Difference between revisions of "Statistical test"

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==Definition==
 
==Definition==
We use the term ''statistical test'' for a test, {{M|T}}, with two outcomes (usually "yes"/1 or "no"/0).
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A ''statistical test'', {{M|T}}, is characterised by two (a [[ordered pair|pair]]) of [[probability (object)|probabilities]]:
Let {{M|P\eq 1}} denote that the unit under test actually has this property we're testing for, and {{M|P\eq 0}} if it does not.
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* {{M|T\eq(u,v)}}, where:
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** {{M|u}} is the probability of the test yielding a ''[[true-positive]]'' result
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** {{M|v}} is the probability of the test yielding a ''[[true-negative]]'' result
  
The statistical test is defined by the following four probabilities:
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Tests are usually ''asymmetric'', see: [[#Explanation|below]] and ''[[asymmetry of statistical tests]]'' for more info.
* {{M|\mathbb{P}[T\eq 1\ \vert\ P\eq 1]\eq\alpha}} - "true positive"
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===Notation and Terminology===
* {{M|\mathbb{P}[T\eq 1\ \vert\ P\eq 0]\eq\beta}} - "''false positive''" as the test incorrectly reports positive.
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For a test subject, {{M|s}}, we say the outcome of the test is:
* {{M|\mathbb{P}[T\eq 0\ \vert\ P\eq 1]\eq\gamma}} - "''false negative''" as the test incorrectly reports negative yet the property is present
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# '''Positive: ''' {{M|[T(s)\eq 1]}}, {{M|[T(s)\eq\text{P}]}}, or possibly either of these without the {{M|[\ ]}}
* {{M|\mathbb{P}[T\eq 0\ \vert\ P\eq 0]\eq\delta}} - "true negative"
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# '''Negative: ''' {{M|[T(s)\eq 0]}}, {{M|[T(s)\eq\text{N}]}}, or possibly either of these without the {{M|[\ ]}}
 
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====Power and Significance====
==Todo==
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The ''power'' of the test is
Add an example where 1 in 10,000 actually has the property, but the test has a non-zero false-positive rate, actually meaning that if the test returns positive it's unlikely you actually have the property, relate to statistical power.
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==Explanation==
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Let {{M|R}} denote the result of the test, here this will be {{M|1}} or {{M|0}}, and let {{M|P}} be whether or not the subject actually has the property being tested for.
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As claimed above the test is characterised by two probabilities

Revision as of 17:05, 4 October 2017

Definition

A statistical test, T, is characterised by two (a pair) of probabilities:

  • T=(u,v), where:
    • u is the probability of the test yielding a true-positive result
    • v is the probability of the test yielding a true-negative result

Tests are usually asymmetric, see: below and asymmetry of statistical tests for more info.

Notation and Terminology

For a test subject, s, we say the outcome of the test is:

  1. Positive: [T(s)=1], [T(s)=P], or possibly either of these without the [ ]
  2. Negative: [T(s)=0], [T(s)=N], or possibly either of these without the [ ]

Power and Significance

The power of the test is

Explanation

Let R denote the result of the test, here this will be 1 or 0, and let P be whether or not the subject actually has the property being tested for. As claimed above the test is characterised by two probabilities