Given the long follow-up periods that are often required for treatment or intervention studies, the potential to use surrogate markers to decrease the required follow-up time is a very attractive goal. However, previous studies have shown that using inadequate markers or making inappropriate assumptions about the relationship between the primary outcome and surrogate marker can lead to inaccurate conclusions regarding the treatment effect. Currently available methods for identifying and validating surrogate markers tend to rely on restrictive model assumptions and/or focus on uncensored outcomes. The ability to use such methods in practice when the primary outcome of interest is a time-to-event outcome is difficult due to censoring and missing surrogate information among those who experience the primary outcome before surrogate marker measurement. In this paper, we propose a novel definition of the proportion of treatment effect explained by surrogate information collected up to a specified time in the setting of a time-to-event primary outcome. Our proposed approach accommodates a setting where individuals may experience the primary outcome before the surrogate marker is measured. We propose a robust nonparametric procedure to estimate the defined quantity using censored data and use a perturbation-resampling procedure for variance estimation. Simulation studies demonstrate that the proposed procedures perform well in finite samples. We illustrate the proposed procedures by investigating two potential surrogate markers for diabetes using data from the Diabetes Prevention Program.

A surrogate marker is often defined as a physical measurement such as a biomarker, clinical measurement, or psychological test that can be “used in therapeutic trials as a substitute for a clinically meaningful endpoint that is a direct measure of how a patient feels, functions, or survives and is expected to predict the effect of the therapy.”[

While the proportion of treatment effect explained by a surrogate marker is intuitively appealing, a number of other quantities to assess surrogate markers have been proposed. For example, relative effect and adjusted association [

For a survival outcome,

In this paper, we generalize the work of Parast _{0}, in the survival setting. In addition, we propose a robust nonparametric procedure to estimate the defined quantity using censored data and a perturbation-resampling procedure for inference. To increase efficiency, we also propose parallel augmented estimates that take advantage of baseline covariate information. Our proposed definition and estimation procedure is the only available method to accommodate survival settings where individuals may experience the primary outcome of interest or be censored before the surrogate marker is measured (other than methods that suggest disregarding individuals), a situation that is quite common in practice. We perform a simulation study to examine the finite sample performance of our proposed procedures and illustrate the proposed procedures by investigating two potential surrogate markers for diabetes using data from the Diabetes Prevention Program.

Let _{0}. Without loss of generality, we assume that ^{(}^{g}^{)} and ^{(}^{g}^{)} denote the survival time and the surrogate marker value under treatment ^{(}^{A}^{)}^{(}^{B}^{)}^{(}^{A}^{)} and ^{(}^{B}^{)} denote the survival time under treatment A, survival time under treatment B, surrogate marker value under treatment A and surrogate marker value under treatment B, respectively. We assume that ^{(}^{B}^{)} and ^{(}^{A}^{)} have the same support. In practice, we can only potentially observe (^{(}^{A}^{)}^{(}^{A}^{)}) or (^{(}^{B}^{)}^{(}^{B}^{)}) for each individual depending on whether

_{0}. We consider a setting where individuals may be censored or experience the primary outcome before _{0} and thus,

Our analytic objective is to study the extent to which the surrogate information available at time _{0} captures the true treatment effect Δ(_{0} should be considered as part of the surrogate information available at _{0}. We argue that this information should indeed be considered as part of the surrogate information. That is, in this paper, we define surrogate information at _{0} as the combination of primary outcome information before _{0} and surrogate marker measurements collected at _{0} for those still being observed. We take this approach because even in the highly optimistic situation where one were to identify _{0}, as a valid surrogate marker that can be used to estimate and test for a treatment effect, it is unlikely that one would completely disregard primary outcome information that is observed up to _{0}. It is more sensible to envision that one uses _{0} and surrogate marker measurements at _{0} to estimate the treatment effect, thus this combination of information is our definition of surrogate information at _{0} throughout this paper. Specifically, we consider the surrogate information available at _{0} as

Further motivating this definition is the fact that it is difficult to consider a reasonable alternative to this approach. For example, one potential alternative would be to restrict estimation to only those who are still under observation at _{0} (i.e. removing individuals who experience the primary outcome before _{0}) [_{0} when the goal is to quantify the treatment effect on

We aim to define the proportion of treatment effect explained by _{t}_{0} by contrasts between the actual treatment effect and the residual treatment effect that would be observed if the surrogate information available at _{0} under treatment A was equal to the surrogate information available at _{0} under treatment B. That is, we define the residual treatment effect by noting that

_{t}_{0} = {_{t}_{0}, _{t}_{0}} = {_{0}, _{0})}, and

Thus,
_{0}_{t}_{0}, in both treatment groups was identical to _{t}_{0}. Interestingly, Δ_{S}_{0}_{t}_{0}) only depends on _{t}_{0} through _{t}_{0}. An equivalent interpretation of Δ_{S}_{0}_{t}_{0}) would be the hypothetical difference in survival at _{0} and the distribution of the surrogate marker at _{0} among those who survived to _{0} were the same in the two treatment groups. This quantity summarizes the residual treatment effect that cannot be explained by the surrogate information available at _{0} and would be expected to equal zero for a perfect surrogate marker.

However,
^{(}^{A}^{)} and ^{(}^{B}^{)} cannot both be observed simultaneously. To overcome this difficulty, we further assume that

Under assumptions (

To consider the residual treatment effect in a population, we may consider _{t}_{0} as a realization from a random distribution 𝒮_{t}_{0} and define the residual treatment effect as

The choice of the distribution of 𝒮_{t}_{0} depends on the specific context. For example, if treatment B is a placebo, then we may be interested in examining the residual treatment effect quantity when treatment A has no effect on the surrogate marker information at _{0}, i.e., when the distribution of the surrogate information at _{0} under treatment

_{g}_{0}) is the cumulative distribution function of ^{(}^{g}^{)} conditional on ^{(}^{g}^{)}
_{0}. Here, treatment group _{S}_{0}). Alternatively, when neither treatment A nor treatment B is a natural reference group, one may be interested in examining the residual treatment effect when the distributions of the surrogate information at _{0} in both groups are identical to that of a mixture population from the two groups. For example, in this case we may let

For a given choice of Δ_{S}_{0}), the proportion of treatment effect explained by the surrogate marker can be expressed using a contrast between Δ(_{S}_{0}):

In this paper, we focus on nonparametrically estimating this proportion using censored data. Informally, we use _{S}_{0}) to measure the extent to which the surrogate marker captures information about the treatment effect on survival by comparing the total treatment effect with the hypothetical treatment effect when there is no difference in surrogate information up at _{0}. The approach to define the proportion of treatment effect explained based on contrasts between the actual treatment effect and the residual treatment effect was proposed in a non-survival setting in Wang & Taylor [

This definition of the proportion of the treatment effect explained by the surrogate marker does not guarantee that the resulting _{S}_{0}) is always between 0 and 1. However, one set of sufficient conditions similar to those given in Wang & Taylor[

_{A}_{0}) is a monotone increasing function of

^{(}^{A}^{)}
^{(}^{A}^{)}
_{0}) ≥ ^{(}^{B}^{)}
^{(}^{B}^{)}
_{0}) for all

_{A}_{0}) ≥ _{B}_{0}) for all

where the first condition implies that the surrogate marker at time _{0} is “positively” related to the survival time; the second condition implies that there is a positive treatment effect on the surrogate marker and the third condition suggests that there is a non-negative residual treatment effect beyond that on the surrogate marker. For (C1), 1/_{S}_{0}) ≤ Δ(_{S}_{0}) ≤ 1.

The quantity _{S}_{0}) depends on the selection of the reference distribution 𝒮_{t}_{0}. When the treatment group _{S}_{0}) definition (_{S}_{0}) definition (

We assume that data are collected from a randomized clinical trial (RCT) and _{gi}, δ_{gi}, S_{gi}_{g}_{gi}_{gi},C_{gi}_{gi}_{gi} < C_{gi}_{gi}_{gi}_{0}, for _{gi}, S_{gi}_{gi}^{(}^{A}^{)}
^{(}^{B}^{)}

To estimate Δ_{S}_{0}) as defined in (

Note that _{A}_{0}) = _{Ai} > t_{Ai} > t_{0}_{Ai}_{gi}, S_{gi}_{gi}_{A}_{0}) as _{A}_{0}) = exp{−Λ̂_{A}_{0})}, where

_{A}_{0}) = −log[_{A}_{0})]_{Ai}_{Ai}_{Ai}_{Ai}_{i}, K_{h}_{opt}_{0} ∈ (1/20_{0} = 0.11. Since _{B}_{0}) = _{Bi}_{Bi} > t_{0}), we empirically estimate _{B}_{0}) using all subjects with _{Bi} > t_{0} as

Subsequently, we may construct an estimator for Δ_{S}_{0}) as

_{S}_{0}) = 1 − Δ̂_{S}_{0})/Δ̂(_{S}_{0}) is a consistent estimator of Δ_{S}_{0}) and that as _{A}, n_{B}

_{A}_{B}_{S}_{0}) is a consistent estimator of _{S}_{0}) and, by the delta method,

Recent work has shown that augmentation can lead to improvements in efficiency by taking advantage of the association between baseline information,

_{gi}, i_{g}^{AUG},_{S}_{0})^{AUG}_{12})(Ξ_{22})^{−1} where

^{AUG}_{S}_{0}), Δ̂(_{S}_{0}). Alternatively, one could consider augmenting _{S}_{0}) directly using a single basis transformation.

Since our definition of Δ_{S}_{0}) considers the surrogate information as a combination of both _{0}, a logical inquiry would be how to assess the incremental value of the _{0}. If the quantity _{S}_{0}) reveals that a large proportion of the treatment effect is explained by information at _{0}, it would be important to know how much of that quantity is attributable to _{0}, then it may not be necessary to measure and incorporate _{S}_{0}) in Section 2, we define the proportion of the treatment effect explained by _{0} only as _{T}_{0}) = 1 − Δ_{T}_{0})/Δ(

As with Δ_{S}_{0}), the choice of 𝔽 depends on the specific context. We will continue to assume, without loss of generality, that treatment ^{(}^{B}^{)}
_{0}) as the reference distribution. It follows that

Although one would generally expect that the proportion of treatment effect explained by both _{0} to be at least as big as the proportion of treatment effect explained by _{0} alone (i.e. _{S}_{0}) ≥ _{T}_{0})) implying Δ_{T}_{0}) ≥ Δ_{S}_{0})), this is only guaranteed to hold under certain conditions. Specifically, we note that

_{T}_{0}) − Δ_{S}_{0}) ≥ 0 if and only if

Sufficient conditions for the inequality above are (C1) and

These conditions are also required to ensure that we are not in a situation known as the surrogate paradox [

To estimate _{T}_{0}), we may employ the IPW estimator _{T}_{0}) = 1 − Δ̂_{T}_{0})_{T}_{0}) = _{B}_{0}) _{A}_{A}_{0}) − _{B}_{S}_{0}) by replacing Δ(_{S}_{0})_{T}_{0}) with Δ̂(_{S}_{0})_{T}_{0})

The methods described in this paper focus on a setting where the treatment effect is defined as the difference in survival rates at time t, Δ(

_{S}_{0}) would be replaced by

Consequently, the proportion of treatment effect explained by the surrogate marker can be defined based on Δ_{RMST}_{,S}_{0}) and Δ _{RMST} (_{RMST}_{,S}_{0}) = 1 − Δ _{RMST}_{,S}_{0})_{RMST} (_{RMST} (_{RMST}_{,S}_{0}) can be estimated by

We propose to estimate the variability of our proposed estimators and construct confidence intervals using a perturbation-resampling method to approximate the distribution of the estimators. Specifically, let

That is, one can approximate the variance of (_{S}_{0}), one can calculate the 100(^{th} and 100(1 − ^{th} empirical percentile of
_{S}_{0}) − _{S}_{0}) by the empirical variance of
_{S}_{0}) as

_{ij}_{1≤i,j≤2} and _{α}

The theoretical justification for the perturbation-resampling procedure is provided in

The perturbed samples can also be used to construct the augmented estimators Δ̂(^{AUG}_{S}_{0})^{AUG}_{12})(Ξ̂_{22})_{−1}, where Ξ̂_{12} is the empirical covariance of

_{22} is the empirical variance of

The estimator _{0})^{AUG}

We conducted simulation studies under two main settings to assess the performance and validity of our proposed estimators and inference procedures. In both settings, data were generated such that individuals may experience the primary outcome or be censored before _{0} and thus, _{0}. Within each setting we examined results where _{A}_{B}_{A}_{B}_{0} = 0.5, and the results summarize 1000 replications. For all estimates, we estimate variance using our proposed perturbation approach and construct confidence intervals using the normal approximation, quantiles of the perturbed values and Fieller’s method (for _{S}_{0}) only).

In the first simulation setting, Setting (i), data were generated as:
^{(}^{A}^{)} and ^{(}^{B}^{)} were generated from a N(0^{(}^{g}^{)} is only observable if ^{(}^{g}^{)}
_{0} and _{0}. In this setting, Δ(_{S}_{0}) = 0.05, _{S}_{0}) = 0.75, ^{(}^{A}^{)}
^{(}^{B}^{)}
^{(}^{A}^{)}
_{0}) = 0.69, ^{(}^{B}^{)}
_{0}) = 0.56, ^{(}^{A}^{)}^{(}^{A}^{)}
_{0}) = 3.35, ^{(}^{B}^{)}^{(}^{B}^{)}
_{0}) = 4.27, and 29% and 25% of individuals in treatment group A and treatment group B are censored before _{A}_{B}

In the second simulation setting (ii), data were generated as:
^{(}^{A}^{)} and ^{(}^{B}^{)} were generated from a N(0_{1} + (1 − _{2}, where _{1} ~ Exp(0.5)_{2} ~ Exp(0.3). In this setting, Δ(_{S}_{0}) = 0.11, _{S}_{0}) = 0.60, ^{(}^{A}^{)}
^{(}^{B}^{)}
^{(}^{A}^{)}
_{0}) = 0.93, ^{(}^{B}^{)}
_{0}) = 0.75, ^{(}^{A}^{)}^{(}^{A}^{)}
_{0}) = 4.16, ^{(}^{B}^{)}^{(}^{B}^{)}
_{0}) = 4.56, 30% and 25% of individuals in treatment group A and treatment group B are censored before _{A}_{B}_{A}_{B}

_{A}_{B}

We illustrate our proposed procedures using data from the Diabetes Prevention Program (DPP), an RCT designed to investigate the efficacy of various treatments on the prevention of type 2 diabetes in high-risk adults. At randomization, participants were randomly assigned to one of four groups: metformin, troglitazone, lifestyle intervention or placebo. The troglitazone arm of the study was discontinued due to medication toxicity. The primary endpoint was time to diabetes as defined by the protocol at the time of the visit: fasting glucose ≥ 140 mg/dL (for visits through 6/23/1997, ≥ 126 mg/dL for visits on or after 6/24/2007) or 2-hour post challenge glucose ≥ 200 mg/dL. DPP results showed that both lifestyle intervention and metformin prevented or delayed development of type 2 diabetes in high risk adults [

For this illustration, we focus on the comparison of the lifestyle intervention group (N=1024) vs. placebo (N=1030) and we aim to examine the proportion of treatment effect explained by two potential surrogate markers: change in log-transformed hemoglobin A1c (HBA1C) from baseline to _{0} and change in fasting plasma glucose from baseline to _{0}, where we first let _{0} = 1 year. We define the treatment effect, Δ(_{0} as including diabetes incidence at _{0}, it is interesting to note that 3.7% and 11.5% of participants in the lifestyle intervention group and placebo group were diagnosed with diabetes before _{0} = 1 year, respectively. The cumulative incidence functions by treatment group, as shown in

Results from estimating the proportion of treatment effect explained by each surrogate are shown in _{S}_{0}), is 0.077 when the surrogate information at 1 year post-baseline consists of information about the change in HBA1C from baseline to 1 year and diabetes incidence up to 1 year and the proportion of treatment effect explained by this surrogate information, _{S}_{0}), is 48.2%. Examining change in fasting plasma glucose, the estimated residual treatment effect is 0.046 when the surrogate information at 1 year post-baseline consists of information about the change in fasting plasma glucose from baseline to 1 year and diabetes incidence up to 1 year and the proportion of treatment effect explained by this surrogate information is 68.7%. To determine the incremental value of the information about change in HBA1C and fasting plasma glucose, we examined the proportion of treatment effect explained by diabetes incidence information only up to 1 year post-baseline which was estimated to be 47.8%. Therefore, the incremental value of change in HBA1C was negligible, while the incremental value of change in fasting plasma glucose was 21.0% (SE= 5.9%). Our application of the proposed procedures to examine surrogate markers shows that fasting plasma glucose appears to capture more of the treatment effect than HBA1C, particularly when considered in terms of incremental value when added to diabetes incidence information at 1 year post-baseline.

To investigate how the surrogacy assessment may vary over _{0}, we present the results for _{0} = 2 years in _{S}_{0}), is almost 90%. When examining fasting plasma glucose, this quantity is 96.9%. As when _{0} = 1 year, the incremental value of HBA1C information is smaller than that for fasting plasma glucose, 1.9% versus 8.8%. Note that we would expect the _{S}_{0}) and _{T}_{0}) to generally increase as _{0} increases, but not necessarily for _{S}_{0}).

To examine whether efficiency could be gained through augmentation, we also calculated our proposed augmented estimates when _{0} = 1 year using the available baseline covariates: age group (less than 40, 40–44,45–49,50–54,55–59,60–64,65 and older), body mass index category (km/m_{2} units, ^{AUG}_{S}_{0})^{AUG}_{S}_{0})^{AUG}^{AUG}_{S}_{0})^{AUG}_{S}_{0})^{AUG}

To illustrate the use of an alternative treatment effect quantity, the restricted mean survival time (RMST), we estimated _{RMST}_{,S}_{0}) as described in Section 3.4 when _{0} = 1 and _{RMST}_{,S}_{0}) = 0.69 while for change in fasting plasma glucose _{RMST}_{,S}_{0}) = 0.75. Thus, this quantity similarly identifies fasting plasma glucose to be slightly better, in terms of surrogacy, compared to HBA1C.

Our application of the proposed procedures to examine surrogate markers shows that fasting plasma glucose appears to capture more of the treatment effect than HBA1C, particularly when considered in terms of incremental value when added to diabetes incidence information at 1 year post-baseline.

The identification and validation of surrogate markers is an important and challenging area of research. Valid surrogate markers that could be used to replace the primary outcome or used in combination with primary outcome information have the potential to lead to gains in efficiency in terms of design, implementation, estimation and testing. In this paper we have proposed a novel model-free framework for quantifying the proportion of treatment effect explained by surrogate information collected up to a specified time in the survival setting and a robust nonparametric procedure for making inference. Our proposed methods also have the advantage of allowing the surrogate marker _{0}. An R package implementing the methods described here, called Rsurrogate, is available on CRAN.

The simulation study shows that the proposed inference procedure has satisfactory empirical performance for moderate sample sizes. When the sample size becomes much smaller, these procedures which are based on asymptotic normality approximations would still lead to reliable inference for Δ_{S}_{0}) and Δ(_{S}_{0}), which involves the ratio of Δ_{S}_{0}) and Δ(

In this paper, we consider the “surrogate information at _{0}” to be a combination of both primary outcome information observed up to _{0}
_{0} for those who have not yet experienced the event. This decision warrants further discussion. There are generally two quite different motivations behind examining the surrogacy of a biomarker _{0} may not be desirable. In this case, we may quantify the “surrogacy” based on the incremental value quantity measure proposed in Section 3.3. It is important to also note that directly measuring the surrogacy of _{0} may be a valuable component of the “surrogate information” used to gauge the true treatment effect in a future trial. Furthermore, within our framework, one is able to overcome the challenge of _{0}.

When

In addition, our proposed definitions require Assumptions (_{S}_{0}) ≤ 1. These conditions are parallel to conditions required for identifiability in most existing surrogate marker literature, discussed in detail in Vander Weele [

Lastly, a limitation of our proposed approach is the theoretical condition that the supports of ^{(}^{A}^{)} and ^{(}^{B}^{)} are equivalent. In practice, the empirical supports may not completely overlap and some type of transformation or extrapolation of the relevant nonparametric estimators may be needed. However, when there is substantial non-overlap between two supports, caution is needed in interpreting the results.

Support for this research was provided by National Institutes of Health grant R21DK103118 and U54 HG007963. The Diabetes Prevention Program (DPP) was conducted by the DPP Research Group and supported by the National Institute of Diabetes and Digestive and Kidney Diseases (NIDDK), the General Clinical Research Center Program, the National Institute of Child Health and Human Development (NICHD), the National Institute on Aging (NIA), the Office of Research on Women’s Health, the Office of Research on Minority Health, the Centers for Disease Control and Prevention (CDC), and the American Diabetes Association. The data from the DPP were supplied by the NIDDK Central Repositories. This manuscript was not prepared under the auspices of the DPP and does not represent analyses or conclusions of the DPP Research Group, the NIDDK Central Repositories, or the NIH.

The reader is referred to the

Cumulative incidence distribution by treatment group (lifestyle versus placebo) in (a) and further stratified by whether fasting glucose had increased or not from baseline to 1 year in (b) for the DPP data.

Performance of the proposed estimates, the estimated treatment effect, Δ̂(_{S}_{0}), the estimated proportion of treatment effect explained by surrogate information at _{0}, _{S}_{0}), the estimated augmented treatment effect, Δ̂(^{AUG}_{S}_{0})^{AUG}_{0}, _{S}_{0})^{AUG}_{A}_{B}

Setting (i) | ||||||
---|---|---|---|---|---|---|

| ||||||

Δ̂( | Δ̂_{S}_{0}) | _{S}_{0}) | Δ̂(^{AUG} | Δ̂_{S}_{0})^{AUG} | _{S}_{0})^{AUG} | |

| ||||||

Bias | −0.0002 | 0.0020 | −0.0045 | 0.0000 | 0.0021 | −0.0051 |

ESE | 0.0254 | 0.0215 | 0.0962 | 0.0231 | 0.0213 | 0.0952 |

ASE | 0.0249 | 0.0210 | 0.0988 | 0.0237 | 0.0208 | 0.0972 |

MSE | 0.0006 | 0.0005 | 0.0093 | 0.0005 | 0.0005 | 0.0091 |

Coverage (normal) | 0.951 | 0.945 | 0.959 | 0.958 | 0.947 | 0.955 |

Coverage (quantile) | 0.945 | 0.943 | 0.942 | 0.953 | 0.944 | 0.941 |

Coverage (Fieller) | – | – | 0.952 | – | – | 0.953 |

| ||||||

Setting (ii) | ||||||

| ||||||

Δ̂( | Δ̂_{S}_{0}) | _{S}_{0}) | Δ̂(^{AUG} | Δ̂_{S}_{0})^{AUG} | _{S}_{0})^{AUG} | |

| ||||||

Bias | 0.0017 | 0.0017 | −0.0041 | 0.0013 | 0.0017 | −0.0043 |

ESE | 0.0206 | 0.0140 | 0.0439 | 0.0176 | 0.0139 | 0.0430 |

ASE | 0.0208 | 0.0141 | 0.0436 | 0.0178 | 0.0139 | 0.0429 |

MSE | 0.0004 | 0.0002 | 0.0019 | 0.0003 | 0.0002 | 0.0019 |

Coverage (normal) | 0.948 | 0.954 | 0.954 | 0.953 | 0.949 | 0.947 |

Coverage (quantile) | 0.943 | 0.946 | 0.948 | 0.949 | 0.942 | 0.942 |

Coverage (Fieller) | – | – | 0.953 | – | – | 0.947 |

Performance of the proposed estimates, the estimated treatment effect, Δ̂(_{S}_{0}), the estimated proportion of treatment effect explained by surrogate information at _{0}, _{S}_{0}), the estimated augmented treatment effect, Δ̂(^{AUG}_{S}_{0})^{AUG}_{0}, _{S}_{0})^{AUG}_{A}_{B}

Setting (i) | ||||||
---|---|---|---|---|---|---|

| ||||||

Δ̂( | Δ̂_{S}_{0}) | _{S}_{0}) | Δ̂(^{AUG} | Δ̂_{S}_{0})^{AUG} | _{S}_{0})^{AUG} | |

| ||||||

Bias | −0.0009 | 0.0034 | −0.0049 | −0.0006 | 0.0036 | −0.0062 |

ESE | 0.0398 | 0.0329 | 0.1607 | 0.0362 | 0.0325 | 0.1552 |

ASE | 0.0395 | 0.0326 | 0.1550 | 0.0375 | 0.0324 | 0.1517 |

MSE | 0.0016 | 0.0011 | 0.0258 | 0.0013 | 0.0011 | 0.0241 |

Coverage (normal) | 0.944 | 0.946 | 0.95 | 0.954 | 0.95 | 0.95 |

Coverage (quantile) | 0.944 | 0.944 | 0.942 | 0.952 | 0.944 | 0.942 |

Coverage (Fieller) | – | – | 0.944 | – | – | 0.947 |

| ||||||

Setting (ii) | ||||||

| ||||||

Δ̂( | Δ̂_{S}_{0}) | _{S}_{0}) | Δ̂(^{AUG} | Δ̂_{S}_{0})^{AUG} | _{S}_{0})^{AUG} | |

| ||||||

Bias | 0.0025 | 0.001 | −0.0006 | 0.0014 | 0.0011 | −0.0012 |

ESE | 0.0470 | 0.0315 | 0.0988 | 0.0405 | 0.0310 | 0.0965 |

ASE | 0.0465 | 0.0313 | 0.1003 | 0.0400 | 0.0309 | 0.0973 |

MSE | 0.0022 | 0.0010 | 0.0097 | 0.0016 | 0.001 | 0.0093 |

Coverage (normal) | 0.949 | 0.949 | 0.960 | 0.943 | 0.953 | 0.959 |

Coverage (quantile) | 0.947 | 0.951 | 0.953 | 0.939 | 0.952 | 0.947 |

Coverage (Fieller) | – | – | 0.955 | – | – | 0.955 |

Proposed estimates examining two potential surrogate markers in the DDP: the estimated treatment effect, Δ̂(_{0}, Δ̂_{S}_{0}), the estimated proportion of treatment effect explained by surrogate information at _{0}, _{S}_{0}), the estimated residual treatment effect using survival information at _{0} only, Δ̂_{T}_{0}), the estimated proportion of treatment effect explained by survival information only at _{0}, _{T}_{0}), the incremental value of the surrogate marker information, _{S}_{0}), with standard error (SE) estimates obtained using the perturbation-resampling procedure and 95% confidence intervals (CI) based on a normal approximation, a quantile-based calculation, and Fieller’s method for _{0} = 1 or (b) _{0} = 2 years.

(a) _{0} = 1 year | ||||||
---|---|---|---|---|---|---|

| ||||||

Difference in HBA1C from baseline to _{0} | ||||||

| ||||||

Δ( | Δ_{S}_{0}) | _{S}_{0}) | Δ_{T}_{0}) | _{T}_{0}) | _{S}_{0}) | |

| ||||||

Estimate | 0.1483 | 0.0769 | 0.4815 | 0.0774 | 0.4779 | 0.0036 |

SE | 0.0191 | 0.0167 | 0.1004 | 0.0162 | 0.0664 | 0.0984 |

95% CI (normal) | (0.11,0.19) | (0.04,0.11) | (0.28,0.68) | (0.05,0.11) | (0.35,0.61) | (−0.19,0.2) |

95% CI (quantile) | (0.11,0.18) | (0.05,0.11) | (0.25,0.65) | (0.05,0.11) | (0.35,0.62) | (−0.23,0.17) |

95% CI (Fieller) | – | – | (0.27,0.68) | – | (0.36,0.62) | – |

| ||||||

Difference in fasting plasma glucose from baseline to _{0} | ||||||

| ||||||

Δ( | Δ_{S}_{0}) | _{S}_{0}) | Δ_{T}_{0}) | _{T}_{0}) | _{S}_{0}) | |

| ||||||

Estimate | 0.1483 | 0.0464 | 0.6873 | 0.0774 | 0.4779 | 0.2094 |

SE | 0.0191 | 0.0181 | 0.098 | 0.0162 | 0.0664 | 0.0585 |

95% CI (normal) | (0.11,0.19) | (0.01,0.08) | (0.5,0.88) | (0.05,0.11) | (0.35,0.61) | (0.09,0.32) |

95% CI (quantile) | (0.11,0.18) | (0.01,0.08) | (0.52,0.9) | (0.05,0.11) | (0.35,0.62) | (0.11,0.35) |

95% CI (Fieller) | – | – | (0.52,0.91) | – | (0.36,0.62) | – |

(b) _{0} = 2 years | ||||||
---|---|---|---|---|---|---|

| ||||||

Difference in HBA1C from baseline to _{0} years | ||||||

| ||||||

Δ( | Δ_{S}_{0}) | _{S}_{0}) | Δ_{T}_{0}) | _{T}_{0}) | _{S}_{0}) | |

| ||||||

Estimate | 0.1483 | 0.0149 | 0.8997 | 0.0177 | 0.8808 | 0.0189 |

SE | 0.0191 | 0.0129 | 0.0856 | 0.0125 | 0.0791 | 0.0768 |

95% CI (normal) | (0.11,0.19) | (−0.01,0.04) | (0.73,1.07) | (−0.01,0.04) | (0.73,1.04) | (−0.13,0.17) |

95% CI (quantile) | (0.11,0.18) | (−0.01,0.04) | (0.72,1.06) | (−0.01,0.04) | (0.74,1.05) | (−0.13,0.15) |

95% CI (Fieller) | – | – | (0.74,1.08) | – | (0.74,1.06) | – |

| ||||||

Difference in fasting plasma glucose from baseline to _{0} years | ||||||

| ||||||

Δ( | Δ_{S}_{0}) | _{S}_{0}) | Δ_{T}_{0}) | _{T}_{0}) | _{S}_{0}) | |

| ||||||

Estimate | 0.1483 | 0.0046 | 0.9687 | 0.0177 | 0.8808 | 0.088 |

SE | 0.0191 | 0.0135 | 0.0927 | 0.0125 | 0.0791 | 0.0348 |

95% CI (normal) | (0.11,0.19) | (−0.02,0.03) | (0.79,1.15) | (−0.01,0.04) | (0.73,1.04) | (0.02,0.16) |

95% CI (quantile) | (0.11,0.18) | (−0.02,0.03) | (0.8,1.17) | (−0.01,0.04) | (0.74,1.05) | (0.03,0.16) |

95% CI (Fieller) | – | – | (0.81,1.18) | – | (0.74,1.06) | – |