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Regression - Equality of Slopes Model




For testing equality of intercepts, the coefficients on the dummy variables test for equal intercepts. 

For equality of slopes, we need the interaction between the dummy variable and the explanatory variable whose slope (coefficient) is of interest.
 
 
For only two groups, we could use a single two-level dummy variable D
D = 0 is the reference level (group)
D = 1 is the other level (group)

 
 
Equality of intercepts
 
y = b0 + b1*x + b2*D
 
If D = 0, then y = b0 + b1*x
If D = 1, then y = b0 + b1*x + b2   ......   group like terms: y = (b0 + b2) + b1*x
 
If coefficient b2 = 0, then we might fail to reject the null hypothesis that the intercepts are equal
If coefficient b2 <> 0, then we would reject the null hypothesis that the intercepts are equal
 
 
Equality of slopes model
 
y = b0 + b1*x + b2*D + b3*x*D   
 
(we added the interaction between x and D)
 
 
If D = 0, then y = b0 + b1*x
If D = 1, then y = b0 + b1*x + b2 + b3*x  ......   group like terms: y = (b0 + b2) + (b1 + b3)*x
 
If coefficient b3 = 0, then we might fail to reject the null hypothesis that the slopes are equal
If coefficient b3 <> 0, then we would reject the null hypothesis that the slopes are equal
 
 
For a model with three groups, assuming that lm / glm / etc. would really do this for you, the explicit dummy variable coding might look like:
 
                 D1      D2
group 1       0         0    (reference level ... can usually choose)
group 2       1         0
group 3       0         1
 
I believe that this is called a sigma-restricted model, as opposed to an overparameterized model where three groups would have three dummy variables.
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