Table 4. Fixed vs. random factors.

	Fixed Factor	Random Factor
Characteristics	Factors could have some unique level values (male, female) or experimenters could assign that level (treatment A, treatment B). Some can be randomized.	Level values are picked among many possible values. Those are not necessarily randomized.
Example	Treatment, Sex, Ethnicity, Season as an idealized one, Relatively permanent and small number of machines	Each patient (subject), Hospitalization date, Drug administration date, Drug bottle, Source barrel, Temporary machines, Some of many machines
Level means and differences after ANOVA (post hoc analysis)	Those can be estimated and tested.	Those should not be estimated nor tested. Only the size of variability (variance) is a concern and should be estimated.
Expectation of a level (ai)	E(ai) = ai	E(ai) = 0
Variance of a level (ai)	Var(ai) = 0	Var(ai) ≠ 0
Summation of level effects	Σai = 0, a̅ = 0	Σai ≠ 0, a̅ ≠ 0
Variability among k levels, Variability of ai	${\sigma }_{\mathit{A}}^{\mathrm{2}}=\sum _{\mathit{i}=\mathrm{1}}^{\mathit{k}}{\mathit{a}}_{\mathit{i}}^{\mathrm{2}}/\left(\mathit{k}-\mathrm{1}\right)$	${\sigma }_{\mathit{A}}^{\mathrm{2}}=\left[\sum _{\mathit{i}=\mathrm{1}}^{\mathit{k}}{\left(a_{\mathit{i}}-\stackrel{-}{\mathit{a}}\right)}^{\mathrm{2}}/\left(\mathit{k}-\mathrm{1}\right)\right]$