**Subsections**

## Consensus nucleotide calculation

The consensus nucleotide calculation is performed following the method described in [Hiatt et al., 2013]. The consensus base is chosen so that the posterior probability of the observed read bases is maximized.

In order to maximize the posterior probability of calling a base, i.e.,

where O_{i} is the observed base at a given position, C the base in question, and where all possible bases are summed up in the denominator, e.g. B=A,T,C,G.

Assuming that the prior for observing any base is equal, i.e., P(A)=P(T)=P(C)=P(G), then the posterior probability is:

And by assuming each read base observation is independent,

To obtain the consensus base we only need to maximize the numerator.

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Consensus Q-score

The Hiatt Q-score is of the probability of making a wrong call, i.e.

which means that the Hiatt Q-score is

Q-scores are capped at 60.

The probabilistic model outlined above and used in the Hiatt Q-score, assumes the only source of errors are independent sequencing errors. While PCR errors are typically rarer than sequencing errors, PCR errors are not independent and they can affect a large fraction of the reads in an UMI group. For this reason, the Hiatt quality scores will often attain the maximum value of 60, even in situations where the reads constituting the UMI group do not unanimously agree on the base call.

The Fixed Ploidy and Low Frequency Variant Detection tools both rely on statistical models for the sequencing error rates, which is estimated for each value of the Q-score and each substitution type, for details see http://resources.qiagenbioinformatics.com/manuals/clcgenomicsworkbench/current/index.php?manual=Variant_Detection_error_model_estimation. If most quality scores are 60, the variant callers can not differentiate between, i.e. reads with unanimous agreement and reads without, or between small groups with unanimous agreement and large groups with unanimous agreement.

MAGERI Q-scores does not have a probabilistic interpretation as Hiatt Q-scores, but they a more distributed in the set of possible Q-scores, allowing the variant callers to differentiate between qualities. The MAGERI Q-scores is an adaption of the method described in [Shugay et al., 2017].

First, the frequency, f, of the consensus base is computed, only bases with a Q-score above 25 contribute to the frequency computation. A pseudo-count is applied to the denominator, so that larger groups automatically get higher Q-scores:

The MAGERI Q-score is then computed as