On the properties of the toxicity index and its statistical efficiency

Abstract

Cancer clinical trials typically generate detailed patient toxicity data. The most common measure used to summarize patient toxicity is the maximum grade among all toxicities and it does not fully represent the toxicity burden experienced by patients. In this article, we study the mathematical and statistical properties of the toxicity index (TI), in an effort to address this deficiency. We introduce a total ordering, (T-rank), that allows us to fully rank the patients according to how frequently they exhibit toxicities, and show that TI is the only measure that preserves the T-rank among its competitors. Moreover, we propose a Poisson-Limit model for sparse toxicity data. Under this model, we develop a general two-sample test, which can be applied to any summary measure for detecting differences among two population of toxicity data. We derive the asymptotic power function of this class as well as the asymptotic relative efficiency (ARE) of the members of the class. We evaluate the ARE formula empirically and show that if the data are drawn from a random Poisson-Limit model, the TI is more efficient, with high probability, than the maximum and the average summary measures. Finally, we evaluate our method on clinical trial toxicity data and show that TI has a higher power in detecting the differences in toxicity profile among treatments. The results of this article can be applied beyond toxicity modeling, to any problem where one observes a sparse array of scores on subjects and a ranking based on extreme scores is desirable.

Keywords: Poisson-Limit model; T-rank preservation; adverse events; toxicity index; two-sample test.

FIGURE 1

TI preserves the T‐rank while the mx and avg do not. The x‐axis is an array, with 37 patients with each column representing a RVF for an individual, ordered in increasing T‐rank. Each cell in the array is the number of AEs experienced at a particular grade for a given cycle of treatment. The TI exhibits strict monotonicity with respect to this ordering while the mx is nondecreasing and avg is neither monotone nor nondecreasing [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 2

Comparison of the asymptotic relative efficiency, as the ratio of test slopes, for the TI vs mx (left), and TI vs avg (right). The plots are the histograms of the ARE under a random Poisson‐Limit model, described in Section 5. The portions colored red correspond to models where the ARE of TI is lower than the competitors [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 3

ROC curves (left) and the simulated and asymptotic power plots at significance level $\alpha =.05$ (right). The dashed lines, in the right panel, denote the asymptotic power functions [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 4

Power comparisons for detecting treatment differences in the NSABP R0‐4 clinical trial. The plots are generated based on the Poisson‐Limit models fitted to each treatment data [Colour figure can be viewed at wileyonlinelibrary.com]