ABSTRACTThis paper shows that the single-risk duration model with two event types is a limiting case of bivariate dependent competing risks model, where the joint distribution of event times are degenerate. Then a new test is proposed for the null hypothesis of single risk against dependent competing risks model under the proportional hazard model assumption.

Partially linear single index Cox regression model in nested case-control studies

The nested case-control (NCC) design is widely used in epidemiologic studies as a cost-effective subcohort sampling method to study the association between a disease and its potential risk factors. NCC data are commonly analyzed using Thomas’ partial likelihood approach under the Cox proportional hazards model assumption. However, the linear modeling form in the Cox model may be insufficient for practical applications, especially when there are a large number of risk factors under investigation. In this paper, we consider a partially linear single index proportional hazards model, which includes a linear component for covariates of interest to yield easily interpretable results and a nonparametric single index component to adjust for multiple confounders effectively. We propose to approximate the nonparametric single index function by polynomial splines and estimate the parameters of interest using an iterative algorithm based on the partial likelihood. Asymptotic properties of the resulting estimators are established. The proposed methods are evaluated using simulations and applied to an NCC study of ovarian cancer.

Testing for Single Against Competing Risks Models in Survival Analysis

In the competing risks model the type and time of the first event are observed and the other latent event is unobserved. This model is widely used in engineering, biology and, recently, social sciences. However, if types of an event are inessential, time to an event can be analyzed more accurately using a single risk model than a competing risks model. Specification testing for single and competing risks model would be of importance, since these two models are difficult to identify empirically. We then propose the Lagrange multiplier test for the null hypothesis of single risk against dependent competing risks model under the proportional hazard model assumption. The test utilizes the fact that single risk model is a limiting case of competing risks model where bivariate survival distributions are perfectly correlated.

Sample size calculation for the proportional hazards cure model

In clinical trials with time-to-event endpoints, it is not uncommon to see a significant proportion of patients being cured (or long-term survivors), such as trials for the non-Hodgkins lymphoma disease. The popularly used sample size formula derived under the proportional hazards (PH) model may not be proper to design a survival trial with a cure fraction, because the PH model assumption may be violated. To account for a cure fraction, the PH cure model is widely used in practice, where a PH model is used for survival times of uncured patients and a logistic distribution is used for the probability of patients being cured. In this paper, we develop a sample size formula on the basis of the PH cure model by investigating the asymptotic distributions of the standard weighted log-rank statistics under the null and local alternative hypotheses. The derived sample size formula under the PH cure model is more flexible because it can be used to test the differences in the short-term survival and/or cure fraction. Furthermore, we also investigate as numerical examples the impacts of accrual methods and durations of accrual and follow-up periods on sample size calculation. The results show that ignoring the cure rate in sample size calculation can lead to either underpowered or overpowered studies. We evaluate the performance of the proposed formula by simulation studies and provide an example to illustrate its application with the use of data from a melanoma trial.

Survival estimation through the cumulative hazard function with monotone natural cubic splines

In this paper we explore the estimation of survival probabilities via a smoothed version of the survival function, in the presence of censoring. We investigate the fit of a natural cubic spline on the cumulative hazard function under appropriate constraints. Under the proposed technique the problem reduces to a restricted least squares one, leading to convex optimization. The approach taken in this paper is evaluated and compared via simulations to other known methods such as the Kaplan Meier and the logspline estimator. Our approach is easily extended to address estimation of survival probabilities in the presence of covariates when the proportional hazards model assumption holds. In this case the method is compared to a restricted cubic spline approach that involves maximum likelihood. The proposed approach can be also adjusted to accommodate left censoring.

90Y/177Lu-DOTATATE therapy: survival of the fittest?

90Y/177Lu-DOTATATE therapy: survival of the fittest?

Exponential tilt models for two-group comparison with censored data

We study application of the Exponential Tilt Model (ETM) to compare survival distributions in two groups. The ETM assumes a parametric form for the density ratio of the two distributions. It accommodates a broad array of parametric models such as the log-normal and gamma models and can be sufficiently flexible to allow for crossing hazard and crossing survival functions. We develop a nonparametric likelihood approach to estimate ETM parameters in the presence of censoring and establish related asymptotic results. We compare the ETM to the Proportional Hazards Model (PHM) in simulation studies. When the proportional hazards assumption is not satisfied but the ETM assumption is, the ETM has better power for testing the hypothesis of no difference between the two groups. And, importantly, when the ETM relation is not satisfied but the PHM assumption is, the ETM can still have power reasonably close to that of the PHM. Application of the ETM is illustrated by a gastrointestinal tumor study.

Graphical comparison of cumulative hazards for two populations

SUMMARY We consider graphs, confidence procedures and tests that can be used with censored survival data to compare the hazard experience of a treatment group with that of a control group. In particular, we consider the relative change A(t) in a cumulative rate function which is used to define a generalized proportional hazard model. This model is equivalent to the model obtained by introducing a gamma distributed frailty term in the proportional hazard model. We introduce an estimate A(t) or A(t) based on the Kaplan-Meier and Nelson-Aalen estimates, and obtain the asymptotic distribution of the process A(t) - A(t) appropriately scaled. This result is used to construct asymptotic pointwise confidence intervals for A(t). Under the proportional hazard model assumption, Doob's transformation of Gaussian processes is used to derive asymptotic simultaneous confidence bands for A(t) from a general class of test statistics defined as sup norms of weighted differences between A(t) and its estimate A(t). We also consider procedures based on estimates r(u) of the relative trend function r(u) between cumulative hazards. The methods are illustrated on a data set from a study of the effect of radiation on mice, and on a data set from a study of the effect of tumour size on the survival of cancer patients.