text: The computed*f*2-statistics are fed into ADMIXTOOLS270 to com- 
  pute derived statistics. ADMIXTOOLS2 implements computation of 
  genome-wide*f*2-,*f*3- and*f*4-statistics, as well as qpgraph and qpAdm 
  models. We implement the sample size correction as detailed in ref. 21. 
  The*f*2-statistics are computed in blocks, typically of prespecified cen- 
  timorgan size or of prespecified physical distance. These blocks are 
  used downstream in ADMIXTOOLS2 to compute standard errors using 
  a block-jackknife approach. By default, we compute*f*-statistics only 
  on internal branches and exclude singleton tip branches to increase 
  robustness against sample age. 
  
  ***f*4-ratio admixture simulation with genotype and phasing errors.**
  We emulate the data quality we expect in imputed ancient genomes 
  (Extended Data Fig. 3b). We implement a simple error model in which 
  every haploid genotype at any segregating site can switch with a certain 
  error probability. We can theoretically compute the predicted squared 
  correlation coefficient (*r*2) between the true simulated genotypes and 
  the genotypes that include error, stratified by minor allele frequency, 
  to generate a plot similar to those used for evaluating imputation accu- 
  racy using downsampled high-coverage ancient genomes72 (Extended 
  Data Fig. 3a). As imputation accuracy varies for each individual in real 
  settings, we randomly sample the error probability for each individual 
  uniformly between 1 × 10−4 and 1 × 10−3 (errors per SNP per haplotype). 
  This yields*r*2 curves that are comparable to those observed in real data. 
  We additionally simulate a high error case, for which we sample error 
  probabilities between 1 × 10−3 and 1 × 10−2. 