Message from discussion
A test for discrete versus continuous?
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From: aruzinsky <aruzin...@general-cathexis.com>
Newsgroups: sci.stat.math
Subject: Re: A test for discrete versus continuous?
Date: Thu, 2 Jul 2009 08:27:39 -0700 (PDT)
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On Jul 2, 3:46=A0am, Gary <LanceG...@gmail.com> wrote:
> This is a somewhat poorly formed question but the problem it
> represents has troubled me several times in my career. Essentially
> many theories may state that a particular phenomenon exists either as
> a dimension or as a set of discrete categories. For example Reversal
> Theory in psychology states (amongst other things) that a person is
> either in a telic (purpose driven, arousal avoiding, future oriented)
> mode or in a paratelic (activity driven, arousal seeking, present
> oriented) mode and that people can't be somewhere "inbetween" the two
> modes. So Reversal theory is positing a set of two discrete categories
> and strongly claims that all people are in one or other of the two
> states and that there is no continuum between them. Similarly Fulda
> developed a mathematical model of the pull of temptation and asserts
> that the model works on discrete moments of thought (in other words
> the probabilities of temptation that are being modelled are discrete
> and not continuous). In my experience it is really hard to devise
> tests for claims of this kind. I wondered whther there are any
> existing statistical tests designed to to test hypotheses of this
> kind, or whether there are procedures and designs suitable for testing
> such claims?
>
> Lance
In some instances a phenomenon is finely discrete, e.g., an integer,
instead of continuous, e.g., a real number. Let's just call both
cases "practically continuous." The first thing that you need to do
define a practically continuous metric, make measurements with this
metric, and plot a histogram. If the histogram has gaps larger than
the discretization of the metric, the phenomenon is not practically
continuous.
For example, to measure schizophrenia, one might construct a
practically continuous metric Yj =3D sum Ai/Bi, i =3D 1 to N
where
N =3D Number of false beliefs that a jth individual has
Ai =3D Strength of conviction (certainty) for ith false belief
Bi =3D Proportion of population with ith (identical) false belief (to
reduce the influence of mass delusions from popular religions and
Obama)
Then you make measurements on many individuals and plot the
histogram. If the histogram has big gaps, the schizophrenia is not
practically continuous. You can bet it would be bell shaped like that
of IQ. Of course, you have to know which beliefs are false to measure
their number. So, dumb F psychologists without thermometers classify
things as hot or cold.
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