# Hypothesis testing and significance for time series

A usual test of significance when looking a two populations is the t-test, paired t-test if possible. This assumes that the distribution is normal.

Are there similar simplifying assumptions that produce a significance test for a time series? Specifically we have two fairly small populations of mice that are being treated differently, and we are measuring weight once a week. Both graphs display smoothly increasing functions, with one graph definitely above the other. How do we quantify "definiteness" in this context?

The null hypothesis should be that the weights of the two populations "behave in the same way" as time passes. How can one formulate this in terms of a simple model that's fairly common (just as normal distributions are common) with only a small number of parameters? Once one has done that, how can one measure significance or something analogous to p-values? What about pairing the mice, matching as many characteristics as possible, with each pair having one representative from each of the two populations?

I would welcome a pointer to some relevant well-written and easily understood book or article about time series. I start as an ignoramus. Thanks for your help.

David Epstein

• You might wish to cast a broader net, because this is not necessarily a time series question. Indeed, perhaps the most fundamental question here concerns the best or at least correct way to quantify a treatment "endpoint": is it mean growth in a population after a certain time, average growth rates over time, etc? If you didn't know this before starting the experiment and are suddenly noticing consistent differences in growth curves, then you are working in an exploratory mode, not a confirmatory one, and hypothesis-testing p-values will be deceptively good. – whuber Nov 3 '11 at 14:35
• The result is qualitatively as expected, and a one-sided test seems appropriate. The reason I asked about time series, is that if one measures only the final weight (which is the most relevant measurement), then one is throwing away all the information from earlier time points, and that seems wrong. – David Epstein Nov 4 '11 at 23:06
• You're right: you don't want to throw away those data. But time series techniques come to the fore for models of the data where temporal correlations of deviations from idealized curves are important, either for their own interest or because they could interfere with good estimation. Your situation isn't likely to fall into either of these cases. Simpler, more scientifically meaningful methods are available. – whuber Nov 5 '11 at 2:30
• @whuber, isn't the weight over time of the control set of mice an "idealised curve" in some sense? Or at least, a theoretical model fitted to that data? – naught101 Apr 20 '12 at 3:21
• Yes, @naught, that is a reasonable way to look at it. But "curve" is not the same as "time series." For instance, linear regression can be (and often is) viewed as fitting curves to data, but is separate from time series analysis, which emphasizes the structure of correlations among deviations between the data and the idealized curve. – whuber Apr 20 '12 at 15:19

## 2 Answers

There are many ways to do it if you think of the weight variations as a dynamical process.

For example, it can be modeled as an integrator $\dot x(t) = \theta x(t) + v(t)$

where $x(t)$ is the weight variation, $\theta$ relates to how fast the weight changes and $v(t)$ is a stochastic disturbance that may affect the weight variation. You could model $v(t)$ as $\mathcal N(0,Q)$, for a known $Q$ (you can also estimate it).

From here, you can try to identify the parameter $\theta$ for the two populations (and their covariance), using, e.g., a prediction error method. If the Gaussian assumption holds, prediction error methods will give that the estimate of $\theta$ is also Gaussian (asymptotically) and you can therefore build a hypothesis testing to determine whether the estimate of $\theta_1$ is statistically close to that of $\theta_2$.

For a reference, I can suggest this book.

I would suggest identifying an ARIMA model for each mice separately and then review them for similarities and generalization. For example if the first mice has an AR(1) and the second one has an AR(2), the most general (largest) model would be an AR(2). Estimate this model globally i.e. for the combined time series. Compare the error sum of squares for the combined set with the sum of the two individual error sum of squares to generate an F value to test the hypothesis of constant parameters across groups. I you wish you can post your data and I will illustrate this test precisely.

ADDITIONAL COMMENTS:

Since the data set is auto-correlated normality does not apply. If the observations are independent over time then one might apply some of the well-known non-time series methods. IN terms of your request about an easy to read book about time series, I suggest the Wei text by Addison-Wesley. Social scientists will find the non-mathematical approach of Mcleary and Hay (1980) to be more intuitive but lacking rigor.

• This really doesn't appear to address the fundamental issues. (1) Why is such a model appropriate? (2) Why should each mouse be modeled and not, say, the mean population weights or gains in weights? (3) Why is a test of constant parameters relevant? The question begs for a one-tailed test. Most of the parameters you mention do not appear scientifically relevant, nor do they directly quantify a sense of one graph being consistently above the other. (4) How do you control for possible differences in characteristics of the two populations at the beginning of the experiment? – whuber Nov 3 '11 at 14:30
• :whuber Thetest for constancy of parametersis relevant because you have aset ofcoefficients forthe first group ofreadings formouse 1 & a second set of coefficients for the 2nd mouse.The question is"is there collectively asignificant difference between the coefficients".Now continuing with your comment , since one of the model coefficients might be a constant and if it is then the difference between the coefficients mightbe due tothe constants being statistically different from one another.Note that the underlying ARIMA model maynot necessarily have a constantas it might be a difference model . – IrishStat Nov 3 '11 at 19:06
• I think you're partly right, but you need to refine your characterization of the problem. Many of the ARIMA coefficients may be scientifically irrelevant. For instance, if one of them acts like a quadratic term over time, a difference might say something about the shape of the growth curves but that could be of little use. If one chooses coefficients to reflect the experimental endpoint(s) and tests only them, some good might be achieved thereby. In general, though, time series models introduce coefficients (e.g., autocorrelation) unlikely to be of direct scientific interest here. – whuber Nov 3 '11 at 19:16
• whuber: "If one chooses coefficients to reflect the experimental endpoint(s) and tests only them, some good might be achieved thereby" doesn't make much sense to me as it ignores the intermediate points. On the contrary to your comment, the time series mode and it's accompanying coefficients are of significant scientific interest as it characterizes the distribution of readings and converts them to a random process ( the error term ) which is free of autocorrelative structure and then amenable to tests requiring normality. The test I propose requires that assumption to hold. – IrishStat Nov 3 '11 at 19:34
• Autocorrelation may be of little importance here. Interest explicitly focuses on the trends: how do the underlying growth curves tend to differ between the two populations? Autocorrelation parameters are nuisance parameters, to be introduced and dealt with only insofar as they might help improve the estimation of those growth curves. The first priority is to adopt a scientific model of the growth, represent that model with parameters that are interpretable and of interest, and estimate them. Automatic application of time series techniques is unlikely to accomplish that. – whuber Nov 3 '11 at 19:40