Extended Data Fig. 2: The effect of time series duration on turnover rates (change in dissimilarity/yr) and the statistical challenges when time series are compared.

a) Duration affects turnover rates partly because there is a 0-1 constraint on dissimilarity, such that longer duration time series (blue) are constrained to a shallower slope than shorter duration time series (green). b) Turnover rates show strong heteroskedasticity with higher variance and faster rates among shorter time series. The red line shows mean turnover rate estimated from LOESS smoothing. c) Temperature changes (°C/yr) also showed strong heteroskedasticity with higher variance among shorter time series. The red line shows a fit from LOESS smoothing. d) Slopes calculated from Gaussian white noise time series also show strong heteroskedasticity with higher variance among shorter time series. The durations of the white noise time series matched the durations in the species composition dataset. The red line shows a fit from LOESS smoothing. e) A comparison of Type I (false positive) error rates shows that one-stage (i.e. fit directly to dissimilarities) generalized linear mixed models (GLMMs) with ordered beta errors have an acceptably low false positive rate when time series of different durations are analyzed together, while other common analytical methods (Pearson correlations of time series slopes, meta-analysis of time series slopes, or one-stage mixed effect models with Gaussian errors fit to time series data) have unacceptably high false positive rates if time series differ in duration (range of durations > 0). All methods have low false positive rates when time series are all the same duration (range of durations = 0). Data are presented as means with error bars for the 95% binomial confidence bounds. f) Example of a time series with a negative turnover rate. Data are demersal marine taxa from the Northeast Fisheries Science Center Bottom Trawl Survey. Beta regression trend line is shown with shading for +/− one standard error.