Optimal Alpha Spending for Sequential Analysis with Binomial Data

Abstract

Summary For sequential analysis hypothesis testing, various alpha spending functions have been proposed. Given a prespecified overall alpha level and power, we derive the optimal alpha spending function that minimizes the expected time to signal for continuous as well as group sequential analysis. If there is also a restriction on the maximum sample size or on the expected sample size, we do the same. Alternatively, for fixed overall alpha, power and expected time to signal, we derive the optimal alpha spending function that minimizes the expected sample size. The method constructs alpha spending functions that are uniformly better than any other method, such as the classical Wald, Pocock or O’Brien–Fleming methods. The results are based on exact calculations using linear programming. All numerical examples were run by using the R Sequential package.

Monte Carlo Test for Stochastic Trend in Space State Models for the Location-Scale Family

“…However, there is a number of problems where the distribution of the data is known due to axiomatic construction and/or derived from the properties behind the sampling scheme. For example, with instrumental variables regression, the endogenous covariates are written as a linear combination of instrumental variables and a white noise following a normal distribution for each fixed time (Nascimento, Abanto-Valle, & Mendonça, 2018); In clinical trials based on a self-control design, the binomial distribution represents the number of patients presenting adverse reactions to a new treatment over time (Silva, Kulldorff, & Yih, 2020); In post-market drug and vaccine safety surveillance, the Poisson distribution counts the number of adverse events from an exposure period window when the vaccine/drug was administered in the population (Kulldorff et al, 2011); The Erlang distribution has been used to model the time series of COVID-19 cases (Arino & Portet, 2020); In spatial statistic, the number of individuals with a certain disease in a region, conditioned on the total number of cases in the map, follows a Poisson distribution (Duczmal & Assunção, 2004); In a mark-recapture design for estimating the size of a population, the distribution of the tagged recaptured items is negative binomial under a with-replacement design (Mukhopadhyay & Bhattacharjee, 2018); In survival analysis, the time between two consecutive failures is well-fitted through a Weibull distribution (Zhang, 2016). Besides, one of the most prominent fields of the statistical inference that assumes knowledge about the distribution of the data is the class of the so-called 'generalized linear models', where the user selects one of distributions in the exponential family to model the data (Nelder & Wedderburn, 1972).…”

Exact sequential test for clinical trials and post‐market drug and vaccine safety surveillance with Poisson and binary data

“…For binary and Poisson counting data, as demonstrated by References 10, 11, and 13, the signaling thresholds can always be redefined in the scale of the original counting scale. For the pair of hypotheses in (), the signaling threshold in the scale of is represented by an upper boundary, that is, is rejected for large values of .…”

“…Although the exact calculation can be performed by running a Markov Chain in , the specific analytical expression for the probability in (), and so in (), are somewhat intricate. For the detailed power functions in each case, we indicate expression (13) by Reference 11 for Poisson data, expressions (8) and (29) by Reference 13 for binary data, and expressions (16) and (22) by Reference 10 for conditional Poisson data.…”

“…The calculations for the illustrative examples are run with the R Sequential package 12 . R Sequential is an easy‐to‐use tool for both the design and the practical implementation of sequential analysis.…”

“…This is done for both with or without an added requirement on the maximum length of surveillance. The optimal solution is obtained using the method proposed by Reference 13. Statistical performance metrics: Exact calculations are illustrated for statistical power, expected time of surveillance given that the null hypothesis is rejected, expected time of surveillance, and maximum maximum sample size. The latter three are calculated in the unit of sample size or number of events.…”