Is it reasonable to calculate a polynomial regression using days but showing month as on plot?

Question

My data concern vegetative relative cover of nettle plants recorded at up to thirty sites at irregular intervals over a calendar year (17/1/2009 - 16/1/2010) and different dates for different groups of sites. I want to be able to describe the changes over the year by a regression equation and calculate the F-statistic and the probability that the variation is related to season.

For the dependent x variable I have used number of days from the start of the study (rather than a date) in order to give a smaller value for the y-intercept. However in showing the results graphically I would like to have x-axis intervals indicating months. This will show the relationship to growth season more immediately. However I realize that using first day of month would give slightly (imperceivably?) uneven intervals.

Is this reasonable approach, or is there another approach that would achieve these objectives better?

I'd appreciate any advice.

I am now adding the results of the polynomial regression I have previously completed:

 colMeans(mse)
[1] 1521.902 1312.779 1283.366 1250.781 1272.761
> best = lm(cover ~ poly(Days,2, raw=T), data=frpd)
> summary(best)

Call:
lm(formula = cover ~ poly(Days, 2, raw = T), data = frpd)

Residuals:
    Min      1Q  Median      3Q     Max 
-63.814 -32.384  -3.897  30.337  57.270 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)             98.4301664 14.8925413   6.609 6.74e-09 ***
poly(Days, 2, raw = T)1 -0.5816455  0.1771272  -3.284  0.00161 ** 
poly(Days, 2, raw = T)2  0.0015144  0.0004444   3.408  0.00110 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 36.29 on 69 degrees of freedom
Multiple R-squared:  0.1443,    Adjusted R-squared:  0.1195 
F-statistic: 5.819 on 2 and 69 DF,  p-value: 0.00462

> lm(formula=cover~poly(Days,2,raw=T),data=frpd)

Call:
lm(formula = cover ~ poly(Days, 2, raw = T), data = frpd)

Coefficients:
            (Intercept)  poly(Days, 2, raw = T)1  poly(Days, 2, raw = T)2  
              98.430166                -0.581645                 0.001514  

Answer 1

I disagree with the suggestion of using days elapsed given by wjktrs. Unlike medical studies of human patients, starting day of year is very important in ecological studies of seasonal patterns. As seasonality of plant growth patterns is strong, patterns observed on 183 days elapsed since a beginning in January would carry completely different implications from those with a beginning day in July. Therefore, consolidating to days elapsed since a study began discards useful information that day of year retains.

It is okay to use different units of measurements in modelling and plotting. They are simple linear transformations. In fact, you do not even need any transformation but a change in axis labelling. You can see tutorials of plotting time series at https://otexts.com/fpp3/graphics.html.

How to represent the seasonality is important. If you use day of year, you will need to include many high-order polynomials such as x + I(x^2) + I(x^3) + I(x^4) + I(x^5) or Fourier terms using sine and cosine functions. But the coefficients will be difficult to interpret intuitively. You could benefit by using 11 indicators of month or three indicators of quarter, then you can state that compared with January or winter, July or summer vegetation increase by this amount.

Since you have longitudinal data, an ordinary least squares estimator may not suffice and suffer from spatial and temporal serial correlation in errors. You may instead need to learn generalized least squares and mixed-effect models.

Since your response variable is vegetative relative cover of nettle plants, it may be a percentage, strictly positive, or nonnegative value. You may need to use fractional model, beta regression potentially with zero and one inflation, gamma regression, or ordinal regression. See

• Stata tutorial on fractional response https://www.stata.com/features/overview/fractional-outcome-models. Michael Clark (2019) Fractional Regression: A quick primer regarding data between zero and one, including zero and one https://m-clark.github.io/posts/2019-08-20-fractional-regression/. Papke, L. E., & Wooldridge, J. M. (2008). Panel data methods for fractional response variables with an application to test pass rates. Journal of Econometrics, 145(1), 121–133. https://doi.org/10.1016/j.jeconom.2008.05.009
• A gamma regression tutorial with shape and scale parameter meaning and calculation https://data.library.virginia.edu/getting-started-with-gamma-regression/
• Christensen, R. H. B. (2022). Cumulative link models for ordinal regression with the R package ordinal. https://cran.r-project.org/web/packages/ordinal/vignettes/clm_article.pdf

Answer 2

Most software packages have a day-based differencing function for which you get the exact number of days between the start of study at each site, and specific date of sample capture. Thus, that will give you the numdays for each measurement.

Now, with regard to using the first day of the month, you should drop that idea. Rather, use a plot that has nettle relative cover vs. day of measurement (over the entire x-axis).

Many failure time studies drop any notion of displaying calendar date on the x-axis and instead just use #days since start of study that each measurement was made on.

Something you never mentioned was whether you were concerned about some months having 28, 30, 31 days? That concern shouldn't matter either, since e.g. Excel will make such a plot with calendar date on the x-axis and your nettle coverage on the y-axis. However, you have multiple collections sites, so simply tell the sites to report, for each sample, the number of days since a global study start date for each measurement, and then just plot the nettle values (or means) per #days since global date representing start of study.