# Statistical test, significance of change in average value

I have past 12 month data (a ratio of occurence: ranges between 0%-100%) and I have most recent data (January, so 13th observatios).

I would calculate the average of the first 12 months (first 12 obs.) of ratios and average of last 12 months (last 12 obs) ratios.

I would like to test significance of change between these two measures. Well, these two samples are sliding windows. So they are not simply two completely different (independent) datasets. Basicaly, only the first and last observations change so the corresponding sample average value changes.

I would like to test significance of increase as compared to past observations.

What test to use here? Would the simple t-test of difference between two means appropriate here?

$\sigma_{M_{1}-M_{2}}=\sqrt{\frac{\sigma^2_{1}}{n_{1}+\frac{\sigma^2_{2}}{n_{2}}$

$*t*=\frac{M_{1}-M_{2}}{s_{M_{1}-M_{2}}$


EDIT: Possibly the Wilcoxon signed rank test to use? Which is the equivalent of the dependent t-test, would that be correct?

• What is your model? – HStamper Feb 26 '16 at 18:40
• Thanks for looking into this. I don't have model. I would like to get simple statistics (sort of p-value) of the significance of change as compared to average value over the first 12 months (out of 13obs.) – Maximilian Feb 26 '16 at 18:42
• It doesn't seem like this is possible without some further assumptions. – HStamper Feb 26 '16 at 18:54

Let $X_{1},\dots,X_{n}$ be your data. You have the following two sample means,
$M_{1} = \frac{1}{n}\sum_{t=1}^{n-1} X_{t}$.
$M_{2} = \frac{1}{n}\sum_{t=2}^{n} X_{t}$.
You want to know whether the difference, $M_{2} - M_{1}$ is statistically significantly different from $0$.
$M_{2} - M_{1} = \frac{1}{n} \times \left[ X_{n+1} - X_{1}\right]$.
To construct a hypothesis test, you need to have a distributional theory for $X$ right? Without one, there is no way to understand the null distribution of $X_{n+1} - X_{1}$. If $X$ is iid for example, then $E[M_{2} - M_{1}] = 0$. But your problem does not sound iid. The main problem is that there is no asymptotics here (no central limit theorem to appeal too). The difference between $M_{2}$ and $M_{1}$ always depends on just two observations regardless of sample size.