Use power(N) in index_settings$q_form to allow catchability to change
with abundance. Ordinary formula terms specify baseline catchability.
N inside this marker is the model's abundance at survey time,
\(A=N\exp(-sZ)\), not an observation column or the observed index.
Details
With baseline \(\log q_0=X\beta\) (including other q effects), $$\log q_{effective}=\log q_0+(p-1)\log A,\qquad \log I\sim N(\log q_0+p\log A,\sigma^2).$$ The exponent is positive, \(p=\exp(\gamma)\); it starts at one. At one, the index is proportional to abundance and q is constant; above one q increases with abundance, and below one q increases as abundance declines. Error is multiplicative on the observation scale. Abundance is not automatically centered or rescaled; baseline q therefore depends on its units and on the exponent.
One additive power term per observation is supported, with the log q link
only. Disjoint for_survey() scopes can each contain a power term. It can be
combined with ordinary q terms and existing catchability curves and Gaussian
effects. Its coefficients are fixed effects, with no additional process SD.
Do not place it in interactions or numeric transformations.
Without by, one exponent is shared across all index observations. With
by, labels define separate exponents, even when labels are numeric. Missing
labels mean no adjustment (p = 1); these observations remain in the likelihood.
Use a separate grouping column to apply powers to selected survey ages.
Groups need fitted positive observations. Limited abundance variation and
flexible q effects can make exponents weakly identified; inspect uncertainty.
Projections and simulations use their own current abundance, not terminal
effective q. tidy_obs_pred() reports effective q, q_base and q_power;
tidy_par() reports the exponent with log-scale SEs. For fixed-effect
coefficient tables, baseline q units depend on p; effective q retains the
index-unit/population-unit ratio.
Compare q_base with a source baseline-q table; baseline q alone is not an
effective-q reference when p differs from one.
Examples
~ q_key + power(N, by = q_key)
#> ~q_key + power(N, by = q_key)
#> <environment: 0x5edf822965c0>
~ survey + power(N)
#> ~survey + power(N)
#> <environment: 0x5edf822965c0>
# Set q_power_group to NA for survey ages without an estimated power:
~ q_key + power(N, by = q_power_group)
#> ~q_key + power(N, by = q_power_group)
#> <environment: 0x5edf822965c0>