Can't find loglinear model's corresponding logistic regression model

Question

I have the loglinear model with parameters x, y, z, v, xy, xv, and z*v. As far as i understand there should exist a logistic regression model that essentially is equivalent to this, using v as response variable. How do i find it, and how does it look like?

I have tried to derive it using the relationship described in http://teaching.sociology.ul.ie/SSS/lugano/node58.html. I end up with the parameters x, z and x*y for the logistic regression model which turns out to be incorrect when testing in R.

I have also tried many other combinations of parameters in R but neither of the parameters in these models has the same values as the parameters in my loglinear model.

The loglinear model and it's results looks like:

Call:

glm(formula = n ~ x * y + x * v + v * z, family = poisson(link = log),

data = data41)

Deviance Residuals:

Min 1Q Median 3Q Max

-0.87421 -0.32788 0.08769 0.38924 1.64946

Coefficients:

 Coefficient      Estimate     Std. Error   z value       Pr(>|z|) 
(Intercept)        4.01862      0.11901    33.767         < 2e-16 ***
     x             -0.35889    0.16723    -2.146          0.03187 *
     y             -2.14736    0.04661    -46.068         < 2e-16 ***
     v              1.78281    0.12707     14.030         < 2e-16 ***
     z             -0.83773    0.17843     -4.695           2.67e-06 ***
    x:y            -0.40431    0.09936     -4.069           4.72e-05 ***
    x:v            -0.55058    0.16924     -3.253           0.00114 **
    v:z             3.32798    0.18425     18.062         < 2e-16 ***

---

Signif. codes: 0 '' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 20311.0677 on 15 degrees of freedom

Residual deviance: 7.7197 on 8 degrees of freedom

AIC: 115.69

Number of Fisher Scoring iterations: 4

The logistic regression model(using same data):

Call:

glm(formula = v ~ x + z + x * y, family = binomial(link = logit),

data = data41, weights = n)

Deviance Residuals:

Min 1Q Median 3Q Max

-15.7143 -8.4149 -0.6557 4.6727 9.6823

Coefficients:

  Coefficient  Estimate   Std.Error       z value   Pr(>|z|)
  (Intercept)   1.8298     0.1383         13.232    < 2e-16 ***
      x         -0.5058    0.1909         -2.650     0.00806 **
      z         3.3089     0.1846         17.922    < 2e-16 ***
      y        -0.5234     0.3058         -1.712     0.08693 .
     x:y        0.3586     0.5977          0.600     0.54854

---

Signif. codes: 0 '' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 1435.5 on 15 degrees of freedom

Residual deviance: 1084.4 on 11 degrees of freedom

AIC: 1094.4

Number of Fisher Scoring iterations: 7

I would expect that for example the parameter x*v in the loglinear model would have equivalent estimate and variance as the x parameter in the logistic regression model, however this is not the case.

I am thankfull for help!

Answer 1

Although every logistic regression model might have a corresponding log-linear model (Poisson regression with categorical variables), the converse doesn't necessarily hold.

Your models involves 4 variables, with v having 2 levels. Work backwards from the logistic regression model. According to the page you link (where $N$ = 4 in this case, counting all variables in the log-linear model):

The general rule is the model should contain the $N-1$ and lower order interactions between the independent variables, and for every term in the logistic model formula, an interaction between in and the dependent variable. If Y is the dependent variable, and A, B and C are independent, with an interaction B:C the loglinear model will have the following design:

Y + A + B + C + A:B + A:C + B:C + A:B:C   <-- Nuisance terms

+Y:A + Y:B + Y:C + Y:B:C               <-- Model terms

This is exactly the situation in your logistic regression, with Y corresponding to your v, the B:C interaction corresponding to your x:y interaction, and A corresponding to your z. So the log-linear model corresponding to your logistic regression model is:

 n ~ v + x + y + z + x:z + x:y + y:z + x:y:z + v:z + v:x + v:y + v:x:y, family = poisson(link = log)

Your log-linear model, when expanded to this form, lacks the x:z and v:y two-way interactions and both 3-way interactions, x:y:z and v:x:y.

Note the requirement for constructing the log-linear equivalent to a logistic regression: "the model should contain the $N-1$ and lower order interactions between the independent variables," where $N-1$ is actually the number of independent variables in the logistic regression. Thus a log-linear model equivalent to a logistic regression model will include all interactions among the independent variables of the logistic regression. If a log-linear model doesn't include all of those interactions, I don't know that you can construct an equivalent logistic regression.