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An overview of the three-factor analysis of variance (anova) model, which is used to analyze the effects of three factors (a, b, c) with different levels. The concept of main effects, second order interactions, and third order interactions, as well as the importance of orthogonal designs and the differences between type i, ii, and iii sums of squares. It also discusses the implications of unbalanced designs.
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Three factor AOV model (factors A, B, C, with + , -, , levels, respectively)
] 3457 œ. &! 3 & " 4 & # 5 & a!" b 34 &a!# b 35
& a "# b 45 & a !"# b 345 &% 3457
. grand mean ! " # 3 4 5 main effects
a !" b 34 a !# b 35 a "#b 45 second order interactions
a !"# b 345 third order interaction
Effects & interactions summed on any subscript add to zero
AOV table, balanced design, 8 obs. per cell
source SS df MS f
A SS Aa b a + , 1 b SS Aa +,^ a^1 bb^ MS errorMS Aa a^ bb
B SS Ba b a , , 1 b SS Ba ,,^ a^1 bb^ MS errorMS Ba a^ bb
C SS Ca b a - , 1 b SS Ca -,^ a^1 bb^ MS errorMS Ca a^ bb
AB SS ABa b a + , 1 ba , , 1 b SS ABa +,^ a 1 ba ,, 1 b^ b^ MS errorMS ABaa bb
AC SS ACa b a + , 1 ba - , 1 b SS ACa +,^ a 1 ba -, 1 b^ b^ MS errorMS ACaa bb
BC SS BCa b a , , 1 ba - , 1 b SS BCa , ,^ a 1 ba -, 1 bb^ MS errorMS BCaa bb
ABC SS ABCa b a + , 1 ba , , 1 ba - , 1 b (^) aSS ABC +, 1 aba ,, 1 ba -,^ b 1 b^ MS ABCMS erroraa^ bb
error SS errora b +,- 8 ,a 1 b +,- 8,SS error^ aa^1 bb
total SS totala b 8+,- , 1
Type II SS compares the particular effect with all other terms that do not contain that particular effect
source type II SS A B C SSE A, B, ABa b ,SSE A, B, AB, Ca b
AB AC SSE A, B, C, AB, BCa b ,SSE A, B, C, AB, BC, ACa b BC
ABC
Type III SS enters the particular effect last.
source type III SS A B C SSE A, B, AB, AC, BC, ABCa b ,SSE A, B, C, AB, AC, BC, ABCa b
AB AC SSE A, B, C, AB, BC, ABCa b ,SSE A, B, C, AB, AC, BC, ABCa b BC
ABC
(Type IV SS: same as type III if there are no empty cells)
