Fig. 4: Calculating soil response deviation from null model prediction and net interaction type classification for 150 multi-factor treatments.

a Treatments in the experimental design. Single factor treatments are shown in blue ovals. Each multi-factor treatment is shown by a red oval. The subscript of a multi-factor treatment indicates the component factors. b Interaction type classification workflow for multi-factor treatments. The workflow includes two parts: (1) estimating the joint response distributions of component factors of multi-factor treatments; (2) identifying the net interaction type for multi-factor treatments. For illustration purposes, one two-factor treatment (includes factor A and B) is taken as an example. In Step 1, we resampled from each control, single factor A and B treatment with replacement to generate \({C}_{i}\), \({T}_{{Ai}}\) and \({T}_{{Bi}}\). Then, in Step 2, mean values of each resampled treatment (\({c}_{i}\), \({t}_{{ai}}\) and \({t}_{{bi}}\)) are calculated. In Step 3, absolute effect sizes from control (\({Z}_{{ai}}\) and \({Z}_{{bi}}\)) for A and B single factor treatments are calculated. In Step 4, combined effect size of A and B (\({Z}_{i}\)) are calculated depending on different null model assumptions (additive, multiplicative or dominative). Then the control mean is added to \({Z}_{i}\) to generate predicted joint response (\({T}_{i}\)). Steps 1-4 are repeated \(K\) times to generate the distribution of the predicted joint response of factor A and B. Then in Step 5, we compared the actual joint response of factor A and B (\({T}_{{Mab}}\)) to the predicted response distribution. If the actual observation fitted within the 95% confidence intervals (CIs) of prediction distribution, then it was regarded as no net interaction. If it did not fit, then we calculated the rescaled Deviation from Null model prediction (DN). Then we classified the net interaction type based on the rescaled DN. c Visualization of the rescaled DN and net interaction types of 150 multi-factor treatments. d Statistical analysis of rescaled DN of soil response across factor dissimilarity index and in three different number of factor groups. Two-sided t-tests were performed for three factor groups. Asterisks represent the statistically significant difference of treatment group from zero (n = 50): ***P < 0.001, **P < 0.01, *P < 0.05. Boxplots showing the distribution of deviations across factor groups. The box spans the interquartile range (IQR) with the median indicated by the line inside. Whiskers extend to the minimum and maximum within 1.5 times the IQR. Outliers are shown as dots.