Estimating Haplotype Effects for Survival Data

Summary Genetic association studies often investigate the effect of haplotypes on an outcome of interest. Haplotypes are not observed directly, and this complicates the inclusion of such effects in survival models. We describe a new estimating equations approach for Cox's regression model to assess haplotype effects for survival data. These estimating equations are simple to implement and avoid the use of the EM algorithm, which may be slow in the context of the semiparametric Cox model with incomplete covariate information. These estimating equations also lead to easily computable, direct estimators of standard errors, and thus overcome some of the difficulty in obtaining variance estimators based on the EM algorithm in this setting. We also develop an easily implemented goodness‐of‐fit procedure for Cox's regression model including haplotype effects. Finally, we apply the procedures presented in this article to investigate possible haplotype effects of the PAF‐receptor on cardiovascular events in patients with coronary artery disease, and compare our results to those based on the EM algorithm.     

Fitting Survival Data to a Piecewise Linear Hazard Rate in the Presence of Covariates

This paper extends the work of KODLIN (1967), who proposed a method for analyzing patient survival data wherein the hazard rate was linearly related to the survival time. The present paper extends Kodlin's model to permit maximum likelihood estimation of the parameters so that covariate effects are included and the slope and intercept parameters are allowed to change over fixed intervals of the time domain of study. An illustration of the method using multiple myeloma data is given and the results are compared with those of Kodlin's model, the Feigl-Zelen, Zippin-Armitage model, the exponential model, and Cox's proportional hazards model.     

Regression on Quantile Residual Life

Summary A time‐specific log‐linear regression method on quantile residual lifetime is proposed. Under the proposed regression model, any quantile of a time‐to‐event distribution among survivors beyond a certain time point is associated with selected covariates under right censoring. Consistency and asymptotic normality of the regression estimator are established. An asymptotic test statistic is proposed to evaluate the covariate effects on the quantile residual lifetimes at a specific time point. Evaluation of the test statistic does not require estimation of the variance–covariance matrix of the regression estimators, which involves the probability density function of the survival distribution with censoring. Simulation studies are performed to assess finite sample properties of the regression parameter estimator and test statistic. The new regression method is applied to a breast cancer data set with long‐term follow‐up to estimate the patients' median residual lifetimes, adjusting for important prognostic factors.     

Analysis of Dichotomized Factorial Data Using Cox's Proportional Hazards Model and Related Tests

This paper describes how Cox's Proportional Hazards model may be used to analyze dichotomized factorial data obtained from a right-censored epidemiological study where time to response is of interest. Exact maximum likelihood estimates of the relative mortality rates are derived for any number of prognostic factors, but for the sake of simplicity, the mathematical details are presented for the case of two factors. This method is not based on the life table procedure. Kaplan-Meier estimates are obtained for the survival function of the internal control population, Which are in turn used to determine the expected number of deaths in the study population. The asymptotic (large sample) joint sampling distribution of the relative mortality rates is derived and some relevant simultaneous and conditional statistical tests are discussed. The relative mortality rates of several prognostic factors may be jointly considered as the multivariate extension of the familiar standard mortality ratio (SMR) of epidemiological studies. A numerical example is discussed to illustrate the method.     

A Comparison Between Several Goodness of Fit Tests for Cox's Model

A comparison is made between two approaches to testing goodness of fit of Cox's regression model for survival data. The first approach is based on the inclusion of time dependent covariates, whereas the second one is based on the autocovariance of successive contributions to the derivative of the loglikelihood. It appears that the second test is most appropriate for testing in situations where the structure of the departure from proportional hazards is not known a priori. An approximate expression for the relative efficiency of the two test procedures is presented.     

Cox Regression in Nested Case–Control Studies with Auxiliary Covariates

Summary Nested case–control (NCC) design is a popular sampling method in large epidemiological studies for its cost effectiveness to investigate the temporal relationship of diseases with environmental exposures or biological precursors. Thomas' maximum partial likelihood estimator is commonly used to estimate the regression parameters in Cox's model for NCC data. In this article, we consider a situation in which failure/censoring information and some crude covariates are available for the entire cohort in addition to NCC data and propose an improved estimator that is asymptotically more efficient than Thomas' estimator. We adopt a projection approach that, heretofore, has only been employed in situations of random validation sampling and show that it can be well adapted to NCC designs where the sampling scheme is a dynamic process and is not independent for controls. Under certain conditions, consistency and asymptotic normality of the proposed estimator are established and a consistent variance estimator is also developed. Furthermore, a simplified approximate estimator is proposed when the disease is rare. Extensive simulations are conducted to evaluate the finite sample performance of our proposed estimators and to compare the efficiency with Thomas' estimator and other competing estimators. Moreover, sensitivity analyses are conducted to demonstrate the behavior of the proposed estimator when model assumptions are violated, and we find that the biases are reasonably small in realistic situations. We further demonstrate the proposed method with data from studies on Wilms' tumor.     

Statistical Methods for Analyzing Right‐Censored Length‐Biased Data under Cox Model

Summary Length‐biased time‐to‐event data are commonly encountered in applications ranging from epidemiological cohort studies or cancer prevention trials to studies of labor economy. A longstanding statistical problem is how to assess the association of risk factors with survival in the target population given the observed length‐biased data. In this article, we demonstrate how to estimate these effects under the semiparametric Cox proportional hazards model. The structure of the Cox model is changed under length‐biased sampling in general. Although the existing partial likelihood approach for left‐truncated data can be used to estimate covariate effects, it may not be efficient for analyzing length‐biased data. We propose two estimating equation approaches for estimating the covariate coefficients under the Cox model. We use the modern stochastic process and martingale theory to develop the asymptotic properties of the estimators. We evaluate the empirical performance and efficiency of the two methods through extensive simulation studies. We use data from a dementia study to illustrate the proposed methodology, and demonstrate the computational algorithms for point estimates, which can be directly linked to the existing functions in S‐PLUS or R .     

Estimating a Treatment Effect in Residual Time Quantiles Under the Additive Hazards Model

For randomized clinical trials where the endpoint of interest is a time-to-event subject to censoring, estimating the treatment effect has mostly focused on the hazard ratio from the Cox proportional hazards model. Since the model’s proportional hazards assumption is not always satisfied, a useful alternative, the so-called additive hazards model, may instead be used to estimate a treatment effect on the difference of hazard functions. Still, the hazards difference may be difficult to grasp intuitively, particularly in a clinical setting of, e.g., patient counseling, or resource planning. In this paper, we study the quantiles of a covariate’s conditional survival function in the additive hazards model. Specifically, we estimate the residual time quantiles, i.e., the quantiles of survival times remaining at a given time t, conditional on the survival times greater than t, for a specific covariate in the additive hazards model. We use the estimates to translate the hazards difference into the difference in residual time quantiles, which allows a more direct clinical interpretation. We determine the asymptotic properties, assess the performance via Monte-Carlo simulations, and demonstrate the use of residual time quantiles in two real randomized clinical trials.     

On Estimating Stratified PH Model with Single Covariate from Sparse Data with Application to Brain Metastases Study

The estimation of the unknown parameters in the stratified Cox's proportional hazard model is a typical example of the trade‐off between bias and precision. The stratified partial likelihood estimator is unbiased when the number of strata is large but suffer from being unstable when many strata are non‐informative about the unknown parameters. The estimator obtained by ignoring the heterogeneity among strata, on the other hand, increases the precision of estimates although pays the price for being biased. An estimating procedure, based on the asymptotic properties of the above two estimators, serving to compromise between bias and precision is proposed. Two examples in a radiosurgery for brain metastases study provide some interesting demonstration of such applications.     

Estimation of a Joint Distribution Function with Multivariate Interval‐Censored Data when the Nonparametric MLE is not Unique

A nonparametric estimator of a joint distribution function F₀ of a d‐dimensional random vector with interval‐censored (IC) data is the generalized maximum likelihood estimator (GMLE), where d ≥ 2. The GMLE of F₀ with univariate IC data is uniquely defined at each follow‐up time. However, this is no longer true in general with multivariate IC data as demonstrated by a data set from an eye study. How to estimate the survival function and the covariance matrix of the estimator in such a case is a new practical issue in analyzing IC data. We propose a procedure in such a situation and apply it to the data set from the eye study. Our method always results in a GMLE with a nonsingular sample information matrix. We also give a theoretical justification for such a procedure. Extension of our procedure to Cox's regression model is also mentioned.     

A latent class model to multiply impute missing treatment indicators in observational studies when inferences of the treatment effect are made using propensity score matching

Analysts often estimate treatment effects in observational studies using propensity score matching techniques. When there are missing covariate values, analysts can multiply impute the missing data to create m completed data sets. Analysts can then estimate propensity scores on each of the completed data sets, and use these to estimate treatment effects. However, there has been relatively little attention on developing imputation models to deal with the additional problem of missing treatment indicators, perhaps due to the consequences of generating implausible imputations. However, simply ignoring the missing treatment values, akin to a complete case analysis, could also lead to problems when estimating treatment effects. We propose a latent class model to multiply impute missing treatment indicators. We illustrate its performance through simulations and with data taken from a study on determinants of children's cognitive development. This approach is seen to obtain treatment effect estimates closer to the true treatment effect than when employing conventional imputation procedures as well as compared to a complete case analysis.     

Estimation in the Cox survival regression model with covariate measurement error and a changepoint

The Cox regression model is a popular model for analyzing the relationship between a covariate vector and a survival endpoint. The standard Cox model assumes a constant covariate effect across the entire covariate domain. However, in many epidemiological and other applications, the covariate of main interest is subject to a threshold effect: a change in the slope at a certain point within the covariate domain. Often, the covariate of interest is subject to some degree of measurement error. In this paper, we study measurement error correction in the case where the threshold is known. Several bias correction methods are examined: two versions of regression calibration (RC1 and RC2, the latter of which is new), two methods based on the induced relative risk under a rare event assumption (RR1 and RR2, the latter of which is new), a maximum pseudo-partial likelihood estimator (MPPLE), and simulation-extrapolation (SIMEX). We develop the theory, present simulations comparing the methods, and illustrate their use on data concerning the relationship between chronic air pollution exposure to particulate matter PM₁₀ and fatal myocardial infarction (Nurses Health Study (NHS)), and on data concerning the effect of a subject's long-term underlying systolic blood pressure level on the risk of cardiovascular disease death (Framingham Heart Study (FHS)). The simulations indicate that the best methods are RR2 and MPPLE.     

Randomization and Additivity in the Two-Period Crossover Clinicial Trial

In certain areas of medical research, the two-period crossover design is a frequent choice for comparing treatments A and B in a randomized clinical trial. Earlier work by Grizzle and by Brown was based upon a parametric theory linear model. Recently, the present authors employed D. R. Cox's additive randomization models and, for the case of zero residual effect, found a discrepancy between it and the parametric model with respect to the precision of period effects. In the present note, this divergence is accounted for by allowing for the possibility of non-additivity through the use of a completely general randomization model. It is concluded that the structure of the crossover design is such that use of the parametric theory linear model is required if a single, consistent model is desired.     

Estimating Black Bear Density Using DNA Data From Hair Snares

ABSTRACT DNA-based mark-recapture has become a methodological cornerstone of research focused on bear species. The objective of such studies is often to estimate population size; however, doing so is frequently complicated by movement of individual bears. Movement affects the probability of detection and the assumption of closure of the population required in most models. To mitigate the bias caused by movement of individuals, population size and density estimates are often adjusted using ad hoc methods, including buffering the minimum polygon of the trapping array. We used a hierarchical, spatial capture-recapture model that contains explicit components for the spatial-point process that governs the distribution of individuals and their exposure to (via movement), and detection by, traps. We modeled detection probability as a function of each individual's distance to the trap and an indicator variable for previous capture to account for possible behavioral responses. We applied our model to a 2006 hair-snare study of a black bear (Ursus americanus) population in northern New York, USA. Based on the microsatellite marker analysis of collected hair samples, 47 individuals were identified. We estimated mean density at 0.20 bears/km². A positive estimate of the indicator variable suggests that bears are attracted to baited sites; therefore, including a trap-dependence covariate is important when using bait to attract individuals. Bayesian analysis of the model was implemented in WinBUGS, and we provide the model specification. The model can be applied to any spatially organized trapping array (hair snares, camera traps, mist nests, etc.) to estimate density and can also account for heterogeneity and covariate information at the trap or individual level.     

Cox regression with survival-time-dependent missing covariate values

Analysis with time-to-event data in clinical and epidemiological studies often encounters missing covariate values, and the missing at random assumption is commonly adopted, which assumes that missingness depends on the observed data, including the observed outcome which is the minimum of survival and censoring time. However, it is conceivable that in certain settings, missingness of covariate values is related to the survival time but not to the censoring time. This is especially so when covariate missingness is related to an unmeasured variable affected by the patient's illness and prognosis factors at baseline. If this is the case, then the covariate missingness is not at random as the survival time is censored, and it creates a challenge in data analysis. In this article, we propose an approach to deal with such survival-time-dependent covariate missingness based on the well known Cox proportional hazard model. Our method is based on inverse propensity weighting with the propensity estimated by nonparametric kernel regression. Our estimators are consistent and asymptotically normal, and their finite-sample performance is examined through simulation. An application to a real-data example is included for illustration.     

The Effect of Temperature on Growth and Mineral Nutrition of Fruits of Apple, with Special Reference to Calcium

Growth and mineral composition of fruits in potted trees were studied at two temperatures (19 and 24°C) in two experiments with Golden Delicious and one with Cox's Orange Pippin under a controlled environment. In Golden Delicious the high temperature promoted fruit growth markedly, but in Cox's Orange Pippin the stimulation was much less pronounced. The levels of K, N, Mg, and P (expressed as amounts per fruit) were increased at the higher temperature. This increase was almost completely the result of differences in fruit growth. With respect to Ca the high temperature reduced Ca influx in one Golden Delicious experiment and had hardly any effect in the other, but favoured influx in Cox's Orange Pippin. In one Golden Delicious experiment the temperature was raised 7 weeks after blossoming, which resulted in a temporary drop of Ca. When the same temperature change was applied 4 weeks earlier, Ca continued to increase. It is assumed that mineral influx occurs via the xylem during the first few weeks after fruit set and via the phloem throughout. The results are discussed in terms of differences in mobility in the phloem for the various minerals and of a shift in the ratio between xylem and phloem movement induced by differences in the growth rate of the fruits.     

On the Estimation of Relative Risks and Longevities of a Class of Cancer Patients Based on Randomly Censored Data

An attempt has been made to measure the relative risks and longevities of a group of cancer patients using a Weibull model whose parameters are function of covariatcs (viz treatment and condition of the patient). The methodology is Cox's Partial Likelihood and the Maximum Likelihood techniques for randomly censored data. The data consists of the records of 134 patients under a clinical trial in the treatment of Carcinoma of the Oro Pharynx as quoted by Kalbfleisch and Prentice (1980).     

Validation of the alternating conditional estimation algorithm for estimation of flexible extensions of Cox's proportional hazards model with nonlinear constraints on the parameters

Standard optimization algorithms for maximizing likelihood may not be applicable to the estimation of those flexible multivariable models that are nonlinear in their parameters. For applications where the model's structure permits separating estimation of mutually exclusive subsets of parameters into distinct steps, we propose the alternating conditional estimation (ACE) algorithm. We validate the algorithm, in simulations, for estimation of two flexible extensions of Cox's proportional hazards model where the standard maximum partial likelihood estimation does not apply, with simultaneous modeling of (1) nonlinear and time‐dependent effects of continuous covariates on the hazard, and (2) nonlinear interaction and main effects of the same variable. We also apply the algorithm in real‐life analyses to estimate nonlinear and time‐dependent effects of prognostic factors for mortality in colon cancer. Analyses of both simulated and real‐life data illustrate good statistical properties of the ACE algorithm and its ability to yield new potentially useful insights about the data structure.     

Using aggregate data to estimate the standard error of a treatment–covariate interaction in an individual patient data meta‐analysis

Subgroup analyses are important to medical research because they shed light on the heterogeneity of treatment effectts. A treatment–covariate interaction in an individual patient data (IPD) meta‐analysis is the most reliable means to estimate how a subgroup factor modifies a treatment's effectiveness. However, owing to the challenges in collecting participant data, an approach based on aggregate data might be the only option. In these circumstances, it would be useful to assess the relative efficiency and power loss of a subgroup analysis without patient‐level data. We present methods that use aggregate data to estimate the standard error of an IPD meta‐analysis’ treatment–covariate interaction for regression models of a continuous or dichotomous patient outcome. Numerical studies indicate that the estimators have good accuracy. An application to a previously published meta‐regression illustrates the practical utility of the methodology.     

The response of apples cv. Cox's Orange Pippin to different concentrations of oxygen in the storage atmosphere

The response of apples cv. Cox's Orange Pippin to storage in flowing streams of air and nitrogen mixed to generate atmospheres ranging from 0.5% to 2% oxygen was examined in two seasons. The benefits of reducing store oxygen concentrations below 1.25% were small but storage in 0.75% may be advantageous for Cox's Orange Pippin apples in seasons when flesh firmness is low at harvest. Ethanol-induced flesh breakdown was recorded in samples stored in 0.5% O₂ in one season and the risk of low temperature breakdown was increased in oxygen levels below 1.25%. Ethylene production was reduced by lowering the oxygen concentration.