inferred positions and numbers of those changes are affected by the thoroughness of the
taxon sampling. Sanderson (1990), for example, demonstrated that decreased taxon
sampling oftenleads to a dramatic decrease in the estimates of branch lengths. In the
terminal clades in our tree, taxon sampling is sparse and families such as Poaceae,
Cyperaceae, Rutaceae, and Araliaceae are only represented by a few accessions. We
would expect an extended sample of these groups to have the effect of pushing their age
estimates closer in line with the fossil-based estimates.
Sources of errorErrors in analyses of this kind arise from several sources, including: (1) noise introduced
from the stochastic nature of the substitution process; (2) rate variations that invalidate
the assumptions of the method; (3) the calibration point obtained from the fossil record;
and (4) use of an incorrect tree. There is, however, no reasonable way of quantifying the
effect they have all taken together. Sanderson and Doyle (2001) recently addressed this
issue when they analysed the effect of various factors on point estimates for the age of
angiosperms. They also conducted a series of resampling experiments designed to provide
a statistical estimate of the relative magnitude of errors due to these factors. Their
analyses clearly indicated that several factors may introduce large errors in our divergence
time estimates (Sanderson and Doyle 2001). However, all their point estimates of the
angiosperm age were based on strict molecular clock models, and it is not clear how
methods trying to deal with unequal rates such as NPRS would have been affected by the
factors analysed.
Following Sanderson (1997, 1998), we estimated errors introduced from the stochastic
nature of the substitution process using a bootstrap resampling procedure (Efron and
Tibshirani 1993). The bootstrap estimates of standard error resulting from these analyses
are comparatively small (on average c. 5 myr), indicating that stochastic errors can be
reduced by including sufficient data (Wikström et al. 2001).
Perhaps a more serious source of error involves rate variations and an inability to infer
shifts in substitution rates correctly (Sanderson 1997). If any amount or any type of
change is allowed, the estimates from NPRS analyses will be associated with large errors
(Sanderson 1997). There is simply no way to avoid making assumptions about the nature
of both rate changes and rates themselves. The NPRS approach allows substitution rates to
change and assumes that such changes are autocorrelated, but there is not much empirical
support for this assumption (but see Harvey et al. 1991, for a discussion of autocorrelation
and heritability of cladogenesis). A proper evaluation would require knowledge of
absolute rates, which itself requires knowledge of absolute divergence times (Springer
1995). The assumption is, however, intuitively reasonable, and a different examination of
its validity could perhaps be accomplished by looking at how rate changes are inferred and
by trying to corroborate these changes through different kinds of analyses.
Our choice of calibration point may incorporate errors into our analyses, and as
discussed above, our choice may be too conservative. If Antiquacupula and Protofagaceae
were shown to have a more derived position within the Fagales clade, or if we were to
recalibrate our results using Normapolles pollen records, all our estimates would become
older.
162 ANGIOSPERM DIVERGENCE