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I have conducted a variance partitioning in R with the vegan package to test how weather variables partition in the variance of the green-up day.

library(readr)
library(vegan)

data = read_csv('C:/path/data.csv)
out = varpart(data$greenup, ~ data$CAPE, ~ data$Temp, ~ data$precip, ~ data$WS)
out

Partition table:
                            Df R.square Adj.R.square Testable
[aeghklno] = X1              1  0.00873      0.00514     TRUE
[befiklmo] = X2              1  0.00357     -0.00004     TRUE
[cfgjlmno] = X3              1  0.03279      0.02929     TRUE
[dhijkmno] = X4              1  0.00029     -0.00333     TRUE
[abefghiklmno] = X1+X2       2  0.01935      0.01221     TRUE
[acefghjklmno] = X1+X3       2  0.03534      0.02832     TRUE
[adeghijklmno] = X1+X4       2  0.00878      0.00157     TRUE
[bcefgijklmno] = X2+X3       2  0.03652      0.02951     TRUE
[bdefhijklmno] = X2+X4       2  0.00621     -0.00102     TRUE
[cdfghijklmno] = X3+X4       2  0.03341      0.02638     TRUE
[abcefghijklmno] = X1+X2+X3  3  0.04303      0.03255     TRUE
[abdefghijklmno] = X1+X2+X4  3  0.02096      0.01024     TRUE
[acdefghijklmno] = X1+X3+X4  3  0.03547      0.02491     TRUE
[bcdefghijklmno] = X2+X3+X4  3  0.04023      0.02972     TRUE
[abcdefghijklmno] = All      4  0.04579      0.03181     TRUE

I would like to find if the variance partitioning of each variable is significant. I understand this is not feasible for the shared and residual parts.

Can I just look up critical values for Pearson R with 275 degrees of freedom at a 0.05 significance level? Or should I compute the p-value for each variable? If the latter, how?

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You cannot have P-values directly from the varpart table. However, the table has column Testable which is just that: it tells you that it is possible to construct an rda model that can be tested. See the function help page (?varpart) that explains how to do this. Read also the warning about trying to test all models simultaneously.

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