Analysis of longitudinal binary data from multiphase sampling

The efficient use of surrogate or auxiliary information has been investigated within both model-based and design-based approaches to data analysis, particularly in the context of missing data. Here we consider the use of such data in epidemiological studies of disease incidence in which surrogate measures of disease status are available for all subjects at two time points, but definitive diagnoses are available only in stratified subsamples. We briefly review methods for the analysis of two-phase studies of disease prevalence at a single time point, and we discuss the extension of four of these methods to the analysis of incidence studies. Their performance is compared with special reference to a study of the incidence of senile dementia.     

Assessment of the information theory approach to evaluating time-to-event surrogate and true endpoints in a meta-analytic setting

In many disease areas, commonly used long-term clinical endpoints are becoming increasingly difficult to implement due to long follow-up times and/or increased costs. Shorter-term surrogate endpoints are urgently needed to expedite drug development, the evaluation of which requires robust and reliable statistical methodology to drive meaningful clinical conclusions about the strength of relationship with the true long-term endpoint. This paper uses a simulation study to explore one such previously proposed method, based on information theory, for evaluation of time to event surrogate and long-term endpoints, including the first examination within a meta-analytic setting of multiple clinical trials with such endpoints. The performance of the information theory method is examined for various scenarios including different dependence structures, surrogate endpoints, censoring mechanisms, treatment effects, trial and sample sizes, and for surrogate and true endpoints with a natural time-ordering. Results allow us to conclude that, contrary to some findings in the literature, the approach provides estimates of surrogacy that may be substantially lower than the true relationship between surrogate and true endpoints, and rarely reach a level that would enable confidence in the strength of a given surrogate endpoint. As a result, care is needed in the assessment of time to event surrogate and true endpoints based only on this methodology.     

Potential surrogate endpoints in cancer research – some considerations and examples

We present an introductory survey of the use of surrogates in cancer research, in particular in clinical trials. The concept of a surrogate endpoint is introduced and contrasted with that of a biomarker. It is emphasized that a surrogate endpoint is not universal for an indication but will depend on the mechanism of treatment. We discuss the measures of validity of a surrogate and give examples of both cancer surrogates and biomarkers on the path to surrogacy. Circumstances in which a surrogate endpoint may actually be preferred to the clinical endpoint are described. We provide pointers to the recent substantive literature on surrogates. Copyright © 2009 John Wiley & Sons, Ltd.     

Criteria for surrogate end points based on causal distributions

Summary.  When a treatment has a positive average causal effect (ACE) on an intermediate variable or surrogate end point which in turn has a positive ACE on a true end point, the treatment may have a negative ACE on the true end point due to the presence of unobserved confounders, which is called the surrogate paradox. A criterion for surrogate end points based on ACEs has recently been proposed to avoid the surrogate paradox. For a continuous or ordinal discrete end point, the distributional causal effect (DCE) may be a more appropriate measure for a causal effect than the ACE. We discuss criteria for surrogate end points based on DCEs. We show that commonly used models, such as generalized linear models and Cox's proportional hazard models, can make the sign of the DCE of the treatment on the true end point determinable by the sign of the DCE of the treatment on the surrogate even if the models include unobserved confounders. Furthermore, for a general distribution without any assumption of parametric models, we give a sufficient condition for a distributionally consistent surrogate and prove that it is almost necessary.     

Validation of a longitudinally measured surrogate marker for a time-to-event endpoint

The objective of this paper is to extend the surrogate endpoint validation methodology proposed by Buyse et al. (2000) to the case of a longitudinally measured surrogate marker when the endpoint of interest is time to some key clinical event. A joint model for longitudinal and event time data is required. To this end, the model formulation of Henderson et al. (2000) is adopted. The methodology is applied to a set of two randomized clinical trials in advanced prostate cancer to evaluate the usefulness of prostate-specific antigen (PSA) level as a surrogate for survival.     

The validation of surrogate end points by using data from randomized clinical trials: a case-study in advanced colorectal cancer

Summary.  In many therapeutic areas, the identification and validation of surrogate end points is of prime interest to reduce the duration and/or size of clinical trials. Buyse and co-workers and Burzykowski and co-workers have proposed a validation strategy for end points that are either normally distributed or (possibly censored) failure times. In this paper, we address the problem of validating an ordinal categorical or binary end point as a surrogate for a failure time true end point. In particular, we investigate the validity of tumour response as a surrogate for survival time in evaluating fluoropyrimidine-based experimental therapies for advanced colorectal cancer. Our analysis is performed on data from 28 randomized trials in advanced colorectal cancer, which are available through the Meta-Analysis Group in Cancer.     

The Surrogate Henderson Filters in X-11

This paper explains the surrogate Henderson filters that are used in the X-11 variant of the Census Method II seasonal adjustment program to obtain trends at the ends of time series. It describes a prediction interpretation for these surrogate filters, justifies an approximation to the filters, proposed by Kenny & Durbin (1982), and proposes a further interpretation of the results. The starting point for the paper is unpublished work by Musgrave (1964a, 1964b). His work has continuing relevance to current seasonal adjustment practice. This paper makes that work generally available for the first time, and reviews and extends it.     

Analyzing survival data in conjunction with time-dependent surrogate endpoints in clinical trials

Current survival techniques do not provide a good method for handling clinical trials with a large percent of censored observations. This research proposes using time-dependent surrogates of survival as outcome variables, in conjunction with observed survival time, to improve the precision in comparing the relative effects of two treatments on the distribution of survival time. This is in contrast to the standard method used today which uses the marginal density of survival time, T. only, or the marginal density of a surrogate, X, only, therefore, ignoring some available information. The surrogate measure, X, may be a fixed value or a time-dependent variable, X(t). X is a summary measure of some of the covariates measured throughout the trial that provide additional information on a subject's survival time. It is possible to model these time-dependent covariate values and relate the parameters in the model to the parameters in the distribution of T given X. The result is that three new models are available for the analysis of clinical trials. All three models use the joint density of survival time and a surrogate measure. Given one of three different assumed mechanisms of the potential treatment effect, each of the three methods improves the precision of the treatment estimate.     

When exposure is subject to nondifferential misclassification,are validation data helpful in testing for an exposure–disease association?

Consider assessing the evidence for an exposure variable and a disease variable being associated, when the true exposure variable is more costly to obtain than an error‐prone but nondifferential surrogate exposure variable. From a study design perspective, there are choices regarding the best use of limited resources. Should one acquire the true exposure status for fewer subjects or the surrogate exposure status for more subjects? The issue of validation is also central, i.e., should we simultaneously measure the true and surrogate exposure variables on a subset of study subjects? Using large‐sample theory, we provide a framework for quantifying the power of testing for an exposure–disease association as a function of study cost. This enables us to present comparisons of different study designs under different suppositions about both the relative cost and the performance (sensitivity and specificity) of the surrogate variable. We present simulations to show the applicability of our theoretical framework, and we provide a case‐study comparing results from an actual study to what could have been seen had true exposure status been ascertained for a different proportion of study subjects. We also describe an extension of our ideas to a more complex situation involving covariates. The Canadian Journal of Statistics 47: 222–237; 2019 © 2019 Statistical Society of Canada     

Cases for the nugget in modeling computer experiments

Most surrogate models for computer experiments are interpolators, and the most common interpolator is a Gaussian process (GP) that deliberately omits a small-scale (measurement) error term called the nugget. The explanation is that computer experiments are, by definition, “deterministic”, and so there is no measurement error. We think this is too narrow a focus for a computer experiment and a statistically inefficient way to model them. We show that estimating a (non-zero) nugget can lead to surrogate models with better statistical properties, such as predictive accuracy and coverage, in a variety of common situations.     

Predicting analysis time in events‐driven clinical trials using accumulating time‐to‐event surrogate information

For clinical trials with time‐to‐event endpoints, predicting the accrual of the events of interest with precision is critical in determining the timing of interim and final analyses. For example, overall survival (OS) is often chosen as the primary efficacy endpoint in oncology studies, with planned interim and final analyses at a pre‐specified number of deaths. Often, correlated surrogate information, such as time‐to‐progression (TTP) and progression‐free survival, are also collected as secondary efficacy endpoints. It would be appealing to borrow strength from the surrogate information to improve the precision of the analysis time prediction. Currently available methods in the literature for predicting analysis timings do not consider utilizing the surrogate information. In this article, using OS and TTP as an example, a general parametric model for OS and TTP is proposed, with the assumption that disease progression could change the course of the overall survival. Progression‐free survival, related both to OS and TTP, will be handled separately, as it can be derived from OS and TTP. The authors seek to develop a prediction procedure using a Bayesian method and provide detailed implementation strategies under certain assumptions. Simulations are performed to evaluate the performance of the proposed method. An application to a real study is also provided. Copyright © 2015 John Wiley & Sons, Ltd.     

Testing Time Series for Nonlinearity

The technique of surrogate data analysis may be employed to test the hypothesis that an observed data set was generated by one of several specific classes of dynamical system. Current algorithms for surrogate data analysis enable one, in a generic way, to test for membership of the following three classes of dynamical system: (0) independent and identically distributed noise, (1) linearly filtered noise, and (2) a monotonic nonlinear transformation of linearly filtered noise.We show that one may apply statistics from nonlinear dynamical systems theory, in particular those derived from the correlation integral, as test statistics for the hypothesis that an observed time series is consistent with each of these three linear classes of dynamical system. Using statistics based on the correlation integral we show that it is also possible to test much broader (and not necessarily linear) hypotheses.We illustrate these methods with radial basis models and an algorithm to estimate the correlation dimension. By exploiting some special properties of this correlation dimension estimation algorithm we are able to test very specific hypotheses. Using these techniques we demonstrate the respiratory control of human infants exhibits a quasi-periodic orbit (the obvious inspiratory/expiratory cycle) together with cyclic amplitude modulation. This cyclic amplitude modulation manifests as a stable focus in the first return map (equivalently, the sequence of successive peaks).     

An investigation into the two‐stage meta‐analytic copula modelling approach for evaluating time‐to‐event surrogate endpoints which comprise of one or more events of interest

Clinical trials of experimental treatments must be designed with primary endpoints that directly measure clinical benefit for patients. In many disease areas, the recognised gold standard primary endpoint can take many years to mature, leading to challenges in the conduct and quality of clinical studies. There is increasing interest in using shorter‐term surrogate endpoints as substitutes for costly long‐term clinical trial endpoints; such surrogates need to be selected according to biological plausibility, as well as the ability to reliably predict the unobserved treatment effect on the long‐term endpoint. A number of statistical methods to evaluate this prediction have been proposed; this paper uses a simulation study to explore one such method in the context of time‐to‐event surrogates for a time‐to‐event true endpoint. This two‐stage meta‐analytic copula method has been extensively studied for time‐to‐event surrogate endpoints with one event of interest, but thus far has not been explored for the assessment of surrogates which have multiple events of interest, such as those incorporating information directly from the true clinical endpoint. We assess the sensitivity of the method to various factors including strength of association between endpoints, the quantity of data available, and the effect of censoring. In particular, we consider scenarios where there exist very little data on which to assess surrogacy. Results show that the two‐stage meta‐analytic copula method performs well under certain circumstances and could be considered useful in practice, but demonstrates limitations that may prevent universal use.     

Marker processes in survival analysis

In the development of many diseases there are often associated variables which continuously measure the progress of an individual towards the final expression of the disease (failure). Such variables are stochastic processes, here called marker processes, and, at a given point in time, they may provide information about the current hazard and subsequently on the remaining time to failure. Here we consider a simple additive model for the relationship between the hazard function at time t and the history of the marker process up until time t. We develop some basic calculations based on this model. Interest is focused on statistical applications for markers related to estimation of the survival distribution of time to failure, including (i) the use of markers as surrogate responses for failure with censored data, and (ii) the use of markers as predictors of the time elapsed since onset of a survival process in prevalent individuals. Particular attention is directed to potential gains in efficiency incurred by using marker process information.     

Bayesian testing for independence of two categorical variables under two-stage cluster sampling with covariates

We consider Bayesian testing for independence of two categorical variables with covariates for a two-stage cluster sample. This is a difficult problem because we have a complex sample (i.e. cluster sample), not a simple random sample. Our approach is to convert the cluster sample with covariates into an equivalent simple random sample without covariates, which provides a surrogate of the original sample. Then, this surrogate sample is used to compute the Bayes factor to make an inference about independence. We apply our methodology to the data from the Trend in International Mathematics and Science Study [30] for fourth grade US students to assess the association between the mathematics and science scores represented as categorical variables. We show that if there is strong association between two categorical variables, there is no significant difference between the tests with and without the covariates. We also performed a simulation study to further understand the effect of covariates in various situations. We found that for borderline cases (moderate association between the two categorical variables), there are noticeable differences in the test with and without covariates.     

Resolving paradoxes involving surrogate end points

Summary.  We define a surrogate end point as a measure or indicator of a biological process that is obtained sooner, at less cost or less invasively than a true end point of health outcome and is used to make conclusions about the effect of an intervention on the true end point. Prentice presented criteria for valid hypothesis testing of a surrogate end point that replaces a true end point. For using the surrogate end point to estimate the predicted effect of intervention on the true end point, Day and Duffy assumed the Prentice criterion and arrived at two paradoxical results: the estimated predicted intervention effect by using a surrogate can give more precise estimates than the usual estimate of the intervention effect by using the true end point and the variance is greatest when the surrogate end point perfectly predicts the true end point. Begg and Leung formulated similar paradoxes and concluded that they indicate a flawed conceptual strategy arising from the Prentice criterion. We resolve the paradoxes as follows. Day and Duffy compared a surrogate-based estimate of the effect of intervention on the true end point with an estimate of the effect of intervention on the true end point that uses the true end point. Their paradox arose because the former estimate assumes the Prentice criterion whereas the latter does not. If both or neither of these estimates assume the Prentice criterion, there is no paradox. The paradoxes of Begg and Leung, although similar to those of Day and Duffy, arise from ignoring the variability of the parameter estimates irrespective of the Prentice criterion and disappear when the variability is included. Our resolution of the paradoxes provides a firm foundation for future meta-analytic extensions of the approach of Day and Duffy.     

Surrogate threshold effect: an alternative measure for meta-analytic surrogate endpoint validation

In many therapeutic areas, the identification and validation of surrogate endpoints is of prime interest to reduce the duration and/or size of clinical trials. Buyse et al. [Biostatistics 2000; 1:49-67] proposed a meta-analytic approach to the validation. In this approach, the validity of a surrogate is quantified by the coefficient of determination Rtrial2 obtained from a model, which allows for prediction of the treatment effect on the endpoint of interest ('true' endpoint) from the effect on the surrogate. One problem related to the use of Rtial2 is the difficulty in interpreting its value. To address this difficulty, in this paper we introduce a new concept, the so-called surrogate threshold effect (STE), defined as the minimum treatment effect on the surrogate necessary to predict a non-zero effect on the true endpoint. One of its interesting features, apart from providing information relevant to the practical use of a surrogate endpoint, is its natural interpretation from a clinical point of view.     

The relationship between osteoporotic fracture risk and a surrogate: Apparent discrepancies between analyses based on individual patient data and summary statistics

There is debate within the osteoporosis research community about the relationship between the risk of osteoporotic fracture and the surrogate measures of fracture risk. Meta‐regression analyses based on summary data have shown a linear relationship between fracture risk and surrogate measures, whereas analyses based on individual patient data (IPD) have shown a nonlinear relationship. We investigated the association between changes in a surrogate measure of fracture incidence, in this case a bone turnover marker for resorption assessed in the three risedronate phase III clinical programmes, and incident osteoporosis‐related fracture risk using regression models based on patient‐level and trial‐level information. The relationship between osteoporosis‐related fracture risk and changes in bone resorption was different when analysed on the basis of IPD than when analysed on the basis of a meta‐analytic approach (i.e., meta‐regression) using summary data (e.g., treatment effect based on treatment group estimates). This inconsistency in our findings was consistent with those in the published literature. Meta‐regression based on summary statistics at the trial level is not expected to reflect causal relationships between a clinical outcome and surrogate measures. Analyses based on IPD make possible a more comprehensive analysis since all relevant data on a patient level are available. Copyright © 2004 John Wiley & Sons Ltd.     

Novel criteria to exclude the surrogate paradox and their optimalities

When the primary outcome is hard to collect, a surrogate endpoint is typically used as a substitute. However, even when a treatment has a positive average causal effect (ACE) on a surrogate endpoint, which also has a positive ACE on the primary outcome, it is still possible that the treatment has a negative ACE on the primary outcome. Such a phenomenon is called the surrogate paradox and greatly challenges the use of surrogates. In this paper, we provide criteria to exclude the surrogate paradox. Our criteria are optimal in the sense that they are sufficient and “almost necessary” to exclude the paradox: If the conditions are satisfied, the surrogate paradox is guaranteed to be absent, whereas if the conditions fail, there exists a data-generating process with surrogate paradox that can generate the same observed data. That is, our criteria capture all the observed information to exclude the surrogate paradox.     

Validation of surrogate end points in multiple randomized clinical trials with failure time end points

Before a surrogate end point can replace a final (true) end point in the evaluation of an experimental treatment, it must be formally 'validated'. The validation will typically require large numbers of observations. It is therefore useful to consider situations in which data are available from several randomized experiments. For two normally distributed end points Buyse and co-workers suggested a new definition of validity in terms of the quality of both trial level and individual level associations between the surrogate and true end points. This paper extends this approach to the important case of two failure time end points, using bivariate survival modelling. The method is illustrated by using two actual sets of data from cancer clinical trials.