Difference between revisions of "Statistical test"

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==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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Tests are usually ''asymmetric'', see: [[#Explanation|below]] and ''[[asymmetry of statistical tests]]'' for more info.
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* {{M|\mathbb{P}[T\eq 1\ \vert\ P\eq 1]\eq\alpha}} - "true positive"
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===Notation and Terminology===
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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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For a test subject, {{M|s}}, we say the outcome of the test is:
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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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# '''Positive: ''' {{M|[T(s)\eq 1]}}, {{M|[T(s)\eq\text{P}]}}, or possibly either of these without the {{M|[\ ]}}
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* {{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====
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==Todo==
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The ''power'' of the test is
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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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==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, [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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