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- Summary
- 1. Introduction
- 2. Methods
- 3. Simulation
- 4. Example
- 5. Discussion
- Supplementary Material
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Biometrics. Author manuscript; available in PMC 2012 September 28.

Published in final edited form as:

Published online 2012 February 2. doi: 10.1111/j.1541-0420.2011.01722.x

PMCID: PMC3417090

NIHMSID: NIHMS373252

The publisher's final edited version of this article is available at Biometrics

See other articles in PMC that cite the published article.

Treatment-selection markers are biological molecules or patient characteristics associated with one’s response to treatment. They can be used to predict treatment effects for individual subjects and subsequently help deliver treatment to those most likely to benefit from it. Statistical tools are needed to evaluate a marker’s capacity to help with treatment selection. The commonly adopted criterion for a good treatment-selection marker has been the interaction between marker and treatment. While a strong interaction is important, it is, however, not suffcient for good marker performance. In this paper, we develop novel measures for assessing a continuous treatment-selection marker, based on a potential outcomes framework. Under a set of assumptions, we derive the optimal decision rule based on the marker to classify individuals according to treatment benefit, and characterize the marker’s performance using the corresponding classification accuracy as well as the overall distribution of the classifier. We develop a constrained maximum-likelihood method for estimation and testing in a randomized trial setting. Simulation studies are conducted to demonstrate the performance of our methods. Finally, we illustrate the methods using an HIV vaccine trial where we explore the value of the level of pre-existing immunity to Adenovirus serotype 5 for predicting a vaccine-induced increase in the risk of HIV acquisition.

In many clinical contexts today, the hope is to improve the effectiveness of interventions by targeting therapy to those most likely to benefit. Some therapies are thought to only benefit a subset of individuals, while others are thought to be particularly harmful for some subjects. Markers that are predictive of the effect of one treatment over another, termed treatment-selection markers, can potentially be used to deliver a treatment only to individuals likely to benefit, and to avoid the burden, cost, and potential toxic effects of providing it to others. These markers may be biological measures, other patient or disease characteristics, or combinations of the above. Well-known examples of existing markers are K-RAS gene expression for selecting against anti-epidermal growth factor receptor treatment of colorectal cancer (Karapetis et al., 2008; Allegra et al., 2009); estrogen-receptor (ER) positivity for selecting endocrine therapy for breast cancer treatment (Harris et al., 2007); and Oncotype-DX, a multi-gene score for selecting adjuvant chemotherapy for the treatment of ER-positive breast cancer (Paik et al., 2006; Harris et al., 2007; Albain et al., 2010).

A fundamental question in the development of treatment-selection markers is what statistical measures should be used for marker evaluation. Answering this question is essential for understanding the clinical utility of a given marker and for comparing candidate markers, as well as for performing more complex tasks such as developing marker combinations, assessing the incremental value of a marker, and evaluating covariate effects. Most of the methodologic literature on treatment-selection markers assumes that the statistical interaction between marker value and treatment assignment in the context of a randomized trial is the primary measure of marker performance (eg Sargent et al. (2005); Freidlin and Simon (2005); Buyse (2007); Simon (2008); Simon et al. (2009)). However, a strong interaction is important but not suffcient for adequate marker performance (Janes et al., 2011). Specifically, two markers can have the same interaction but very different performance. In Web supplementary figure 1, we present an example where two markers have the same interaction coefficient but different capacity in terms of classifying a subject according to treatment effectiveness. Details on our approach to assessing a marker’s treatment-selection capacity will be presented in Section 2.

Additional limitations of the interaction approach include the fact that the scale of the interaction coefficient depends on the functional form of the model relating marker and treatment to outcome. The interaction approach also does not facilitate more sophisticated tasks such as deriving marker combinations.

In this paper, our goal is to develop a measure for assessing the treatment-selection performance of a marker (or risk model) which puts different markers (models) on the same scale to facilitate their comparison. We focus on the setting of a randomized trial, and use a potential outcomes framework to develop novel measures of the performance of a continuous treatment-selection marker. While most of the existing methodologic literature focuses on binary markers (Mandrekar et al., 2005; Lijmer and Bossuyt, 2009; Wang and Zhou, 2010), commonly markers are continuous. Under a set of assumptions, we derive the optimal decision rule for using the marker to classify individuals according to the benefit of one treatment over another. We then characterize the accuracy with which this classifier distinguishes between individuals who do and do not benefit, as well as the overall distribution of the classifier. We develop a constrained maximum-likelihood method for estimation and testing. Simulation studies are conducted to demonstrate the finite-sample performance of the estimators. Finally, we illustrate the methods using an HIV vaccine trial where we explore the value of the level of pre-existing immunity to Adenovirus serotype 5 for identifying individuals with a vaccine-induced increase in the risk of HIV acquisition.

We consider a study setting where individuals are randomly assigned to a placebo arm or an active treatment arm. Let *S* denote the continuous candidate baseline biomarker of interest, which could be a vector. Let *Z* be a binary treatment indicator, with values 0 and 1 indicating placebo and active treatment arm respectively. Let *Y* be the binary disease outcome of interest, with value 0 and 1 indicating nondiseased and diseased respectively. Without loss of generality (WLOG), suppose every subject has *S* and *Y* measured. Moreover, we adopt a potential outcomes framework and use *Y* (0) and *Y* (1) to indicate the potential outcome for *Y* if an individual is assigned to placebo or active treatment respectively.

We make the following assumptions required for identifying and estimating the measures that will be defined later:

- (A1) The stable unit treatment value and consistency assumption (SUTVA):
*Y*(1),*Y*(0) for a subject are independent of the treatment assignments of other subjects, and a subject’s potential outcome*Y*(*z*) under the assigned treatment*Z*=*z*equals the observed outcome*Y* - (A2) The ignorable treatment assignments assumption:
*Y*(0),*Y*(1),*S**Z* - (A3) The monotone treatment effect assumption: treatment either won’t do any harm or won’t have any benefit. WLOG here we assume
*Y*(0) ≥*Y*(1). That is, treatment will not do any harm to an individual where this particular clinical outcome is concerned - (A4) We assume the probability of
*Y*(*Z*) given*S*and*Z*follows a generalized linear structural model:*Risk*_{(Z)}(*S*) =*P*{*Y*(*Z*) = 1|*S*} =*G*(*S, Z, θ*).

A1 is commonly adopted in causal inference literature, and is plausible when participants do not interact with one another in a trial. A2 is satisfied by randomization. The risk model assumption A4 provides interpretable parameters for inference and can be tested using the study data with standard statistical tools. For example, a Hosmer-Lemeshow test can be performed to compare the expected and observed risk quantiles. More powerful empirical likelihood based techniques can also be utilized (Qin and Zhang, 1997; Gilbert, 2003). The monotonicity assumption A3 is the key for the definition and identifiability of our classification accuracy measure defined next. Although it cannot be confirmed, it can be rejected based on statistical tests in a randomized trial. Throughout this manuscript, A1,A2 and A4 are assumed; we assume A3 in developing our framework and estimation methods and relax it later in section 2.4 in developing a sensitivity analysis method.

Let *D* = *Y* (0) – *Y* (1) denote the individual treatment effect on the disease outcome. Under the monotone treatment effect assumption A4, *D* takes values 0 or 1. For those subjects with *D* = 0, *Y* (0) = *Y* (1), having the active treatment or not does not change their disease outcome, whereas for those with *D* = 1, *Y* (0) > *Y* (1), the active treatment prevents the disease. Therefore, we can divide individuals into a treatment-effective group (*D* = 1) and a treatment-ineffective group (*D* = 0).

We characterize the marker’s treatment-selection ability in two ways. First, researchers will be interested in the discriminatory power of a marker in terms of correctly identifying an individual’s treatment-effectiveness status. Suppose in practice a decision rule is constructed based on *Q*(*S*), a function of *S*, with larger value of *Q*(*S*) indicating better chance of benefitting from the active treatment. Given a particular threshold value *q*, suppose an individual is classified as treatment-effective if *Q*(*S*) ≥ *q*, and treatment-ineffective if *Q*(*S*) < *q*. We define the true positive fraction as the probability of correctly identifying a treatment-effective individual, TPF(*q*) = *P*{*Q*(*S*) > *q*|*D* = 1}, and the false positive fraction as the probability of incorrectly classifying a treatment-ineffective individual to be treatment-effective, FPF(*q*) = *P*{*Q*(*S*) > *q*|*D* = 0}. An ROC curve for varying thresholds can be constructed using ROC(*t*) = TPF{FPF^{−1}(*t*)}.

While TPF and FPF tell us about the marker’s underlying accuracy of classifying a subject according to treatment response, health policy makers may also be interested in gauging the population impact of the marker in helping make informed treatment decisions. This can be addressed by examining the distribution of *Q*(*S*), namely the CDF of *Q*(*S*) where CDF_{Q}(*q*) = *P*{*Q*(*S*) ≤ *q*}. In practice suppose we adopt a low and a high thresholds *q _{L}* and

For a related task of evaluating a diagnostic testing marker, the accuracy of correctly classifying a subject according to disease status is typically defined conditional on the observed disease status *Y* . In the treatment-selection problem, TPF and FPF defined in section 2.2 under assumption A3 are defined conditional on treatment-effectiveness status *D* = *Y* (0) – *Y* (1), which is not completely observable. Hence these classification accuracy measures cannot be estimated directly conditional on *D*. However, under A3 they both can be represented as functions of disease risk:

$$\begin{array}{cc}\hfill & \mathrm{TPF}\left(q\right)=\frac{P\{Y\left(0\right)-Y\left(1\right)=1,Q\left(S\right)>q\}}{P\{Y\left(0\right)-Y\left(1\right)=1\}}=\frac{E\left[\{Y\left(0\right)-Y\left(1\right)\}I\{Q\left(S\right)>q\}\right]}{E\{Y\left(0\right)-Y\left(1\right)\}}\hfill \\ \hfill & =\frac{E\left[E\{Y\left(0\right)-Y\left(1\right)\mid S\}I\{Q\left(S\right)>q\}\right]}{E\left[\{Y\left(0\right)-Y\left(1\right)\mid S\}\right]}=\frac{E\left[\{Ris{k}_{\left(0\right)}\left(S\right)-Ris{k}_{\left(1\right)}\left(S\right)\}I\{Q\left(S\right)>q\}\right]}{E[Ris{k}_{\left(0\right)}\left(S\right)-Ris{k}_{\left(1\right)}\left(S\right)]},\hfill \\ \hfill & \mathrm{FPF}\left(q\right)=\frac{P\{Y\left(0\right)-\left(1\right)=0,Q\left(S\right)>q\}}{P\{Y\left(0\right)-Y\left(1\right)=0\}}=\frac{E\left[\{1-Y\left(0\right)+Y\left(1\right)\}I\{Q\left(S\right)>q\}\right]}{E\{1-Y\left(0\right)+Y\left(1\right)\}}\hfill \\ \hfill & =\frac{E\left[E\{1-Y\left(0\right)+Y\left(0\right)\mid S\}I\{Q\left(S\right)>q\}\right]}{E\left[\{1-Y\left(0\right)+Y\left(1\right)\mid S\}\right]}=\frac{E\left[\{1-Ris{k}_{\left(0\right)}\left(S\right)+Ris{k}_{\left(1\right)}\left(S\right)\}I\{Q\left(S\right)>q\}\right]}{E[1-Ris{k}_{\left(0\right)}\left(S\right)+Ris{k}_{\left(1\right)}\left(S\right)]}.\hfill \end{array}$$

(1)

Therefore, as will be shown later, TPF and FPF can be calculated on the basis of a risk model estimated from the observed data.

Having introduced measures of treatment-selection capacity using the decision rule *Q*(*S*), we now consider the choice of *Q*(*S*). Examples of *Q*(*S*) could simply be *S* itself, the difference between *Risk*_{(0)}(*S*) and *Risk*_{(1)}(*S*), or the ratio of *Risk*_{(1)}(*S*) over *Risk*_{(0)}(*S*). Under A3, there does in fact exist an optimal choice in terms of maximizing classification accuracy. Let *H*_{0} and *H*_{1} be the point hypotheses that a subject is treatment-ineffective (*D* = 0) and treatment-effective (*D* = 1). According to the Neyman Pearson Lemma, the most powerful test rejecting *H*_{0} favoring *H*_{1} [i.e., the test with smallest type-II error (1-TPF) at a fixed type-I error rate (FPF)] is based on comparing the likelihood ratio *P*(*S*|*D* = 1)/*P*(*S*|*D* = 0) with a threshold. By Bayes’ theorem, the likelihood ratio

$$\frac{P(S\mid D=0)}{P(S\mid D=1)}=\frac{P(D=0\mid S)P(D=1)}{P(D=1\mid S)P(D=0)}=\frac{P(D=1\mid S)}{1-P(D=1\mid S)}\frac{P(D=1)}{P(D=0)},$$

is a monotone increasing function of *P*(*D* = 1|*S*). Therefore a decision rule comparing *P*(*D* = 1|*S*) = *Risk*_{(0)}(*S*) – *Risk*_{(1)}(*S*) to a threshold maximizes classification accuracy for identifying treatment effect. It follows that *P*(*D* = 1|*S*) has the highest ROC curve among all functions of *S* for classifying subjects according to their treatment-effectiveness.

An argument similar in flavor was made by McIntosh and Pepe (2002), where they pointed out that for classification with respect to a binary disease outcome, the risk of disease as a function of marker value is the optimal function in terms of maximizing the ROC curve at every point. In our setting, the risk difference between treatment arms, *Risk*_{(0)}(*S*) – *Risk*_{(1)}(*S*), can be viewed as the ‘risk’ of the binary outcome *D* = 1 under A3.

We therefore employ *Q*(*S*) = *Risk*_{(0)}(*S*) – *Risk*_{(1)}(*S*) in simulation studies and in the data example. Next we investigate the identifiability of the performance measures and propose methods for estimation and testing.

In this section, we demonstrate the identifiability of the distribution of *Y* (0) and *Y* (1) based on risk modeling, under assumption A3. Given *P*{*Y* (0) = 0, *Y* (1) = 1|*S*} = 0, there are only three possible combinations of *Y* (0) and *Y* (1). Let *C* = *k* indicate the event that an individual’s pair of potential outcomes falls into the *k ^{th}* category as shown below,

$$\begin{array}{cc}\hfill & C=1:\phantom{\rule{1em}{0ex}}Y\left(0\right)=1\&Y\left(1\right)=1\hfill \\ \hfill & C=2:\phantom{\rule{1em}{0ex}}Y\left(0\right)=1\&Y\left(1\right)=0\hfill \\ \hfill & C=3:\phantom{\rule{1em}{0ex}}Y\left(0\right)=0\&Y\left(1\right)=0.\hfill \end{array}$$

(2)

For categorical outcomes, alternative models can be used for the distribution of *C* conditional on *S*. Here we adopt a cumulative link model because its parameters can be interpreted in terms of a generalized linear risk model, as will be shown next.

For *k* = 1, 2, suppose we model the cumulative probabilities of *C* conditional on *S* by

$$g\left\{P(C\le k\mid S)\right\}={\alpha}_{k}+{\beta}_{k}^{T}\gamma \left(S\right),$$

(3)

where *g* is a link function such as logit or normal CDF and *γ*(*S*) is a vector of pre-specified functions of *S* such that *P*(*C* ≤ 2|*S*) ≥ *P*(*C* ≤ 1|*S*) holds for a practical range of *S*. We take *γ*(*S*) = *S* later in the simulation studies and in the data example, but *γ*(*S*) can be made flexible by including transformed versions of *S* such as polynomial terms and regression spline.

Note that subjects with *Y* (1) = 1 belong to category *C* = 1 and subjects with *Y* (0) = 1 belong to either category *C* = 1 or category *C* = 2. As a result, there is a one-to-one relationship between the risk model and the joint distribution of *Y* (0) and *Y* (1), here based on the cumulative link model. Specifically, a risk model including main effects for *Z* and *γ*(*S*) and their interaction,

$$g\left[P\{Y\left(Z\right)=1\mid S\}\right]={\theta}_{0}+{\theta}_{1}Z+{\theta}_{2}^{T}\gamma \left(S\right)+{\theta}_{3}^{T}\gamma \left(S\right)Z,$$

(4)

corresponds to a cumulative link model (3) with *α*_{2} = *θ*_{0}, *β*_{2} = *θ*_{2}, *α*_{1} = *θ*_{0} + *θ*_{1}, and *β*_{1} = *θ*_{2} + *θ*_{3}.

With a little abuse of notation, let *Risk*(*S, Z*) denote the probability of disease conditional on *S* and the assigned treatment *Z*. Under (A2) the ignorable treatment assignment assumption, *Risk*(*S, Z*) *P*(*Y* = 1|*Z, S*) = *P*{*Y* (*Z*) = 1|*S*} *Risk*_{(Z)}(*S*). Thus parameters in (3) can be uniquely identified based on observed data. Next we propose methods for estimating and comparing biomarkers’ treatment-selection capacity. Estimation can be carried out in two steps, the estimation of parameters in model (3) and the estimation of measures of interest as defined in section 2.2.

As shown above, with the one-to-one correspondence between (3) and (4), applying an ordinary risk model such as a logistic regression model to the data will lead to consistent estimates for parameters in the cumulative link model. However, this procedure does not explicitly exploit the monotonicity assumption A3 during estimation. Note that *P*(*C* ≤ 1|*S*) ≤ *P*(*C* ≤ 2|*S*) *Risk*_{(0)}(*S*) ≥ *Risk*_{(1)}(*S*) for every *S* value, which is not guaranteed by fitting of an ordinary risk model.

To address this issue, we propose to maximize a constrained observed likelihood. Suppose there are *n*_{0} subjects in placebo arm and *n*_{1} subjects in active treatment arm with *n* = *n*_{0}+*n*_{1}. Let *i* be the subject indicator. We maximize

$$l=\mathrm{log}\phantom{\rule{thickmathspace}{0ex}}L=\sum _{i=1}^{n}{Y}_{i}\mathrm{log}\left\{P({Y}_{i}=1\mid {Z}_{i},{S}_{i})\right\}+(1-{Y}_{i})\mathrm{log}\left\{P({Y}_{i}=0\mid {Z}_{i},{S}_{i})\right\}$$

(5)

subject to the constraint that *Risk*(*S _{i}*,

Let $\widehat{\theta}$ be the constrained MLE for *θ* in risk model (4). We calculate estimates for *Risk*_{(0)} and *Risk*_{(1)} given each biomarker value observed in the sample as ${\widehat{Risk}}_{\left(0\right)}\left({S}_{i}\right)=G({S}_{i},Z=0,\widehat{\theta})$ and ${\widehat{Risk}}_{\left(1\right)}\left({S}_{i}\right)=G({S}_{i},Z=1,\widehat{\theta})$. Assumption A2 implies the equal distribution of marker within each treatment arm: *S*|*Z* = 0 =^{d}
*S*|*Z* = 1. Thus *F* (*S*), the common cumulative distribution function of *S*, can be estimated empirically with $\widehat{F}\left(s\right)={\sum}_{i=1}^{n}I({S}_{i}\le s)\u2215n$. Then we estimate TPF(*q*) and FPF(*q*) by entering ${\widehat{Risk}}_{\left(z\right)}$, $\widehat{F}$, and $\widehat{Q}\left(s\right)$ into (2):

$$\begin{array}{cc}\hfill \widehat{\mathrm{TPF}}\left(q\right)& =\frac{\int \left\{{\widehat{Risk}}_{\left(0\right)}\left(s\right)-{\widehat{Risk}}_{\left(1\right)}\left(s\right)\right\}I\left\{\widehat{Q}\left(s\right)>q\right\}d\widehat{F}\left(s\right)}{\int \left\{{\widehat{Risk}}_{\left(0\right)}\left(s\right)-{\widehat{Risk}}_{\left(1\right)}\left(s\right)\right\}d\widehat{F}\left(s\right)}\hfill \\ \hfill \widehat{\mathrm{FPF}\left(q\right)}& =\frac{\int \left\{1-{\widehat{Risk}}_{\left(0\right)}\left(s\right)+{\widehat{Risk}}_{\left(1\right)}\left(s\right)\right\}I\left\{\widehat{Q}\left(s\right)>q\right\}d\widehat{F}\left(s\right)}{\int \left\{1-{\widehat{Risk}}_{\left(0\right)}\left(s\right)+{\widehat{Risk}}_{\left(1\right)}\left(s\right)\right\}d\widehat{F}\left(s\right)},\hfill \end{array}$$

where for *Q*(*S*) being a pre-specified function of *S*, $\widehat{Q}\left(s\right)$ denotes the estimator of *Q*(*s*) based on $\widehat{\theta}$ when *Q*(*S*) involves *θ* (e.g. when *Q*(*S*) *Risk*_{(0)}(*S*) – *Risk*_{(1)}(*S*)), and $\widehat{Q}\left(s\right)$ equals *Q*(*s*) otherwise (e.g. when *Q*(*S*) *S*). Observe that because of the constraints guaranteeing that ${\widehat{Risk}}_{\left(0\right)}\left({S}_{i}\right)\ge {\widehat{Risk}}_{\left(1\right)}\left({S}_{i}\right)$ for *i* = 1, , *n*, $\widehat{\mathrm{TPF}}\left(q\right)$ and $\widehat{\mathrm{FPF}}\left(q\right)$ will always fall within (0,1), whereas estimates for TPF and FPF in (1) could go beyond (0,1) if an ordinary risk model were applied.

Finally, the ROC curve can be estimated by $\widehat{\mathrm{ROC}}\left(t\right)=\widehat{\mathrm{TPF}}\left\{{\widehat{\mathrm{FPF}}}^{-1}\left(t\right)\right\}$ and the distribution of *Q*(*S*) by ${\widehat{\mathrm{CDF}}}_{Q}\left(q\right)=\int Q\left(s\right)d\widehat{F}\left(s\right)$.

Under standard regularity conditions, asymptotic normality of the constrained MLE for risk model coefficients and measures of treatment-selection performance can be established based on standard techniques (Web Supplementary Appendix B). However, in practical sample sizes, applying the constraints leads to a substantial drop in the variances, which are over-estimated by asymptotic formulas (results omitted). Instead we use bootstrap resampling for variance estimation and construction of confidence intervals.

Using estimates of the measures developed above such as a point on the ROC curve, area under the ROC curve (AUC), or the distribution of *Q*(*S*), we can characterize and test for the treatment-selection performance of a marker (or risk model). For example, whether or not *AUC* > 0.5 in our setting corresponds to whether the biomarker distinguishes patients who benefit from the treatment versus patients who do not. A more relevant question in biomarker research is to compare treatment-selection performance between markers (or risk models), which can also be addressed by comparing these summary measures using a Wald-test, as will be demonstrated in the simulation studies.

So far we have assumed the monotonicity assumption (A3), which is key to the definition and identification of TPF and FPF defined in section 2.2. In settings where A3 is likely to hold, e.g., when the outcome is specific to a particular disease such as HIV and the treatment cannot plausibly cause the disease, adopting this assumption simplifies the problem and allows efficiency gain in estimating CDF_{Q} by restricting the parameter space during estimation. There are other occasions, however, where A3 could be violated. For example, if the outcome of interest is survival, even if the treatment is effective at treating the disease of interest, it might increase the risk of other adverse events and therefore result in a nonzero probability of *Y* (1) > *Y* (0). When this happens, *D* = *Y* (0) – *Y* (1) takes three values, 1, 0, −1, which we define as the *Benefitted, Unaffected*, and *Harmed* groups respectively. We can define the probability of *Q*(*s*) > *q* conditional on *D* = 1, 0, −1 as the true positive fraction (*Benefitted*), false positive fraction (*Unaffected*), and false positive fraction (*Harmed*) respectively. That is: TPF_{B}(*q*) = *P*{*Q*(*S*) > *q*|*D* = 1}, FPF_{U}(*q*) = *P*{*Q*(*S*) > *q*|*D* = 0}, and FPF_{H}(*q*) = *P*{*Q*(*S*) > *q*|*D* = −1}. Then two ROC curves can be constructed as ${\mathrm{ROC}}_{U}\left(t\right)={\mathrm{TPF}}_{B}\left\{{\mathrm{FPF}}_{U}^{-1}\left(t\right)\right\}$ and ${\mathrm{ROC}}_{H}\left(t\right)={\mathrm{TPF}}_{B}\left\{{\mathrm{FPF}}_{H}^{-1}\left(t\right)\right\}$. Alternatively, we can categorize an individual according to whether *D* = 1 or *D* < 1. The quantities of interest would be the true positive fraction (*Benefitted*) TPF_{B}(*q*) as defined before and the false positive fraction (*Non-benefitted*) FPF_{N}(*q*) = *P*{*Q*(*S*) > *q*|*D* < 1}, with the corresponding ROC curve being ${\mathrm{ROC}}_{N}\left(t\right)={\mathrm{TPF}}_{B}\left\{{\mathrm{FPF}}_{N}^{-1}\left(t\right)\right\}$. In Web Supplementary Appendix C, we relate the quantities TPF_{B}, FPF_{U}, FPF_{H}, and FPF_{N} to the TPF and FPF defined in (1). When the probability of *Y* (0) < *Y* (1) or *D* = −1 is small, *TPF* approximates TPF_{B}, and FPF approximates FPF_{U} or FPF_{N}:

$$\begin{array}{cc}\hfill & \mathrm{TPF}\left(q\right)-{\mathrm{TPF}}_{B}\left(q\right)=[P\{Q\left(S\right)>q\mid D=1\}+P\{Q\left(S\right)>q\mid D=-1\}]P(D=-1)\u2215E\left(D\right),\hfill \\ \hfill & \mathrm{FPF}\left(q\right)-{\mathrm{FPF}}_{U}\left(q\right)=[P\{Q\left(S\right)>q\mid D=-1\}-2P\{Q\left(S\right)>q\mid D=0\}]P(D=-1)\u2215\{1-E\left(D\right)\},\hfill \\ \hfill & \mathrm{FPF}\left(q\right)-{\mathrm{FPF}}_{N}\left(q\right)=[P\{Q\left(S\right)>q\mid D=-1\}-P\{Q\left(S\right)>q\mid D<1\}]P(D=-1)\u2215\{1-E\left(D\right)\}.\hfill \end{array}$$

This is intuitive since small *P*(*D* = −1) corresponds to small violation of A3. Therefore, under a small violation of A3, our proposed ROC curve defined under A3 still approximates some ROC curves with meaningful interpretations (*ROC*_{U} and *ROC*_{N})

In practice, although the monotonicity assumption (A3) cannot be confirmed, it does have testable implications in a randomized trial. That is, evidence for a greater probability of Y(1)=1 than of Y(0)=1 within any subgroup rejects the monotonicity assumption. In addition we can conduct a sensitivity analysis in order to relax A3.

For a sensitivity analysis, we use a similar type of selection model as used by Gilbert et al. (2003) and Shepherd et al. (2008), and specify the following two model assumptions:

- S.1:
*P*{*Y*(0) = 0|*Y*(1) = 1,*S*} =*w*(*S*;*δ*,*γ*), where*w*(*S*;*δ*,*γ*) = Φ {*δ*+*h*(*γ, S*)},*γ*is fixed and known, Φ (·) is a known cumulative distribution function,*σ*is an unknown parameter, and for each*γ*,*h*(*S*;*γ*) is a known function of*S*. - S.2:
*P*{*Y*(0) = 0|*Y*(1) = 1} =*ϕ*for a fixed constant*ϕ*.

The parameters *γ* and *ϕ* are unknown and not identified by the observed data; they are sensitivity parameters, treated as fixed and known in a single analysis and then varied over a plausible range of values to form a sensitivity analysis (Scharfstein et al., 1999). The parameter *ϕ* is constrained between max(0, *ρ*_{1} – *ρ*_{0})/*ρ*_{1} and min(*ρ*_{1}, 1 – *ρ*_{0})/*ρ*_{1} for *ρ*_{z} = *P*{*Y* (*z*) = 1}, *z* = 0, 1. The parameter *γ* varies between −∞ and ∞. Setting *γ* to either extreme and also setting *ϕ* to its extremes provides the nonparametric bounds for the sensitivity analysis.

Following Gilbert et al. (2003), we specify *w*(*S*; *δ, γ*) with the anti-logit function, *w*(*S*; *δ, γ*) = {1 + *exp*(−δ – *γ**S*)}^{−1}, under which *γ* is the change in the log odds that a subject who experiences disease under active treatment *Z* = 1 would be harmed by treatment – i.e., would have been spared disease had placebo been assigned – per unit increase of the marker. The parameter *ϕ* is interpreted as the conditional probability that the active treatment causes harm given that a subject gets disease under assignment to active treatment.

S.1 together with A1 and A2 imply the following relationships:

$$\begin{array}{cc}\hfill P\{Y\left(0\right)=0,Y\left(1\right)=1\mid S\}=& w(S;\delta ,\gamma )Ris{k}_{\left(1\right)}\left(S\right)\hfill \\ \hfill P\{Y\left(0\right)=1,Y\left(1\right)=1\mid S\}=& \{1-w(S;\delta ,\gamma )\}Ris{k}_{\left(1\right)}\left(S\right)\hfill \\ \hfill P\{Y\left(0\right)=1,Y\left(0\right)=1\mid S\}=& Ris{k}_{\left(0\right)}\left(S\right)-\{1-w(s;\delta ,\gamma )\}Ris{k}_{\left(1\right)}\left(S\right)\hfill \\ \hfill P\{Y\left(0\right)=0,Y\left(1\right)=0\mid S\}=& 1-w(s;\delta ,\gamma )Ris{k}_{\left(1\right)}\left(S\right)-Ris{k}_{\left(0\right)}\left(S\right).\hfill \end{array}$$

(6)

In addition, straightforward calculation shows that

$$\int w(s;\delta ,\gamma )Ris{k}_{\left(1\right)}\left(s\right)dF\left(s\right)=P\{Y\left(0\right)=0,Y\left(1\right)=1\}=\varphi {\rho}_{1}.$$

(7)

Equation (7) determines *δ* given fixed *γ* and *ϕ*. In earlier work (Gilbert et al., 2003), estimation of *Risk*_{(z)}(*S*) and estimation of *δ* were performed separately. That is, *Risk*_{(1)}(*S*) was estimated based on an ordinary MLE method, and together with estimates of *F* and *ρ*_{1}, (7) was used to estimate *δ*. However, this separate estimation procedure will not guarantee that the estimates of *P*{*Y* (0) = 1, *Y* (0) = 0|*S*} and *P*{*Y* (0) = 0, *Y* (0) = 0|*S*} in (6) are nonnegative. We propose to extend the constrained MLE method under A3 to estimate the risk model and *δ* together. Specifically for given *γ* and *ϕ*, we maximize the log-likelihood (5) under the constraints, *Risk*(*S*_{i}, *Z* = 0) ≥ {1 – *w*(*S*_{i}; *δ, γ*)} *Risk*(*S*_{i}, *Z* = 1), and 1 – *w*(*S*_{i}; *δ, γ*)*Risk*(*S*_{i}, *Z* = 1) – *Risk*(*S*_{i}, *Z* = 0) ≥ 0 for *i* = 1, … , *n*. Note that the constrained MLE under A3 is a special case where *ϕ* = 0 and *w* = 0. For an application, one might vary *γ* between −log(10) and log(10), assuming that selection odds ratios greater than 10 are unlikely, and vary *ϕ* between $\mathrm{max}(0,{\widehat{\rho}}_{1}-{\widehat{\rho}}_{0})\u2215{\widehat{\rho}}_{1}$ and $\mathrm{min}({\widehat{\rho}}_{1},1-{\widehat{\rho}}_{0})\u2215{\widehat{\rho}}_{1}$, where ${\widehat{\rho}}_{z}$ is an empirical estimate based on disease outcomes in treatment arm *Z* = *z*. Note that once and *ϕ* in S.1-S.2 are specified, assumptions (A1, A2, A4) and the observed data identify all of the conditional probabilities in (6).

We consider a simulation setting where we compare two markers with respect to their treatment-selection capacity. Two markers, *S*_{1} and *S*_{2}, are sampled from truncated bivariate normal distribution with correlation *ρ* equal to 0.0 or 0.5. For each marker, the potential outcomes follow the three-category cumulative link model (3) with logit link and *γ*(*S*) = *S*. Consider a randomized trial with 1:1 randomization to *Z* = 0 or *Z* = 1. The risk model parameters are chosen such that the disease prevalence is 0.12 and 0.06 in the placebo and active arm respectively. For total sample size *n* varying from 200 to 1,000, we evaluate performance of the constrained MLE for TPF(*q*), FPF(*q*), and CDF_{Q}(*q*) for *q*_{L} = 0.035 and *q _{H}* = 0.085, ROC(t) for

We present results for a setting where *S*_{1}, *S*_{2} are each simulated from a normal distribution truncated at zero with mean 3, with standard deviation 0.5 and 0.8 respectively. The AUCs for marker 1 and 2 are 0.61 and 0.73 respectively. Estimates of the risk model parameters (Table 1), TPF, FPF, CDF_{Q}, ROC(*t*), and AUC (Table 2 and Web Supplementary Table 1) have minimal bias and good converge using percentile bootstrap confidence intervals. Note that in our setting, estimates of ROC(*t*) and AUC have much smaller variability compared to estimates of TPF, FPF, or CDF_{Q}. This is intuitive, considering the small difference in disease prevalence between treatment arms. The curves of TPF(*q*), FPF(*q*) or CDF_{Q}(*q*) versus *q* thus are very steep, making it hard to estimate points on these curves precisely.

Enforcing the constraints on risk model estimation tends to increase efficiency. We looked at the efficiency of the constrained MLE relative to the MLE from ordinary logistic model fitting (i.e. ratio of variance for unconstrained MLE over variance of constrained MLE) for estimation of risk model parameters and CDF_{Q}. Enforcing the constraints appears to have a larger effect on model parameter estimates than on CDF_{Q} estimates. For example, with a sample size 1,000, the efficiency gains achieved by constrained model are around 46% for marker 1 and 50% for marker 2 in estimating *α*_{1} and *β*_{1}, and around 32% for marker 1 and 28% for marker 2 in estimating *α*_{2} and *β*_{2}; for CDF_{Q}(*q _{L}*) the efficiency gains are 10% and 18% for markers 1 and 2 respectively.

Our method has a small to medium power for comparing the two markers with an AUC difference of around 10% as sample size varies from 200 to 1000 (Table 3). Power increases when the two markers are positively correlated. In another example, we compare marker 1 with another marker that comes from a normal distribution truncated at zero with mean 3, variance 1, and AUC value 0.81. With a 20% difference in AUC between the two markers, we achieve reasonable power: 58.4, 81.3, and 94.8 respectively for *n* = 200, 500, 1, 000 for *ρ* =0.0, and 74.3, 92.0, and 98.9 for *ρ* = 0.5 (results not shown). Thus comparing treatment-selection performance between markers is a challenging task. Satisfactory power requires either a large sample size or a big difference in marker performance.

Table 4 presents the results of a sensitivity analysis based on a sample size of 500. We compare the area under ROC estimated under A3 with the areas under ROC_{U} and ROC_{N} for different ranges of the sensitivity parameter *ϕ* and for a range of values for *γ* from −log(10) to log(10). For ROC_{U} (and similarly for ROC_{N}), as deviation from A3 increases, we see an increase in bias of the AUC estimated under A3 and an increase in length of the sensitivity interval (the union of confidence intervals across the range of *ϕ* and *γ*) as expected.

We illustrate our methods using data from the ‘Step’ HIV vaccine efficacy trial, a study with the primary objective of evaluating the ability of the MRKAd5 HIV-1 gag/pol/nef vaccine to prevent HIV infection. In this study 3000 HIV-seronegative participants were 1:1 randomized to receive the candidate vaccine or placebo, pre-stratified by sex, baseline adenovirus type 5 (Ad5) neutralization titer, and study site (Buchbinder et al., 2008). Since only one female was infected during the study, we restricted our analysis to men with 922 in the placebo arm and 914 in the vaccine arm.

Among male participants, HIV acquisition rates within 3 years of randomization were similar in vaccine (0.060) and placebo (0.051) recipients with baseline Ad5 titers < 18, but for those with Ad5 ≥ 18, the infection rate in vaccinees (0.058) appeared to be more than twice as high as that in placebo recipients (0.030). Since this particular vaccine appears to have either a harmful effect or no effect in various subgroups, we adopt the model assuming the vaccine has no benefit to an individual, i.e., *Y* (0) ≤ *Y* (1). Log-transformed baseline Ad5 is evaluated as a treatment-selection marker for the vaccine-induced increase in risk of HIV infection using the constrained MLE method with logistic risk model.

Let *Y* (0) and *Y* (1) be potential HIV infection status at the end of the study for a subject assigned to placebo and vaccine respectively. The estimated probabilities of *Y* (0), *Y* (1) conditional on Ad5 and their 95% percentile bootstrap confidence intervals are presented in figure 1. Specifically, *P*{*Y* (0) = 0, *Y* (1) = 0} is the probability of avoiding infection under both assignments, *P*{*Y* (0) = 0, *Y* (1) = 1} is the probability of being harmed by vaccination, and *P*{*Y* (0) = 1, *Y* (1) = 1} is the probability of being infected no matter what assignment a subject receives. Note that the probability of being harmed by vaccine appears to increase with baseline Ad5 titer, which is consistent with the result based on dichotomizing baseline Ad5 at value 18.

On the basis of a decision rule *Q*(*S*) = *Risk*_{(1)}(*S*) – *Risk*_{(0)}(*S*), plots of CDF_{Q}, TPF and FPF at various risk difference thresholds for baseline Ad5 are displayed in figure 2(a), with the ROC curve displayed in figure 2(c). For comparison purposes, the corresponding curves for log-transformed age are presented in figures 2(b) and (c). The sample correlation between log(Ad5) and log(age) is −0.15. Variability in risk difference appears to be slightly larger for baseline Ad5 compared to age. Baseline Ad5 seems to have better discriminatory power in terms of the ROC curve for classifying an individual into treatment-effectiveness groups compared to age, with AUC estimate 0.72 and 0.63 respectively.

Plot of CDF_{Q}(*q*), TPF(*q*), FPF(*q*) versus *q* = *Risk*_{(1)} – *Risk*_{(0)} for baseline Ad5 (a) and age (b), and corresponding ROC curves for classifying a subject into treatment effective or ineffective groups (c).

Web Supplementary Table 2 shows the treatment-selection capacity of Ad5 and age with respect to TPF, FPF, and CDF_{Q} for a given low risk difference threshold *q _{L}* = 0.005 and a high risk difference threshold

We also conducted sensitivity analysis under violation of the monotonicity assumption. Without monotonicity, baseline Ad5 could have substantially lower classification accuracy. For example, estimates of the AUC under the ROC_{N} curve (based on *D* < 1 versus *D* = 1) range from 0.64 to 0.75 for *ϕ* ≤ 0.1, from 0.58 to 0.75 for *ϕ* ≤ 0.2, and from 0.53 to 0.75 for *ϕ* ≤ 0.3 when *γ* varies from −log(10) to log(10).

In this paper we propose characterizing a marker’s capacity for predicting individual treatment effect using a potential-outcomes based framework. Under this framework with a monotone treatment effect assumption, we can identify the joint distribution of the pair of the potential outcomes under the two treatment assignments, which is meaningful to individual patients. We propose measures that quantify a marker’s ability to classify a subject according to potential treatment-effectiveness — the ROC measure, as well as a marker’s ability to inform treatment decisions — the CDF_{Q} measure. While the CDF_{Q} measure depends on the scale of response rate, the ROC measure can be thought of an intrinsic property of a biomarker since it does not depend on disease prevalence within treatment group and thus is potentially useful to compare markers across different populations or studies. Under a randomized trial design, we develop a constrained maximum likelihood method for estimation of these measures. The procedure is closely related to common risk model estimation and is easy to implement. R code is available for downloading at http://labs.fhcrc.org/huang/index.html.

In practice, we recommend the method with monotonicity assumption be used when its plausibility is supported by data and ideally also by the scientific knowledge. If no major subgroups of interest have an overall treatment effect estimate in the ‘wrong direction’, then monotonicity is supported. When this assumption holds, the method we use to incorporate it is desirable for the simple interpretation and the efficiency gain. We have shown through simulation studies that when violation of the monotonicity assumption is minor, we still get meaningful estimates that approximate the measures defined in a general setting that allows for a non-monotone treatment effect. We further propose a sensitivity analysis as a way to relax the monotonicity assumption.

Our proposed approach facilitates a more-informed assessment of a marker’s value for treatment selection than does the classical approach of testing for an interaction between marker and treatment in an ordinary risk model. As we have demonstrated in the example of Web Supplementary Appendix A, a strong interaction coefficient is important for a marker to have value for treatment selection but is not useful for summarizing performance because it depends on other coefficients in the risk model as well as the functional form of the model. Therefore the interaction coefficient is not directly comparable between markers (models). In practice, a common approach is to plot disease risk versus marker value for each treatment group. This provides useful information to individual patients who have marker results in hand about their expected benefit of treatment given their marker measure. In this manuscript, we address a different question which is more relevant to biomarker researchers and policy makers: i.e., how to characterize and compare markers (risk models) with respect to their treatment-selection capacity. The ROC and CDF_{Q} measures proposed here are suitable for this task. For a univariate marker, a plot of *q* versus CDF_{Q}(*q*) is actually just the difference in risk between treatment groups as a function of marker percentile.

We want to point out that good treatment selection performance as characterized by our proposed measures corresponds to large variability in risk difference as a function of marker value. The actual impact of measuring the marker on treatment decisions depends on the risk difference threshold. Our measures provide an overview of treatment-selection capacity allowing the risk difference threshold to vary. This is helpful in situations where there does not exist a well-established decision threshold and the choice relies on other factors such as the cost and side-effects of the active treatment. This is oftentimes true in practice.

Song and Pepe (2004) proposed two measures to characterize a univariate marker’s treatment-selection performance: (i) the population response rate (*θ**v*) given a treatment policy that assigns a subject to active treatment if one’s marker value exceeds the *v ^{th}* quantile in the population; (ii) individual difference in risk with and without treatment,

We studied a univariate marker in our example, but the methodology is general and applies to models with multiple biomarkers by entering the joint distribution of markers into (1). Our measures can be used to compare two general risk models. For example, we can compare two biomarkers with respect to their treatment-selection capacity as demonstrated in this paper, and we can look at the gain by adding a new marker into a baseline model.

The methods we propose summarize the treatment-selection capacity of a marker in the whole population. In practice, a marker’s performance might vary with other baseline covariates such as age or gender and its performance conditional on those covariates is often of interest. Our framework can be easily extended to allow the assessment of a marker’s treatment-selection capacity within subpopulations defined by other covariates. Specifically, for estimation of the proposed performance measures conditional on specific covariates value, we need to estimate the risk model with the additional covariates included and estimate the marker’s distribution conditional on covariates. Again constraints can be enforced on the risk model. This is currently under investigation.

In this manuscript we focus on a binary endpoint, but the ROC and CDF_{Q} measures proposed can be naturally extended to handle an event time endpoint. Specifically, for given time *t* of interest, these measures can be constructed from the risk difference between treated and untreated groups, where risk is defined as the probability of developing the disease before time *t*. Existing techniques for estimating risk using censored data can be employed.

The authors are grateful to reviewers for constructive comments. This work was supported by NIH grants P30CA015704, R01CA152089, 2R37AI05465-08, R01GM054438, P01CA053996, and U24CA086368.

**Supplementary Materials**

Web Appendices referenced in Sections 1, 2.3.2, 2.4, 3, and 4 are available under the Paper Information link at the Biometrics website http://www.biometrics.tibs.org.

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