The log-linear model used in the analysis produces two sets of parameter estimates: one for the logged ratio of not dependent to ADL dependent, and one for the logged ratio of IADL dependent to ADL dependent. These equations produced from our analysis can be written as:
log (P1/P3) = 2.51 - 0.178 - 0.41R - 0.59A - 0.11A2 - 0.35P
log (P2/P3) = 0.44 + 0.718 - 0.02R - 0.20A - 0.07A2 - 0.15P
P2 = the probability of being independent;
P2 = the probability of being IADL dependent;
P3 = the probability of being ADL dependent;
S = sex (coded 1 if female and 0 if male);
R = race (coded 1 if nonwhite and 0 if white);
A = age group (coded -2 if 65-69, -1 if 7074, 0 if 75-79, 1 if 80-84, and 2 if 85 or over);
A2 = the square of the variable “A”; and
P = the percent of elderly in poverty (coded -1 if <8%, 0 if between 8-15% and 1 if >15%).
For ease of illustration, let
E1 = log (P1/P3) and E2 = log (P2/P3).
Taking the exponent of both sides of both equations yields
eE1 = P1/P3 and eE2 = P2/P3
As, by definition P1+P2+P3 = 1, the three equations can be solved simultaneously for P1, P2 and P3. The result is:
P1 = eE1/(1+eE1+eE2)
P2 = eE2/(1+eE1+eE2) and
P3 = 1/(1+eE1+eE2).