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

Content Provider	Springer Nature Link
Author	Mukhopadhyay, Nitis Solanky, Tumulesh K. S.
Copyright Year	2011
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.
Ending Page	522
Page Count	22
Starting Page	501
File Format	PDF
ISSN	13875841
e-ISSN	15737713
Journal	Methodology and Computing in Applied Probability
Issue Number	3
Volume Number	14
Language	English
Publisher	Springer US
Publisher Date	2011-07-27
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Fixed-size confidence intervals Simulations One-sided confidence intervals Statistical tables Treatments vs. control Economics general Two-sided confidence intervals Business/Management Science Tolerance and confidence regions Two-stage sampling Statistics Life Sciences Paired and multiple comparisons Slepian’s inequality Real data illustration Sequential estimation Simultaneous confidence intervals Dunn’s inequality Electrical Engineering
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability

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