Difference between revisions of "Statistical test"
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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)}} | ||
| + | __TOC__ | ||
| + | ==Definition== | ||
| + | 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 | ||
| + | |||
| + | 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: | ||
| + | {| class="wikitable" border="1" | ||
| + | |- | ||
| + | ! ''Outcomes:'' | ||
| + | ! Truly present<br/>{{M|[P\eq 1]}} | ||
| + | ! Truly absent<br/>{{M|[P\eq 0]}} | ||
| + | |- | ||
| + | ! Test positive<br/>{{M|[T\eq 1]}} | ||
| + | | {{M|\Pcond{T\eq 1}{P\eq 1}\eq u}} | ||
| + | * '''''[[true positive]]''''' | ||
| + | * '''''[[Power of statistical test|power]]''''' | ||
| + | | {{M|\Pcond{T\eq 1}{P\eq 0}\eq 1-v}} | ||
| + | * '''''[[false positive]]''''' | ||
| + | * '''''[[Significance level of a statistical test|significance level]]''''' | ||
| + | |- | ||
| + | ! Test negative<br/>{{M|[T\eq 0]}} | ||
| + | | {{M|\Pcond{T\eq 0}{P\eq 1}\eq 1-u}}<br/> | ||
| + | '''''[[false negative]]''''' | ||
| + | | {{M|\Pcond{T\eq 0}{P\eq 0}\eq v}}<br/> | ||
| + | '''''[[true negative]]''''' | ||
| + | |} | ||
| + | =OLD PAGE= | ||
==Definition== | ==Definition== | ||
A ''statistical test'', {{M|T}}, is characterised by two (a [[ordered pair|pair]]) of [[probability (object)|probabilities]]: | A ''statistical test'', {{M|T}}, is characterised by two (a [[ordered pair|pair]]) of [[probability (object)|probabilities]]: | ||
Latest revision as of 13:32, 14 December 2017
\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} }
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)
Contents
[hide]Definition
A statistical test, T, is characterised by two (a pair) of probabilities:
- T\eq(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
If we write [T\eq 1] for the test coming back positive, [T\eq 0] for negative and let P denote the actual correct outcome (which may be unknowable), denoting [P\eq 1] if the thing being tested for is, in truth, present and [P\eq 0] if absent, then:
| Outcomes: | Truly present [P\eq 1] |
Truly absent [P\eq 0] |
|---|---|---|
| Test positive [T\eq 1] |
\Pcond{T\eq 1}{P\eq 1}\eq u | \Pcond{T\eq 1}{P\eq 0}\eq 1-v |
| Test negative [T\eq 0] |
\Pcond{T\eq 0}{P\eq 1}\eq 1-u |
\Pcond{T\eq 0}{P\eq 0}\eq v |
OLD PAGE
Definition
A statistical test, T, is characterised by two (a pair) of probabilities:
- T\eq(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:
- Positive: [T(s)\eq 1], [T(s)\eq\text{P}], or possibly either of these without the [\ ]
- Negative: [T(s)\eq 0], [T(s)\eq\text{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
