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I have annual temperature data from a variety of weather stations in the Caribbean and I want to be able to show statistically that the trends for each station are significant, either positive or negative or no trend.

I can obviosly plot the data and fit a line, but I want to make a stronger argument than just eyeballing it. I was thinking about a runs's test or a Mann-Kendall test. Most everything I've read talks about ARIMA models, concernes about autocorrelation, etc. but I feel like those are for econometric predictions. I'm not trying to predict anything. I just want to be able show whether or not the increase or decrease in temperature is due to random chance, or not. (my n=49) I may have larger n values for other stations, but for right now, working with 1 station, I have 49 observations. It would be nice to be give some idea of how far above or below random chance it is.

I don't know this helps, but I actually have monthly data, but to avoid periodicity issues I've so far been playing around with July values to represent N. Hemisphere summers.

In terms of hypothesis testing, if the Caribbean is not warming, then none of the weather stations would show increasing trends in temperature.

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    $\begingroup$ If you're going to step into the climate change controversies, you had better be prepared to do excellent statistical work, or you will be slaughtered by the critics. In this case such work has to include (among other things) proper assessment of temporal correlation and careful justification of the time ranges chosen for analysis. That's why you need ARIMA models or their equivalent and that's why a simple runs test or M-K is not going to pass muster. $\endgroup$ – whuber Feb 14 '13 at 19:22
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    $\begingroup$ This is just for a class so I don't need to worry too much about critics. I can justify the time ranges to my professor. Its the assessment of temporal correlation that has me bothered. Since I'm not doing any predicting, I just would need to show that my ARIMA model for the observed data is accurate, and that the ARIMA model contains an increasing, or decreasing trend. Effectively I would need to detrend the model, but in doing so, I'm justifying that there was a trend to begin with? $\endgroup$ – Noah Feb 14 '13 at 21:31
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    $\begingroup$ Not at all: detrending works even in the absence of trends; this operation is not an implicit admission that there's any trend. BTW, I added the homework tag to this question to reflect its classroom nature--that ought to help deflect other comments about real-world applications. $\endgroup$ – whuber Feb 14 '13 at 21:40
  • $\begingroup$ So maybe this is where I need to do more reading. How does an ARIMA model indidcate any trend is significant? Maybe I am confusing decompostion and detrending. $\endgroup$ – Noah Feb 14 '13 at 22:05
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    $\begingroup$ I'm not necessarily advocating ARIMA; what is most important from your point of view is the assumption of independence of the regression residuals. Yours likely have strong serial correlation. That could completely change the hypothesis tests concerning significance of the slope; no matter what procedure you use, it would be invalid to ignore such correlation. ARIMA is one of many methods to model and explicitly account for serial correlation. $\endgroup$ – whuber Feb 14 '13 at 22:30
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I use a method for temperature trends that relies on actual measurements rather than computed values such as monthly or annual means. First I de-diurnalize and de-seasonalize the data with a general linear model against dummy coded months and times of day. The residuals of this regression form the deseasonalized series in which I look for long term trends with a simple OLS linear regression against time measured in years. The regression coefficient for time is usually very small and it explains a very small portion of the sum of squares; so even if the regression coefficient is statistically significant, it is necessary to confirm the trend with some kind of robust methodology that is not sensitive to OLS assumptions. For an example of these computations please see: https://www.academia.edu/14038781/A_ROBUST_TEST_FOR_OLS_TRENDS_IN_DAILY_TEMPERATURE_DATA

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