By P. McCullagh, J. A. Nelder (auth.)
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32 AN OUTLINE OF GENERALIZED LINEAR MODELS A convenient feature of generalized linear models is that they have a simple starting procedure necessary to allow the iteration to get under way. This consists of using the da ta themselves as the first estimate ofPo and from this deriving ~o, (dl1/ dJl)o and ~. Adjustments may be required to the da ta to prevent, for example, our trying to evaluate In (0) for the log link applied to counts when the datum value is zero. These will be described under the appropriate chapters, as will various complexities sometimes associated with the convergence of the iterative process.
In which area = constant x length x breadth), with length and breadth being measured as weil as area. 3) where C is the logarithm of the shape constant. Hence 1/ = PO+PlXl +P2X2+Pa(C+X l +X2) = Po + Pa C+ (fJl + Pa)X l + (fJ2 + Pa)X 2' Thus we can distinguish the three quantities Po + Pa C, Pl+ Pa, P2 + Pa' but not the four components Po, Pl' P2' Pa separately. 3) we have Xa as a linear combination of X O, Xl and X 2, so that the subspace for X a (with one dimension) is contained in the sum of the subspaces for X O, Xl and X 2• Notice that not only have we assumed that the leaves are of the same shape in the sense that the area is proportional to length x breadth, but we have also assumed that length and breadth show some independent variation between leaves.
If such a set is denoted by Xl' X 2, ... , Xp , the xs being vectors of length N, the number of units, then the xs can be thought of as defining p directions in an N-dimensional Euclidean space. e. ifno quantities Sj,j = 1, .. p, not all zero, can be found such that If k such independent linear relations exist then the X vectors span a subspace of dimension p-k. Generally the individual tenns in a model fonnula willfonn subspaces ofmaximum dimensionality, with loss of dimensionality occurring only when we consider the joint subspaces covered by more than one tenn.
Generalized Linear Models by P. McCullagh, J. A. Nelder (auth.)