Measuring the individualization potential of treatment individualization rules: application to rules built with a new parametric interaction model for parallel-group clinical trials

Abstract

For personalized medicine, we propose a general method of evaluating the potential performance of an individualized treatment rule in future clinical applications with new patients. We focus on rules that choose the most beneficial treatment for the patient out of two active (non-placebo) treatments, which the clinician will prescribe regularly to the patient after the decision. We develop a measure of the individualization potential (IP) of a rule. The IP compares the expected effectiveness of the rule in a future clinical individualization setting vs. the effectiveness of not trying individualization. We illustrate our evaluation method by explaining how to measure the IP of a useful type of individualized rules calculated through a new parametric interaction model of data from parallel-group clinical trials with continuous responses. Our interaction model implies a structural equation model we use to estimate the rule and its IP. We examine the IP both theoretically and with simulations when the estimated individualized rule is put into practice in new patients. Our individualization approach was superior to outcome weighted machine learning according to simulations. We also show connections with crossover and N-of-1 trials. As a real data application, we estimate a rule for the individualization of treatments for diabetic macular edema and evaluate its IP.

1. Introduction

The strong influence of the ideas of personalized medicine in current biomedical research has awakened an interest in developing statistical methods that assist clinicians in individualizing medical treatments. The question of how to use data from superiority or, more generally, comparative effectiveness clinical trials to create such methods has motivated the development of parametric or semiparametric methods for the individualization of competing treatments,^1,2,3,4,5,6 methods for dynamic treatment regimes concerned with the individualization of treatment sequences,^7,8,9 and regression models with random effects for drug dosage or treatment individualization,^10,11,12 among others.

Intricately connected with treatment individualization are also methods of measuring individual treatment effects. If we can predict future effects of alternative treatments in a patient, in principle we know which treatment is better for the patient. Regressions with interactions^13,14 and machine learning¹⁵ have been used to examine individual effects. Mayer¹⁶ and Mayer et al.,^17,18 showed that individual effects can be determined with latent variable models, and Diaz¹⁹ proposed using random effects linear models to measure individual benefits.^20,21 Here, we propose a novel, general measure of the performance of an individualized treatment rule (ITR) and put the measure into practice with a model that lives in these two parametric worlds (latent variables and random effects), depending on the trial design.

We are not concerned here with dynamic treatment regimes. Rather, we focus on ITRs that try to choose the best treatment for a new patient out of two active (non-placebo) treatments, which the clinician will administer regularly to the patient afterwards. We will also study the ITRs of this type that recommend one treatment over the other if a special continuous measure calculated with patient’s individual data is $\ge 0$, using a parametric approach; we call these rules monotonic individualized treatment rules (MITRs). Several methods to develop MITRs have been proposed in the statistical literature. For example, Xu et al.³ proposed a regularized outcome weighted method that produces an optimal (parametric or semiparametric) linear individualized rule that incorporates interactions between treatment and covariates. Liang & Yu⁶ investigated a semiparametric treatment decision rule based on variables interacting with treatment. Chen et al.⁵ proposed a general method for subgroup identification that minimizes a convex loss function, encompassing parametric and semiparametric models. Zhao et al.¹ developed an individualized rule based on outcome weighted learning. Lu et al.² proposed a penalized regression method that simultaneously identify moderators of treatment effects and estimate an individualized rule; this approach was further refined by Shi et al.⁴ to include larger sets of variables. And Diaz¹² implemented a partial empirical Bayes approach to develop an MITR for individual N-of-1 trials that combine patient data with information from crossover trials to predict the post-treatment disease severity.

Our goals are: (1) to propose a general method of evaluating the potential performance of an ITR in future clinical applications with new patients; we propose a measure of performance called the individualization potential (IP) of the ITR. (2) To develop a new parametric interaction model for parallel-group clinical trials and propose an MITR calculated with the model. We will illustrate our evaluation method using this MITR. Our model implies a structural equation model (SEM) that we will use to estimate the MITR and its IP. (3) To derive novel formulas for calculating and evaluating the IP of MITRs developed with data from clinical trials. (4) To investigate the statistical properties of the estimate of the IP when the proposed MITR is used in new patients. The investigation in new patients is conducted both theoretically and with simulations. (5) To compare the MITR with outcome weighted machine learning.¹ And (6) to illustrate our methods by estimating an MITR for the individualization of treatments for diabetic macular edema (DME) using real data.²² We also make connections with crossover and N-of-1 trials.¹² Proofs are provided in Supplemental Material (Supp. Mat.).

2. Motivation for the proposed measure of IP

The impact of any ITR on the patient population should be evaluated by comparing the approach with treatment without individualization (TWI).¹² We should check whether the rule is better than not attempting any individualization. Our measure of IP enables this comparison. TWI is defined as any clinical decision approach that prescribes a treatment to a patient in a way independent of any variable determining the patient’s response to treatment including the patient’s clinical features.¹² Several forms of TWI are common. They include prescribing to all patients the treatment that previous clinical trials suggested was superior on average (the “one-size-fits-all” approach), or prescribing medicines based only on costs or insurance plans, or guesswork when clinical trials found that the two alternative treatments were equivalent on average.¹² If statisticians are to convince clinicians and insurance companies to use their ITRs instead of TWI, they need to show evidence that their methods will be more beneficial for the patient population than not attempting any individualization, because individualization usually requires additional patient evaluations, costly measurements and expertise and, in the case of individualization based on statistical models, specialized numerical computations.

However, a review of the literature reveals that investigators rarely compare the performance of statistical ITRs with the performance of TWI. Except for the work in Diaz,¹² none of the studies reviewed in the introduction compared their ITRs against TWI. They compared the ITRs with other ITRs instead. Even more, there are reports of sophisticated individualization rules that prescribe the treatment superior on average to all patients for some medical conditions and treatments, even when the optimal rule was trained with dozens of variables. There, the “one-size-fits-all” approach would be equally effective and cheaper. Examples are the rule in Diaz¹² for treating hypertension with Lopressoretic vs. Lopressor; and the rule in Zhao et. al.¹ for treating depression with Nefazodone plus psychotherapy vs. Nefazodone alone. However, we want not only to know whether an ITR is superior to TWI but also to measure the extent of its superiority.

Shi et al.²³ proposed a statistical test of the null hypothesis that no ITR developed with the available data will be superior to the one-size-fits-all approach, provided it is developed by maximizing the “value function”, defined as the unconditional expected value of the response to the treatment recommended by the ITR. The alternative hypothesis is the existence of a superior ITR. However, the test does not allow comparing the performance of specific ITRs with TWI, the only form of TWI that considers is the one-size-fits-all approach, and its application is limited to rules that maximize the value function.

A popular method to evaluate ITRs is to calculate conditional average treatment effects (CATEs), one for each competing treatment.^3,5,6,24 The CATE of a specific treatment is the achieved expected gain in the response after administering the treatment compared to the counterfactual response produced by the alternative treatment, calculated with only the subgroup of patients recommended the former by the ITR. Researchers have used CATEs to compare different ITRs with each other, but not with TWI. If both CATEs are positive, they conclude in qualitative evaluations that the ITR improves clinical outcomes. Only separate quantitative evaluations are possible, one for each treatment. Our proposed IP measure is a unified measure of performance that both accounts for the CATEs and allows comparisons with TWI.

3. Measuring treatment individualization performance

Thus, we will compare the performance of an individualization rule with that of TWI. In this section, the setting is not a clinical trial but a clinical practice in which a clinician decides which of two treatments should they prescribe to a patient. Hereafter, the letter $G$ will generically represent any clinical decision approach, regardless of whether the process attempts individualization, and $Q_G$ represents the treatment recommended by $G$ to the patient. Let $A_1$ and $A_2$ denote the alternative non-placebo treatments considered, which we view as specific realizations of the stochastic entity $Q_G$. Here, “not treating” is not an option. The clinician will implement $G$ to decide.

Here, we treat “a patient” as a stochastic entity $U$ that has a probability distribution. This point of view was adopted by Steyer,²⁵ Mayer,¹⁶ and Diaz.^12,19,20 The symbol $U$ represents a patient’s information. We view $U$ as a family of random variables that contains all the known and unknown patient’s characteristics determining, mediating or moderating the clinical response to treatment. This view of a patient as a stochastic entity is crucial in the derivation of a computational formula for the IP, as seen below [Eq. (2)]. In the parametric model in Section 4, $U$ is a random vector.

The random vector $(U,Q_G)$ has a joint probability distribution that depends on both the composition of the patient population and $G$. By definition, if $G=\text{TWI}$, then $Q_G$ is statistically independent of the patient $U$. For theoretical or simulation purposes, this means that $P\left(Q_{TWI}=A_k\right)$ does not depend on the patient for any $k$. For instance, if the clinician flips a coin to prescribe a treatment, $P\left(Q_{TWI}=A_k\left|U\right.\right)=P\left(Q_{TWI}=A_k\right)=1/2,k=1,2$. But if the clinician individualizes the treatment based on patient’s characteristics and/or some responses by implementing an ITR $G,P\left(Q_G=A_k\left|U\right.\right)$ will be a non-constant random variable.

3.1. The average effect on response of a clinical decision

Diaz¹² found that the average effect on response (AER) of an individualization approach is useful for evaluating the performance of individual N-of-1 trials in clinical settings. Here, we generalize Diaz’s definition of AER.

Suppose a clinical decision approach $G$ is to be implemented in a new patient $U$ and let $Y\equiv Y\left(A\right)\equiv Y(A,U,ϵ)$ denote the potential response of the patient to treatment $A=A_1$ or $A_2$. Here, $ϵ$ represents known or unknown factors independent of $U$ and $Q_G$ that may affect the observed response. We define the AER of $G$ by

$$AE{R}_G=\left\{E\left[Y\left(Q_G\right)\right]-E\left[Y\left(Q_G^C\right)\right]\right\},$$

where $Q_G^C$ is the treatment alternative to $Q_G$. In words, $AE{R}_G$ compares the expected potential response of a patient treated with $Q_G$ with the expected counterfactual response should the patient be treated with the alternative therapy instead.¹² $AE{R}_G$ measures the performance of $G$ in non-relative terms, without comparing $G$ with TWI. Ideally, if higher values of $Y$ indicate higher treatment benefits, $AE{R}_G$ should be $>0$. Otherwise, $G$ is not effective or even detrimental to the patient population health.

3.2. Individualization potential

We now explain how to measure the superiority of an ITR over any TWI approach. If the therapy tries to lower the value of $Y$ and the therapeutic target is to achieve $Y<y$ for some $y$ prespecified by the clinician, the target is called decremental.¹⁹ But if the goal is to achieve $Y>y$, the target is incremental. If it is decremental, $AE{R}_G<AE{R}_{TWI}$ indicates $G$ will be fruitful in treatment individualization.¹² We conclude the same for incremental targets when $AE{R}_G>AE{R}_{TWI}$. Thus, we define the individualization potential of $G$ as $I{P}_G=\left(AE{R}_G-AE{R}_{TWI}\right)$ and propose using $I{P}_G$ as a measure of the performance of $G$ relative to TWI. If $G$ is superior to TWI, $I{P}_G<0$ when the target is decremental, and $I{P}_G>0$ when incremental.

$I{P}_G$ determines the CATEs of the treatments under $G$. Denote $m_{i,j}\equiv E\left[Y\left(A_i\right)|Q_G=A_j\right]$. By definition, the CATE of treatment $A_i$ under $G$ is the effect of $A_i$ among the subjects prescribed $A_i$ by $G$, and is calculated as ${ℰ}_i=m_{i,i}-m_{j,i},j\ne i$. Therefore, in general,

$${\text{AER}}_G=\left(1-p\right){ℰ}_2+p{ℰ}_1,$$ (1)

where $p\equiv P(Q_G=A_1)$ is the proportion of patients prescribed $A_1$ by $G$. In particular, when $G=\text{TWI},AE{R}_{TWI}=\left(m_2-m_1\right)\left(1-2p\prime \right)$, where $m_i\equiv E\left[Y\left(A_i\right)\right]$ and $p^{\prime }=P\left(Q_{TWI}=A_1\right)$. The reason is that, by definition, the treatment prescribed by TWI is independent of $U$ and $ϵ$ and, therefore, $m_{i,j}=m_i$. Thus, when $G$ is an ITR,

$$I{P}_G\equiv I{P}_G\left(p,p\prime \right)=2\left\{\left(1-p\right)p\prime {ℰ}_2+p\left(1-p\prime \right){ℰ}_1\right\}.$$ (2)

Thus, $I{P}_G$ depends on both $p$ and $p\prime$. But $I{P}_G$ has information on the CATEs, which do not depend on $p$ (nor $p\prime$). When $p^{\prime }\ne 1/2$, the formulas for recovering the CATEs from $I{P}_G$ are:

$${ℰ}_1=\frac{1}{2p}\left(I{P}_G\right)+\frac{p\prime }{p\left(2p^{\prime }-1\right)}\left({\mathit{\text{AER}}}_{\mathit{\text{TWI}}}\right)\text{and}{ℰ}_2={\left(1-p\right)}^{-1}\left\{\frac{1}{2}\left(I{P}_G\right)-\frac{1-p\prime }{2p^{\prime }-1}\left({\mathit{\text{AER}}}_{\mathit{\text{TWI}}}\right)\right\}.$$ (3)

In the common situation when TWI prescribes the same treatment $A_2$ to all patients, plug in $p^{\prime }=0$ and $AE{R}_{TWI}=\left(m_2-m_1\right)$. As another possible situation, if $A_2$ is superior on average to $A_1$ but more expensive, clinicians may still favor $A_2$ but in some cases need to prescribe $A_1$; in this case, $0<p^{\prime }<1/2$.

But a unified ITR performance measure that does not depend on $p$ and $p\prime$ is the global potential. For incremental targets, we define it as:

$${\stackrel{-}{IP}}_G={\left(\int {\int }_{\left\{AE{R}_{TWI}\ge 0\right\}}dpd{p}^{\prime }\right)}^{-1}\left(\int {\int }_{\left\{AE{R}_{TWI}\ge 0\right\}}I{P}_G\left(p,p^{\prime }\right)dpd{p}^{\prime }\right).$$

${\stackrel{-}{IP}}_G$ is the average IP across all TWI implementations producing $AE{R}_{TWI}\ge 0$, and all populations with comparable CATEs for each treatment. When ${ℰ}_1>0$ and ${ℰ}_2\ge 0$,

$${\stackrel{-}{IP}}_G=\frac{4^{-1}{ℰ}_1^2+{ℰ}_1{ℰ}_2+4^{-1}{ℰ}_2^2}{{ℰ}_1+{ℰ}_2}.$$ (4)

Note the above definitions of AER and IP assume no specific parametric or nonparametric model of $Y$ nor a particular clinical decision strategy. $Y$ may be continuous, binary, etc. For instance, if $Y$ indicates whether or not a patient survived 5 years, $AE{R}_G$ is a difference in survival rates. Hereafter, we will focus on continuous responses.

4. The model

To build the ITR that will decide in a clinical practice which treatment ($A_1$ or $A_2$) is more beneficial for the patient, we will use data from a randomized clinical trial that investigated the superiority of $A_2$ over $A_1$ or vice versa, or their equivalence. Although the main contribution of this section is a new interaction model for parallel-group trials and an MITR derived from it, our formulation also allows calculating IPs for MITRs built from both parallel-group and crossover designs.¹²

4.1. Model of the response

Let $\omega$ denote a subject of the trial. Before the trial, the subject has $n_{\omega }$ (possibly transformed) potential responses to the treatments, denoted $Y_{\omega ,j},j=1,\dots ,n_{\omega }$, which correspond to the same study outcome. But after the trial, some or all these responses are available as measures, depending on the design. $Y_{\omega ,j}$ is the subject’s response to either $A_1$ or $A_2$, but not both. Let $Y_{\omega ,0}$ denote a baseline measure. We assume the subject belongs to a population that satisfies the linear model

$$Y_{\omega ,j}={\alpha }_{\omega }+{\beta }_{\omega ,1}{T}_{\omega ,1,j}+{\beta }_{\omega ,2}{T}_{\omega ,2,j}+{\theta }^T{X}_{\omega }+{ϵ}_{\omega ,j},j=0,1,\dots ,n_{\omega },$$ (5)

where $T_{\omega ,k,j}=1$ if $Y_{\omega ,j}$ was a response to treatment $A_k$, or $T_{\omega ,k,j}=0$ otherwise. Since no treatment is given at baseline, $T_{\omega ,k,0}=0$ for all $k$. We assume that ${\left({\alpha }_{\omega },{\beta }_{\omega ,1},{\beta }_{\omega ,2}\right)}^T$ is a vector of characteristic constants of patient $\omega$, which varies from patient to patient according to ${\left({\alpha }_{\omega },{\beta }_{\omega ,1},{\beta }_{\omega ,2}\right)}^T~N\left(\mathit{\mu },D\right)$, where $\mathit{\mu }={\left({\mu }_{\alpha },{\mu }_1,{\mu }_2\right)}^T$. The ${ϵ}_{\omega ,j}\prime s$ are normal with mean 0 and variance ${\sigma }_{{ϵ}_j}^2$, are independent of ${\left({\alpha }_{\omega },{\beta }_{\omega ,1},{\beta }_{\omega ,2}\right)}^T$ and may be correlated and heteroscedastic. Here, ${\mathit{X}}_{\omega }={\left(x_{\omega ,1},\dots ,x_{\omega ,q}\right)}^T$ is a vector of non-stochastic, patient-level time-independent covariates including demographics, comedications or clinical characteristics. Assume $\mathit{\theta }$ is a population-constant vector, that is, it has the same value in all patients. We view ${\beta }_{\omega ,k}$ as the effect of treatment $A_k$ on patient $\omega$ (relative to baseline).¹² Since the patients were randomized, treatment assignments $T_{\omega ,k,j}$ are independent of the covariates ${\mathit{X}}_{\omega }$.

The usefulness of Eq. (5) in personalized medicine rests with the interpretation of the variability of the random effects ${\beta }_{\omega ,1}$ and ${\beta }_{\omega ,2}$. Following Diaz,¹⁹ this variability is “not just a mathematical artifact to control for patients’ heterogeneity but also the result of real variation in the biological and environmental factors that have made humans develop as individuals”. Thus, it has information on known and unknown variables interacting with the treatments. We extract this information through an empirical Bayesian approach when there are repeated measures,^12,19,20 or by considering variables that explain that variability as in the approach for parallel-group trials developed below.

Although we view ${\left({\alpha }_{\omega },{\beta }_{\omega ,1},{\beta }_{\omega ,2}\right)}^T$ as a random vector, our model is not always the usual random effects model because not all $n_{\omega }$ responses need be measured. For instance, in a parallel-group trial, subject $\omega$ may provide a measured baseline response $Y_{\omega ,0}$ along with a measured response $Y_{\omega ,1}$ to its assigned treatment, and we reserve $Y_{\omega ,2}$ for the unobserved (potential) response to the alternative treatment (here, $n_{\omega }=2$). We cannot apply the usual random effects methods to estimate the matrix $D$ in this situation.^26,27 However, as seen below, the variability of the random effects is crucial to estimate the AER of any ITR built with Eq. (5). Our solution to this problem will be to calculate maximum likelihood estimates (MLEs) through a SEM²⁸ after reparameterizing Eq. (5), finding explanatory variables for the variability of the random effects, and viewing random effects as latent variables.

We can estimate all parameters of Eq. (5) with standard methods for random effects only if the clinical trial implements special crossover sequences of three or more periods and carry-over effects are prevented.^12,29,30

4.2. New vs. non-new patients

Here, we use the letter $\omega$ only to represent a patient who participated in the clinical trial that provided data for both estimating model parameters and designing the ITR. But we envision the application of ITRs to new patients, that is, those who did not participate in the clinical trial.^12,20 Thus, we assume that Eq. (5) is also valid for measured or potential responses of new patients. And, like Diaz^12,19,20 and Diaz et al.,¹⁰ when we need to emphasize that the equation also applies to new patients, we drop $\omega$ and write $Y_j,\alpha ,{\beta }_k,T_{k,j},{ϵ}_j$, and $\mathit{X}={\left(x_1,\dots ,x_q\right)}^T$ in place of the corresponding notations with $\omega$, where $k=1,2$. In accordance with the formulation of a patient as a stochastic entity given in Section 3, we interpret the vector ${U=\left(\alpha +{\mathit{\theta }}^T\mathit{X},{\beta }_1,{\beta }_2\right)}^T$ as “the patient”.¹²

The efficacy and statistical properties of an ITR should be investigated in new patients, not in the patients used to estimate the model supporting the ITR, because clinicians usually expect ITRs to be applied to new patients in clinical practice and because individualization through community-based trials is usually unrealistic.¹² In community-based trials, the patients of a group of patients undergo separate trials in possibly different health institutions with possibly different physicians. The decision on which treatment is more beneficial for one patient is based on the analysis of the combined data from all patients.^31,32,33 This approach is limited by the fact that it conflates clinical research with clinical practice, which burdens practicing physicians and regular patients with the added stringent regulations and protocols of clinical trials.¹² Patients’ reluctance to broadly share personal information may also make it impractical in many cases. Our position is that an ITR should be ready for application to new patients in clinical practice without the patients or clinicians necessarily sharing their data with other clinicians, needing to sign up to a clinical trial, or coordinating with other health centers. The ITR developed here applies directly to new patients once estimated, not necessarily to only the patients used to estimate it.

In N-of-1 trials, Diaz¹² formally defined a new patient as that for whom both $U$ and all errors ${ϵ}_j$’s are statistically independent of the model parameter estimates obtained with other patients. He showed that the distinction between new and non-new patients has profound consequences for the statistical properties of an ITR. Here we generalize this definition and define a new patient as one whose measured, and unmeasured variables and characteristics are independent of the population parameter estimates.

When attempting treatment individualization in a new patient using information from a clinical trial, the number and nature of the measured responses from the patient are not necessarily the same as those provided by the subjects of the clinical trial. For example, for an ITR designed with a parallel-group trial, we may not need to measure any response from the patient before making any clinical decision, although any decision better than TWI requires patient’s pretreatment variables. As another example, in individual N-of-1 trials, the patient may need to undergo different treatment sequences than those implemented in the crossover trial that informed the ITR, and the patient may even need to provide more responses than a typical subject of the crossover trial.¹²

4.3. Individual benefits

Implementing an ITR on a specific patient aims at finding the treatment producing the greatest benefit. Diaz¹⁹ defines a treatment’s benefit as the reduction in disease severity produced by the treatment. Disease severity is the probability that the patient’s response $Y$ is outside the therapeutic target. Thus, individual benefits are measured on an extended probability scale that yields a positive value if the treatment is beneficial, or negative if the treatment is detrimental.^19,20,21 For a decremental target, the benefit of treatment $A_k$ for a new individual patient, achieved at occasion $j$, is $b_{k,j}=\left\{\Phi \left(\left(y-\alpha -{\mathit{\theta }}^T\mathit{X}-{\beta }_k\right)/{\sigma }_{{ϵ}_j}\right)-\Phi \left(\left(y-\alpha -{\mathit{\theta }}^T\mathit{X}\right)/{\sigma }_{{ϵ}_0}\right)\right\},$ where $\Phi$ is the standard normal cumulative distribution function (CDF). [For an incremental target, multiply this formula by $\left(-1\right)$.] Insert the ${\omega }^{\prime }s$ for non-new patients. When applied to non-new patients, $b_{k,j}$ helps describe and compare treatments’ personal benefits in patient populations using data from clinical trials.^19,20 The formula has also been used to simulate individual benefits in new patients.^12,20,21,34

4.4. Model reparameterization

The individual benefit $b_{k,j}$, however, is not directly estimable in new patients because they rarely provide enough data to estimate their $\alpha$.¹² Like Diaz,¹² we resolve this problem reparameterizing Eq. (5) to

$$Y_{\omega ,j}={\gamma }_{\omega ,1}+{\gamma }_{\omega ,2}{T}_{\omega ,2,j}+{\theta }^T{X}_{\omega }+{ϵ}_{\omega ,j},j=1,\dots ,n_{\omega },$$ (6)

where ${\gamma }_{\omega ,1}={\alpha }_{\omega }+{\beta }_{\omega ,1}$ and ${\gamma }_{\omega ,2}={\beta }_{\omega ,2}-{\beta }_{\omega ,1}$. For a new patient, ${\gamma }_1=\alpha +{\beta }_1,{\gamma }_2={\beta }_2-{\beta }_1$. When the target is decremental, $b_{2,j}>b_{1,j}$ if and only if ${\gamma }_2<0$ (or ${\gamma }_2>0$ when incremental), regardless of the value of $y$. Thus, with a decremental target, the clinician may want to recommend $A_2$ as a therapy for a specific patient if the patient’s value of ${\gamma }_2$ is $<0$. Diaz¹² applied the above reparameterization to developing an ITR for N-of-1 trials using crossover designs. The ITR recommends the treatment that reduces the individual disease severity. The ITR, however, still needs parameter estimates obtained by fitting Eq. (5) to the crossover data because ${\alpha }_{\omega }$ affects the variability of the severity.¹²

But there is another reparameterization that leads to an equation analogous to Eq. (6), which is useful for parallel-group trials when the outcome of interest is the change in the response from baseline. If we calculate $\left(Y_{\omega ,j}-Y_{\omega ,0}\right)$ using Eq. (5), we obtain Eq. (6) with $\mathit{\theta }=0,{\gamma }_{\omega ,1}={\beta }_{\omega ,1}$ and ${\gamma }_{\omega ,2}=\left({\beta }_{\omega ,2}-{\beta }_{\omega ,1}\right)$. In this case, we reinterpret $Y_{\omega ,j}$ as a response change and ${ϵ}_{\omega ,j}$ as $\left({ϵ}_{\omega ,j}-{ϵ}_{\omega ,0}\right)$. But we still can interpret ${\gamma }_2$ as a proxy of the individual benefit of one treatment relative to the other. We use below this reinterpretation of Eq. (6) in the construction of MITRs with data from parallel-group trials, relabeling $\left(Y_{\omega ,1}-Y_{\omega ,0}\right)$ as $Y_{\omega ,1}$.

The variance of ${\gamma }_{\omega ,2}$ is usually called the patient-by-treatment (or subject-by-formulation) interaction variance.^30,35,12 The reason is that ${\gamma }_{\omega ,2}$ is viewed as the treatment effect on the response of patient $\omega$. A positive variance means that patients modify the treatment effect, that is, they interact with the treatment. Thus, this variance is essential to model individual effects and is therefore crucial to develop ITRs. Usually, it is estimated with random effects linear models. Not every clinical trial design, however, allows estimating it with these models. Only crossover designs with three or more periods that administer each treatment in at least two periods, and the two treatments in different periods, are useful.^26,30,35 Therefore, we cannot use random effects models to estimate the variance of ${\gamma }_{\omega ,2}$ with data from parallel-group trials because subjects do not crossover from one treatment to another in these trials. Here, we propose estimating the variance of ${\gamma }_{\omega ,2}$ by modeling ${\gamma }_{\omega ,2}$ as a function of baseline covariates that interact with the treatments, using SEMs.

4.5. A new interaction model for parallel-group trials

Next, we explain how to model ${\gamma }_{\omega ,2}$ with data from parallel-group trials. In these trials, subjects are randomized into two or more treatment groups, but here we focus on two groups. The number of subjects assigned to each group is fixed before the trial starts, all subjects in a group receive the same treatment, and subjects from different groups receive different treatments.³⁶ Here, $Y_{\omega ,1}$ measures the response change from baseline and we let $\mathit{\theta }=0$ in Eq. (6). A limitation of parallel-group designs is that they do not allow estimating the variance of ${\gamma }_{\omega ,2}$ using methods for random effects linear models.^14,26 But this variance is indispensable in the understanding of how individuals react differentially to treatments.³⁰ Thus, to develop MITRs with parallel-group trials, next we introduce assumptions on the variability of ${\gamma }_{\omega ,2}$ that allow overcoming this limitation using SEMs.

Suppose $N$ independent subjects were randomized to either $A_1$ or $A_2.$ Here, we focus only on the measured responses, $Y_{\omega ,1},\omega =1,\dots ,N$, and, therefore, simplify the notation: $Y_{\omega ,j}=Y_{\omega }$, $T_{\omega ,2,j}=T_{\omega ,2}$ and ${ϵ}_{\omega ,j}={ϵ}_{\omega }$. Suppose that, at baseline, subject $\omega$ additionally provided subject-level (possibly transformed) stochastic continuous variables $Z_{\omega ,i},i=1,\dots ,m$, which may potentially modify the response to treatment ($m\ge 2$). The $Z_{\omega ,i}$’s may be “repeated measures”; for instance, measures of a relatively stable dimension obtained with the same instrument at two or more different time points before treatment, subject only to measurement error variations. Or they may be conceptually related variables obtained simultaneously from subject $\omega$ (in our application, we use the retinal thicknesses of the left and right eye); or different item scores composing the final scores of one or more validated questionnaires. They can also be biomarkers such as gene expressions. Our model of ${\gamma }_{\omega ,2}$ assumes that

$${\gamma }_{\omega ,2}={\psi }_{1,i}+w_i\left(Z_{\omega ,i}-{\xi }_{\omega ,i}\right),i=1,\dots ,m,\text{and}{\gamma }_{\omega ,1}=\tau +{\psi }_2{\gamma }_{\omega ,2}\text{a}.\text{s}.,$$ (7)

where all the $\psi$’s, $w$’s and $\tau$ are population constants, with $w_i\ne 0$ for all $i$. The ${\xi }_{\omega ,i}$’s are zero-mean normal random variables with (possibly different) variances ${\sigma }_{{\xi }_i}^2$, which are mutually independent and independent of both ${\gamma }_{\omega ,2}$ and ${ϵ}_{\omega }$. The $Z_{\omega ,i}$’s may have different means and variances and may be correlated, but they may not be linearly dependent.

We can interpret these assumptions using the original model in Eq. (5). Note that Eqs. (7) imply

$${\gamma }_{\omega ,2}={\psi }_1+{\tilde{Z}}_{\omega }-{\tilde{\xi }}_{\omega }\text{and}{\gamma }_{\omega ,1}={\psi }_0+{\psi }_2\left({\tilde{Z}}_{\omega }-{\tilde{\xi }}_{\omega }\right),$$ (8)

where ${\psi }_1=m^{-1}{\sum }_{i=1}^m{\psi }_{1,i};{\psi }_0=m^{-1}{\sum }_{i=1}^m\left({\psi }_2{\psi }_{1,i}+\tau \right);{\tilde{Z}}_{\omega }=m^{-1}{\sum }_{i=1}^m{w}_i{Z}_{\omega ,i}$; and ${\tilde{\xi }}_{\omega }=m^{-1}{\sum }_{i=1}^m{w}_i{\xi }_{\omega ,i}$. But, since $Y_{\omega }$ is a response change from baseline, ${\gamma }_{\omega ,1}={\beta }_{\omega ,1}$ and ${\gamma }_{\omega ,2}=\left({\beta }_{\omega ,2}-{\beta }_{\omega ,1}\right)$, as explained in Section 4.4. That is, ${\gamma }_{\omega ,1}$ is the effect of $A_1$ on the original response in Eq. (5) relative to baseline, and ${\gamma }_{\omega ,2}$ is the effect of $A_2$ relative to $A_1$. So, ${\tilde{Z}}_{\omega }$ explains these two effects on the original response, but in different proportions when ${\psi }_2\ne 1.$

But our formulation is also an explanation of the interactions between the $Z_{\omega ,i}$’s and the treatments. In fact, Eq. (6) with $\mathit{\theta }=0$ and Eqs. (8) determine the interaction equation

$$Y_{\omega }={\psi }_0+{\psi }_1{T}_{\omega ,2}+{\tilde{Z}}_{\omega }{T}_{\omega ,2}+{\psi }_2{\tilde{Z}}_{\omega }+{ϵ}_{\omega }^{\left(1\right)},\omega =1,\dots ,N,$$ (9)

where ${ϵ}_{\omega }^{\left(1\right)}=\left({ϵ}_{\omega }-\left({\psi }_2+T_{\omega ,2}\right){\tilde{\xi }}_{\omega }\right)$.

4.6. An individualized rule calculated with parallel-group trials (the intersection rule)

The interaction model in Section 4.5 allows building MITRs with data from parallel-group trials and calculating their IPs. Consider the linear relationship between $Y_{\omega }$ and ${\tilde{Z}}_{\omega }$ in Eq. (9). The intersection of the lines corresponding to $\left(T_{\omega ,2}=1\right)$ and $\left(T_{\omega ,2}=0\right)$ occurs at $\left(-{\psi }_1\right)$. Thus, we propose the following individualization rule, which the clinician applies to a new patient: for a decremental therapeutic target, administer $A_2$ if ${\gamma }^{*}={\psi }_1+\tilde{Z}<0$. Otherwise, administer $A_1$. Reverse the inequality if the target is incremental. (We dropped $\omega$ from ${\tilde{Z}}_{\omega }$ because the patient is new.) In practice, we refine the rule by using ${\gamma }_{\Delta }^{*}={\gamma }^{*}-\Delta$ in place of ${\gamma }^{*}$, where $\Delta$ is a constant that optimizes the AER of the rule, given by Theorem 1 below. When the individualization rule is based on ${\gamma }_{\Delta }^{*}=\left({\psi }_1+\tilde{Z}-\Delta \right)$, we call it the intersection rule.

But using ${\gamma }^{*}$ (or ${\gamma }_{\Delta }^{*}$) does not always guarantee individualization. If the range of possible values of $\tilde{Z}$ does not include the two lines’ intersection, then the rule based on ${\gamma }^{*}$ will always prescribe the same treatment to all patients. For instance, the range will not include the intersection point if the target is incremental, $\tilde{Z}>0$ with probability 1 and ${\psi }_1>0$. In such a case, the rule is a version of TWI, so implementing it in a clinical setting is unnecessary because IP $=0$. But although the equivalence of the individualization rule and TWI is clear in this case, in most cases we need to calculate the IP to measure the superiority of the rule over TWI.

From another perspective, significant interactions between subject-level variables and medical treatments are not necessarily helpful in personalized medicine, a fact usually underappreciated. All this highlights the need for estimating IP.

5. Estimating the intersection rule and its IP

As shown in Section 6.1 below, the parameters needed to calculate the intersection rule and its IP are ${\psi }_1,{\mu }_{{\gamma }_2},{\sigma }_{{\gamma }_2}^2,w_i$ and ${\sigma }_{{\xi }_i}^2$, where ${\mu }_{{\gamma }_2}=E\left[{\gamma }_2\right]$ and ${\sigma }_{{\gamma }_2}^2=\text{Var}\left[{\gamma }_2\right]$. It is tempting to apply standard least squares to Eq. (9) to estimate both ${\psi }_1$ and the interaction parameters, the $w_i$’s. However, the error ${ϵ}_{\omega }^{\left(1\right)}$ may be correlated with ${\tilde{Z}}_{\omega }$ and, therefore, least squares are not proper here. Moreover, Eq. (9) alone does not have information on the other parameters. Our interaction model, however, implies the following SEM, which allows estimating all needed parameters.

Viewing ${\gamma }_{\omega ,2}$ as a latent variable and combining Eq. (6) (with $\mathit{\theta }=0$) and Eqs. (7), we obtain a measurement model that depends on the treatment group:

$$Y_{\omega }=\left\{\tau +\left({\psi }_2+T_{\omega ,2}\right){\gamma }_{\omega ,2}+{ϵ}_{\omega }\right\}\text{and}{Z}_{\omega ,i}=\left\{w_i^{-1}{\gamma }_{\omega ,2}-w_i^{-1}{\psi }_{1,i}+{\xi }_{\omega ,i}\right\},i=1,\dots ,m,$$ (10)

where all ${ϵ}_{\omega }$’s are independent due to randomization, with variance ${\sigma }_{ϵ}^2$, say. We fit Eqs. (10) simultaneously using SEM software. The coefficient of ${\gamma }_{\omega ,2}$ in the equation for $Y_{\omega }$ is either ${\psi }_2$ or ${\psi }_2+1$, depending on whether subject $\omega$ received $A_1$ or $A_2$, respectively. All other parameters are constrained to be the same for the two groups. Thus, if $c_{0,i}$ and $c_{1,i}$ denote the intercept and slope in the equation for $Z_{\omega ,i}$ vs. ${\gamma }_{\omega ,2}$, we calculate the intersection rule using $w_i=c_{1,i}^{-1}$ and ${\psi }_1=-m^{-1}{\sum }_{i=1}^m\left(c_{0,i}{c}_{1,i}^{-1}\right)$. Fitting Eqs. (10) gives direct estimates of ${\mu }_{{\gamma }_2},{\sigma }_{{\gamma }_2}^2$ and ${\sigma }_{{\xi }_i}^2$. To fix the scale of ${\gamma }_2$, we set $w_1=1$ and test its significance by testing the corresponding standardized coefficient. We obtained parameter MLEs^27,28 (Table 1).

Table 1.

Parameter estimates and estimated characteristics of the intersection individualization rule $G$ based on ${\gamma }_{\Delta }^{*}=\{{\psi }_1+2^{-1}\left(w_1{Z}_1+w_2{Z}_2\right)-\Delta \}$, for the individualization of aflibercept vs. bevacizumab in treating diabetic macular edema, where $Z_1$ and $Z_2$ are respectively the transformed baseline retinal thickness of the treated and non-treated eye.^a A new patient should be prescribed bevacizumab in place of aflibercept if ${\widehat{\gamma }}_{\Delta }^{*}\le 0$. The estimates were obtained with a structural equation model applied to data from a parallel-group clinical trial.

Parameter or Characteristic	Estimate	95% Confidence intervalb
Treated eye weight, ${\widehat{w}}_1$	1c	---
Non-treated eye weight, ${\widehat{w}}_2$	0.49660	(0.39478, 0.598418)
${\widehat{\psi }}_1$	−39.162	(−42.611, −35.712)
$\widehat{\Delta }$	−2.3008	(−3.8939, −0.70773)
$AE{R}_{\widehat{G}}$	6.231d	(4.739, 7.723)
${\widehat{AER}}_{TWI}={\widehat{\mu }}_{{\gamma }_2}$	4.067d,e	(1.844, 6.29)
Individualization potential, ${\widehat{IP}}_G$	2.164d,f	(0.561, 3.767)
$\widehat{p}=\widehat{P}\left({\gamma }_{\Delta }^{*}\le 0\right)$	0.269g	(0.147, 0.390)
Global potential, ${\stackrel{-}{IP}}_G$	4.048h	(3.090, 5.005)
Bevacizumab CATE, ${\widehat{ℰ}}_1$	4.027i	(2.664, 5.390)
Aflibercept CATE, ${\widehat{ℰ}}_2$	7.041j	(5.358, 8.723)
Bias factor, ${\widehat{\rho }}_Z$	0.639	(0.540, 0.737)

AER: Average effect on response; TWI: Treatment without individualization (prescribing aflibercept to all patients); IP: Individualization potential.

^aThe transformation was $Z=100\times \left(g\left(RT\right)-g\left(209\right)\right)/(g\left(\mathrm{1,290}\right)-g(209\left)\right)$, with $g\left(x\right)=\left(1-1/x\right)$, where RT is the retinal thickness of one eye. RT was measured in $\mu m$. This transformation produces values within 0–100.

^bStandard errors were calculated with the delta method.

^cThe parameter $w_1$ was constrained to 1. According to the standardized SEM model, $Z_1$ was a significant variable at a 0.05 significance level (standardized coefficient=0.595995, p<0.001); this justifies the inclusion of $Z_1$ in the individualization rule.

^dThe AERs and IP are measured in visual acuity units, which range from 0 to 100.

^eEstimate of $AE{R}_{TWI}$ when TWI prescribes to all patients the treatment superior on average, namely aflibercept. So, $p^{\prime }=0$.

^fIP that compares the rule $G$ versus TWI with $p^{\prime }=0$.

^gEstimated probability that the individualization rule prescribes bevacizumab.

^hGlobal potential under an incremental target (Eq. (4)).

ⁱConditional average treatment effect of bevacizumab among patients recommended bevacizumab by the intersection rule (in visual acuity units).

^jConditional average treatment effect of aflibercept among patients recommended aflibercept by the intersection rule (in visual acuity units).

6. Evaluation of monotonic individualized rules

Now we return to the clinical practice setting and focus on the calculation of IPs of MITRs using the parametric formulation in Sections 4.1–4.4. Then we will illustrate the general formulae by calculating the IP of the intersection rule introduced in Sections 4.5–4.6. Suppose the clinician implements an MITR that uses a continuous random variable ${\gamma }^{*}\equiv {\gamma }^{*}\left(\mathit{\vartheta }\right)$. We stress that the clinician calculates ${\gamma }^{*}$ with data from a new patient. ${\gamma }^{*}$ may or may not depend on an unknown vector $\mathit{\vartheta }$ of population parameters that needs to be estimated with data from a previously conducted clinical trial. For a decremental target, the clinician prescribes $A_2$ to the patient as therapy if and only if ${\gamma }^{*}<0$; they prescribe $A_1$ otherwise. Reverse the inequality if the therapeutic target is incremental. We call this individualization approach an MITR based on ${\gamma }^{*}$.

6.1. A formula to calculate AERs

In this section, we assume that Eq. (6) is valid and, except when we illustrate the results with the intersection rule, $Y_{\omega ,j}$ is not necessarily a response change. Let $Q_G$ be the treatment recommended by decision approach $G$ to a specific new patient, where $G$ can be an ITR or TWI. We also assume that after recommending the treatment, Eq. (6) continues to be valid for any future (unrealized) potential response from the patient, denoted by $Y_{j^{*}}$. That is, assume that $Y_{j^{*}}={\gamma }_1+{\gamma }_2{T}_{2,j^{*}}+{\mathit{\theta }}^T\mathit{X}+{ϵ}_{j^{*}}$, where ${ϵ}_{j^{*}}$ is a future unrealized error. (We drop $\omega$ since the patient is new.) We also assume that $Q_G$ is independent of ${ϵ}_{j^{*}}$; this is reasonable because $Q_G$ is a decision made only with the data currently provided by the patient. Theorem 1 below is valid in general for any MITR, not only for the intersection rule.

Theorem 1. (i) Let $G$ be an MITR based on ${\gamma }^{*}$. Assume ${\gamma }^{*}$ and ${\gamma }_2$ are jointly normally distributed. If the therapeutic target is decremental,

$$AE{R}_G=2{\mu }_{{\gamma }_2}\Phi \left(\frac{-{\mu }_{{\gamma }^{*}}}{{\sigma }_{{\gamma }^{*}}}\right)-2\frac{\text{Cov}\left({\gamma }_2,{\gamma }^{*}\right)}{{\sigma }_{{\gamma }^{*}}}{\Phi }^{\prime }\left(\frac{-{\mu }_{{\gamma }^{*}}}{{\sigma }_{{\gamma }^{*}}}\right)-{\mu }_{{\gamma }_2},$$ (11)

where ${\sigma }_{{\gamma }^{*}}=\sqrt{\text{Var}\left({\gamma }^{*}\right)},{\mu }_{{\gamma }^{*}}=E\left[{\gamma }^{*}\right]$. If the target is incremental, $AE{R}_G$ is the negative of the above quantity.

(ii) If ${\gamma }_{\Delta }^{*}={\gamma }^{*}-\Delta$ for a constant number $\Delta$ and $G_{\Delta }$ is an MITR based on ${\gamma }_{\Delta }^{*}$, then $AE{R}_{G_{\Delta }}$ is obtained by replacing ${\mu }_{{\gamma }^{*}}$ with ${\mu }_{{\gamma }^{*}}-\Delta$ in Eq. (11). Thus, for a decremental (incremental) target, the value of $\Delta$ that minimizes (maximizes) $AE{R}_{G_{\Delta }}$ is $\Delta =\left\{{\mu }_{{\gamma }^{*}}-{\mu }_{{\gamma }_2}\left(\frac{{\sigma }_{{\gamma }^{*}}^2}{\text{Cov}\left({\gamma }_2,{\gamma }^{*}\right)}\right)\right\}.$ So, we can improve the outcome of an MITR at the population level by subtracting $\Delta$.

(iii) For both decremental and incremental targets, if $G=\text{TWI}$, then $AE{R}_G={\mu }_{{\gamma }_2}\left\{2P\left(Q_G=A_2\right)-1\right\}$.

We use Theorem 1 to calculate IPs. Note that only Eq. (6) plays a role in its proof. Thus, it applies to both crossover and parallel-group designs. As an application, let us consider the intersection rule. By the first equation in Eqs. (8), ${\mu }_{{\gamma }^{*}}={\mu }_{{\gamma }_2},{\sigma }_{{\gamma }^{*}}^2=\left({\sigma }_{{\gamma }_2}^2+m^{-2}{\sum }_{i=1}^m{w}_i^2{\sigma }_{{\xi }_i}^2\right),\text{Cov}\left({\gamma }_2,{\gamma }^{*}\right)={\sigma }_{{\gamma }_2}^2,\Delta =-{\mu }_{{\gamma }_2}\left({\sum }_{i=1}^m{w}_i^2{\sigma }_{{\xi }_i}^2\right)/\left(m^2{\sigma }_{{\gamma }_2}^2\right)$, and $E\left[{\gamma }_{\Delta }^{*}\right]={\mu }_{{\gamma }_2}-\Delta$, with ${\sigma }_{{\gamma }_2}^2=\text{Var}\left({\gamma }_2\right)$. To calculate the IP of the intersection rule, we replace these quantities in the formulas of Theorem 1.

6.2. The AER of common TWI approaches

In some familiar TWI approaches, $\text{AER}=\pm {\mu }_{{\gamma }_2}$. One of them is when clinicians prescribe to all new patients the treatment found to be superior on average in clinical trials; in such case, by Theorem 1, ${AER}_{TWI}={\mu }_{{\gamma }_2}$ when $A_2$ is superior, or ${AER}_{TWI}=-{\mu }_{{\gamma }_2}$ when $A_1$ is superior. If the superior treatment is labeled $A_2$, this $AE{R}_{TWI}$ corresponds to the classic definition of the average effect of a treatment compared to another; that is, $AE{R}_{TWI}=\left(E\left[{\beta }_2\right]-E\left[{\beta }_1\right]\right)$.¹² In contrast, if data shows that a fraction of patients $p^{\prime }>0$ receives $A_1$ because it is cheaper than the superior treatment, then $AE{R}_{TWI}={\mu }_{{\gamma }_2}\left(1-2p^{\prime }\right)$. In another familiar setting, if research found that the two competing treatments were equivalent on average (i.e., ${\mu }_{{\gamma }_2}=0$), the clinician may choose treatments based on non-clinical considerations (costs or random pick, for example); in this case, $AE{R}_{TWI}=0$.

6.3. IP of the rule vs. IP of the estimated rule

When $\mathit{\vartheta }$ is unknown, we need to distinguish between the theoretical performance of the MITR $G$ based on ${\gamma }^{*}\left(\mathit{\vartheta }\right)$ from the performance of the estimated rule $\widehat{G}$ based on ${\widehat{\gamma }}^{*}={\gamma }^{*}\left(\widehat{\mathit{\vartheta }}\right)$, where $\widehat{\mathit{\vartheta }}$ is an estimate of $\mathit{\vartheta }$. In the real world, we can only use $\widehat{G}$ and the clinician will use the same estimates in all decisions. (Updating estimates with new patient data may not be ethically workable and may conflict with current clinical practice standards.¹²) Thus, here we distinguish between the IP of the rule $G$ (i.e., $I{P}_G$) and the IP of the rule $\widehat{G}$, defined as $I{P}_{\widehat{G}}=AE{R}_{\widehat{G}}-AE{R}_{TWI}$. Note $I{P}_{\widehat{G}}$ is different from an estimator ${\widehat{IP}}_G$ of $I{P}_G$. Both $I{P}_G$ and $I{P}_{\widehat{G}}$ are population constants and, in general, $I{P}_G\ne I{P}_{\widehat{G}}$ because the estimate $\widehat{\mathit{\vartheta }}$ will rarely coincide with $\mathit{\vartheta }$. Whereas $I{P}_G$ is the average relative clinical improvement in the population of patients should the individualized rule be ideally implemented with an exact estimate of $\mathit{\vartheta },I{P}_{\widehat{G}}$ is the actual relative improvement we can achieve. Some MITRs do not require parameter estimation for their implementation. In such cases, $I{P}_{\widehat{G}}$ is undefined. An example is the “averaging responses” approach to individualization in N-of-1 trials,¹² whose IP can be estimated by fitting Eq. (5) to crossover data. See Supp. Mat. B for details.

6.4. Calculating and estimating the IP of the estimated intersection rule

Next, we prove two useful facts about new patients: (1) when we estimate the intersection rule with large samples and apply it to new patients, $I{P}_G$ accurately measures the estimated rule’s actual performance, namely $I{P}_{\widehat{G}}$. And (2) we can use an estimate of $I{P}_G$ as an estimate of $I{P}_{\widehat{G}}$. When $G$ is the intersection rule, we still can use Theorem 1 to calculate $I{P}_{\widehat{G}}$. However, the quantities to replace in Eq. (11) are now different and given in Theorem 2.

Theorem 2. Suppose that $\widehat{\mathit{\vartheta }}=\left({\widehat{\psi }}_1,\widehat{\Delta },{\widehat{w}}_1,\dots ,{\widehat{w}}_m\right)$ is an estimator of $\mathit{\vartheta }=\left({\psi }_1,\Delta ,w_1,\dots ,w_m\right)$, and that an MITR based on ${\widehat{\gamma }}_{\Delta }^{*}=\left({\widehat{\psi }}_1+\widehat{\tilde{Z}}-\widehat{\Delta }\right)$ is implemented in a new patient, where $\widehat{\tilde{Z}}=m^{-1}{\sum }_{i=1}^m{\widehat{w}}_i{Z}_i$. Then, $E\left[{\widehat{\gamma }}_{\Delta }^{*}\left|\widehat{\mathit{\vartheta }}\right.\right]=\left\{{\widehat{\psi }}_1+m^{-1}{\sum }_{i=1}^m{\widehat{w}}_i{w}_i^{-1}\left({\mu }_{{\gamma }_2}-{\psi }_{1,i}\right)-\widehat{\Delta }\right\}$,

$$\text{Cov}\left({\gamma }_2,{\widehat{\gamma }}_{\Delta }^{*}\left|\widehat{\mathit{\vartheta }}\right.\right)={\sigma }_{{\gamma }_2}^2\left(m^{-1}{\sum }_{i=1}^m{\widehat{w}}_i{w}_i^{-1}\right),\text{and}$$

$$\text{Var}\left({\widehat{\gamma }}_{\Delta }^{*}|\widehat{\mathit{\vartheta }}\right)=m^{-2}\left\{{\sigma }_{{\gamma }_2}^2{\sum }_{i=1}^m{\sum }_{j=1}^m\left({\widehat{w}}_i{w}_i^{-1}\right)\left({\widehat{w}}_j{w}_j^{-1}\right)+{\sum }_{i=1}^m{\widehat{w}}_i^2{\sigma }_{{\xi }_i}^2\right\}.$$

The assumption that the patient is new is crucial for the validity of Theorem 2, as seen from its proof. To calculate ${IP}_{\widehat{G}}$, where $\widehat{G}$ is the estimated intersection rule, we replace ${\mu }_{{\gamma }^{*}},\text{Cov}\left({\gamma }_2,{\gamma }^{*}\right)$, and ${\sigma }_{{\gamma }^{*}}^2$ in Eq. (11) with the conditional mean, covariance, and variance given by Theorem 2, respectively. We stress that we need conditional quantities to calculate ${IP}_{\widehat{G}}$ because the clinician will keep the estimates ${\widehat{\psi }}_1,\widehat{\Delta },{\widehat{w}}_1$, …, ${\widehat{w}}_m$ constant when applying the rule $\widehat{G}$ to the patient population.

By Theorem 2, if parameter estimates are consistent, ${IP}_{\widehat{G}}$ converges in probability to $I{P}_G$, as $N\to \infty .$ In that case, the conditional quantities in the theorem converge to the corresponding quantities used to calculate IP. That is, $E\left[{\widehat{\gamma }}_{\Delta }^{*}\left|\widehat{\mathit{\vartheta }}\right.\right]\to {\mu }_{{\gamma }^{*}},\text{Cov}\left({\gamma }_2,{\widehat{\gamma }}_{\Delta }^{*}\left|\widehat{\mathit{\vartheta }}\right.\right)\to \text{Cov}\left({\gamma }_2,{\gamma }^{*}\right)$ and $\text{Var}\left({\widehat{\gamma }}_{\Delta }^{*}\left|\widehat{\mathit{\vartheta }}\right.\right)\to {\sigma }_{{\gamma }^{*}}^2$ in probability. Therefore, when we estimate the rule with large samples and apply it to new patients, $I{P}_G$ accurately measures its actual performance $I{P}_{\widehat{G}}$.

Until now, we have been concerned with “calculating”, not with estimating $I{P}_G$ and $I{P}_{\widehat{G}}$ for the intersection rule. As described in Section 5, we estimate model parameters (and the IP) via maximum likelihood, which produces consistent estimators.

The question now arises as to whether we can estimate the $I{P}_{\widehat{G}}$ of the intersection rule directly instead of estimating it indirectly via the $I{P}_G$. The answer is that, for new patients, we cannot. Suppose we estimate ${\psi }_{1,i}$ with ${\widehat{\psi }}_{1,i}$, define ${\widehat{\psi }}_1=m^{-1}{\sum }_{i=1}^m{\widehat{\psi }}_{1,i}$, and replace ${\psi }_{1,i}$ and $w_i$ in the conditional expressions of Theorem 2 with their estimators. Note that this produces the same quantities needed to calculate $P_G$. Thus, we have met the surprising fact that, for the intersection rule, point estimates of $I{P}_G$ and $I{P}_{\widehat{G}}$ are equal, regardless of the accuracy of parameter estimates and sample size. Consequently, we can view an estimator ${\widehat{IP}}_G$ of $I{P}_G$ as an estimator of $I{P}_{\widehat{G}}$ too. Therefore, we need to evaluate the performance of ${\widehat{IP}}_G$ as an estimator of both $I{P}_G$ and $I{P}_{\widehat{G}}$, a task undertaken through simulations in Section 8.

7. Investigating the bias of the MLE of the intersection rule’s IP

Standard linear regression with Eq. (9) produces biased least squares estimates (LSEs) of ${\psi }_1$ and $w_i$ due to the dependency of the error and ${\tilde{Z}}_{\omega }$. This is one reason we calculated their MLEs using the SEM in Eqs. (10). It is well known, however, that finite-sample MLEs are also biased.²⁸ In fact, as discovered through simulations below, the MLE of IP inherits bias from the LSEs. Specifically, we found that the quantity ${\rho }_Z=\left({\sigma }_{{\gamma }_2}^2/{\sigma }_{{\gamma }^{*}}^2\right)$, which measures LSE bias, also governs the bias of the MLEs. Theorem 3 shows that ${\rho }_Z$ indeed measures LSE bias.

Theorem 3. Suppose we fit two simple linear regression models of $Y_{\omega }$ versus $Z_{\omega ,i},i=1,\dots ,m$, one with the subjects of a randomized clinical trial who received $A_1$, the other with those who received $A_2$, both with i.i.d. errors. That is, for $j=0,1$, fit $Y_{\omega }=c_j+{\sum }_{i=1}^m{b}_{j,i}{Z}_{\omega ,i}+{ϵ}_{\omega }^{\left(1\right)}$ with only the subjects for whom $T_{\omega ,2}=j$. Let ${\widehat{c}}_j,{\widehat{\mathit{b}}}_j$ be the least squares estimates of $c_j,{\mathit{b}}_j={\left(b_{j,1},\dots ,b_{j,m}\right)}^T$. Then $E\left[{\widehat{c}}_1-{\widehat{c}}_0\right]={\mu }_{{\gamma }_2}\left(1-{\rho }_Z\right)+{\rho }_Z{\psi }_1\text{and}E\left[m{\widehat{\mathit{b}}}_1-{m\widehat{\mathit{b}}}_0\right]={\rho }_Z\mathit{w},$ where $\mathit{w}={\left(w_1,\dots ,w_m\right)}^T$, with ${\rho }_Z=\left({\sigma }_{{\gamma }_2}^2/{\sigma }_{{\gamma }^{*}}^2\right)=\left[{\sigma }_{{\gamma }_2}^2/\left({\sigma }_{{\gamma }_2}^2+m^{-2}{\sum }_{i=1}^m{w}_i^2{\sigma }_{{\xi }_i}^2\right)\right]$.

We call ${\rho }_Z$ the bias factor. Noting that $0<{\rho }_Z<1$, Theorem 3 says that the LSEs of ${\psi }_1$ and the $w_i$’s are unbiased only if ${\rho }_Z$ is $1$. In our simulations, we use the bias factor to further assess the bias and reliability of the IP estimates calculated through the SEM approach, especially with smaller samples. Higher ${\rho }_Z$ values give more confidence in the estimates.

8. Simulations with the intersection rule

8.1. Simulation scenarios and procedures

We evaluated the accuracy of ${\widehat{IP}}_G$ as an estimator of both $I{P}_G$ and $I{P}_{\widehat{G}}$ with simulations. We focused on the case $A_2$ being superior to $A_1$ on average, with TWI prescribing $A_2$ to all patients ($p^{\prime }=0$). Our therapeutic target will be incremental. Thus, $AE{R}_{TWI}={\mu }_{{\gamma }_2}>0$. The MITRs prescribed $A_2$ when ${\gamma }_{\Delta }^{*}>0$, where ${\gamma }_{\Delta }^{*}=\left({\psi }_1+\tilde{Z}-\Delta \right)$ and $\tilde{Z}=\left(m^{-1}{\sum }_{i=1}^m{w}_i{Z}_i\right)$ with $m=2$. We created 31,104 simulation scenarios, each corresponding to a specific combination of parameter values. We assumed ${\sigma }_{{\xi }_1}^2={\sigma }_{{\xi }_2}^2$. The algorithm for creating the parameter grid, and its rationale, are in Supp. Mat. C. To guarantee the existence of patients for whom $A_1$ was a better choice, the grid included scenarios with probabilities of 0.1, 0.2, 0.3, 0.4, that $A_1$ was more beneficial than $A_2$. These probabilities occurred in equal amounts across the scenarios. The grid also featured true bias factors of 0.3, 0.4, 0.6, 0.9, in equal parts. Table S.1 in Supp. Mat. describes parameter values used.

We simulated parallel-group trials with a 1:1 randomization ratio, and total patient sample sizes of $N=100$, 150, 200, 300, 400, 500, 600. For each $N$, each scenario produced one random sample of size $N$. Once a sample was simulated, the MITR and its IP were estimated. For each scenario and $N$, we also calculated the true IP of the estimated rule, that is, the $I{P}_{\widehat{G}}$, following a Monte Carlo approach that simulated 10,000 new patients, to each of whom the estimated rule recommended a treatment. We calculated $I{P}_{\widehat{G}}$ using the general definition of AER in Section 3.1 and the patients’ simulated responses to the two treatments.

For each simulated sample, we calculated the relative bias (RB) of an estimate as $\left\{\left(\text{Estimate}-\text{Parameter}\text{True}\text{Value}\right)/\left|\text{Parameter}\text{True}\text{Value}\right|\right\}(\times 100)$, and its square error (SQE) as ${\left\{\text{Estimate}-\text{Parameter}\text{True}\text{Value}\right\}}^2$. See figure captions for the specific estimates and parameters examined in the figures.

Figure 1, and Figure S.1 in Supp. Mat., examine the statistical properties of ${\widehat{IP}}_G$ when used as an estimate of $I{P}_G$; and Figure S.2 in Supp. Mat. its properties when used as an estimate of $I{P}_{\widehat{G}}$. Each curve corresponds to a fixed bias factor and is therefore calculated with 7,776 scenarios. For a particular sample size, a point on a curve represents a summary of the 7,776 simulated samples (one sample per scenario). We used medians and interquartile ranges (IQRs) as summaries.

Figure 1.

Investigation of the statistical properties of ${\widehat{IP}}_G$ when used as an estimator of $I{P}_G$, using simulated samples of different sizes and hypothetical bias factors of 0.3, 0.4, 0.6, 0.9. Each curve shows results for a fixed bias factor and 7,776 scenarios. The left-hand side of the curves shows their corresponding bias factors. (A) Median relative bias (RB) of ${\widehat{IP}}_G$. (B) Interquartile range (IQR) of relative biases. (C) Median square error (SQE). (D) IQR of SQE. Here, $RB=\left\{\left({\widehat{IP}}_G-I{P}_G\right)/\left|I{P}_G\right|\right\}(\times 100)$, and ${SQE=\left({\widehat{IP}}_G-I{P}_G\right)}^2$. We assumed the TWI approach that prescribes the treatment superior on average to all patients.

8.2. Using ${\widehat{IP}}_G$ as an estimate of $I{P}_G$

Figures 1A and 1B show the medians and IQRs of the RBs of ${\widehat{IP}}_G$. The median RBs were small, not larger than 1.5% in absolute value, regardless of sample size (Figure 1A). They approached zero with larger sample sizes. The IQRs decreased as the sample size increased, suggesting that larger sample sizes produced less biased IP estimates (Figure 1B). Figure 1A shows a tendency for ${\widehat{IP}}_G$ to be smaller than $I{P}_G,$ but the wide IQRs in Figure 1B compared to the medians in Figure 1A show that both positive and negative biases are possible. Figures 1A and 1B together also show that scenarios with higher bias factors produced less biased ${\widehat{IP}}_G$’s. The median and IQR of the SQEs decreased with the sample size, suggesting that larger sample sizes produced more accurate ${\widehat{IP}}_G$ estimates (Figures 1C and 1D). However, the role of the bias factor on the behavior of the SQEs has now reversed: scenarios with higher bias factors tended to produce more inaccurate ${\widehat{IP}}_G$’s.

Figure S.1 shows coverage probabilities across simulation scenarios of the regular 95% confidence intervals (CIs) of IP, namely ${\widehat{IP}}_G\pm 1.96\times \text{se}\left({\widehat{IP}}_G\right)$, where $\text{se}\left({\widehat{IP}}_G\right)$ is the standard error of ${\widehat{IP}}_G$. For a particular sample size and bias factor, we calculated the coverage probability as the proportion of scenarios, among the 7,776 scenarios corresponding to the bias factor, whose $I{P}_G$ was included in the CI calculated with the unique sample of that size. As expected, coverage probabilities approached 95% as the sample size increased, but scenarios with higher bias factors produced more accurate CIs. Overall, as an estimate of $I{P}_G,{\widehat{IP}}_G$ tended to be less biased, less variable and more accurate for larger sample sizes, as desired, and the influence of the bias factor on the estimate waned with larger samples.

8.3. Using ${\widehat{\mathit{I}\mathit{P}}}_{\mathit{G}}$ as an estimate of $\mathit{I}{\mathit{P}}_{\widehat{\mathit{G}}}$

Overall, when used as an estimate of $I{P}_{\widehat{G}},{\widehat{IP}}_G$ behaved similarly to when used as an estimate of $I{P}_G$ (Supp. Mat. D and Figure S.2). The simulations show that ${\widehat{IP}}_G$ and its regular CI are precise estimators of the $I{P}_{\widehat{G}}$’s, even for smaller sample sizes.

8.4. Comparing the intersection rule with outcome weighted learning.

Using the same simulations scenarios, sample sizes and intersection rule investigated in Sections 8.1 to 8.3, we compared the IP of the estimated intersection rule with the IPs of two estimated ITRs built through outcome weighted learning (OWL).¹ OWL is a machine learning method that builds a treatment classifier by weighting a subject’s contribution to the loss function with the subject’s response to his/her assigned treatment.¹ Using our notation, the OWL classifier is learned by minimizing the penalized cost function

$$Q=\sum _{\omega =1}^N{R}_{\omega }L\left(T_{\omega ,2},{\mathit{V}}_{\omega }^{\prime }\mathit{\tau }\right)+\lambda {‖\mathit{\tau }‖}_2^2,$$

where ${\mathit{V}}_{\omega }$ is a vector of standardized transformations of $Z_{\omega ,1}$ and $Z_{\omega ,2}$; $\mathit{\tau }$ is a vector of regression coefficients; $L$ is the loss incurred when predicting the value of $T_{\omega ,2}$ using ${\mathit{V}}_{\omega }$; $\lambda$ is the penalization parameter; and $R_{\omega }=\frac{\left(Y_{\omega }-c\right){\left\{P\left(T_{\omega ,2}=1\right)\right\}}^{-1}}{{\sum }_{\nu =1}^N\left(Y_{\nu }-c\right){\left\{P\left(T_{\nu ,2}=1\right)\right\}}^{-1}}$, where $c=\text{min}\left\{y_{\omega },\omega =1,\dots ,N\right\}$. In principle, any loss function for binary responses can be used. Zhao et al.¹ used the hinge loss, which produces a support vector machine (SVM).³⁷ However, the hinge loss and logistic loss have very similar shapes as functions of the margin,^37,38 and penalized logistic regression and SVMs produce similar binary classifiers.^38,39 Thus, for computational convenience, we used here the logistic loss.

If $\widehat{\mathit{\tau }}$ is the estimate of $\mathit{\tau }$ and $\mathit{V}$ is a vector of standardized transformations from a simulated new patient, the OWL rule prescribed treatment $A_2$ if $1/(1+\text{exp}\left(-{\mathit{V}}^{\prime }\widehat{\mathit{\tau }}\right))>0.5$, or $A_1$ otherwise. The means and standard deviations used in standardizations were provided by the same patient sample used to minimize $Q$. We learned $\lambda$ through 10-fold cross-validation.

We built two OWL rules. In one rule, called here the linear rule, ${\mathit{V}}_{\omega }^{\prime }\mathit{\tau }$ was a linear combination of standardized $Z_{\omega ,1}$ and $Z_{\omega ,2}$ plus a constant. In the other, called the cubic rule, ${\mathit{V}}_{\omega }^{\prime }\mathit{\tau }$ contained the standardized elements of a cubic polynomial in $Z_{\omega ,1}$ and a cubic polynomial in $Z_{\omega ,2}$ with one constant, plus a standardized product $Z_{\omega ,1}{Z}_{\omega ,2}$. We learned the rules using STATA’s elasticnet command (Stata Corp LLC, College Station, TX).

In these comparisons, we considered the important probability that treatment $A_1$ is more beneficial than $A_2$ for an individual patient, calculated as $𝒫=P\left(b_1>b_2\right)=\Phi \left(-{\mu }_{{\gamma }_2}/{\sigma }_{{\gamma }_2}\right),$ where $b_k$ is the benefit of treatment $A_k$. ($b_k=b_{k,j}$ for $j=1$ or $2$.) Note that $𝒫$ is a constant of the simulation scenario. Since $A_2$ was superior to $A_1$ on average in each scenario, we prefer ITRs with an IP $>0$ in the presence of small $𝒫\prime \text{s}$, because that means that the ITR identifies more easily patients for whom $A_1$ is more beneficial when the population has only a few of them. In the extreme situation that $𝒫$ is small, if an ITR cannot easily detect patients from the minority, the ITR is essentially the one-size-fits-all approach and is therefore not an individualization approach at all. In each of the 31,104 simulation scenarios, $𝒫=0.1,0.2,0.3,$ or $0.4$, where a specific value of $𝒫$ was represented in one quarter of the scenarios. Thus, each $𝒫$ occurred in 7,776 scenarios.

For fixed values of $𝒫$ and $N$, each simulation scenario with that $𝒫$ provided one random sample of size $N$. After estimating the intersection rule and the linear and cubic OWL rules with this sample, we calculated the true IPs of the estimated rules with the simulated responses to the two treatments from 10,000 simulated new patients. Thus, each of the 7,776 scenarios produced a value of $I{P}_{\widehat{G}}$ for the intersection rule $G$, and values of $I{P}_{\widehat{OWL}}$ for both the linear and cubic OWL rules. We thus calculated the proportion of scenarios yielding an estimated intersection rule with $I{P}_{\widehat{G}}>0$ over the 7,776 scenarios, and analogous proportions for the OWL rules (Table 2). Since the therapeutic target was incremental, the best ITR for a fixed sample size and $𝒫$ is that with the highest proportion of IPs $>0$, because IP $\le 0$ means that the rule was equivalent or inferior to TWI, where, here, TWI is the one-size-fits-all approach.

Table 2.

Proportion of simulation scenarios that produced an estimated individualized treatment rule with an $IP>0$, stratified by the probability $𝒫$ that treatment $A_1$ is more beneficial than $A_2$ and by the sample size used to estimate the rule. The table shows that the intersection rule was superior to the OWL rules at every combination of $𝒫$ and sample size, and that the performance of estimated OWL rules was seriously impaired by smaller $𝒫$’s and smaller sample sizes. Each proportion summarizes results from 7,776 scenarios from a total of 31,104 scenarios.

		Total Sample Size (N)
$\mathcal{P}$ a	Rule	100	150	200	300	400	500	600
0.4	Intersection	100.0%	100.0%	100.0%	100.0%	100.0%	100.0%	100.0%
	Linear OWL	71.1%	76.4%	80.5%	85.3%	87.8%	90.2%	91.5%
	Cubic OWL	64.5%	69.7%	75.3%	81.1%	83.2%	86.8%	88.2%
0.3	Intersection	99.6%	99.9%	100.0%	100.0%	100.0%	100.0%	100.0%
	Linear OWL	65.6%	71.9%	76.3%	80.6%	84.1%	86.8%	88.8%
	Cubic OWL	56.3%	62.4%	67.3%	73.1%	77.6%	80.5%	83.2%
0.2	Intersection	98.3%	99.2%	99.5%	99.8%	99.9%	100.0%	100.0%
	Linear OWL	56.5%	62.3%	66.7%	72.8%	77.0%	79.8%	82.3%
	Cubic OWL	43.7%	49.5%	54.1%	61.4%	66.1%	69.1%	72.5%
0.1	Intersection	89.5%	91.9%	93.1%	95.9%	96.5%	97.2%	97.4%
	Linear OWL	38.7%	44.6%	49.4%	56.2%	59.9%	63.4%	65.4%
	Cubic OWL	25.2%	30.2%	33.6%	38.8%	43.5%	46.6%	49.6%

^aFraction of patients for whom treatment $A_1$ is more beneficial than $A_2$, calculated as $𝒫=P\left(b_1>b_2\right)=\Phi \left(-{\mu }_{{\gamma }_2}/{\sigma }_{{\gamma }_2}\right)$. This fraction is a constant in a simulation scenario.

For both the intersection and OWL rules, Table 2 shows the proportion of simulation scenarios producing an estimated rule with IP $>0$, stratified by $𝒫$ and $N$. The table shows three interesting patterns:

(1) For fixed $𝒫$ and $N$, the intersection rule always exhibited a considerably higher proportion compared to the linear or cubic OWL rules, especially for smaller $N$’s and $𝒫$’s. For instance, for $N=100$ and $𝒫=0.1$, the proportions were 89.5, 38.7 and 25.2% for the intersection, linear OWL rule and cubic OWL rule, respectively.

(2) For the OWL rules, the proportion went down dramatically with decreasing values of $𝒫$, even under the largest sample sizes. For example, for the linear OWL rule with $N=600$, the proportion decreased from 91.5% ($𝒫=0.4$) to 65.4% ($𝒫=0.1$); when $N=100$, it decreased from 71.1% to 38.7%. In contrast, for the intersection rule with $N=600$, the proportion slightly decreased from 100.0% ($𝒫=0.4$) to 97.4% ($𝒫=0.1$); and from 100.0% to 89.5% when $N=100$.

(3) For all three rules, the proportion of IPs $>0$ went up with the sample size used to estimate the rule. However, for the intersection rule, the proportion was stably high for all sample sizes and almost insensitive to sample size, especially for $𝒫\ge 0.2$.

In conclusion, under the assumptions of our proposed interaction model for parallel group trials, the estimated intersection rule was markedly superior to the investigated OWL rules in the sense that the former is more likely to have an IP $>0$. Moreover, the OWL rules demanded much larger sample sizes for their reliable estimation than the intersection rule. Finally, the OWL rules were undesirably more likely to be equivalent or inferior to TWI in the presence of small fractions of patients benefitting more from the inferior-on-average treatment.

9. Real data application

9.1. individualization of therapies for DME

Next, we use data from a randomized parallel-group trial that compared the efficacies of intravitreous aflibercept (2.0 mg/every four weeks), bevacizumab (1.25 mg), and ranibizumab (0.3 mg) as treatments of DME.²² The primary outcome was a change in visual acuity (VA) of the randomized eye from baseline to 1 year of treatment. VA ranges from 0 to 100, with higher scores showing better VA. On average, aflibercept was superior to both ranibizumab and bevacizumab, but there were no significant differences in efficacy between the latter. Thus, the ideal TWI approach would prescribe aflibercept to all patients (it produces the largest $AE{R}_{TWI}$).

Presently, however, aflibercept is expensive, and bevacizumab is the cheapest. Bevacizumab is not inferior to ranibizumab for related eye conditions.⁴⁰ Thus, it is of clinical and economic interest to design an ITR that prescribes bevacizumab instead of aflibercept to patients who benefit more or equally from the former. So, we will construct an MITR that chooses between aflibercept and bevacizumab, and compare its performance with TWI’s performance, where TWI consists of prescribing aflibercept to all patients. Labeling aflibercept as $A_2$ and bevacizumab as $A_1$, this TWI corresponds to $p^{\prime }=0$.

The eye’s retinal thickness (RT) of the central subfield is a variable that may determine differences in the efficacies of aflibercept and bevacizumab because, in contrast with bevacizumab, which is a whole antibody, aflibercept is an antibody fragment that may penetrate the retina better. The original study²² and the author did not find variables interacting with treatment other than RT. Thus, we used both the treated and non-treated eye baseline RTs ($\mu m$) to build the MITR ($m=2$). To achieve approximate normality and linearity, we transformed the RTs with $Z_{\omega ,i}=\left(g\left({\left[RT\right]}_{\omega ,i}\right)-g\left(209\right)\right)/(g\left(\mathrm{1,290}\right)-g(209\left)\right)\times 100$, with $g\left(x\right)=\left(1-1/x\right)$, and ${\left[RT\right]}_{\omega ,1}$ and ${\left[RT\right]}_{\omega ,2}$ are respectively the RT’s of the treated and non-treated eye. Here, $Y_{\omega }=$ VA after one year of treatment minus baseline VA, which cancels out the effects of ${\mathit{X}}_{\omega }$. The therapeutic target was incremental.

Thus, we fitted the interaction model introduced in Section 4.5 to 224 subjects on Aflibercept and 218 on Bevacizumab, who had at least one measure in $Z_{\omega ,1}$ and $Z_{\omega ,2}$. Then we estimated the intersection rule described in Section 4.6. We estimated model parameters through maximum likelihood as described in Section 5, implementing Stata’s sem command for structural equation modeling (StataCorp LLC, College Station, TX). We treated missing values as random.

Table 1 shows the estimated intersection rule and its properties. On average, the VA improvement produced by aflibercept was 4.067 units higher than that produced by bevacizumab. This difference is the AER if the clinician prescribes aflibercept to all patients. However, if the clinician implements the intersection rule instead, the AER is 6.231, which has 2.164 VA units more. This improvement in prescription performance is the estimated IP for the MITR when compared with the one-size-fits-all approach, which differed significantly from 0 at a 0.05 significance level (p-value=0.008).

We warn against the temptation of interpreting an IP of 2.164 as an unimportant increase in prescription performance. This increase is in VA units but, compared to TWI, it is a 53.2% increase. The intersection rule will help identify patients for whom TWI will be unnecessarily expensive, including those for whom bevacizumab is only slightly more beneficial than aflibercept. To further understand this, Table 1 shows the estimated probability that the rule recommends bevacizumab to a specific patient, calculated as $p=P\left({\gamma }_{\Delta }^{*}\le 0\right)=\Phi \left(-{\mu }_{{\gamma }_2}{\sigma }_{{\gamma }_2}^{-2}{\sigma }_{{\gamma }^{*}}\right)$. The intersection rule will recommend bevacizumab for 26.9% of the patients. Although bevacizumab may not produce substantially better VA than aflibercept in some of these patients, we expect economic savings for all patients because of the considerable difference in costs between the two drugs, without compromising the expected health improvement from the ophthalmological treatment.

In general, when $\text{IP}>0$, the individualization rule may reduce the economic burden of medical therapy for many patients, besides producing better clinical outcomes in some, if the treatment superior on average is costly compared to the alternative treatment.

But we would also like to examine the intersection rule in populations with retinal thickness distributions other than that of the population supplying the sample, and in populations undergoing TWI practices that also prescribe bevacizumab. The global potential, 4.048, shows that, when averaging across all populations and all TWI implementations with $AE{R}_{TWI}\ge 0$, VA improves by 4 units using the intersection rule instead of TWI (Table 1). Another tool to extrapolate to other populations is a contour plot of $IP$ as a function of $p$ and $p\prime$ (Figure 2). We use this plot to predict the IP of the intersection rule in populations with different values of $p$ and $p\prime$. For instance, suppose the rule will be used in a clinical practice whose catchment area includes a higher proportion of patients with thinner retinal thickness so the rule will recommend bevacizumab to $p=40\%$ of the patients, say. Figure 2 shows the rule will improve VA by an average of at least 3.22 compared to TWI, no matter how frequent TWI prescribes bevacizumab, provided $AE{R}_{TWI}\ge 0$. And if TWI prescribes bevacizumab to $\approx 30\%$ of the patients, $IP\approx 4.8$. Interestingly, if $p\approx \frac{{ℰ}_2}{{ℰ}_1+{ℰ}_2}=64\%$, then $IP\approx 5.12$, no matter how TWI is done; that is, for that $p$, IP does not depend on $p\prime$. This is also the largest possible IP achieved by the intersection rule when $AE{R}_{TWI}\ge 0$ with $p^{\prime }\le \frac{{ℰ}_1}{{ℰ}_1+{ℰ}_2}=36\%$.

Figure 2.

Contour plot of the individulization potential of the estimated intersection rule for the individualization of aflibercept versus bevacizumab, as a function of the proportions of patients recommended bevacizumab by the rule (y axis) and by TWI (x axis). The curves are labeled with their corresponding IPs. Solid and dashed parts of the curves correspond to $AE{R}_{TWI}>0$ and $AER{T}_{TWI}<0$, respectively. The dotted horizontal and vertical lines correspond to $AE{R}_{TWI}=0$. The plot depicts Eq. (2) (See Supp. Mat. E for more details).

Finally, in our application to patients with DME, both the linear and cubic OWL rules recommended aflibercept to all patients disregarding their retinal thicknesses (data not shown). Thus, the OWL rules were equivalent to the one-size-fits-all approach and were inferior to the intersection rule according to their IPs.

9.2. Goodness of fit of the estimated SEM

The likelihood ratio test comparing the fitted model vs. the saturated model suggested a good fit (${\chi }^2=13.42$, df=9, p-value=0.144). Both the comparative fit index (CFI) and Tucker-Lewis index (TLI) were close to 1, also indicating a good fit (CFI=0.943, TLI=0.962).⁴¹ We examined the assumption of normality for the latent variable ${\gamma }_{\omega ,2}$ by plotting a normal quantile-quantile (Q-Q) plot of ${\widehat{\gamma }}_{\omega ,2}={\widehat{\tilde{Z}}}_{\omega }+{\widehat{\psi }}_1$, using the data from the subjects who provided retinal thickness measures from both eyes, where ${\widehat{\tilde{Z}}}_{\omega }=2^{-1}{\sum }_{i=1}^2{\widehat{w}}_i{Z}_{\omega ,i}$. [By Eq. (8), ${\widehat{\gamma }}_{\omega ,2}$ is a reasonable predictor of ${\gamma }_{\omega ,2}$ for subject $\omega$.] Figure S.3 in the Supp. Mat. shows the Q-Q plot, which suggests the assumption of normality is likely to hold for our ophthalmological application.

9.3. Robustness of the estimated individualization rule for DME treatment

Next, we examine the performance of the estimated intersection rule in simulated populations of new patients that do not thoroughly satisfy the assumptions of the interaction model. Here, we consider two model deviations for the population of new patients: (1) the interaction model is still valid but the true model parameters differ from the corresponding estimates, and (2) the distribution of ${\gamma }_2$ is not normal. To investigate robustness under inaccurate parameter estimates, we implemented the following Monte Carlo approach that we have used to evaluate drug dosage individualization methods, individual benefit predictors, and individualization rules for N-of-1 trials.^10,12,19,21 Note first that the interaction model had 9 parameters, since we assumed ${\sigma }_{{\xi }_1}={\sigma }_{{\xi }_2}$ and $w_1$ was fixed to 1. Parameters estimates are shown in Table S.2 of Supp. Mat along with their standard errors. But we do not expect these estimates to be exactly the true parameter values. We used Eqs. (6) and (7) to simulate the responses and the $Z_i$’s for the new patients (with $\mathit{\theta }=0$) using realistic true values calculated as follows. For fixed $\delta =0,0.2,0.6,1.0,1.4,1.8,2.0$, we chose all parameter values that are at a distance of $\delta$ standard errors from their corresponding parameter estimates. Specifically, denoting the true parameter values as ${\phi }_1,\dots ,{\phi }_9$, we choose ${\phi }_i={\widehat{\phi }}_i\pm \delta \times \text{se}\left({\widehat{\phi }}_i\right)$, where $\text{se}\left({\widehat{\phi }}_i\right)$ is the standard error of the estimator ${\widehat{\phi }}_i,i=1,\dots ,9$.¹² Larger values of $\delta$ reflect more inaccurate parameter estimates in the context of the population of new patients. And since the intersection rule relies on the parameter estimates, we expect larger $\delta$’s to be associated with poorer rule performances. For $\delta >0$, there are a total of $2^9=512$ possible parameter combinations. The case $\delta =0$ is the ideal situation in which the parameter estimates are the true parameters.¹²

We considered three distributions for ${\gamma }_2$: normal, logistic and Gumbel.⁴² If ${\mu }_{{\gamma }_2}$ and ${\sigma }_{{\gamma }_2}^2$ are the mean and variance of ${\gamma }_2$ for a specific true parameter combination, we always generated ${\gamma }_2$ as a random variable with mean ${\mu }_{{\gamma }_2}$ and variance ${\sigma }_{{\gamma }_2}^2$ regardless of its distribution. The logistic distribution is symmetric, but its tails are heavier than the tails of the normal distribution (Table 3). Thus, under this distribution, the population of new patients will usually have atypical subjects more often than the patient population that produced the intersection rule. The Gumbel distribution is right skewed and has heavier tails than the logistic distribution,⁴² so it will produce a patient population even less similar to the population assumed by the intersection rule.

Table 3.

Performance of the estimated intersection rule for the individualizing of DME treatments in populations of new patients following the normal, logistic or Gumbel distribution for ${\mathit{\gamma }}_2$, under various levels of inaccuracy for the parameter estimates (measured by $\mathit{\delta }$). When $\mathit{\delta }>0$, the table summarizes the actual IPs of the rule under all 512 combinations of “true” parameter values that are possible at a distance of $\mathit{\delta }$ standard errors from their corresponding parameter estimates. The only IP possible for $\mathit{\delta }=0$ is also described in the table.

(A) Median (10th and 90th percentiles) of IPs of estimated rule.
	Probability distribution of ${\mathit{\gamma }}_2$
$\mathit{\delta }$	Normalb	Logisticc	Gumbeld
0.0	2.165	2.014	1.325
0.2	2.161 (1.915, 2.412)	2.008 (1.793, 2.241)	1.322 (1.096, 1.565)
0.6	2.143 (1.471, 2.885)	1.996 (1.381, 2.684)	1.313 (0.676, 2.031)
1.0	2.098 (1.074, 3.367)	1.977 (1.009, 3.143)	1.284 (0.329, 2.542)
1.4	2.051 (0.722, 3.866)	1.939 (0.671, 3.625)	1.197 (0.026, 3.092)
1.8	2.003 (0.407, 4.357)	1.910 (0.369, 4.099)	1.112 (−0.196, 3.731)
2.0	1.964 (0.250, 4.625)	1.897 (0.247, 4.344)	1.098 (−0.305, 4.055)
(B) Proportion of IPs that were $>0$.
	Probability distribution of ${\mathit{\gamma }}_2$
$\mathit{\delta }$ a	Normalb	Logisticc	Gumbeld
0.0	100.0%	100.0%	100.0%
0.2	100.0%	100.0%	100.0%
0.6	100.0%	100.0%	100.0%
1.0	100.0%	100.0%	97.9%
1.4	98.4%	97.5%	91.0%
1.8	95.1%	94.3%	84.4%
2.0	92.8%	92.4%	80.9%

^bNormal distribution with mean ${\mu }_{{\gamma }_2}$ and variance ${\sigma }_{{\gamma }_2}^2$.

^cLogistic distribution with CDF, $F\left(x\right)=1/(1+\text{exp}(-(x-a)/b\left)\right),$ $-\infty <x<\infty ,$ with mean ${\mu }_{{\gamma }_2}=a$ and variance ${\sigma }_{{\gamma }_2}^2={\pi }^2{b}^2/3$.

^dGumbel distribution with CDF, $F\left(X\right)=\text{exp}(-\text{exp}(-(x-a)/b\left)\right)$, $-\infty <x<\infty$, with mean ${\mu }_{{\gamma }_2}=a+bc$ and variance ${\sigma }_{{\gamma }_2}^2={\pi }^2{b}^2/6$, where $c\approx 0.5772$ is the Euler-Mascheroni constant.

For a fixed combination of parameter values and a specific distribution for ${\gamma }_2$, we simulated a population of 100,000 values of ${\gamma }_2$ (each corresponding to a new patient) and simulated responses to the two treatments from each patient. After applying the estimated rule $\widehat{G}$ in Table 1 to each patient, we calculated $I{P}_{\widehat{G}}$ using both the simulated responses and the general definition of AER. For a fixed $\delta >0$, we calculated the median and the 10^th and 90^th percentiles of the 512 IPs corresponding to that $\delta$ (Table 3A), and the proportion of IPs $>0$ (Table 3B).

Table 3A shows that, as the inaccuracy of parameter estimates increased, the median $I{P}_{\widehat{G}}$ decreased under the three investigated distributions of ${\gamma }_2$. For instance, when the inaccuracy raised from $\delta =0.2$ to $\delta =2.0$, the median IP decreased from 2.161 to 1.964 under the normal distribution, from 2.008 to 1.897 (logistic), and from 1.322 to 1.098 (Gumbel). Interestingly, although greater inaccuracies determined smaller 10^th percentiles for the IPs, they also determined greater 90^th percentiles. In fact, for a particular distribution and $\delta >0$, the 90^th percentiles were always higher than the corresponding IP for $\delta =0$. This means that the intersection rule may not necessarily lose potential when the model parameters are estimated without accuracy. In fact, it may gain potential for some combinations of parameter values.

Table 3A also suggests that, for fixed $\delta$, the intersection rule had just a slightly lower IP when applied to a patient population with logistic ${\gamma }_2$ instead of a population with normal ${\gamma }_2$ with the same mean and variance. For instance, for $\delta =1.0$, the median IP (10^th and 90^th percentiles) for the normal and logistic distributions were 2.098 (1.074, 3.367) and 1.977 (1.009, 3.143), respectively. Thus, the presence of a few atypical subjects in the population may not affect substantially the performance of the estimated intersection rule. But the intersection rule may lose potential substantially if it is applied to a population with ${\gamma }_2$ following a Gumbel distribution, compared to a population with a normal ${\gamma }_2$ of comparable mean and variance. For instance, for $\delta =1.0$, the median IP (10^th and 90^th percentiles) under the Gumbel distribution was 1.284 (0.329, 2.542), which is a 39% reduction compared to the normal case.

However, even though the intersection rule may lose some potential if administered to patient populations with logistic or Gumbel distributions for ${\gamma }_2$, the rule still enjoys a high tendency to have $I{P}_{\widehat{G}}>0$ (Table 3B). In fact, under both $\delta \le 1.0$ and the Gumbel distribution, at least 97.9% of all possible parameter combinations produced patient populations for which $I{P}_{\widehat{G}}>0$. And the proportion of IPs $>0$ was 100% for $\delta \le 1.0$ with the normal and logistic distributions. Thus, the estimated intersection rule for the individualization of aflibercept versus bevacizumab seems to be remarkably robust to both deviations from the distributional assumptions of the interaction model and inaccurate parameter estimates in that under these deviations it is frequently better than TWI.

10. Discussion

The main contributions of this article are a general method of examining the performance of ITRs, applicable to both parametric and nonparametric settings, and a novel interaction model for data from parallel-group trials which we use to design a useful MITR. Our model implies a SEM that we used to estimate the MITR and its IP. Other SEMs potentially useful in personalized medicine are the multigroup SEMs, which have been used in the analysis of individual treatment effects and are supported by the stochastic theory of causal effects.^16,17,18 Our SEM is not a multigroup SEM because it is not group-invariant. Still, it would be interesting to see future research applying multigroup SEMs to developing individualization rules and calculating their IPs. Interestingly, our SEM can be viewed as the limit of a sequence of multigroup SEMs with two latent variables whose correlation converges to 1. Laubender et al.¹⁴ also used a SEM to resolve the problem of estimating the variance of the subject-by-treatment interaction (${\gamma }_{\omega ,2}$ in our formulation) with parallel-group trials. Their model incorporated a relationship between a predictive biomarker and the sum of the two potential outcomes; however, it included only one biomarker.

Some comments about $p$ and $p\prime$ in Eq. (2) are in order. For a specific clinical decision approach $G$ applied to a specific patient population, we view the proportion of patients recommended $A_1$ by $G$ as a characteristic parameter determined by both the patient population and the decision process, regardless of whether $G$ is an ITR or a form of TWI. For instance, if $G$ prescribes $A_1$ to patients positive for a specific genetic marker and prescribes $A_2$ to those negative, the proportion $p$, and therefore $AE{R}_G$, will depend on the genetic composition of the population. A population with a different composition is a different population that will produce a different $p$ (and a different $AE{R}_G$). As another example, in a hypothetical TWI, 15% of the patients receive the cheapest treatment for economic reasons and the other patients receive the alternative, more expensive but superior-on-average treatment. In this case, we view $p^{\prime }=0.15$ as a fixed parameter unless the population’s economic composition changes. We do not view $p$ (or $p\prime$) as a random characteristic of a sample of patients but as a constant parameter that either is known or estimated as in the application in Section 9.

We designed our simulation scenarios so the theoretical $I{P}_G$ of the intersection rule was $>0$ for each scenario (See Table S.1 in Supp. Mat.). Our simulations showed that the estimated intersection rules performed remarkably well in the majority or all the scenarios in that $I{P}_{\widehat{G}}>0$, including scenarios in which only a small fraction $𝒫$ of patients can be benefitted more by the treatment alternative to the treatment superior on average (Table 2). The rules performed well even when estimated with the smallest sample sizes ($N=50$ per group).

Compared to the intersection rule, the OWL rules performed poorly in most situations, except perhaps when they were estimated with the largest sample sizes in the presence of the largest $𝒫$’s when it is easier to find a patient that can be benefitted more by the non-superior treatment (Table 2). The OWL rules tended more to equal the one-size-fits-all approach (that is, to have $IP=0$), or even to be inferior to TWI ($IP<0$). Note also that the simulated cubic OWL rule performed even more poorly than the linear OWL rule in simulations (Table 2). This suggests that regularization through a penalty parameter, which is usually viewed as a method to prevent both overfitting and numerical instability due to collinearity,³⁷ may not necessarily impede calculating suboptimal ITRs in the presence of collinearity or more monomial terms than needed. Machine learning methods such as OWL are usually introduced as nonparametric or semiparametric, with the common implication they do not need or need little assumptions.³⁷ But knowing the assumptions is more productive than not having assumptions, as our simulations show so vividly.

Diaz¹⁹ definition of treatment’s individual benefits, which in this paper assumes Eq. (5) and is reviewed in Section 4.3, played an illuminating role in our evaluation of the intersection rule through simulations.^20,21,34,43 We found that the fraction $𝒫$ of patients who benefit more by the treatment inferior on average needs to be controlled in the evaluation of ITRs (Table 2). For example, the performance of the intersection rule decreased slowly as $𝒫$ decreased. But it continued having an excellent performance under smaller $𝒫\prime s$ and $N$’s. By contrast, the investigated machine learning rules performed very poorly under small $𝒫$’s, even with the largest samples (Table 2). This proved the intersection rule’s superiority when it is difficult to identify the patients who are helped less by the one-size-fits-all approach. Moreover, we designed the simulation scenarios so values of $𝒫$ occurred uniformly across the scenarios. This made sure the scenarios did not artificially favor or disfavor the intersection rule or did not introduce artifacts in the evaluation of IP estimates in Figures 1, S.1 and S.2 (see Supp. Mat. C).

In addition to standard linear regression, several tests have been proposed to detect or confirm interactions between patient covariates and treatments. For example, Gail and Simon⁴⁴ test for qualitative interactions can be used to examine whether the direction of the treatment differences varies between the two subgroups of patients determined by an MITR not equivalent to the one-size-fits-all approach (i.e. ${\gamma }^{*}>0$ or ${\gamma }^{*}\le 0$), which suggests that the rule is classifying the patients in some useful clinical way.⁴⁵ Using the CATE notation, this test contrasts the null hypothesis $H_0:{ℰ}_1{ℰ}_2\le 0$ versus the alternative $H_1:{ℰ}_1{ℰ}_2>0$. The test was significant for our intersection rule individualizing bevacizumab vs. aflibercept (test statistic=33.5, p-value<0.001). However, note that qualitative interaction (${ℰ}_1{ℰ}_2>0$) is not a sufficient or necessary condition for $I{P}_G>0$ or ${\stackrel{-}{IP}}_G>0$. It is only a sufficient (not necessary) condition for $AE{R}_G\ne 0$ or $I{P}_G\ne 0$. Thus, researchers must consider both TWI and the therapeutic target (incremental or decremental) to interpret the test correctly, and a lack of significance does not necessarily mean the rule has no individualization potential.

The test of Shi et al.²³ can also be used to investigate the existence of qualitative interactions, especially with large datasets. The test, however, is sensitive to only interactions useful in the development of ITRs that maximize the value function. As in all statistical tests for interactions, more analyses need to be done to specify the nature of the interactions when the test is significant. The nonparametric tests of Chang et al.⁴⁶ and Hsu⁴⁷ examine whether the CATE of a specific treatment given a covariate vector is $>0$ (or $<0$) for at least one value of the vector in a subset of its range. These tests examine average treatment effects adjusting for potential confounders. A significant test, however, does not rule out the possibility that the CATE is $>0$ (or $<0$) across the entire range of values of the vector and, therefore, that the one-size-fits-all approach is most appropriate. Thus, complementary analyses such as applying the test to different subsets, and covariate stratification along with plotting kernel-smoothed responses versus continuous covariates,⁴⁶ are still needed to find useful interactions with these tests. In addition to expert opinion, systematic methods for selecting variables interacting with treatments can be used when there is a large number of candidate variables.^45,48

But a qualitative interaction is not a necessary condition for $I{P}_G>0$, suggesting that non-qualitative interactions may also be useful. To see a hypothetical example of an ITR with $I{P}_G>0$ in the absence of qualitative interaction, consider $p=p^{\prime }\ne 0$ with ${ℰ}_1=-{ℰ}_2/d,{ℰ}_2>0,d>1$, replace in Eq. (2), and assume an incremental target. Here, drug $A_2$ benefits more each individual patient than drug $A_1$. However, the ITR administers $A_1$ to the fraction $p$ of patients who get fewer benefits from $A_2$; the ones who get the largest benefits are administered $A_2$. The fact that $I{P}_G>0$ indicates that this ITR is more effective than randomly administering $A_1$ to all patients with probability $p$. This individualization strategy makes sense, for example, if the clinician can administer $A_2$ only to $\left(1-p\right)\times 100\%$ of the patients due to cost or drug-availability constraints. Prescribing $A_2$ to all patients is the ideal strategy, but only if the health system can afford it. This also shows that non-qualitative interactions may be useful for personalized medicine, contrary to some authors’ opinion (for example, references 23, 44, 45). Research is needed to investigate the IP measure as a tool to evaluate ITRs under cost constraints.

Although a significant part of the statistical literature has focused on MITRs, there are ITRs that are not MITRs. Examples are qualitative interaction trees that prescribe the optimal treatment according to chains of decision rules based on baseline covariates not combined in a one-dimensional statistic.⁴⁸ Future research can also evaluate these approaches with our IP measure.

Our theoretical formulation of the IP of an ITR is in line with the stochastic theory of potential outcomes.^18,25 This theory differs from Rubin’s theory of potential outcomes.⁴⁹ In Rubin’s theory, a patient’s potential outcome is a fixed number that depends only on the treatment potentially administered to the patient. The stochastic theory essentially postulates that a potential outcome is a random variable with a probability distribution affected not only by the potential treatment but by the current history of the patient, their clinical and biological background and measurement errors. The advantages of this perspective over Rubin’s theory are:^18,25 (1) it allows modeling treatment causal effects in individual patients, not only average effects; (2) it is consistent with the standard formulations of random effects and latent variable models; and (3) it is consistent with all concepts and methods of classic probability theory. The latter is illustrated by our proofs of Eqs. (1) and (2) and Theorem 1.

We showed how to calculate the IP of MITRs under normality conditions and how to estimate it for the intersection rule. But our definition of IP in Section 3.2 is general and includes many other forms of ITRs. The estimation of the IP for ITRs built with nonparametric or semiparametric approaches requires further research. Calculating nonparametric point estimates of the IP or the global potential seems straightforward: estimate the CATEs as differences of conditional averages, and $p$, and plug them into Eqs. (2) or (4). However, calculating confidence intervals may be more complex because the CATE and $p$ estimates will be unavoidably dependent, and the dependency will probably depend on the particular ITR. Alternatively, we can always compare the IPs of competing ITRs through simulations, as we did in Section 8.4. Simulations, however, always require assuming a parametric model for data generation.

Finally, our proposed measure of IP, interaction model for parallel group trials, and intersection rule for treatment individualization only apply to two alternative treatments. Future research should investigate how to extend these ideas to more than two treatments.

Data Availability Statement.

The ophthalmology data is publicly available at the Jaeb Center for Health Research (https://public.jaeb.org/drcrnet).