reality to some extent if the measuring scale is nonlinear. Confidence intervals are useful, however, because the semilinearity of most empirical scales of measurement means they are usually approximately correct.
To improve scales of measurement is a continuing goal of empirical science. Among the many aspects of this endeavor, closer approach to linearity is an important one. Although the problem of linearity is basically substantive and extrastatistical, Anova-regression can provide unique assistance as shown in the discussion of psychological measurement theory in Chapter 21.
19.1.1a Few statistical issues have generated as much controversy as Fisher's concept of fiducial probability. In their concluding discussion comparing the three concepts of fiducial interval, confidence interval, and Bayesian belief interval, Kendall and Stuart (1979, p. 165) state
There has been so much controversy … that, at this point, we shall have to leave our customary objective standpoint and descend into the arena ourselves. The remainder of this chapter is an expression of our personal views. We think that it is the correct viewpoint; and it represents the result of many years' silent reflexion on the issues involved, a serious attempt to understand what the protagonists say, and an even more serious attempt to divine what they mean.
19.1.2a Usefulness of confidence as personal, extrastatistical belief seems to be ignored by Bayesians, who contrast the statistical concept of confidence with the Bayesian concept of personal, statistical belief.
The qualifications noted in the second sentence of the quotation from Mosteller and Tukey deserve comment. “Typicality, ” I take it, rules out samples considered to be biased. “Absence of selection” refers to the diverse dangers of sample selection that capitalize on chance outcomes listed in the later subsection, Bayesian Theory and Randomization. Similar qualifications apply to Bayesian belief intervals.
Lehmann (1993) considers that the Fisher and Neyman–Pearson theories are complementary, not contradictory (aside from fiducial probability). The main difference is whether tests of composite hypotheses should be conditional, as Fisher later maintained, or unconditional, as Neyman and Pearson claimed. This issue, he says, cannot be decided by abstract principles, but depends on the empirical context of application.
. Examples in which the confidence interval may be known to be incorrect after the data have been collected are given by Jaynes (1976, p. 198) and by Mosteller and Tukey (1968, p. 181). These examples are not realistic, but they show that the concept of confidence interval is no panacea.
Questia, a part of Gale, Cengage Learning. www.questia.com
Book title: Empirical Direction in Design and Analysis.
Contributors: Norman H. Anderson - Author.
Publisher: Lawrence Erlbaum Associates.
Place of publication: Mahwah, NJ.
Publication year: 2001.
Page number: 637.
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