For each of the previous estimates of distance the mutational process was assumed to be the IAM. As discussed previously, microsatellites are more likely evolving via the SMM or TPM, and high levels of homoplasy exist. This homoplasy will lead to underestimation of the true genetic distance in all of the above distance estimates. Slatkin (1995) and Goldstein et al. (1995a) have developed equivalent distance measures which take into account the consequences of the SMM. In the SMM the alleles themselves contain information about the previous mutation events. For example, under the SMM, you can assume that two alleles that differ by three repeat units must have had at least three mutational events occur since their most common ancestor. As such,
where a is the number of repeats of an allele, 
is the number of
mutations that have occurred, and 
is the increment in repeat
length. The expected mean of the distribution of 
, i.e. 
, is
zero with a variance 
. This model directly relates to the
TPM mutational model
described by
Di Rienzo et al. (1994).
As mentioned previously, 
when there were no
mutations of large effect, i.e. the SMM.
Slatkin (1995) developed a genetic
distance measure based on the average sum of squares of the difference
in allele size. Average within population distance 
is
calculated by
where 
is the number of subpopulations j examined, n is
the sample size of each population, and 
is the number of
repeats of the ith allele in the jth subpopulation. Average
between population distance is
and the average distance for the entire population is
The first two measures, 
and 
, are equivalent to the
distance measures, 
and 
, developed by
Goldstein et al. (1995a).
A comparison of 
, 
, and 
was conducted by
Goldstein et al. (1995a) using
data generated via a SMM simulation. This study showed that both the
distance measures 
and 
asymptote faster than 
which remained linear for a longer period of time. However the measure
was not superior in all respects, as 
has a lower variance.
Goldstein et al. (1995a)
concluded that for relatively short periods of time 
or 
was a better measure, but as time increased 
would become
superior. When populations have only been separated for a short period
of time (
300 generations) the effect of mutation is minimal, most of
the genetic difference should be the result of drift, and the IAM based
estimates of distance are superior. As time increases (> 500
generations), mutations become an important force leading to genetic
differentiation, and a method that bases a distance on these events
becomes superior (Goldstein et al. 1995a; Forbes et al. 1995).
The estimation of generation times is based on a moderate population
size and a mutation rate of about 
.
Goldstein et al. (1995a) also
studied the length of time that 
is linear with time, under more
realistic conditions. This was achieved by imposing a size constraint
and limiting the range of microsatellite repeats at each loci. They
found that the greater the number of alleles a locus has the more
useful it will be for the study of distantly related populations.
Distance estimates from these loci are linear with time for a greater
amount of time. They found that estimates from some loci may be linear
for up to 1-2 million years. They point out that this difference in
linearity may be a problem when combining data from different loci, and
are currently working on a weighting scheme that may overcome this
drawback. However, they also note that 
is still linear with
time even when data from a number of loci were combined (by taking the
average 
), and that this measure is still useful in most
situations.
Slatkin (1995) investigated how 
performs under the TPM. He showed that 
is linear with time
even when the model allows for a low level of multi-step mutations.
Not surprisingly, as the model becomes more like the IAM, 
becomes progressively worse. Slatkin(1995) also cautions that if the mutation rate does change as the
number of repeats increases, this measure may no longer be linear.
A modification of the average sum of square distance method has
recently been made by Goldstein et al. (1995b). They have developed a measure 
that is independent
of population size when the populations are in a mutation-drift
equilibrium, and, like 
, is linear for longer periods of time
but has a lower initial variance than 
,
where 
and 
is the mean number of repeats found in the alleles of
population A and B respectively. An analysis of human
microsatellites shows the 
is still superior to 
for closely
related populations. However, the measure may be useful for the
resolution of the deeper ancestral nodes
(Goldstein et al. 1995b).