Difference between revisions of "Statistical test"
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| + | {{ProbMacro}} | ||
| + | __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]]: | ||
Revision as of 08:38, 15 November 2017
Contents
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] |
[ilmath]\Pcond{T\eq 0}{P\eq 0}\eq v[/ilmath] |
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:
- Positive: [ilmath][T(s)\eq 1][/ilmath], [ilmath][T(s)\eq\text{P}][/ilmath], or possibly either of these without the [ilmath][\ ][/ilmath]
- 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
