weights vs. offset (negative binomial regression)

Using an offset of log(Effort) as in your second model is the more 
standard way to approach this problem; it corresponds to assuming that 
catch is strictly proportional to effort. Adding log(Effort) as a 
covariate (as illustrated below) tests whether a power-law model (catch 
propto (Effort)^(b+1), b!=0) is a better description of the data.  (In 
this case it is not, although the confidence intervals on b are very 
wide, indicating that we have very little information -- this is not 
surprising since the proportional range of effort is very small 
(246-258) in this data set.

   In general you should *not* check overdispersion of the raw data 
(i.e., the *marginal distribution* of the data, you should check 
overdispersion of a fitted (e.g. Poisson) model, as below.

   cheers
    Ben Bolker

edata <- data.frame(Catch, Effort, xx1, xx2, xx3)

## graphical exploration

library(ggplot2); theme_set(theme_bw())
library(tidyr)
edata_long <- edata |> pivot_longer(names_to="var", cols =-c("Catch", 
"Effort"))
ggplot(edata_long, aes(value, Catch)) +
     geom_point(alpha = 0.2, aes(size = Effort)) +
     facet_wrap(~var, scale="free_x") +
     geom_smooth(method = "glm", method.args = list(family = 
"quasipoisson"))
#

library(MASS)
g1 <- glm.nb(Catch~xx1+xx2+xx3+offset(log(Effort)), data=edata)
g2 <- update(g1, . ~ . + log(Effort))
g0 <- glm(Catch~xx1+xx2+xx3+offset(log(Effort)), data=edata,
           family = poisson)
performance::check_overdispersion(g0)
summary(g1)
summary(g2)
options(digits = 3)
confint(g2)
summary(g1)

weights vs. offset (negative binomial regression)

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