A user of my Introductory Fisheries Analyses with R book recently asked me how to compute standard errors (SE) for the various PSD indices by using the usual equation for the SE of a proportion (square root of p times (1-p) divided by n). Below is my response using the Inch Lake data from the book.

  • Load required packages.
library(FSA)
library(magrittr)
library(dplyr)
  • Load (and prepare) the Inch Lake data used in the Size Structure chapter of the book (this and next step is from this script).
inchAlls <- read.csv("http://derekogle.com/IFAR/scripts/InchLake1113.csv") %>%
  mutate(gcat=psdAdd(tl,species)) %>%
  filterD(gcat!="substock")
## No known Gabelhouse (PSD) lengths for Iowa Darter
  • Compute the PSD-X values.
( freq <- xtabs(~species+gcat,data=inchAlls) )
##                  gcat
## species           stock quality preferred memorable
##   Black Crappie      33      40         8        81
##   Bluegill          571     199       223        43
##   Largemouth Bass    26      36         0         0
##   Pumpkinseed         5       2         0         0
##   Yellow Perch        4       0         0         1
iPSDs <- prop.table(freq,margin=1)*100
( PSDs <- t(apply(iPSDs,MARGIN=1,FUN=rcumsum)) )
##                  gcat
## species           stock  quality preferred memorable
##   Black Crappie     100 79.62963  54.93827 50.000000
##   Bluegill          100 44.88417  25.67568  4.150579
##   Largemouth Bass   100 58.06452   0.00000  0.000000
##   Pumpkinseed       100 28.57143   0.00000  0.000000
##   Yellow Perch      100 20.00000  20.00000 20.000000
  • Compute the row sums for the freq table to get the total number of fish (that are stock size or greater). These will be n in the SE calculations.
( sums <- rowSums(freq) )
##   Black Crappie        Bluegill Largemouth Bass     Pumpkinseed 
##             162            1036              62               7 
##    Yellow Perch 
##               5
  • Compute the SE for PSD-Q.
SEs <- sqrt(PSDs[,"quality"]*(100-iPSDs[,"quality"])/sums)
round(SEs,1)
##   Black Crappie        Bluegill Largemouth Bass     Pumpkinseed 
##             6.1             1.9             6.3            17.1 
##    Yellow Perch 
##            20.0
  • Repeat for other size indices. For example, for PSD-P.
SEs <- sqrt(PSDs[,"preferred"]*(100-iPSDs[,"preferred"])/sums)
round(SEs,1)
##   Black Crappie        Bluegill Largemouth Bass     Pumpkinseed 
##             5.7             1.4             0.0             0.0 
##    Yellow Perch 
##            20.0

Note that the SE formula is usually used for proportions (and then multiplied by 100 to get a SE for percentages). However, it can also be shown algebraically to be useful for percentages.


I would caution against using these standard errors to produce confidence intervals for the PSD-X values. If confidence intervals are the end-product for this analysis, then I suggest using the methods described in my book.