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How do I work out degrees of freedom for terms in an ANOVA?
For N subjects and g group variables (factors) with respective levels L1, L2, ... , Lg and a within subjects factors, W, the table below gives the degrees of freedom for various types of sources variation in an analysis of variance.
Source of variation |
df |
|
Factor |
L1-1 |
|
2-way interaction |
(L1-1)(L2-1) |
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K-way interaction of factors |
$$\prod_text{k} (L_text{k}-1)$$ |
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Between subjects error terms |
|
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Error (one-way anova between subjects) |
N - L1 |
|
Error (between subjects) |
N - df of terms involving between subjects factors - 1 |
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Within subjects error terms |
|
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Error (subjects x W1), no between subjects factor |
(N-1)(L1-1) |
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Error (subjects x W1 x W2, no between subjects factor ) |
(N-1)(L1-1)(L2-1) |
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Error (subjects x W1 x W2, 1 between subjects factor) |
(N-L1)(L2-1)(L3-1) |
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Error (subjects x Within subjects interaction) |
df of Error (between subjects) x df(Within subjects interaction term) |
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Reference
Boniface DR (1995) Experiment design and statistical methods for behavioural and social research. Chapman and Hall:London. (This book contains further details about computing degrees of freedom and also SS in balanced designs for terms in an ANOVA).