A Note on Lancaster's Procedure for the Combination of Independent Events

Lancaster (1961) generalized Fisher's (1932) nonparametric procedure for combining independent p‐values by transforming P i from the i ‐th experiment to a chi‐squared random variable with d i degrees of freedom, with d i not necessarily equal to 2. We explore the relationship between Lancaster's procedure and a weighted Lipták procedure (Koziol and Tuckwell, 1994) under which P i is transformed to the standard normal scale. We investigate approximations to the null distribution of Lancaster's test procedure, chi‐squared with d degrees of freedom. We find that the Cornish‐Fisher (1960) expansions and the Lugannani‐Rice (1980) saddlepoint approximations are quite accurate, for non‐integral values of d , and for values of d as low as 20.