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I have a simple time series with 5-10 data points per data set at regular intervals. I am wondering what is the best way to determine whether two data sets are different. Should i try t-tests on each data point, or look at the area under the curves or is there some kind of multivariate model that would work better?

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  • $\begingroup$ What do you mean by "different"? $\endgroup$ – Shane Oct 14 '10 at 20:21
  • $\begingroup$ What do you mean with "5-10 data points per data set"? $\endgroup$ – Stephan Kolassa Oct 14 '10 at 20:25
  • $\begingroup$ I think he has a collection of several time series, each one with 5-10 observations. $\endgroup$ – Rob Hyndman Oct 15 '10 at 1:07
  • $\begingroup$ I still think that this question is nearly impossible to answer without understanding what "different" means... $\endgroup$ – Shane Oct 15 '10 at 15:30
  • $\begingroup$ My applogies for the poorly worded question. By different I mean whether over the course of the time series (rather than at individual points) there is a difference between two treatment groups. There would be inter-subject variation (which i guess would need to be accounted for) as well as inter-group variation (which is what I am interested in). $\endgroup$ – Dave Oct 15 '10 at 19:32
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You will need to specify precisely what you mean by "different". You will also need to specify what assumptions you are willing to make about the serial correlation structure within each time series.

With t-tests, you are comparing the mean of each group and you are assuming that the groups consist of independent observations with equal variances (the latter is sometimes relaxed). When testing time series, the assumption of independence is usually not reasonable, but then you need to replace it with a specified correlation structure -- e.g., you might assume that the time series follow AR(1) processes with equal autocorrelation. Consequently, even comparing the means of two or more time series is considerably more difficult than with independent data.

I would carefully specify what assumptions I was willing to make about each time series, and what I was wishing to compare, and then use a parametric bootstrap (based on the assumed model) to carry out the test.

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Maybe repeated measures anova is what you want. It allows you to compare the subjects (inter subject factors) while taking the correlated structure of the "time series" per subject (intra subject factor). It is an easy but dated method and can be found in the context of "general linear models", it needs some additional features (e.g. sphericity). Another way could be mixed linear models which allow for more general correlations structures (even AR(1) like Rob suggested) and unbalanced data.

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If you want to assume simple linear trend, you can take the difference of each data set at the various time points and test that the slope of the line is zero.

-Ralph Winters

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