On Two-Stage Comparisons with a Control Under Heteroscedastic Normal Distributions

Abstract

New two-stage sampling methodologies are developed for both one-sided and two-sided comparisons between the means from treatment groups and a control. We suppose that we have k _i ( ≥ 1) independent treatments with associated response variables $X_{i1},...,X_{ik_{i}},$ i = 1,...,r( ≥ 2). We let X _ijl denote the lth observation recorded independently and assume that their common distribution is $N(\mu _{ij},\sigma _{i}^{2}),l=1,...,n_{i}(\geq 2),j=1,...,k_{i},i=1,...,r.$ Also, let X _0l denote the lth observation recorded independently from a control and we assume that their common distribution is $N(\mu _{0},\sigma _{0}^{2}),l=1,...,n_{0}(\geq 2).$ The parameters μ _ij ’s, σ _i ’s, μ ₀, and σ ₀ are assumed finite, unknown, and σ _i ’s unequal, j = 1,...,k _i , i = 1,...,r. Denote the treatment-control difference Δ _ij  = μ _ij  − μ ₀ and our goal is to make simultaneous one-sided and two-sided fixed-precision confidence statements regarding all Δ _ij ’s, j = 1,...,k _i , i = 1,...,r. Specifically, given two preassigned numbers d( > 0) and 0 < α < 1, we develop two sets of appropriate two-stage stopping variables N ₀,N ₁,...,N _r via procedures $\mathcal{P}_{\text{I}}^{ \mathbf{N}}$ and $\mathcal{P}_{\text{II}}^{\mathbf{N}}$ respectively such that the simultaneous probability that $(\overline{X}_{ijN_{i}}-\overline{X} _{0n_{0}})-\Delta _{ij}\leq d$ under $\mathcal{P}_{\text{I}}^{\mathbf{N}}$ and that $\left| (\overline{X}_{ijN_{i}}-\overline{X}_{0N_{0}})-\Delta _{ij}\right| \leq d$ under $\mathcal{P}_{\text{II}}^{\mathbf{N}}$ would both exceed or equal 1 − α for all j = 1,...,k _i , i = 1,...,r. We note that the two-stage methodology for our one-sided problem covers more diverse applications than Dudewicz et al.’s (Biom Z 17:13–26, 1975) methodology did since we no longer assume k ₁ = ... = k _r  = 1. With regard to our two-sided problem, no previous two-stage methodology was available in the literature. In this work, mathematical determinations of appropriate design constants with regard to either goal, the proposed methodologies, and the construction of the requisite tables are all new. This paper emphasizes practical implementations of the proposed methodologies.