Computing SE for PSD indices

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5 years ago

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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.

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