Parameterization
Parameterization is the process of transforming a variable with levels so as to estimate their differences.
Reference Parameterization
An arbitrary level, conventionally the first, is singled out as a reference. The effect of this reference level is implicitly the intercept. All other levels receive a distinct parameter, indicating a difference from the reference.
For a variable with k levels, there are k-1 fitted parameters coded as follows:
Case Number |
Parameterized Variable |
Dummy 1 |
Dummy 2 |
Dummy 3 |
1 |
A |
0 |
0 |
0 |
2 |
B |
1 |
0 |
0 |
3 |
C |
0 |
1 |
0 |
4 |
D |
0 |
0 |
1 |
The consequence of this coding is that the significance tests of parameters cannot be interpreted directly. A level may not be significantly different from the reference, but still be significantly different from other levels. Joint tests must be additionally computed.
Effects Parameterization
The first level is coded as the absence of all others as:
Case Number |
Parameterized Variable |
Dummy 1 |
Dummy 2 |
Dummy 3 |
1 |
A |
-1 |
-1 |
-1 |
2 |
B |
1 |
0 |
0 |
3 |
C |
0 |
1 |
0 |
4 |
D |
0 |
0 |
1 |
The average effect is the intercept, and each level has a distinct calculable effect. A positive effect is therefore a greater than average effect.