Difference between revisions of "Statistical test"

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(Created page with "==Definition== We use the term ''statistical test'' for a test, {{M|T}}, with two outcomes (usually "yes"/1 or "no"/0). Let {{M|P\eq 1}} denote that the unit under test actual...")
 
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{{ProbMacro}}{{Stub page|grade=A*|msg=Include [[hypothesis testing]] of which are an instance of statistical tests. Link to [[true/false positive/negative]] and create pages for [[false positive]] and such that redirect to an anchor on that page. Don't forget the {{link|power function|statistics}} - [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 13:32, 14 December 2017 (UTC)}}
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__TOC__
 
==Definition==
 
==Definition==
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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]]:
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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
  
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The statistical test is defined by the following four probabilities:
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If we write {{M|[T\eq 1]}} for the test coming back positive, {{M|[T\eq 0]}} for negative and let {{M|P}} denote the ''actual'' correct outcome (which may be unknowable), denoting {{M|[P\eq 1]}} if the thing being tested for is, in truth, present and {{M|[P\eq 0]}} if absent, then:
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* {{M|\mathbb{P}[T\eq 1\ \vert\ P\eq 1]\eq\alpha}} - "true positive"
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{| class="wikitable" border="1"
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* {{M|\mathbb{P}[T\eq 1\ \vert\ P\eq 0]\eq\beta}} - "''false positive''" as the test incorrectly reports positive.
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|-
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* {{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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! ''Outcomes:''
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* {{M|\mathbb{P}[T\eq 0\ \vert\ P\eq 0]\eq\delta}} - "true negative"
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! Truly present<br/>{{M|[P\eq 1]}}
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! Truly absent<br/>{{M|[P\eq 0]}}
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|-
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! Test positive<br/>{{M|[T\eq 1]}}
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| {{M|\Pcond{T\eq 1}{P\eq 1}\eq u}}
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* '''''[[true positive]]'''''
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* '''''[[Power of statistical test|power]]'''''
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| {{M|\Pcond{T\eq 1}{P\eq 0}\eq 1-v}}
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* '''''[[false positive]]'''''
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* '''''[[Significance level of a statistical test|significance level]]'''''
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|-
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! Test negative<br/>{{M|[T\eq 0]}}
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| {{M|\Pcond{T\eq 0}{P\eq 1}\eq 1-u}}<br/>
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'''''[[false negative]]'''''
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| {{M|\Pcond{T\eq 0}{P\eq 0}\eq v}}<br/>
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'''''[[true negative]]'''''
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|}
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=OLD PAGE=
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==Definition==
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A ''statistical test'', {{M|T}}, is characterised by two (a [[ordered pair|pair]]) of [[probability (object)|probabilities]]:
 +
* {{M|T\eq(u,v)}}, where:
 +
** {{M|u}} is the probability of the test yielding a ''[[true-positive]]'' result
 +
** {{M|v}} is the probability of the test yielding a ''[[true-negative]]'' result
  
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==Todo==
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Tests are usually ''asymmetric'', see: [[#Explanation|below]] and ''[[asymmetry of statistical tests]]'' for more info.
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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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===Notation and Terminology===
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For a test subject, {{M|s}}, we say the outcome of the test is:
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# '''Positive: ''' {{M|[T(s)\eq 1]}}, {{M|[T(s)\eq\text{P}]}}, or possibly either of these without the {{M|[\ ]}}
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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====
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The ''power'' of the test is
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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

Latest revision as of 13:32, 14 December 2017

[ilmath]\newcommand{\P}[2][]{\mathbb{P}#1{\left[{#2}\right]} } \newcommand{\Pcond}[3][]{\mathbb{P}#1{\left[{#2}\!\ \middle\vert\!\ {#3}\right]} } \newcommand{\Plcond}[3][]{\Pcond[#1]{#2}{#3} } \newcommand{\Prcond}[3][]{\Pcond[#1]{#2}{#3} }[/ilmath]
Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Include hypothesis testing of which are an instance of statistical tests. Link to true/false positive/negative and create pages for false positive and such that redirect to an anchor on that page. Don't forget the power function - Alec (talk) 13:32, 14 December 2017 (UTC)

Definition

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

  • [ilmath]T\eq(u,v)[/ilmath], where:
    • [ilmath]u[/ilmath] is the probability of the test yielding a true-positive result
    • [ilmath]v[/ilmath] is the probability of the test yielding a true-negative result

If we write [ilmath][T\eq 1][/ilmath] for the test coming back positive, [ilmath][T\eq 0][/ilmath] for negative and let [ilmath]P[/ilmath] denote the actual correct outcome (which may be unknowable), denoting [ilmath][P\eq 1][/ilmath] if the thing being tested for is, in truth, present and [ilmath][P\eq 0][/ilmath] if absent, then:

Outcomes: Truly present
[ilmath][P\eq 1][/ilmath] 		Truly absent
[ilmath][P\eq 0][/ilmath]
Test positive
[ilmath][T\eq 1][/ilmath] 		[ilmath]\Pcond{T\eq 1}{P\eq 1}\eq u[/ilmath] [ilmath]\Pcond{T\eq 1}{P\eq 0}\eq 1-v[/ilmath]
Test negative
[ilmath][T\eq 0][/ilmath] 		[ilmath]\Pcond{T\eq 0}{P\eq 1}\eq 1-u[/ilmath]

false negative

[ilmath]\Pcond{T\eq 0}{P\eq 0}\eq v[/ilmath]

true negative

OLD PAGE

Definition

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

  • [ilmath]T\eq(u,v)[/ilmath], where:
    • [ilmath]u[/ilmath] is the probability of the test yielding a true-positive result
    • [ilmath]v[/ilmath] 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, [ilmath]s[/ilmath], we say the outcome of the test is:

  1. Positive: [ilmath][T(s)\eq 1][/ilmath], [ilmath][T(s)\eq\text{P}][/ilmath], or possibly either of these without the [ilmath][\ ][/ilmath]
  2. Negative: [ilmath][T(s)\eq 0][/ilmath], [ilmath][T(s)\eq\text{N}][/ilmath], or possibly either of these without the [ilmath][\ ][/ilmath]

Power and Significance

The power of the test is

Explanation

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

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