Sum-to-zero constraints in a two-way ANOVA model
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I'm reviewing my lecture notes on sum-to-zero constraints and I am having a tough time understanding the concept.
Say the constraints are $sum_{i=1}^{a} alpha_i = 0$, $sum_{j=1}^{b} beta_j = 0$, $sum_{i=1}^{a} (alpha beta)_{ij} = 0$ for $j = 1,...,b$ and $sum_{j=1}^{b} (alpha beta)_{ij} = 0$ for $i = 1,...,a$.
Here are the following questions I have:
1) Why does this represent $a + b + 1$ independent constraints and not $a + b + 2$ independent constraints?
2) How do I determine that there are $a-1$ linearly independent $alpha_i$'s, $b-1$ linearly independent $beta_j$'s and $(a-1)(b-1)$ linearly independent $(alpha beta ) _{ij}$'s?
Thanks in advance.
linear-algebra statistics
edited Nov 26 at 22:37
Davide Giraudo
124k16150258
asked Nov 18 at 8:16
angleofaxis
65
add a comment |
up vote
1
down vote
favorite
I'm reviewing my lecture notes on sum-to-zero constraints and I am having a tough time understanding the concept.
Say the constraints are $sum_{i=1}^{a} alpha_i = 0$, $sum_{j=1}^{b} beta_j = 0$, $sum_{i=1}^{a} (alpha beta)_{ij} = 0$ for $j = 1,...,b$ and $sum_{j=1}^{b} (alpha beta)_{ij} = 0$ for $i = 1,...,a$.
Here are the following questions I have:
1) Why does this represent $a + b + 1$ independent constraints and not $a + b + 2$ independent constraints?
2) How do I determine that there are $a-1$ linearly independent $alpha_i$'s, $b-1$ linearly independent $beta_j$'s and $(a-1)(b-1)$ linearly independent $(alpha beta ) _{ij}$'s?
Thanks in advance.
linear-algebra statistics
edited Nov 26 at 22:37
Davide Giraudo
124k16150258
asked Nov 18 at 8:16
angleofaxis
65
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm reviewing my lecture notes on sum-to-zero constraints and I am having a tough time understanding the concept.
Say the constraints are $sum_{i=1}^{a} alpha_i = 0$, $sum_{j=1}^{b} beta_j = 0$, $sum_{i=1}^{a} (alpha beta)_{ij} = 0$ for $j = 1,...,b$ and $sum_{j=1}^{b} (alpha beta)_{ij} = 0$ for $i = 1,...,a$.
Here are the following questions I have:
1) Why does this represent $a + b + 1$ independent constraints and not $a + b + 2$ independent constraints?
2) How do I determine that there are $a-1$ linearly independent $alpha_i$'s, $b-1$ linearly independent $beta_j$'s and $(a-1)(b-1)$ linearly independent $(alpha beta ) _{ij}$'s?
Thanks in advance.
linear-algebra statistics
edited Nov 26 at 22:37
Davide Giraudo
124k16150258
asked Nov 18 at 8:16
angleofaxis
65
I'm reviewing my lecture notes on sum-to-zero constraints and I am having a tough time understanding the concept.
Say the constraints are $sum_{i=1}^{a} alpha_i = 0$, $sum_{j=1}^{b} beta_j = 0$, $sum_{i=1}^{a} (alpha beta)_{ij} = 0$ for $j = 1,...,b$ and $sum_{j=1}^{b} (alpha beta)_{ij} = 0$ for $i = 1,...,a$.
Here are the following questions I have:
1) Why does this represent $a + b + 1$ independent constraints and not $a + b + 2$ independent constraints?
2) How do I determine that there are $a-1$ linearly independent $alpha_i$'s, $b-1$ linearly independent $beta_j$'s and $(a-1)(b-1)$ linearly independent $(alpha beta ) _{ij}$'s?
Thanks in advance.
linear-algebra statistics
linear-algebra statistics
edited Nov 26 at 22:37
Davide Giraudo
124k16150258
asked Nov 18 at 8:16
angleofaxis
65
edited Nov 26 at 22:37
Davide Giraudo
124k16150258
asked Nov 18 at 8:16
angleofaxis
65
edited Nov 26 at 22:37
Davide Giraudo
124k16150258
edited Nov 26 at 22:37
Davide Giraudo
124k16150258
edited Nov 26 at 22:37
Davide Giraudo
124k16150258
124k16150258
asked Nov 18 at 8:16
angleofaxis
65
asked Nov 18 at 8:16
angleofaxis
65
asked Nov 18 at 8:16
angleofaxis
65
65
add a comment |
add a comment |
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