Phillips, D.L. and J.W. Gregg. 2001. Uncertainty in source partitioning using stable isotopes. Oecologia 127:171-179.

Stable isotope analyses are often used to quantify the contribution of
multiple sources to a mixture, such as proportions of food sources in an
animal’s diet, C_{3} vs. C_{4} plant inputs to soil
organic carbon, *etc*. Linear mixing models can be used to partition
two sources with a single isotopic signature (*e.g*., d^{13}C)
or three sources with a second isotopic signature (*e.g*., d^{15}N).
Although variability of source and mixture signatures are often reported,
confidence interval calculations for source proportions typically use only
the mixture variability. We provide examples showing that omission of
source variability can lead to underestimation of the variability of
source proportion estimates. For both two and three source mixing models,
we present formulas for calculating variances, standard errors (SE), and
confidence intervals for source proportion estimates that account for the
observed variability in the isotopic signatures for the sources as well as
the mixture. We then performed sensitivity analyses to assess the relative
importance of: (a) the isotopic signature difference between the sources,
(b) isotopic signature standard deviations (SD) in the source and mixture
populations, (c) sample size, (d) analytical SD, and (e) the evenness of
the source proportions, for determining the variability (SE) of source
proportion estimates. The proportion SE’s varied inversely with the
signature difference between sources, so doubling the source difference
from 2l to 4l
reduced the SE’s by half. Source and mixture signature SD’s had a
substantial linear effect on source proportion SE’s. However, the
population variability of the sources and the mixture are fixed and the
sampling error component can be changed only by increasing sample size.
Source proportion SE’s varied inversely with the square root of sample
size, so an increase from 1 to 4 samples per population cut the SE in
half. Analytical SD had little effect over the range examined since it was
generally substantially smaller than the population SD’s. Proportion SE’s
were minimized when sources were evenly divided, but increased only
slightly as the proportions varied. The variance formulas provided will
enable quantification of the precision of source proportion estimates.
Graphs are provided to allow rapid assessment of possible combinations of
source differences and source and mixture population SD’s that will
allow source proportion estimates with desired precision. In addition, an
ExcelÓ spreadsheet to perform the calculations
for the source proportions and their variances, SE’s, and 95% confidence
intervals for the two source and three source mixing models can be
accessed at http://www.epa.gov/wed/pages/models.htm.