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# How taxonomic bias affects abundance measurements {#abundance-measurement}

## Absolute abundance

Here we consider some practical applications of this method and how their measurements are affected by bias in the MGS measurement and error in $\widehat{\text{abun}}_{r}(a)$.
Appendix [REF] describes the general case where there is a set $R$ of multiple reference species; here we restrict our attention to a single reference species $r$.

### Leveraging information about a reference species

In practice, researchers commonly use a less-direct but equivalent approach to measure species abundances from a reference such as a spike-in.
Set $S$ to be the species of interest excluding the spike-in species $r$.
In this approach, one measures the total abundance of the community $S$ from the ratio of non-spike-in to spike-in reads,
\begin{align}
  (\#eq:total-density-spike)
  \widehat{\text{abun}}_S(a)
  &= \frac{\text{reads}_S(a) \cdot \widehat{\text{abun}}_R(a)}{\text{reads}_r(a)}.
\end{align}
Then, one uses this measurement $\widehat{\text{abun}}_{S}(a)$ along with the MGS proportions in Equation \@ref(eq:density-prop-meas) to determine the abundance of individual species.
This calculation yields measurements that are identical to those from directly applying Equation \@ref(eq:density-ratio-meas) (Appendix \@ref(total-density-ref)).
This observation might seem paradoxical: Equation \@ref(eq:density-ratio-error) indicates proportional error for the direct approach, whereas \@ref(eq:density-prop-error) suggests non-proportional errors for the indirect approach.
This apparent contradiction is resolved by the fact the measurement $\widehat{\text{abun}}_{S}(a)$ by this method has errors that are proportional to $\text{efficiency}_S(a)$ and thus exactly offset the proportionality with $1/\text{efficiency}_S(a)$ in the MGS proportions, such that the measured species abundances have constant fold errors.


## Summary

Taxonomic bias in MGS measurements makes all MGS-based measures of relative and absolute abundance inaccurate.
For some measures---but not all---consistent bias at the species level creates constant FEs in species abundances.
Of two types of relative abundance, the ratio between a pair of species has constant FEs, whereas the proportion of an individual species does not. 
Instead, the FE a species' proportion varies with the mean efficiency across samples.
Methods for measuring species absolute abundance that involve measuring the abundance of the total community likewise have FEs that vary with the sample mean efficiency; 
a notable exception is if the mean efficiency of the total-abundance measurement mirrors that of the MGS measurement.
Absolute-abundance methods that instead involve measuring or spiking individual reference species have constant FEs provided that the abundance of the reference species can be determined up to a constant FE.
The efficiency of higher-order taxa may vary across samples even if the efficiencies of their constituent species are constant; as a result, all standard abundance measures can yield variable FEs at taxonomic resolutions higher than that at which bias is conserved.


# How bias affects DA results {#differential-abundance}

The previous section showed that for some methods of measurement, taxonomic bias creates constant FEs in the (relative or absolute) abundance of a species; but in others, bias creates FEs that vary inversely with the mean efficiency of the sample.
This section describes how these constant or varying FEs affect DA results, finding conditions for significant inferential errors.

The fact that some measures have constant FEs while others do not suggests that these DA analyses based on these measures will be more or less robust to bias.
Here we rigorously evaluate this idea / explore in detail, finding precise conditions for significant error in DA results.