Relative cotton yield for different soil potassium concentrations

Format

A data frame with 24 observations on the following 2 variables.

yield: Relative yield
potassium: Soil potassium, ppm

Details

Cate & Nelson used this data to determine the minimum optimal amount of soil potassium to achieve maximum yield.

Note, Fig 1 of Cate & Nelson does not match the data from Table 2. It sort of appears that points with high-concentrations of potassium were shifted left to a truncation point. Also, the calculations below do not quite match the results in Table 1. Perhaps the published data were rounded?

Source

Cate, R.B. and Nelson, L.A. (1971). A simple statistical procedure for partitioning soil test correlation data into two classes. Soil Science Society of America Journal, 35, 658--660. https://doi.org/10.2136/sssaj1971.03615995003500040048x

Examples


library(agridat)

data(cate.potassium)
dat <- cate.potassium
names(dat) <- c('y','x')

CateNelson <- function(dat){
  dat <- dat[order(dat$x),] # Sort the data by x
  x <- dat$x
  y <- dat$y

  # Create a data.frame to store the results
  out <- data.frame(x=NA, mean1=NA, css1=NA, mean2=NA, css2=NA, r2=NA)

  css <- function(x) { var(x) * (length(x)-1) }
  tcss <- css(y) # Total corrected sum of squares

  for(i in 2:(length(y)-2)){
    y1 <- y[1:i]
    y2 <- y[-(1:i)]

    out[i, 'x'] <- x[i]
    out[i, 'mean1'] <- mean(y1)
    out[i, 'mean2'] <- mean(y2)
    out[i, 'css1'] <- css1 <- css(y1)
    out[i, 'css2'] <- css2 <- css(y2)
    out[i, 'r2'] <-  ( tcss - (css1+css2)) / tcss
  }
  return(out)
}

cn <- CateNelson(dat)
ix <- which.max(cn$r2)
with(dat, plot(y~x, ylim=c(0,110), xlab="Potassium", ylab="Yield"))
title("cate.potassium - Cate-Nelson analysis")
abline(v=dat$x[ix], col="skyblue")
abline(h=(dat$y[ix] + dat$y[ix+1])/2, col="skyblue")

# \dontrun{
  # another approach with similar results
  # https://joe.org/joe/2013october/tt1.php
  libs("rcompanion")
  cateNelson(dat$x, dat$y, plotit=0)
#> ....................
#> 
#> Critical x that maximize sum of squares: 
#>  
#>   Critical.x.value Sum.of.squares
#> 1             46.5       5374.217
#> ........................
#> 
#> Critical y that minimize errors: 
#>  
#>   Critical.y.value Q.i Q.ii Q.iii Q.iv Q.model Q.err Cramer.V
#> 1            82.40   0   15     1    8      23     1   0.9129
#> 2            65.65   2   16     0    6      22     2   0.8165
#> 3            74.90   1   15     1    7      22     2   0.8125
#> 4            85.55   0   14     2    8      22     2   0.8367
#> 
#> n         = Number of observations 
#> CLx       = Critical value of x 
#> SS        = Sum of squares for that critical value of x 
#> CLy       = Critical value of y 
#> Q         = Number of observations which fall into quadrants I, II, III, IV 
#> Q.Model   = Total observations which fall into the quadrants predicted by the model 
#> p.Model   = Percent observations which fall into the quadrants predicted by the model 
#> Q.Error   = Observations which do not fall into the quadrants predicted by the model 
#> p.Error   = Percent observations which do not fall into the quadrants predicted by the model 
#> Fisher.p  = p-value from Fisher exact test dividing data into these quadrants 
#> Cramer.V  = Cramer's V statistic from dividing data into these quadrants 
#> 
#> Final model: 
#>  
#>    n  CLx       SS  CLy Q.I Q.II Q.III Q.IV Q.Model   p.Model Q.Error
#> 1 24 46.5 5374.217 82.4   0   15     1    8      23 0.9583333       1
#>      p.Error Fisher.p.value Cramer.V
#> 1 0.04166667   1.223706e-05   0.9129
# }