Statistical test

[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]

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] true positive power	[ilmath]\Pcond{T\eq 1}{P\eq 0}\eq 1-v[/ilmath] false positive significance level
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