I've got a couple of time series obtained from a small number of replicates,where many variables (all of them continuous) are measured at various time points (longitudinal study). I was looking for a statistical test that could answer whether or not the data supports any type of trend (increasing, decreasing or even oscillations), or alternatively, the actual value stays constant and what I see in my individual or average time series is due to measurement error alone. I've had a look and found the Ljung–Box test - Is is appropriate to answer what I want? If not, what do you suggest?
The Ljung-Box test is a test as to whether or not there are any significant autocorrelations in a single time series. If there are no correlations the conclusion is that the time series is just random noise. So it does test for randomness. But in your problem you have multiple time series that are presumably realizations of the same process.
To test for trends or seasonals you would try fit a model with trends or seasonal components (sine waves for example) and test whether or not the coefficients are significantly different from 0. But this would be done separately on the individual time series.