1. Standardizing catch per unit effort data

2. Standardization of CPUE
Catch = catchability * Effort * Biomass
CPUE = Catch/Effort = U = catchability * Biomass
Ut = qBt
Ut: Catch per unit effort at time t
q : catchability of the whole fleet
catchability: the proportion of the stock caught per one unit of effort
In most fisheries we normally have fleets with different catchabilities
Lets start by looking at a fishery on a stock where we have vessel types (i) that have different catchabilities:
Ut,i: Catch per unit effort of vessel type i at time t
qi : catchability of vessel type i

3. A very simplified artificial case: 2 Fleets
For illustration we create some CPUE data for 6 years for 2 fleets from known stock size, effort and catchability
Catchability is fleet specific, with fleet 2 having 3 times higher catchability than fleet 1
Effort in Fleet 1 declines while it increases in fleet 2. Total effort remains constant

4. Stock size and overall unstandardize CPUE

5. Relative values
lets standardize the overall CPUE series relative to that of the first year:
it is obvious that ignoring the different catchabilities of fleet 1 and 2 would lead wrong conclusion about biomass development
however, if we were to use either Fleet 1 OR 2 we would get accurate representation of the relative change in biomass
CPUE from some “selected” fleet, which is assumed to be homogenous over time is often used in practice
The problem is that the assumption of homogeneity is an assumption in real cases!

6. Relative stock size and overall CPUE
it is obvious that ignoring the different catchabilities of fleet 1 and 2 would lead wrong conclusion about biomass development when using the

7. Standardizing the catch rate of each fleet
lets standardized the CPUE series of each fleet relative to the first year:
if we were to use either Fleet 1 OR 2 we would get accurate representation of the relative change in biomass
CPUE from some “selected” fleet, which is assumed to be homogenous over time are often used in practice

8. Relative stock size and CPUE from Fleet 1 and 2

9. General linear modeling of CPUE data – the math
Relative changes in biomass:
Lets first describe changes in biomass relative to the first year in the data series:
Bt – biomass at time t
B1 – biomass in year 1
at – scaling factor
where:
and hence a1 = 1.00
The aim is to use the at parameter in the relationship Ut = qBt

10. Biomass (Bt) and relative biomass (at)

11. General linear modeling of CPUE data – the math
We have
then
the a92, a93, a94, … are hence (again) a measure of biomass (B) relative to a91 = 1.00 but here we have related it to CPUE
more importantly we gotten rid of the actual biomass (B)!

12. General linear modeling of CPUE data – the math
The relationship:
only applies to homogenous fleet
Lets revisit the imaginary 2 vessel class fisheries (where we “know” that q within a fleet has remained constant):
Here the b2|1 is the efficiency of vessel class 2 relative to vessel class 1.
The mathematical formula is effectively saying:
the CPUE of vessel class 2 at any one time t is just a multiplier of CPUE of vessel class 1 at time t taking into account changes relative changes in biomass (at) since first year

13. General linear modeling of CPUE data (4)
In general for multivessel fisheries we can write
where
i: vessel size class i
Ut,i : CPUE of the for time t and vessel class i
U1,1: CPUE of the 1st vessel class in the 1st time period
bi: The efficiency of vessel class i relative to vessel class 1
at: Relative abundance
Food for thought:
What is the value of at when t = 1?
What is the value of bi when i = 1?

14. General linear modeling of CPUE data (5)
To take into account measurement errors the statistical model becomes:
The error can be normalized by transformation:
We hence have a general linear model which can be used to estimate the parameters. For stock assessment purposed the parameters at is of most interest. However, one could consider that the bi parameters may be of interest in terms of understanding the fishery and for management
What we have here is nothing more than:
Yi = Ŷi i
… and we know how we estimates the parameters of such a simple model 
Food for thought:
What is the value of ln(at) when t = 1?
What is the value of ln(bi) when i = 1?

15. General linear modeling of CPUE data (6)
The GLIM model fit is often done by rescaling all the cpue observation to that of U1,1 (as we have already done) I.e.:

16. Our example
Lets first add some measurement noise (stochasticity) to our artificial deterministic CPUE data:

17. Spreadsheet schematics of the model for the simplified example
The ln-value for t=1 and i=1 is by definition zero
The b value for the reference fleet i=1 is always zero, irrespective of the year (t)
The a value for the reference fleet i=2 is the same as for i=1 within each year
Just moving the parameter values here for clarity/convenience

18. Best fit
Minimized the squared residuals to obtain the best parameter estimates

19. Expanding the GLM model
The expansion of the GLM model to take into account:
Area
Season/month …
is mathematically straightforward:
the model fitting process is the same?

20. Catch rate may not be proportional to abundance
Where it goes wrong …
Catch rate may not be proportional to abundance
Hence the abundance trend from GLM will not be proportional to abundance
Any changes unrelated to quantifiable effects will not be captured in the GLM analysis.
In such cases the change will wrongly be ascribed to changes in abundance
e.g. increase in vessel efficiency within a fleet class due to increased skill
ERGO:
GLM analysis is the best tool available to calculate standardized catch rate
Weather the actual abundance trend form GLM represents true changes in stock abundance will always be a subjective call