Hi Sara,

I am not sure whether this helps, because I cannot check what Mathematica 5 does. I am running version 10.0.2. The syntax of your command is incorrect, but I would require more information to fix it. There are several problems with the function. For example the Poisson for the exponential family needs to go in quotation marks. Also it appears that the input data is in the wrong format. I am not sure which variable is supposed to be explained by which other variables.

One way of making it work is:

GeneralizedLinearModelFit[
 Transpose[{{1, 8, 25, 13, 5, 4, 1, 1, 2, 2, 3, 3, 1, 4, 3, 4, 5, 5, 
    0, 4, 0, 3, 2, 1}, {3418, 5213, 4882, 4994, 4359, 4360, 4159, 
    4568, 3232, 4168, 4232, 3987, 4760, 6692, 4703, 3980, 5187, 4430, 
    1135, 2007, 1736, 1792, 1759, 1618}, {1121, 1546, 1846, 1826, 
    1519, 1538, 1158, 1334, 1433, 982, 1278, 995, 214, 332, 545, 581, 
    396, 526, 175, 351, 262, 293, 287, 322}, {0, 0, 1, 1, 0, 0, 0, 0, 
    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}], {x, y, z}, {x, 
  y, z}, ExponentialFamily -> "Poisson"]

I would need to know more about what means what in this example to give a reasonable answer.

Best wishes, Marco