Height of barley plants in a study of non-normal data

Height of barley plants in a study of non-normal data.

data("eden.nonnormal")

Format

A data frame with 256 observations on the following 3 variables.

pos

position within block

block

block (numeric)

height

height of wheat plant

Details

This data was used in a very early example of a permutation test.

Eden & Yates used this data to consider the impact of non-normal data on the validity of a hypothesis test that assumes normality. They concluded that the skew data did not negatively affect the analysis of variance.

Grown at Rothamsted. Eight blocks of Yeoman II wheat. Sampling of the blocks was quarter-meter rows, four times in each row. Rows were selected at random. Position within the rows was partly controlled to make use of the whole length of the block. Plants at both ends of the sub-unit were measured. Shoot height is measured from ground level to the auricle of the last expanded leaf.

Source

T. Eden, F. Yates (1933). On the validity of Fisher's z test when applied to an actual example of non-normal data. Journal of Agric Science, 23, 6-17. https://doi.org/10.1017/S0021859600052862

References

Kenneth J. Berry, Paul W. Mielke, Jr., Janis E. Johnston Permutation Statistical Methods: An Integrated Approach.

Examples

library(agridat)

data(eden.nonnormal)
dat <- eden.nonnormal
mean(dat$height) # 55.23 matches Eden table 1
#> [1] 55.23242

# Eden figure 2
libs(dplyr, lattice)
# Blocks had different means, so substract block mean from each datum
dat <- group_by(dat, block)
dat <- mutate(dat, blkmn=mean(height))
dat <- transform(dat, dev=height-blkmn)

histogram( ~ dev, data=dat, breaks=seq(from=-40, to=30, by=2.5),
          xlab="Deviations from block means",
          main="eden.nonnormal - heights skewed left")

# \dontrun{

  # calculate skewness, permutation 
  
  libs(dplyr, lattice, latticeExtra)
  
  # Eden table 1
  # anova(aov(height ~ factor(block), data=dat))
  
  # Eden table 2,3. Note, this may be a different definition of skewness
  # than is commonly used today (e.g. e1071::skewness).
  skew <- function(x){
    n <- length(x)
    x <- x - mean(x)
    s1 = sum(x)
    s2 = sum(x^2)
    s3 = sum(x^3)
    k3=n/((n-1)*(n-2)) * s3 -3/n*s2*s1 + 2/n^2 * s1^3
    return(k3)
  }
  # Negative values indicate data are skewed left
  dat <- group_by(dat, block)
  summarize(dat, s1=sum(height),s2=sum(height^2), mean2=var(height), k3=skew(height))
#> `summarise()` ungrouping output (override with `.groups` argument)
#> # A tibble: 8 x 5
#>   block    s1      s2 mean2     k3
#>   <int> <dbl>   <dbl> <dbl>  <dbl>
#> 1     1 1682   95930. 243.  -1269.
#> 2     2 1858  111662. 122.  -1752.
#> 3     3 1810. 108967. 214.  -3173.
#> 4     4 1912  121748. 242.  -2548.
#> 5     5 1722   99026. 205.   -559.
#> 6     6 1339   63077  227.   -801.
#> 7     7 1963  123052.  85.0  -713.
#> 8     8 1854  112366  160.  -1062.
  ##   block     s1       s2     mean2         k3
  ##   <int>  <dbl>    <dbl>     <dbl>      <dbl>
  ## 1     1 1682.0  95929.5 242.56048 -1268.5210
  ## 2     2 1858.0 111661.5 121.97984 -1751.9919
  ## 3     3 1809.5 108966.8 214.36064 -3172.5284
  ## 4     4 1912.0 121748.5 242.14516 -2548.2194
  ## 5     5 1722.0  99026.5 205.20565  -559.0629
  ## 6     6 1339.0  63077.0 227.36190  -801.2740
  ## 7     7 1963.0 123052.5  84.99093  -713.2595
  ## 8     8 1854.0 112366.0 159.67339 -1061.9919

  # Another way to view skewness with qq plot. Panel 3 most skewed.
  qqmath( ~ dev|factor(block), data=dat,
         as.table=TRUE,
         ylab="Deviations from block means",
         panel = function(x, ...) {
           panel.qqmathline(x, ...)
           panel.qqmath(x, ...)
         })

  # Now, permutation test.
  # Eden: "By a process of amalgamation the eight sets of 32 observations were
  # reduced to eight sets of four and the data treated as a potential
  # layout for a 32-plot trial".
  dat2 <- transform(dat, grp = rep(1:4, each=8))
  dat2 <- aggregate(height ~ grp+block, dat2, sum)
  dat2$trt <- rep(letters[1:4], 8)
  dat2$block <- factor(dat2$block)

  # Treatments were assigned at random 1000 times
  set.seed(54323)
  fobs <- rep(NA, 1000)
  for(i in 1:1000){
    # randomize treatments within each block
    # trick from https://stackoverflow.com/questions/25085537
    dat2$trt <- with(dat2, ave(trt, block, FUN = sample))
    fobs[i] <- anova(aov(height ~ block + trt, dat2))["trt","F value"]
  }

  # F distribution with 3,21 deg freedom
  # Similar to Eden's figure 4, but on a different horizontal scale
  xval <- seq(from=0,to=max(fobs), length=50)
  yval <- df(xval, df1 = 3, df2 = 21)
  # Re-scale, 10 = max of historgram, 0.7 = max of density
  histogram( ~ fobs, breaks=xval,
            xlab="F value",
            main="Observed (histogram) & theoretical (line) F values") +
    xyplot((10/.7)* yval ~ xval, type="l", lwd=2)

# }