# Using day length and rate of change of day length in linear regression

I have a time series in which samples were collected approximately monthly over ten years. I have wanted to identify whether a number of bacterial species were seasonally variable and whether they increased or decreased over time. I had been modelling species abundance with linear regression according to the equation

S = a + b*ED +c*DL + d*DDL.

Where S is the abundance of a species.

ED: Number of days elapsed since the beginning of the study.

DL: Day length on the day the samples were collected

DDL: The rate of change in day length, measured here as the day length minus the day length measured thirty days previous.

I then looked at the P values associated with each of these coefficients and the coefficients themselves. If ED was statistically significant, I said there was a long term triend, and if ED's coefficient was positive, that species abundance was increasing. If DL or DDL's coefficients were statistically significant, I said the species' abundance was seasonal. If DL had a statistically significant coefficient, I said the species was most abundant in the summer; if negative, most abundant in the winter. Same pattern for DDL, positive, most abundant in spring, negative most in fall.

A reviewer tells me that "The regression analysis may not be appropriate because day length and day length change are autocorrelated and also covarying." Is this criticism valid? If valid, is there anything I can salvage from this kind of analysis? Is there a better approach? If not, why isn't it valid? I feel like this approach is analogous to building trigonometric models to describe a data set eg y = a + b*t + c*sin(2pi*t/L) + d*cos(2pi*t/L), which people apply to model some time series. Can I just replace the sign and cosine terms with DL and DDL (which are sinusoidal and DDL is basically the derivative of DL) or am I breaking some rule there? Thanks!

• (1)have you looked at the residuals, done a lag1 test (Durbin-Watson), white-noise test (portmanteu) or looked at auto-correlation plots? (2) if this is OLS then an assumption that the residuals are not autocorrelated, in this case where there is high suspicious one should either prove they aren't or correct (3) there are a very large number of methods to correct/account for autocorrelation including praise-winston or cochrane-orcutt (if this is OLS) – charles Mar 25 '14 at 0:03
• In the presence of autocorrelation, the parameter estimate is unbiased, SE is biased (downward), as are t (upward) and p (downward) in favor of significance...I think? – charles Mar 25 '14 at 0:05
• 1) Residuals look reassuringly random. I can't make an autocorrelation plot, I don't think because the results are unevenly spaced and autocorrelation assumes the presence of a time series object, which has spaced data. Furthermore I don't think lag1 is appropriate either for the same reason. 2) it is an OLS. I i'm just not sure how to show residuals are not autocorrelated if the data structure violates the assumptions of autocorrelation tests. 3) I can't figure out if these would work for unevenly spaced data. – ohnoplus Mar 25 '14 at 17:59
• I will add, however, that if I assume a lag of 1 between each data point regardless of actual ing and run a Durbin-Watson test or a Box-Ljug test, I get non significant p-values suggesting no autocorrelation. However, I am skeptical that I can trust this result, given the uneven spacing. Any thoughts? – ohnoplus Mar 25 '14 at 18:27