Information-Theoretic Approach: Individual-Level Surrogacy
Ariel Alonso Abad
KU Leuven, Belgium
Similar to the trial level, at the individual level the mutual information between both endpoints, I(T, S), quantifies the amount of uncertainty in T, expected to be removed if the value of S becomes known, and hence, it seems sensible to use this measure again to quantitatively assess the definition of surrogacy given in Chapter 9. However, unlike at the trial level where both a and в are always continuous random variables, at the individual level, S and T may be both binary, time-to-event, continuous, or have different levels of measurement. This additional complexity brings some problems that need to be addressed in the information-theoretic approach but, at the same time, make this methodology more appealing at this level.
Finding suitable estimators for I(T, S), where T and S are vector-valued true and surrogate endpoints, is a fundamental step when using the information- theoretic approach at the individual level. Importantly, the same line of reasoning employed in Chapter 9 to estimate mutual information at the trial level can be followed at the individual level as well. To illustrate this, let us consider the random vectors T and S with density function f (t, sф) and suppose that the realizations (ti, Sj), i = 1, 2, ...n, are available. In addition, it will be assumed that ф = (ф0, ф1) and let ф0 denote the maximum likelihood estimator of ф0 under the null hypothesis of independence (ф1 = 0). We define the estimator
with ф the full model maximum likelihood estimator. Here again (10.1) is the classical likelihood ratio test statistic for the hypothesis of independence (H0 : ф1 =0) divided by n. Actually, in (10.1), G2 = log(likelihood ratio test) comparing f (tisi, ф) and f (фф0). Moreover, if ф0 ^ ф0, in probability then, under general regularity conditions, the statistic (10.1) will tend to I(T, S), i.e., (10.1) provides a consistent estimator for the mutual information (Kent, 1983; Brillinger, 2004). A confidence interval for I(T, S) can be obtained along the lines presented in Section 9.3.3.
Based on (10.1), Alonso et al. (2004b) proposed to quantify the individual- level surrogacy using the so-called likelihood reduction factor (LRF). Later, Alonso and Molenberghs (2007) introduced a unified framework for the evaluation of surrogate endpoints based on the squared informational coefficient of correlation (SICC) proposed by Linfoot (1957) and Joe (1989) and, along the lines previously discussed, showed that the LRF is a consistent estimator of the SICC.
One of the major problems associated with the assessment of the individual-level surrogacy in the meta-analytic approach, is the need for complex joint models to characterize f (t, s^) and to describe the association between both endpoints. The use of (10.1) completely avoids this issue by only considering the conditional and marginal models f (tisi, ф) and f (ti ф0), respectively. Note further that, although the expression I(T, S) involves a possibly high-dimensional integral, in (10.1) no integrals need to be evaluated.
In the next section the information-theoretic approach will be explained in detail for some relevant and frequently encountered outcome types.