I have pairs of timeseries that are estimating the same quantity over the years. It is some survival data: number of "dead" subjects during the year over the number of total subjects at the beginning of the year. The underlying data is regularly revised, so I need to assess the effect of the changes. Below an example is given.

the two series

I want to specifically test if the mean (which is the parameter of interest) is the same for the timeseries pairs (the analysis will be repeated for a lot of pairs, please do not focus too much on the particular image although it is fairly representative).

I have several questions:

  1. Should I use a paired test or not?
  2. If it is paired, how do I deal with the ties, i.e. zero differences (I will always have a lot of ties)?
  3. I believe that bootstrapping can be usefull in this case. I have bootstrapped the sample mean difference. This gives a distribution centered around the observed mean difference. How can I compute a p-value out of this distribution? Would it be reasonable to bootstrap the t-test instead?

Obviously time is the most important factor so if they are measuring the same thing at the same time rather than at lagged times pair them so that the values for the pairs occurred at the same time. But then you may have a problem using the paired t test or the Wilcoxon signed rank test because the pairs are dependent. (x(t), y(t)) is correlated with (x(t+1), y(t+1)) by virtue of the correlation in the individual series.

Bootstrapping can avoid distributional assumptions and can incorporate the correlation structure. But saying that you "bootstrapped the mean differences" does not explain what you did. You could bootstrap each time series using moving block bootstrap or any other type of block bootstrap or you can fit the time series to ARMA models and bootstrap residuals or vectors. Whichever way you do this you will have taken the correlation into account in forming the bootstrap time series data sets. Each time you bootstrap you get a bootstrap sample mean difference. You can use the bootstrap (Monte Carlo approximate) distribution to generate confidence interval using one of the various bootstrap confidence interval methods and invert the confidence interval to get the hypothesis test. Or you could consider doing the hypothesis test directly by assuming the null hypothesis with the bootstrap distribution (this involves centering the data as describe in the paper by Hall and Wilson) and calculating the probability in the tail of the bootstrap distribution just beyond where your original sample estimate is. That is your bootstrap p-value. This is a better approach than bootstrapping the t test.

  • $\begingroup$ Could you please the reference for the paper? By bootstrapping mean difference, I mean that I have sampled the difference from the timeseries and have taken the mean... $\endgroup$ – teucer Jun 1 '12 at 8:29
  • $\begingroup$ @teucer There are many research papers and books on the bootstrap. Let me address your question by referring you to the books that cover time series applications of the bootstrap. (1)"Resampling Methods for Dependent Data." S. N. Lahiri Springer-Verlag 2003. (2) "Bootstrap Methods. 2nd Edition" M. R. Chernick (2007) Wiley. (3) "An Introduction to the Bootstrap" B. Fron and R. Tibshirani (1994) Chapman and Hall (4) "Bootstrap Methods and Their Applications" A. C. Davison and D. V. Hinkley (1997) Cambridge University Press. $\endgroup$ – Michael R. Chernick Jun 1 '12 at 9:44
  • $\begingroup$ (5) "An Introduction to Bootstrap Methods with Applications to R." M. R. Chernick and R. A. LaBudde (2011) Wiley. For articles related to time series methods try to Google key words like circular block bootstrap, moving block bootstrap, wild bootstrap, stationary bootstrap and authors such as S. N. Lahiri, J. Romano, T. DiCiccio and D. Freedman. $\endgroup$ – Michael R. Chernick Jun 1 '12 at 9:48
  • $\begingroup$ Should I use a paired or unpaired test? I believe if I use a paired test, I can only bootstrap the differences... $\endgroup$ – teucer Jun 14 '12 at 22:08
  • $\begingroup$ Not sure that I know what you mean. You said that you match time points and take paired differences at each time point and then take tha average of these paired differences. Let's say to bootstrap you fit an ARIMA model to each series and then bootstrap the residuals for each series. Then with the bootstrap you match what you did with the original pair of series. $\endgroup$ – Michael R. Chernick Jun 14 '12 at 22:28

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