In the article „Does climate change affect period, available field time and required capacities for grain harvesting in Brandenburg, Germany?“ by A. Prochnow et al. (Agricultural and Forest Meteorology 203, 2015, 43–53), the authors calculate (simple) moving averages of the considered quantities („number of hours per year available for harvesting grain with defined moisture content“; coefficient of variation; total sunshine duration; etc.) for time periods of 30 years in steps of one year (i.e., 1961-1990, 1962-1991, …, 1984-2013) first. After that, they derive the trend of these averaged values and use these values to estimate the significance with the help of the „t-test for the regression slope“ (see their section 2.4 and Table 5 and Table 6). This way most of these trends are proofed to be significant with p
Essentially, you are right: The moving averages can be strongly dependent, especially locally in time, if your ordered sample/time series is small.
At the same time, if you have lots of data and you let your windows size (i.e., the size of time neighborhood) be comparably small (say, a constant times n^{1/5}), you can view your moving average as nonparametric kernel smoothing which is known to asymptotically consistent under appropriate regularity conditions. In this sense, taking moving averages and applying an adequate modification of the t-test (say, under strong mixing conditions) could be justified if your data are abundant.
In the particular paper you are referencing, the significance level is surely much higher than the authors claimed. So I do share your opinion that the trends they observed could be a pure accident.
In my opinion, the "best" solution here would be to apply an nonparametric regression method allowing for a consistent estimation of the first derivative of the conditional mean. By doing this, one could get a good idea of linear trends and hence make a good (local in time) forecast.
> I can not believe that four authors and two or three reviewers/referees did not recognize this pitfall!
I wouldn't be too surprised. Recently, my colleague was attending a rather high-level conference on statistics in US. One guy was giving a talk on climate change, and he made exactly the same mistake by applying a test to a time series (the amount of Arctic ice left) without accounting for possible dependencies among data.
Hi Michael,
thank you very much for your helpful answer. In the particular paper the window size is 30 (at a total of n = 53) and therefore much greater than n^0.2 = 2.2. Hence the p-value is much higher than 0.01 (as I proofed in my review, it is about 0.38).
Dear Klaus, I had not time to read thoroughly, but the fact that you have found many shortcomings in that paper tends to make me in agreement with you, since a correct work has to be tested and be correct all around the paper. Besides that, if I remember well, if we use moving averages, then the E(ei,ej)=0 condition does not hold. It is simple time series analysis.
