18 BIOLOGICAL EFFECTS OF LOW LEVEL EXPOSURES
Figure 1.4. Statistical “weights” vs time data used for the procaine-treated group in the
nonlinear least-squares analysis illustrated in Figure 1.3. Weights (estimates
of the reciprocal of variance) were calculated from Equation 16. See Sacher2
and Boxenbaum et al.14 for further discussions of Gompertz plot weighting. In
unweighted least-squares analysis, one attempts to minimize the sum of
squared deviations by parameter adjustment (iteration). In weighted
least-squares analysis, it is the weighted sum of squared deviations that one
attempts to minimize. The weighting procedure is used to adjust for the
magnitude and “ clout” of each dependent variable. See Boxenbaum et al.75
and Daniel and Wood80 for general discussions.
mesis can be gender specific. Figure 1.4 illustrates statistical weights used in
the regression analysis for the procaine treated population. In conventional
(unweighted) least squares analysis, it is assumed (usually tacitly) that vari
ance estimates about each data point are equal.80 However, in Gompertz
analyses, particularly those employing cohorts, this is hardly ever the case.
Therefore, it is best to weight the dependent variable in accordance with an
estimate of the reciprocal of its variance.81 Taking cognizance of this,
Sacher 2 developed relationships to estimate the sampling variance:
(15)(16)where Nj = the number of survivors at the beginning of an age
interval
dj = the number of deaths over that same interval
w = the statistical weight
v = varianceTime is taken as the midtime of the interval (as is ftx).
For studies in which survival data are not reported but mean and/or
median survival times are, it is not possible to unequivocally explain obser
vations of longevity enhancement. Consider Figure 1.5, which illustrates the