A sequential design for maximizing the probability of a favourable response

Consider the situation in which subjects arrive sequentially for a treatment and in which there are two distinct ways in which the treatment may fail. Treatments are given at different dosages, and the probabilities of the two failure types vary with dose. Assuming that decreasing the chances of one failure type increases the chances of the other, we say the failures oppose each other. Also assume that one failure type is primary in that, if it occurs, it censors the trial, so that observation of the secondary failure type is contingent on no failure of the primary type. We are interested in designs that provide information about the dose that maximizes the probability of success, i.e., the optimal dose, while treating very few subjects at dosages that have high risks of failure. Assuming that dosages belong to a discrete set, we show that a randomized version of the Polya urn scheme causes dose selection to be progressively biased so as to favour those doses that produce success with higher probability. Considerons la situation dans laquelle des sujets arrivent sequentiellement pour un traitement et dans laquelle il y a deux facons distinctes selon lesquelles le traitement peut echouer. Les traitements sont donnes e des dosages difterents, et les probabilite de deux types d'echecs varient avec la dose. Prenant pour acquis qu'en decroissant les chances d'un type d'echec on accroǐt les chances de l'autre, nous disons que les echecs s'opposent. Nous supposons aussi qu'un type d'echec est primaire, en ceci que s'il se produit, il censure l'essai, de telle maniere que l'observation du type secondaire est contigente au non-echec du type primaire. Nous sommes interesses par les plans qui donnent de l'information e propos de la dose qui maximize la probabilite de succes, c'est a dire la dose optimale, alors que l'on traite tres peu de sujets, a des dosages qui ont de hauts risques d'echec. Supposant que les dosages appartiennent a un ensemble discret, nous demontrons qu'une version rendue aleatoire du schema en urne de Polya a pour effet de rendre la selection des doses progressivement biaisee arm de favoriser les doses produisant le succes avec la plus haute probabilite.

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