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Related Concept Videos

Introduction to Functions01:29

Introduction to Functions

Functions are essential mathematical tools used to describe consistent relationships between varying quantities. A function connects each input to a single, corresponding output based on a defined rule. These relationships appear in both everyday contexts and natural phenomena, providing a framework for understanding change and prediction.One common real-life example is a parking garage fee system, where the total cost depends on the amount of time a vehicle remains inside. In this case, the...
Energy Budgets and Reproductive Strategies00:51

Energy Budgets and Reproductive Strategies

Organisms must balance energy intake with the energy required for growth, maintenance, and reproduction. These trade-offs result in a variety of survivorship and reproductive strategies, including semelparity and iteroparity. Semelparous species reproduce only once in their lifetime, often investing most available resources into that single reproductive event. Iteroparous species, by contrast, reproduce multiple times over their lifetimes, typically allocating fewer resources to any single...
Limit Laws I01:25

Limit Laws I

Limit laws provide essential tools for analyzing how functions behave as their input approaches a specific value. These laws are particularly useful when dealing with combinations of functions, provided the individual limits exist. The Sum and Difference Laws state that the limit of the sum or difference of two functions equals the sum or difference of their respective limits:The Product Law asserts that the limit of the product of two functions equals the product of their individual limits:A...
Piecewise-Defined Functions01:28

Piecewise-Defined Functions

Piecewise defined functions are mathematical models where different expressions define a function over distinct intervals of the domain. These functions are useful for representing systems with varying behaviors depending on input values.For example, the function:  uses a linear rule for inputs less than or equal to –1 and a quadratic rule for values greater than –1. Although it has two formulas, it still defines a single function.Another common type is the absolute value function, given...
Types of Functions I01:26

Types of Functions I

Functions are fundamental mathematical tools that capture relationships between variables and describe how one quantity changes in relation to another. Their diverse forms allow them to model various real-world phenomena with precision and flexibility. Among the various categories, algebraic functions are prominent due to their formulation through basic arithmetic operations: addition, subtraction, multiplication, division, and root extraction.Algebraic functions include polynomial, rational,...
Increasing Function01:18

Increasing Function

An increasing function exhibits a rise in output values as input values increase. This behavior is depicted graphically as a curve or line that slopes upward from left to right. Such a function satisfies the condition that if x1 < x2, then f(x1) < f(x2), indicating that the function values grow with increasing inputs. This concept is fundamental in understanding growth trends across various domains, such as population dynamics, financial investments, or resource consumption.The average...

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Updated: Jun 19, 2026

Computerized Adaptive Testing System of Functional Assessment of Stroke
05:21

Computerized Adaptive Testing System of Functional Assessment of Stroke

Published on: January 7, 2019

Fitting spending functions.

Keaven M Anderson1, Jason B Clark

  • 1Clinical Biostatistics and Research Decision Sciences, Merck Research Laboratories, North Wales, PA 19454-2505, U.S.A. keaven_anderson@merck.com

Statistics in Medicine
|October 21, 2009
PubMed
Summary
This summary is machine-generated.

Group sequential monitoring helps stop clinical trials early for efficacy or futility. This study introduces new flexible alpha- and beta-spending functions for more precise trial stopping boundaries.

Related Experiment Videos

Last Updated: Jun 19, 2026

Computerized Adaptive Testing System of Functional Assessment of Stroke
05:21

Computerized Adaptive Testing System of Functional Assessment of Stroke

Published on: January 7, 2019

Area of Science:

  • Biostatistics
  • Clinical Trial Methodology
  • Statistical Inference

Background:

  • Group sequential monitoring enables interim analyses in clinical trials to assess efficacy or safety.
  • Alpha- and beta-spending functions are crucial for defining stopping boundaries in these trials.
  • Existing one-parameter families may lack flexibility for specific interim analysis criteria.

Purpose of the Study:

  • To explore fitting alpha- and beta-spending functions with specific values at defined interim analyses.
  • To develop new flexible one- and two-parameter families for group sequential methods.
  • To enhance the precision of stopping rules in clinical trials.

Main Methods:

  • Investigated fitting alpha- and beta-spending functions to achieve desired critical values at interim analyses.
  • Defined novel one- and two-parameter families of spending functions.
  • Demonstrated the utility of these new families with practical examples.

Main Results:

  • Commonly used one-parameter families may not adequately fit multiple desired critical values.
  • New one- and two-parameter families offer enhanced flexibility in setting stopping boundaries.
  • The logistic family, a two-parameter example, has shown practical application.

Conclusions:

  • Flexible alpha- and beta-spending functions are essential for accurate group sequential monitoring.
  • The proposed new families provide improved adaptability for clinical trial stopping rules.
  • These methods enhance the ability to make timely and informed decisions during clinical trials.