Randomization Procedures for Changing Criterion Designs.
To strengthen the scientific credibility arguments for single-case intervention studies, randomization design-and-analysis methods have been developed for the multiple-baseline, ABAB, and alternating treatment designs, including options for preplanned designs, wherein the series and phase lengths are established prior to gathering data, as well as options for response-guided designs, wherein ongoing visual analyses guide decisions about when to intervene. Our purpose here is to develop randomization methods for another class of single-case design, the changing criterion design. We first illustrate randomization design-and-analysis methods for preplanned changing criterion designs and then develop and illustrate methods for response-guided changing criterion designs. We discuss the limitations associated with the randomization methods and the validity of the corresponding intervention-effect inferences.
Randomization procedures in single-case intervention research contexts: (Some of) "the rest of the story".
Following up on articles recently published in this journal, the present contribution tells (some of) "the rest of the story" about the value of randomization in single-case intervention research investigations. Invoking principles of internal, statistical-conclusion, and external validity, we begin by emphasizing the critical distinction between design randomization and analysis randomization, along with the necessary correspondence between the two. Four different types of single-case design-and-analysis randomization are then discussed. The persistent negative influence of serially dependent single-case outcome observations is highlighted, accompanied by examples of inappropriate applications of parametric and nonparametric tests that have appeared in the literature. We conclude by presenting valid applications of single-case randomization procedures in various single-case intervention contexts, with specific reference to a freely available Excel-based software package that can be accessed to incorporate the present randomization schemes into a wide variety of single-case intervention designs and analyses.
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.