# Background

Crop–weed competition is extensively studied in weed science. The additive design, in which weed density varies and the crop density is kept constant, is the most commonly utilized design in plant competition studies. The additive design is important to calculate economic weed thresholds and improve weed control decision-making.

# Getting started

R is a programming language and free software environment for statistical computing and graphics supported by the R Foundation for Statistical Computing.

RStudio is an integrated development environment for R, a programming language for statistical computing and graphics.

# Create a new R project file

• Click in File -> New project… -> New directory -> New Project -> Save the R work directory anywhere you want in your laptop.

The saved file will have an R project and you should copy and paste all your raw data (excel file) in the same work directory (folder).

• Click in File -> New File -> R script (.R)

or

• Click in File -> New File -> R markdown… (.Rmd)

Although I like working within R markdown, R script is easier to work with in R studio.

# Install packages

In order to accomplish this exercise, you will need to install the following R packages:

install.packages(nlstools)
install.packages(AICcmodavg)
install.packages(broom)
install.packages("tidyverse")
install.packages("RCurl")


Run all codes below by clicking in the Run option in the top right corner of you R script or R markdown. These codes will install the necessary packages for analyzing the weed competition experiment. Once you install these packages you will not need to install them every time you work in Rstudio, unless you update R or Rstudio.

# Data

• Run the packages below
library(tidyverse)
library(RCurl)

The data used for the exercise is from a published manuscript of Commelina benghalensis and Richardia brasiliensis competition in corn.

Run the following codes below to load the data into R:

df_path <- url("https://raw.githubusercontent.com/openweedsci/data/master/posts/additive.csv")

DMT <- read_csv(df_path) 
##
## ── Column specification ────────────────────────────────────────────────────────
## cols(
##   block = col_double(),
##   treat = col_double(),
##   densitycrop = col_double(),
##   densityweed = col_double(),
##   biomass = col_double(),
##   yl = col_double(),
##   weed = col_double()
## )

After running the codes above, the data should appear as DMT (“DMT” was chosen as the name of the dataset; you could have called it “data” or something else). In the spreadsheet, weeds are named 1 (C. benghalensis) and 2 (R. brasiliensis), and desity of weeds varies from 0 to 4 plants per plot, and response variable is yl (% yield loss).

DMT
## # A tibble: 40 x 7
##    block treat densitycrop densityweed biomass     yl  weed
##    <dbl> <dbl>       <dbl>       <dbl>   <dbl>  <dbl> <dbl>
##  1     1     1           1           0   59.4  -16.4      1
##  2     2     1           1           0   34.4   32.7      1
##  3     3     1           1           0   56.7  -11        1
##  4     4     1           1           0   53.8   -5.28     1
##  5     1     2           1           1   12.7   75.1      1
##  6     2     2           1           1   14.0   72.7      1
##  7     3     2           1           1   12.4   75.7      1
##  8     4     2           1           1   23.2   54.6      1
##  9     1     3           1           2    4.55  91.1      1
## 10     2     3           1           2    3.91  92.3      1
## # … with 30 more rows

# Rectangular hyperbola model

The empirical model: $Y=\frac{I * x}{ 1 + (\frac{I}{A})*x}$ is the standard model to describe additive competition studies. $$I$$ represents the slope of $$Y$$ (yield loss) when $$x$$ (weed density) approximate zero. Also, $$A$$ is the asymptote or maximum expected yield loss (%).

## Step 1) Fit a full model, a rectangular hyperbola with 4 parameters

Full is a user-defined name that will contain all information about the fitted model generated by nls (nonlinear least squares) function. The start is used to estimate values of parameter $$I$$ and $$A$$ for the model. Parameters can determine from visual inspection of the data set (plotting data and observing trends). The brackets [weed] for each parameter in the equation tell R to estimate a parameter for each weed species (4 parameters).

nls function

Full = nls(yl ~ (I[weed]*densityweed)/(1+(I[weed]/A[weed])*densityweed), data=DMT,
start=list(I=c(60,30), A=c(80,60)), trace=T)

### Check estimated parameters

The summary command provides the estimated parameters $$I$$ and $$A$$ for each weed species, Commelina benghalensis (species 1) and Richardia brasiliensis (species 2).

summary(Full)
##
## Formula: yl ~ (I[weed] * densityweed)/(1 + (I[weed]/A[weed]) * densityweed)
##
## Parameters:
##    Estimate Std. Error t value Pr(>|t|)
## I1   210.23      88.55   2.374  0.02304 *
## I2    50.25      22.64   2.220  0.03280 *
## A1   108.56      11.15   9.740 1.25e-11 ***
## A2    82.07      23.06   3.559  0.00107 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14 on 36 degrees of freedom
##
## Number of iterations to convergence: 4
## Achieved convergence tolerance: 3.438e-06

## Step 2) Fit a reduced model (Red.1), rectangular hyperbola model with 2 parameters.

Red.1 is a user-defined name that will contain information about the first reduced model generated by the nls function. Notice that we do not include bracket [weed] after each parameter $$I$$ and $$A$$. In this case, we are combining parameter $$I$$ and $$A$$ for both weed species. We hypothesize that a single parameter $$I$$ and $$A$$ for both species is enough to describe the crop-weed relationship (e.g., no difference of $$I$$ and $$A$$ between species).

Red.1 = nls(yl ~ (I*densityweed)/(1+(I/A)*densityweed), data=DMT,
start=list(I=40, A=80), trace=T)

### Check estimated parameters

This command provides the estimated parameters $$I$$ and $$A$$ for both weed species combined.

summary(Red.1)
##
## Formula: yl ~ (I * densityweed)/(1 + (I/A) * densityweed)
##
## Parameters:
##   Estimate Std. Error t value Pr(>|t|)
## I   114.55      55.93   2.048   0.0475 *
## A    92.62      15.93   5.814 1.02e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 22.78 on 38 degrees of freedom
##
## Number of iterations to convergence: 4
## Achieved convergence tolerance: 7.759e-07

### Test the first hypothesis

Hypothesis testing using ANOVA. We test this hypothesis using the Full model ($$I$$ and $$A$$ for each species) to compare with Red.1 (single $$I$$ and $$A$$ for both species). If P-value>0.05, models are similar; therefore we should use the Red.1 model, which means that the simplest model (Red.1) is appropriate to describe crop-weed relationship. If not we should proceed to the next hypothesis.

anova function

anova(Full, Red.1)
## Analysis of Variance Table
##
## Model 1: yl ~ (I[weed] * densityweed)/(1 + (I[weed]/A[weed]) * densityweed)
## Model 2: yl ~ (I * densityweed)/(1 + (I/A) * densityweed)
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)
## 1     36     7056.7
## 2     38    19715.8 -2 -12659   32.29 9.293e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The F-test showed P<0.05. Therefore the Red.1 model is not appropriate to describe the crop-weed relationship.

## Step 3) Fit a reduced model (Red.2), rectangular hyperbola model with 3 parameters

Red.2 is a user-defined name that will contain information about the second reduced model generated by the nls function. Notice that the bracket [weed] is after the parameter $$A$$ only, which means that we are testing a hypothesis of single parameter $$I$$, but different $$A$$ for the species.

Red.2 = nls(yl ~ (I*densityweed)/(1+(I/A[weed])*densityweed), data=DMT,
start=list(I=60, A=c(80,60)),  trace=T)

### Check estimated parameters

This command provides the estimated parameters $$I$$ for both weed species and $$A$$ for each weed species.

summary(Red.2)
##
## Formula: yl ~ (I * densityweed)/(1 + (I/A[weed]) * densityweed)
##
## Parameters:
##    Estimate Std. Error t value Pr(>|t|)
## I   163.850     56.984   2.875  0.00666 **
## A1  115.285     12.779   9.021 7.03e-11 ***
## A2   56.077      5.714   9.813 7.65e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.58 on 37 degrees of freedom
##
## Number of iterations to convergence: 6
## Achieved convergence tolerance: 5.504e-06

### Test a second hypothesis

Hypothesis testing using F-test. We are using the Full model (separated I and $$A$$ for each species) to compare with Red.2 (single $$I$$ and different $$A$$ for both species). If P-value>0.05, models are similar; therefore, we should use the Red.2 model, which means that the simplest model (Red.2) is appropriate to describe crop-weed relationship. If not we should proceed to the next hypothesis.

anova(Full, Red.2)
## Analysis of Variance Table
##
## Model 1: yl ~ (I[weed] * densityweed)/(1 + (I[weed]/A[weed]) * densityweed)
## Model 2: yl ~ (I * densityweed)/(1 + (I/A[weed]) * densityweed)
##   Res.Df Res.Sum Sq Df  Sum Sq F value  Pr(>F)
## 1     36     7056.7
## 2     37     7864.6 -1 -807.88  4.1214 0.04978 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The F-test showed P<0.05. Therefore the Red.2 model is not appropriate to describe the crop-weed relationship.

## Step 4) Fit a reduced model (Red.3), rectangular hyperbola model with 3 parameters

Red.3 is a user-defined name that will contain information about the third reduced model generated by the nls function. Notice that the bracket [weed] is after the parameter $$I$$ only, which means that we are testing a hypothesis of different parameter $$I$$, but single parameter $$A$$ for the species.

Red.3 = nls(yl ~ (I[weed]*densityweed)/(1+(I[weed]/A)*densityweed), data=DMT,
start=list(I=c(30,30), A=70), trace=T)

### Check estimated parameters

This command provides the estimated parameters $$I$$ for each weed species and $$A$$ for both weed species.

summary(Red.3)
##
## Formula: yl ~ (I[weed] * densityweed)/(1 + (I[weed]/A) * densityweed)
##
## Parameters:
##    Estimate Std. Error t value Pr(>|t|)
## I1  228.357    100.178   2.280   0.0285 *
## I2   37.000      6.196   5.972 6.85e-07 ***
## A   106.170     10.318  10.289 2.10e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.95 on 37 degrees of freedom
##
## Number of iterations to convergence: 7
## Achieved convergence tolerance: 1.144e-06

### Test a third hypothesis

Hypothesis testing using F-test. We are using the Full model (separated $$I$$ and $$A$$ for each species) to compare with Red.3 (different $$I$$ and single $$A$$ for both species). If P-value>0.05, models are similar; therefore we should use the Red.3 model, which means that the simplest model (Red.3) is appropriate to describe the crop-weed relationship.

anova(Full, Red.3)
## Analysis of Variance Table
##
## Model 1: yl ~ (I[weed] * densityweed)/(1 + (I[weed]/A[weed]) * densityweed)
## Model 2: yl ~ (I[weed] * densityweed)/(1 + (I[weed]/A) * densityweed)
##   Res.Df Res.Sum Sq Df  Sum Sq F value Pr(>F)
## 1     36     7056.7
## 2     37     7200.0 -1 -143.35  0.7313 0.3981

Results showed that P >0.05. Therefore, the Full model can be simplified to Red.3 model.

## Plotting the Red.3 model

### Rstudio basic figure

The command par is used to define the plot size. The command plot and lines are used to generate the figure, and the averaged points of yield loss at each density (Fig. 5). The command subset is adding each weed species separately in the plot (weed 1) and lines (weed 2).

The x is a user-defined name; it will contain the x-axis sequence of the data set. weed1 and weed2 is also a user-defined name, and this is the equation with the previous parameter estimates $$I$$ and $$A$$ estimated from Red.3 model using the nls function. Notice that the parameters estimated in Red.3 model were inserted in the rectangular hyperbola model for each weed species (Figure 1).

The command lines will insert the previous equation into the plot. Command lty, lwd, and col define the line type, size, and color, respectively.

The command legend will add the legend to the plot area.

par(mar=c(5,6,2,2), mgp=c(3,1.5,0))
plot(yl~densityweed, data=DMT, subset = weed =="1", pch=16, cex=1, las=1,
xlab=expression("Weed Density (plants pot"^-1*")"), ylim=c(-10,110),
ylab = "Yield Loss (%)", cex.axis=1, cex.lab=1)
lines(yl~densityweed, type="p",data=DMT, subset = weed =="2", col=2, cex=1, pch=1)
x=seq(0,4,0.25)
weed1=(228.357*x)/(1+(228.357/106.170)*x)
weed2=(37.000*x)/(1+(37.000/106.170)*x)
lines(x,weed1, lty=1, lwd=1, col=1)
lines(x,weed2, lty=3, lwd=1, col=2)
legend("bottomright", legend=c("C. benghalensis", "R. brasiliensis"), text.font = 3,
col=c(1,2), pch= c(16,1), lty=c(1,3), lwd= c(1,1), bty="n", cex=1)

### High-quality figure in Rstudio

The package ggplot2 (whithin tidyverse), an excellent package for producing high-quality figures in Rstudio.

DMT$weed<-factor(DMT$weed, levels=c("1", "2"),
labels=c("Commelina benghalensis",
"Richardia brasiliensis"))
Red.3 = nls(yl ~ (I[weed]*densityweed)/(1+(I[weed]/A)*densityweed), data=DMT,
start=list(I=c(30,30), A=70), trace=T)
## 53764.77 :  30 30 70
## 26821.46 :  128.12346  55.66148  65.14150
## 13779.51 :  265.38910  15.43032 100.87497
## 7506.945 :  205.09965  30.83279 107.29119
## 7203.363 :  225.88449  36.30998 106.26798
## 7200.048 :  228.19527  36.97054 106.19148
## 7200.044 :  228.35524  36.99872 106.17039
## 7200.044 :  228.35703  36.99971 106.16982
summary(Red.3)
##
## Formula: yl ~ (I[weed] * densityweed)/(1 + (I[weed]/A) * densityweed)
##
## Parameters:
##    Estimate Std. Error t value Pr(>|t|)
## I1  228.357    100.178   2.280   0.0285 *
## I2   37.000      6.196   5.972 6.85e-07 ***
## A   106.170     10.318  10.289 2.10e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.95 on 37 degrees of freedom
##
## Number of iterations to convergence: 7
## Achieved convergence tolerance: 1.144e-06
nd1 = data.frame(densityweed=seq(0, 4, 0.01), weed="Commelina benghalensis")
nd2 = data.frame(densityweed=seq(0, 4, 0.01), weed="Richardia brasiliensis")
nd = rbind(nd1, nd2) %>%
mutate(weed = as.factor(weed))

pred<- augment(Red.3, newdata=nd)
ggplot(DMT, aes(x=densityweed, y=yl, color=weed)) + geom_point(shape=1, size=3) +
geom_line(data = pred, size=1.3, aes(x=densityweed, linetype=weed, y=.fitted)) +
labs(fill="", y="Yield loss (%)",
x=expression(bold(paste("Weed density (plants pot"^"-2",")")))) +
scale_colour_manual(values = c("red", "black"))+
scale_y_continuous(limits=c(-25,110), breaks = c(-25,0,25,50,75,100)) +
theme(axis.text=element_text(size=15, color="black"),
axis.title=element_text(size=17,face="bold"),
panel.background = element_rect(fill="white", color = "white"),
panel.grid.major = element_line(color = "white"),
panel.grid.minor = element_blank(),
panel.border = element_rect(fill=NA,color="black", size=0.5,
linetype="solid"), legend.position=c(0.7,0.15),
legend.text = element_text(size = 12, colour = "black", face="italic"),
legend.key = element_rect(fill=NA), legend.key.height  = unit(1.5, "line"),
legend.key.width = unit(2.2, "line"),
legend.background = element_rect(fill =NA),  legend.title=element_blank())

Notice that the figure is created with Red.3 model (rectangular hyperbola model) using ggplot2 package (Figure 2). This is the Fig. 5 published in the manuscript Additive design: the concept and data analysis.

## AICc model selection and Goodness of fit

According to the AICc criterion, the top model has the lowest AICc value. The AICc calculation can be simplified using R, the first step is loading the package AICcmodavg.

library(broom)
library(AICcmodavg)

The four candidate models using the rectangular hyperbola are compared using AICc.

cand.mods<- list(Full, Red.1, Red.2, Red.3)

Modnames<- c('Full',' Red.1',' Red.2',' Red.3')
aictab(cand.set = cand.mods,modnames = Modnames, sort = TRUE)
##
## Model selection based on AICc:
##
##        K   AICc Delta_AICc AICcWt Cum.Wt      LL
##  Red.3 4 330.38       0.00   0.64   0.64 -160.62
## Full   5 332.19       1.82   0.26   0.89 -160.21
##  Red.2 4 333.91       3.53   0.11   1.00 -162.38
##  Red.1 3 368.19      37.82   0.00   1.00 -180.76

Root mean square error (RMSE) for goodness of fit of the top model (Red.3) selected.

mse <- mean(residuals(Red.3)^2/df.residual(Red.3))
rmse <- sqrt(mse)
rmse
## [1] 2.205651

## Obtaining the Confidence Internals for the Top model (Red.3)

It is needed the package nlstools and the command confint2 to obtain the 95% confidence intervals for parameters $$I$$ and $$A$$ for the Red.3.

library(nlstools)
confint2(Red.3, level=0.95)
##       2.5 %    97.5 %
## I1 25.37778 431.33628
## I2 24.44635  49.55307
## A  85.26278 127.07685

# Extra - Setting a limit to the rectangular hyperbola parameters

Here we demonstrate how to set an upper limit to parameter $$A$$ of Red.3 model. Notice that we have to add alg=“port” and upper command to the function. The upper command has three numbers, the first two set a limit of 10000 to parameter $$I$$ of R. brasiliensis and C. benghalensis. The last upper number set a limit $$A$$=100%, which will lock the upper limit to a biologically meaningful value.

Red.3_lim = nls(yl ~ (I[weed]*densityweed)/(1+(I[weed]/A)*densityweed), data=DMT,
start=list(I=c(30,30), A=70), alg="port",
upper=c(10000, 10000, 100), trace=T)
##   0:     26882.386:  30.0000  30.0000  70.0000
##   1:     18178.029:  42.8369  32.3788  84.6203
##   2:     6263.6597:  102.843  37.2539  100.000
##   3:     4270.9222:  165.478  39.1160  100.000
##   4:     3745.8945:  228.173  39.3120  100.000
##   5:     3641.7888:  285.567  39.3210  100.000
##   6:     3640.0157:  296.138  39.3212  100.000
##   7:     3640.0156:  296.037  39.3212  100.000
##   8:     3640.0156:  296.041  39.3212  100.000
summary(Red.3_lim)
##
## Formula: yl ~ (I[weed] * densityweed)/(1 + (I[weed]/A) * densityweed)
##
## Parameters:
##    Estimate Std. Error t value Pr(>|t|)
## I1  296.041    163.899   1.806    0.079 .
## I2   39.321      6.983   5.631 1.98e-06 ***
## A   100.000      9.337  10.710 6.85e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 14.03 on 37 degrees of freedom
##
## Algorithm "port", convergence message: relative convergence (4)

# Summary

• Fit the rectangular hyperbola model with nls (nonlinear least squares) function

• In the model, use brackets [weed] to generate parameters for each weed species

• Fit a Full and Reduced models and compare them with anova funcion

• Use RMSE to check the model goodness of fit

• Create a figure using plot or ggplot function