Linear Regression to detect between a linear and non-linear trend I have measured the area of spread of a number of plants through time. I'm interested in trying to ascertain whether a linear or a non-linear relationship (i.e. quadratic) best represents the increase in the sqrt of the area occupied by these plants through time 
My first feeling was that I could use a linear regression for this, whereby I fit a linear regression model containing a squared term to each plant's growth through time  i.e. 
y = a + b.x + c.x^2 (where x equals time)
and compare this to 
y = a + b.x
via standard linear regression simplification to see whether a non-linear trend explains significantly more of the variation than just a linear trend. 
However, I'm aware that linear regression probably shouldn't be used for time series data.
It is possible to do it in this way? 
Thanks 
 A: Consider your linear model 
$$
y_t = a + bt + e_t 
$$
where $e_t$ is the error term at time $t$.
If you took the difference
$$
y_t - y_{t-1} = a + bt + e_t -a - b(t-1) - e_{t-1} = b + e_t - e_{t-1}
$$
which is an integrated moving average model (ARIMA(0,1,1)) model.  You can write it as 
$$
\Delta y_t = b + e_t - \theta e_{t-1}
$$
where in this particular case $\theta=1$.  With ARIMA (Autoregressive Integrated Moving Average) models you could investigate the more general ARIMA(p,1,q);
$$
\Delta y_t = b + \sum_{i=1}^p \phi_i \Delta y_{t-i} + e_t + \sum_{j=1}^q \theta_j e_{t-j}$$ 
for which your linear model is a subset 
You can work without differencing $y_t$ but the series must be stationary.  You can check out the Wikipedia entry on ARIMA for more details.
For model selection you can use information criterion like AIC, BIC, or one of its variants (generally speaking, a lower AIC/BIC implies a better predictive model). R has particularly good functions for estimating and selecting among ARIMA models see the Arima and auto.arima functions and documentation.  However, you have to be particularly careful about comparing models that difference/transform $y_t$ with those that do not (AIC and BIC are not appropriate for that comparison).  Using Mean-squared prediction error or something similar to that would be more appropriate. 
Non-linear time series can get hairy, there are methods out there but they tend to be harder to implement and interpret.  One thing that practitioners often do is model $\ln[y_t]$ with an ARIMA, this implies non-linear exponential growth/decay in $y_t$. 
