Testing trend in proportions? Simple question: I have a sequence of 10 proportions $(p_1,\dots,p_{10})$ (indexed by time), $p_j\in(0,1)$, that seems to have an increasing trend. How can I test this trend?
 A: There is a test called the Mann-Kendall test that can assess if a particular (monotonic, or one-directional) trend exists.  If you can conclude that no serial correlation exists, that the observations at time $i$ are representative of the conditions at time $i$, and that the sampling, measurements, etc. are unbiased and representative of the populations over time.
Although the article here doesn't mention testing proportions (just general random variables $x_1,\ldots,x_n$), I am confident that the test still works.  There may be an issue if some of your observations are $0$ or $1$, but you wrote above that $p_i\in(0,1)$ so that should be avoided.
A: Welcome to CV. I've assumed that you've already looked at the graph of the proportions against time. The simplest approach would be to regress the proportions against time as the single predictor, assuming time is coded in an analyzable format, e.g., as in a trend from 1 to 10. This would give you the strength of a linear or deterministic relationship as well as its sign or direction. For curvilinear relationships, e.g., quadratic, introduce time-squared along with time. You don't have many observations so you can't fit too many more parameters with this method.
You don't indicate what the unit of time is for your data...e.g., is it daily, weekly, monthly, quarterly, etc. This would be useful information.
This CV thread discusses a similar question about trends... Describing trend magnitude or detecting trends Among the recommendations was one from @whuber regarding building a Loess model for seasonal decomposition ... http://www.wessa.net/download/stl.pdf 
In addition, with so few observations many of the more advanced models and tests simply won't work, e.g., tests for unit root, stationarity, arima,  etc...
