In general, our TCN exposure ages and luminescence ages from any given site show good agreement and form approximately normal distributions (fig. S7). We calculated χR2 statistics for all sites (except Tolsta Head; see below). In several cases, the experimental χR2 statistic exceeded the value expected (at 95% confidence). We tested for outliers using a uniform-phase Bayesian sequence model in OxCal 4.2 (26) run in an outlier mode (62, 67) to assess for outliers in time (t). Sample sites are arranged in a pseudostratigraphical and spatial sequence of ice-marginal retreat (68), i.e., they are arranged in the order by which they would have been deglaciated (Fig. 1). This prior model is informed by the reconstruction of the flow signature of the MnIS at maximum conditions (17, 49) and mapping of the offshore moraines (Fig. 1). We include all of the data presented in this model with the exception of the site at Stoer as we were unable to confidently determine its relative position within the sequence, vis a vis the sites at Garrabost and Rainish, without reference to its age determinations. In addition to the 10Be and OSL chronology, the Bayesian model is constrained by a suite of radiocarbon dates from an organic horizon underlying till at Tolsta Head, east Lewis (see table S4) (69). These provide terminus post quem for advance of the MnIS towards its LGM limit. All data were assigned a prior probability of 0.05 (i.e., 1 in 20) for being an outlier. The model was set up to assess for outliers in time (t) and uses a Student’s t distribution to define how the outliers are distributed and a scale of 100 to 104 years.

The geochronological data produced a sequence (fig. S7) with overall agreement indices of ~35%, below the >60% threshold commonly applied (26, 68). The modeled posterior outlier probabilities are shown in table S4. To improve the coherence of the sequence model, we iteratively increased the individual prior probabilities of samples where the model gave a posterior probability that exceeded our assigned initial value of 0.05. The refined model produced a conformable sequence with an overall model agreement index of 136 (fig. S7). The results, including modeled boundary ages, are summarized in table S11. These modeled boundary ages document the timing of retreat of the MnIS. Notably, the samples that initially gave posterior outlier probabilities exceeding the initial assigned prior probability of 0.05 came from sample groups (North Rona, North Lewis, and North Raasay), where the experimental χR2 statistic exceeded the value expected at 95% confidence. Excluding these samples would yield an acceptable χR2 for these sites. In other sites where the χR2 statistic exceeded the acceptable value, the Bayesian outlier analysis did not identify any samples as outliers (e.g., Cape Wrath). We attribute this to the fact that χR2 calculations use the analytical uncertainties of the surface exposure ages. Because our full retreat sequence also includes radiocarbon and OSL age determinations, we modeled the geochronology using the full, external uncertainties of the ages. As a test, we used the quality control criteria (52) to assign prior outlier probabilities to all geochronological ages with our dataset. In these cases, ages were manually removed until an acceptable χR2was obtained. This sequence shows good coherence (A = 227) but, more importantly, is in good agreement with our refined-age model. Note that, for discussion purposes, we use our initial refined-age model as it retains a larger number of geochronological dates and thus, we believe, yields a more conservative assessment of the associated uncertainties.

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