Maturity Ogive Estimation with fit_bayesian

Overview

The fit_bayesian_maturity() function estimates length-at-maturity (\(L_{50}\)) and/or age-at-maturity (\(t_{50}\)) using Bayesian probit regression. It supports single-sex and two-sex models with optional partial pooling between sexes.

The Model

\[P(\text{mature}_i) = \Phi[\beta \cdot (x_i - x_{50})]\]

where \(x\) is either length (\(L\)) or age (\(t\)), and \(x_{50}\) is the value at which 50% of individuals are mature.

Derived quantities:

\(x_{05} = x_{50} - 1.645/\beta\) (5% maturity)
\(x_{95} = x_{50} + 1.645/\beta\) (95% maturity)
Transition width \(= x_{95} - x_{05} = 3.29/\beta\)

Basic Usage: Single-Sex Model

library(vitalBayes)
library(data.table)

# Load simulated data
data(growth_data)

# Filter to free-swimming females only
female_data <- growth_data[embryo == FALSE & sex == "female"]

# Fit length-at-maturity
L50_fit <- fit_bayesian_maturity(
 maturity = "mat",
 lt       = "fl",
 data     = female_data,
 chains   = 4,
 parallel = TRUE
)

# View results
L50_fit$summary(c("L50", "slope", "L05", "L95", "transition_width"))

Two-Sex Models

When sex data are available, fit both sexes simultaneously:

# Prepare data (both sexes)
mat_data <- growth_data[embryo == FALSE & !is.na(mat)]

# Two-sex model with partial pooling (recommended)
L50_fit_2sex <- fit_bayesian_maturity(
 maturity    = "mat",
 lt          = "fl",
 sex         = "sex",
 data        = mat_data,
 use_pooling = TRUE,   # Enable partial pooling
 prior_tau   = 0.5     # Half-normal scale for between-sex SD
)

# Sex-specific estimates
L50_fit_2sex$summary("L50")

# Difference between sexes
L50_fit_2sex$summary("L50_diff")

Why Partial Pooling?

Elasmobranch datasets often have imbalanced sex ratios. With 150 females but only 34 males, the male estimate would have very wide credible intervals if fit independently.

Partial pooling addresses this by modeling sex-specific parameters as draws from a common distribution:

\[\log(L_{50,s}) = \mu + \tau \cdot \eta_s, \quad \eta_s \sim \mathcal{N}(0,1)\]

The model learns the appropriate degree of pooling (\(\tau\)) from the data:

Large \(\tau\) → sexes are different → estimates stay separated
Small \(\tau\) → sexes are similar → estimates shrink together
Sparse sex → borrows strength from the other sex

For a comprehensive treatment, see vignette("partial_pooling") or the Partial Pooling section in the Statistical Methods guide.

Fitting Both Length and Age Maturity

# Filter to individuals with both maturity and age data
mat_aged <- growth_data[embryo == FALSE & !is.na(mat) & !is.na(age)]

# Fit both models simultaneously
mat_fits <- fit_bayesian_maturity(
 maturity = "mat",
 lt       = "fl",
 age      = "age",
 sex      = "sex",
 data     = mat_aged,
 use_pooling = TRUE
)

# Access individual fits
mat_fits$length$summary("L50")
mat_fits$age$summary("t50")

Sex Coding

The function auto-detects common sex coding conventions:

# All of these work automatically:
# sex = c("F", "M")           # English
# sex = c("Female", "Male")   # English full
# sex = c("female", "male")   # lowercase (used in simulated data)
# sex = c(1, 2)               # Numeric (1=female, 2=male)
# sex = c("Hembra", "Macho")  # Spanish
# sex = c("Femelle", "Mâle")  # French
# sex = c("メス", "オス")      # Japanese

# For non-standard coding, specify explicitly:
fit <- fit_bayesian_maturity(
 maturity = "mat",
 lt       = "fl",
 sex      = "sex",
 female   = "A",   # Your female code
 male     = "B",   # Your male code
 data     = your_data
)

Prior Specification

Priors use a CV-based approach for intuitive specification:

# Custom priors
L50_fit <- fit_bayesian_maturity(
 maturity   = "mat",
 lt         = "fl",
 data       = mat_data,
 mean_L50   = 75,     # Prior mean (cm)
 cv_L50     = 0.2,    # 20% CV → SD = 15 cm
 mean_slope = 0,      # Log-scale prior for slope
 sd_slope   = 1
)

Visualization

# Basic maturity ogive
plot_maturity_ogive(
 fit  = L50_fit_2sex,
 type = "length",
 data = mat_data,
 x_col = "fl",
 maturity_col = "mat",
 sex_col = "sex"
)

# With French labels
plot_maturity_ogive(
 fit        = L50_fit_2sex,
 type       = "length",
 data       = mat_data,
 sex_labels = c("female" = "Femelle", "male" = "Mâle"),
 x_lab      = "Longueur (cm)",
 y_lab      = "Probabilité de maturité"
)

Posterior Predictive Checks

# Classification accuracy
L50_fit_2sex$summary("prop_correct_rep")

# Mean predicted probability by actual status
L50_fit_2sex$summary(c("mean_p_mature_f", "mean_p_mature_m",
                      "mean_p_immature_f", "mean_p_immature_m"))

Using Output in Growth Models

The \(L_{50}\) and \(t_{50}\) estimates become informative priors for the maturity-based growth parameterization:

# Fit t50 for age-at-maturity
t50_fit <- fit_bayesian_maturity(
 maturity = "mat",
 age      = "age",
 sex      = "sex",
 data     = mat_aged
)

# Pass to growth model
growth_fit <- fit_bayesian_growth(
 lt      = "fl",
 age     = "age",
 sex     = "sex",
 data    = growth_data[embryo == FALSE & !is.na(age)],
 k_based = FALSE,           # Use maturity-based parameterization
 L50_fit = L50_fit_2sex,    # Length-at-maturity fit
 t50_fit = t50_fit          # Age-at-maturity fit
)

Troubleshooting

Issue	Solution
Divergent transitions	Increase `adapt_delta`; check for separation in data
Very wide \(\tau\) estimate	Data may not support pooling; try `use_pooling = FALSE`
Poor classification	Check maturity coding (must be 0/1)
Sex detection fails	Use explicit `female`/`male` arguments

Maturity Ogive Estimation with fit_bayesian_maturity()