Extensions To More Than Two Samples

Follmann et al. (1994) and more recently Hellmich (2001) considered monitoring single endpoint trials which had two or more treatment arms. The spending function approach was generalized and one is allowed to drop treatments which are inferior. Hellmich proved the strong family-wise error control of the sequentially rejective approach proposed by Follmann et al. for the Pocock and O'Brien and Fleming spending function. We observe here that these results would hold whether the trial had a single or multiple endpoints.

Suppose we have A treatment arms which will be compared on multiple endpoints. We test

H0{1,2,...,K}: = M*2 = ' ' ' = M'A against an alternative hypothesis, e.g.,

Hf 2,...,k}: ^S _ Mu = d(>0) for at least one s and all t Ps.

One can envision different schemes for monitoring such a trial combining the methodology of Hellmich and that here. For this alternative hypothesis and multivariate normally distributed data, one could use a group sequential version of the F test proposed by O'Brien (1984). For other alternative hypotheses, one might use the chi-squared or F tests proposed by Jennison and Turnbull (1991) or Lauter (1996).

For any alternative, one could monitor arms as suggested by Hellmich (2001) until the trial is stopped and follow this by a step-down procedure to determine which endpoints differ. Alternately, one could monitor arms-then-endpoints during the course of the trial. The advantages and disadvantages of various schema have not been investigated. When one is allowed to drop either arms or endpoints during the course of a trial, care needs to be taken that the results are interpretable by clinicians as well as statisticians.

There are other limitations to implementing such methodology at present. There are a limited number of test statistics for combining endpoints of different types (continuous, discrete, censored) and properties of different test statistics in complex settings are not well studied. The adequacy of using the group sequential boundaries of the ''normal theory'' as approximations when parameters are estimated may be unknown. Further research into the multiple arm and multiple endpoints problem is needed.

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