Statistical analysis
The prevalence of single-cell protists was determined by counting any PCR positivity at any level. Their presence and quantity were then modeled using generalized estimating equations (GEE) with the subject as the grouping variable, time point (factor with three levels—0, 3 and 6 months) and a first-order autoregressive covariance structure; the predictors were the study allocation (placebo or probiotics) and whether the sample was taken while on intervention with the live mixture.Bacteriome alpha (within-sample) diversity was assessed from the unfiltered rarefied dataset, agglomerated at the genus level, using the observed counts, Chao1, ACE, Shannon, Simpson, inverse Simpson and Fisher indices. The association of alpha diversity indices with the presence or quantity of For the beta diversity (between samples) analysis, Bray-Curtis distance was calculated on Hellinger-transformed abundance data agglomerated at the taxonomic level of the genus, and ordination was performed using metric multi-dimensional scaling (MDS) and visually inspected. Associations of Using GEE, individual bacterial taxa were tested for associations with the positivity of the two abundant protists (The commented R Markdown code and output of the statistical analysis are shown, along with the session information, as Additional file 