On Tukey's Confidence Intervals for the Contrasts in the Means of the Intraclass Correlation Model

Abstract

In the intraclass correlation model when the variance 0'2 and the intraclass correlation coefficient p are both unknown, Tukey's confidence intervals for the contrasts in the means are obtained for the fixed sample and the sequential case. In the latter case confidence intervals of fixed width are obtained and the cost of not knowing 0'2 and p has been studied. From these results parallel results for Scheffe's confidence intervals can easily be obtained. A recommendation for the choice between the two is also made.

Testing Equality of Means and Simultaneous Confidence Intervals in Repeated Measures with Missing Data

“…Simultaneous confidence intervals of Scheffé and Tukey type for linear contrasts a 0 m for all non-null vector a such that a 0 1 ¼ 0, when r is unknown in the intraclass correlation model have been given by Bhargava and Srivastava (1973) and by Scheffé (1959) and Miller (1966) when r is known. However, when n i 's are not all equal, this problem has remained open, see Littell, Milliken, Stroup and Wolfinger (1996) and Kleinbaum, Kupper, Muller, and Nizam (1998).…”

Fixed-width confidence intervals for contrasts in the means

“…For the case when a is unknown but p is known, Scheff~ (1959) and Miller (1966) developed fixed sample size procedures to construct Tukey's confidence intervals for all contrasts {b'# : b E Rp, b'e = 0} having prescribed confidence coefficient a E (0, 1). Bhargava and Srivastava (1973) extended the results to the case when a and p both are unknown. They also proved the non-existence of fixed sample size procedures to construct T~key's confidence intervals having "pre-assigned width and coverage probability" when a and/or p are unknown.…”

“…It has been shown in Bhargava and Srivastava (1973, p. 152) that f'(q)/f(q) > -q/2. Utilizing this result and Lemma 2.1, we obtain from (2.6),…”

“…Bhargava and Srivastava (1973) studied asymptotic (as d -~ 0) properties of the sequential procedure (1.1) and proved that "asymptotic efficiency and consistency" (see, Chow and Robbins (1965) for definitions) hold for it. They also derived upper bounds for the first two moments of the stopping time and along the lines of Simons (1968), they proved that if after stopping, an "additional" number of observations, say K, is taken, then P{I(CI.~2(N+K) --c'tt)] < d} > a for all #, a and p. Wald (1947) pointed out that due to changing nature of sample space at each stage of sampling, the purely sequential procedure is complicated in nature to apply.…”

“…In Section 2, improving the results obtained by Bhargava and Srivastava (1973), we obtain second-order approximations for the first two moments of the stopping time and coverage probability of the sequential procedure (1.1). Moreover, a lower bound for the number of "additional" observations "K" is derived, which guarantees the exact probability of coverage.…”