2.3 Data analyses
A nonlinear mixed-effects model (4-parameter log-logistic function) was fit to the data and compared with the linear mixed-effects model using Akaike’s Information Criterion (AIC). In all cases (except for leaf angle), the linear model provided a similar (AIC values within 10) or better (linear model AIC < nonlinear model AIC) fit to the data than the nonlinear model. The nonlinear model was of the form
\begin{equation} f\left(x\right)=c+\ \frac{d-c}{1+e^{b(\log\left(x\right)-\log\left(e\right))}}\nonumber \\ \end{equation}
where x is the duration of weed-free or weedy GDD after planting;f (x ) is the leaf angle as a function of GDD after planting; d and c are upper and lower limits, respectively, indicating the estimated leaf angle at extreme values ofx ; b is the slope at the inflection point of the curve; and e is the value of x at which the inflection point occurs. The medrc package (Gerhard & Ritz, 2016) in the R statistical language was used to fit nonlinear models, with random components included for the c , d , and eparameters.
A linear mixed-effects ANOVA was performed in R statistical language using the lmer function of the lme4 package (v1.1-23) and convenience functions from the lmerTest package (v.3.1-0) (Kuznetsova, Brockhoff, & Christensen, 2017; R Core Team, 2020). Weed removal timing (early-season series) or addition timing (late-season series) were considered fixed effects, and year was considered a random effect. An indicator variable was included in the analysis for each response variable, with a value of 1 for the weed-free treatment or a value of 0 for one of the addition or removal timings; each fixed effect (linear effect of addition or removal, and the weed-free indicator) was dropped from the model and the full and reduced models were compared with an F-test. If removal of an effect resulted in a significant reduction in model fit compared to the full model (P<0.1), then that effect was retained in the final model. In this way, if there was a significant effect of the indicator variable, but no significant effect of the removal or addition timing, then the effect was assumed to be attributable to only the presence of weeds during early sugar beet development, and not to the duration of weed presence beyond that period. Conversely, if there was an effect of addition or removal timing, but no effect of the weed-free indicator variable, then the response was attributed to the linear duration of weed presence.
Two-sample t-tests were used to compare the season-long weed-free treatment to season-long weedy treatments in the greenhouse study for final number of leaves, number of senesced leaves at harvest, and plant biomass. The 95% confidence intervals at each leaf pair were used as a conservative estimate of statistical difference between the season-long weed-free vs season-long weedy treatments (Austin & Hux, 2002). A non-parametric local regression (loess) was fit to petiole proportion of leaf length as a function of leaf age for the greenhouse study.
3. Results