Multi-Criteria Decision Analysis (MCDA) in Healthcare Decision-Making

Multi-criteria decision analysis (MCDA) is increasingly used to support healthcare decision-making. MCDA involves decision makers evaluating the alternatives under consideration based on the explicit weighting of criteria relevant to the overarching decision—in order to, depending on the application, rank (or prioritize) or choose between the alternatives. A prominent example of MCDA applied to healthcare decision-making that has received a lot of attention in recent years and is the main subject of this article is choosing which health “technologies” (i.e., drugs, devices, procedures, etc.) to fund—a process known as health technology assessment (HTA). Other applications include prioritizing patients for surgery, prioritizing diseases for R&D, and decision-making about licensing treatments. Most applications are based on weighted-sum models. Such models involve explicitly weighting the criteria and rating the alternatives on the criteria, with each alternative’s “performance” on the criteria aggregated using a linear (i.e., additive) equation to produce the alternative’s “total score,” by which the alternatives are ranked. The steps involved in a MCDA process are explained, including an overview of methods for scoring alternatives on the criteria and weighting the criteria. The steps are: structuring the decision problem being addressed, specifying criteria, measuring alternatives’ performance, scoring alternatives on the criteria and weighting the criteria, applying the scores and weights to rank the alternatives, and presenting the MCDA results, including sensitivity analysis, to decision makers to support their decision-making. Arguments recently advanced against using MCDA for HTA and counterarguments are also considered. Finally, five questions associated with how MCDA for HTA is operationalized are discussed: Whose preferences are relevant for MCDA? Should criteria and weights be decision-specific or identical for repeated applications? How should cost or cost-effectiveness be included in MCDA? How can the opportunity cost of decisions be captured in MCDA? How can uncertainty be incorporated into MCDA?

Smooth Polynomial Solutions to a Ternary Additive Equation

Let Fq[T] be the ring of polynomials over the finite field of q elements and Y a large integer. We say a polynomial in Fq[T] is Y-smooth if all of its irreducible factors are of degree at most Y. We show that a ternary additive equation a + b = c over Y-smooth polynomials has many solutions. As an application, if S is the set of first s primes in Fq[T] and s is large, we prove that the S-unit equation u + v = 1 has at least exp solutions.

Approximate Homomorphisms and Derivations on Normed Lie Triple Systems

We investigate the generalized Hyers-Ulam stability of homomorphisms and derivations on normed Lie triple systems for the following generalized Cauchy-Jensen additive equationr0f((s∑j=1pxj+t∑j=1dyj)/r0)=s∑j=1p‍f(xj)+t∑j=1d‍f(yj), wherer0,s, and tare nonzero real numbers. As a results, we generalize some stability results concerning this equation.

On Ulam's Type Stability of the Cauchy Additive Equation

We prove a general result on Ulam's type stability of the functional equationfx+y=fx+fy, in the class of functions mapping a commutative group into a commutative group. As a consequence, we deduce from it some hyperstability outcomes. Moreover, we also show how to use that result to improve some earlier stability estimations given by Isaac and Rassias.

A BINARY ADDITIVE EQUATION INVOLVING FRACTIONAL POWERS

Let c be a real number with 1 < c < 2. We study the representations of a large integer n in the form [Formula: see text] where m is an integer and p is a prime number. We prove that when [Formula: see text], all sufficiently large integers are thus representable.

STABILITY OF MAPPINGS ON MULTI-NORMED SPACES

In this paper, we define multi-normed spaces, and investigate some properties of multi-bounded mappings on multi-normed spaces. Moreover, we prove a generalized Hyers–Ulam–Rassias stability theorem associated to the Cauchy additive equation for mappings from linear spaces into multi-normed spaces.

A Ruler for Interpreting Diagnostic Test Results

Summary Objectives: Bayes’ rule formalizes how the pre-test probability of having a condition of interest is changed by a diagnostic test result to yield the post-test probability of having the condition. To simplify this calculation a geometric solution in form of a ruler is presented. Methods: Using odds and the likelihood ratio of a test result in favor of having the condition of interest, Bayes’ rule can succinctly be expressed as ”the post-test odds equals the pre-test odds times the likelihood ratio”. Taking logarithms of both sides yields an additive equation. Results: The additive log odds equation can easily be solved geometrically. We propose a ruler made of two scales to be adjusted laterally. A different, widely used solution in form of a nomogram was published by Fagan [2]. Conclusions: Whilst use of the nomogram seems more obvious, the ruler may be easier to operate in clinical practice since no straight edge is needed for precise reading. Moreover, the ruler yields more intuitive results because it shows the change in probability due to a given test result on the same scale.