I am supposed to fit a logistic regression model and the find the log- linear model which correspond to it, fit that model and show the correspondence between parameters. But it is not working, I am not getting things right:
I have a 2x2x2x2 contingency table. I should handle the data as if it was a product of 8 binomial distributed RVs:
let X = 0 if (<30), X = 1 if (+30). Y = 0 if (<5) , Y = 1 if (+5) , Z = 0 if (<260),Z=0 if (+260). V= 0 (alive) , V=1 (dead).
The logistic regression model is(model the probability of V=0, with three explanatory variables):
$log\dfrac{P(V=0\mid X=x,Y=y, Z=z)}{P(V=1\mid X=x,Y=y, Z=z)} = \alpha + \beta^X x + \beta^Y y + \beta^Z z $
The ML estimates (by SAS-software) are;
$ \hat{\alpha} = 1.8139 , \hat{\beta}^X = -0.4675, \hat{\beta}^Y = -0.4228, \hat{\beta}^Z = 3.3094$.
Okey, the corresponding log linear model is $(VX, VY, VZ)$:
$log\dfrac{P(V=0\mid X=x,Y=y, Z=z)}{P(V=1\mid X=x,Y=y, Z=z)} = log [\dfrac{\mu_{x,y,z,0}}{\mu_{x,y,z,1}}] = log(\mu_{x,y,z,0}) - log(\mu_{x,y,z,1})= (\lambda + \lambda^X_x + \lambda^Y_y + \lambda^Z_z +\lambda^V_0 + \lambda^{XV}_{x0} +\lambda^{YV}_{y0} +\lambda^{ZV}_{z0} ) - (\lambda + \lambda^X_x + \lambda^Y_y + \lambda^Z_z +\lambda^V_1 + \lambda^{XV}_{x1} +\lambda^{YV}_{y1} +\lambda^{ZV}_{z1} ) = (\lambda^V_0 -\lambda^V_1) + (\lambda^{XV}_{x0} - \lambda^{XV}_{x1})+ (\lambda^{YV}_{y0} - \lambda^{YV}_{y1}) + (\lambda^{ZV}_{z0} - \lambda^{ZV}_{z1})$
In the last equality we first have the constant term ($\lambda^V_0 -\lambda^V_1$) that should correspond to $\alpha$ in the logistic model , and then we have the term that depends on only on $X$ and so on. Now after fitting this model and then comparing parameters with the logistic regression model nothing seems right.
So my question is: am I doing all things right here (as above) ? I just want to confirm with you so that i know I am doing something wrong with the "fitting process" instead of doing something wrong in the theoretical part.
