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Comments
FYI if I recall correctly linear regression is said to be "robust" to violations of the normality assumption, meaning that even when it is violated the coefficients and their hypothesis tests are OK. (This is unlike the homogeneity of residuals assumption, violations to which the regression is NOT robust.)
Also checking assumptions with hypothesis tests (here, the Shapiro-Wilk test) has the same problems associated with any hypothesis test, e.g. likelihood of significance depends on your sample size. A visual check of the QQ plot (residual vs cumulative normal distribution) should be sufficient. (Note that the normality assumptions relates to the residuals, not the distribution of your dependent variable)
I am not sure what nonparametric test would be best. But I think the simplest nonparametric test would be to code your data as "larger" or "smaller" than the previous, and then do a binomial test!
EJ
A more informative nonparametric test could be to transform the sizes into ranks. You then do the linear regression on the ranks (for instance). Transforming to ranks would probably be a good way to start. Note that I am making this up as I go along, but if you Google you will find something similar I'm sure. Please keep us posted!
EJ
Also, the JASP Time Series module offers a test of trend stationarity. From the help file:
"Kwiatkowski-Phillips-Schmidt-Shin: Computes a KPSS test where the null assumes that the time-series is level or trend stationary. The p-values are interpolated from a table of critical values in Kwiatkowski et al. (1992), when the statistic lies outside the range of critical values, a note is added to the table."
Oh that KPSS test is actually not what you need, because it tests whether the series is stationary around a trend
This looks like the standard test: https://help.healthycities.org/hc/en-us/articles/233420187-Mann-Kendall-test-for-trend-overview
It feels similar to doing a linear regression on the ranks, but perhaps it is different
E.J.
sorry for the deluge of shorts posts --- I keep discovering new issues. The Mann-Kendall test assumes that the observations are independent, which I don't think is the case (there is autocorrelation, even apart from any trend). So right now I am thinking that some sort of ARIMA model + trend is needed. This is parametric but it seems to me there is no way around it. I might change my mind though :-)
EJ
Last post on this for a while, promise. check out https://robjhyndman.com/hyndsight/arima-trends/index.html. This seems a reasonable idea.
EJ
Thanks so much for all your suggestions! I really appreciated it.
Now I'll try to figure out how to go on...
If indeed "all dens are different," then I think it would be appropriate to simply compute either Kendall's tau-b or Spearman's rho to assess the rank correlation between year and den height.
R