Methods

Proportion of Treatment Effect Explained (PTE)

First, we describe the proportion of treatment effect explained (PTE). The full phrasing is: the proportion of the treatment effect on the primary outcome that is explained by the treatment effect on the surrogate, but this is often shortened to simply PTE. Let \(Y\) denote the primary outcome, \(S\) denote the surrogate marker, and (\Z\) denote the treatment indicator where \(Z \in \{0,1\}\) (i.e., treatment vs. control) and treatment is assumed to be randomized unless otherwise noted. Let’s assume that higher values of \(Y\) and \(S\) are “better". The observed data consists of \(\{Y_i, S_i, Z_i\}\) for each person \(i\).

Previous work proposed to evaluate a surrogate marker by defining and estimating the PTE via specifying two regression models, for example:

$$E(Y|Z) = \beta_0 + \beta_1Z$$

$$E(Y|Z,S) = \beta_0^* + \beta_1^*Z + \beta_2S$$

where the PTE is defined as \(R_F = 1-\beta_1^*/\beta_1\) and estimated by plugging in the corresponding regression estimates. Intuitively, the idea is that if one fits the first model and observes a large treatment effect (large \(\beta_1\)) and then fits the second model, essentially adding in the surrogate marker, and the treatment effect is now small or close to 0 (small \(\beta_1^*\)), then \(R_F\) will be close to 1, indicating that the surrogate is capturing the effect of the treatment on Y. This approach is extremely appealing and easy to implement. However, numerous studies have pointed out problems with this approach, one of which is that it relies on both models being correctly specified. Notably, not only is the estimate dependent on correct specification, but the definition of the quantity \(R_F\) itself relies on correct specification.

An alternative approach is one that aims to capture the same idea with a model-free definition. To introduce this, we use potential outcomes notation where each person has potential outcomes \( \{Y^{(1)}, Y^{(0)}, S^{(1)}, S^{(0)}\} \) where \( Y^{(g)} \) is the outcome when \( Z = g \) and \( S^{(g)} \) is the surrogate when \( Z = g \). The PTE is defined using contrasts between the actual treatment effect on \( Y \), defined as:

$$ \Delta = E(Y^{(1)} - Y^{(0)}) $$

and the residual treatment effect on \( Y \), defined as:

$$ \Delta_S = E_{S^{(0)}} \left[ E(Y^{(1)} - Y^{(0)} \mid S^{(1)} = S^{(0)} = s) \right] = \int E(Y^{(1)} - Y^{(0)} \mid S^{(1)} = S^{(0)} = s) \, dF_{S^{(0)}}(s) $$

where \( F_{S^{(0)}} \) is the cumulative distribution function of \( S^{(0)} \). The residual treatment effect can be interpreted as the “leftover" treatment effect on \( Y \), after accounting for the treatment effect on \( S \). That is, it is the hypothetical treatment effect on \( Y \) if the distribution of the surrogate in both groups looked like the distribution of the surrogate in the control group. Importantly, the definition of \( \Delta_S \) uses the distribution of \( S^{(0)} \), but in theory, one can select the distribution of \( S^{(1)} \) or some combination of the two. The PTE is then defined as:

$$ R_W = \frac{\Delta - \Delta_S}{\Delta} = 1 - \frac{\Delta_S}{\Delta} $$

where \( \Delta - \Delta_S \) is the treatment effect explained by \( S \). Ideally, this quantity is between 0 and 1, with values close to 0 indicating a poor surrogate (not capturing the treatment effect) and values close to 1 indicating a good surrogate (able to capture the treatment effect). However, \( R_W \) is only guaranteed to be within \( [0,1] \) under certain assumptions, detailed elsewhere. This construction of \( R_W \) highlights a key challenge in surrogate evaluation: when the overall treatment effect, \( \Delta \), is small, identifying a surrogate becomes inherently difficult. In this particular framework, a small treatment effect implies that finding a surrogate capable of explaining a large proportion of such a small effect will be highly challenging. Unlike \( R_F \), the definition \( R_W \) does not involve any model specification and thus, is model-free. In terms of estimation of \( R_W \), there are both parametric and nonparametric options available. Overall, this framework offers a single number, the PTE, that aims to quantify the strength of the surrogate marker with respect to capturing the treatment effect on \( Y \). While there is no agreed upon threshold for a “good enough" surrogate, some have proposed using 0.5 or 0.75 as a threshold for either the point estimate or, more strictly, for the lower bound of the confidence interval.