Statistical test

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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 $[T\eq 1]$	$\Pcond{T\eq 1}{P\eq 1}\eq u$ true positive power	$\Pcond{T\eq 1}{P\eq 0}\eq 1-v$ false positive significance level
Test negative $[T\eq 0]$	$\Pcond{T\eq 0}{P\eq 1}\eq 1-u$ false negative	$\Pcond{T\eq 0}{P\eq 0}\eq v$ true negative

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:

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