BT_temp.qmd

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# Bluetongue Virus: Thermal Performance Curves

**Source:** El Moustaid et al. 2021 Supplement | **Data:** Lysyk & Danyk 2007, Mayo et al. 2020

This analysis recreates key thermal performance curves from the Bluetongue virus transmission study, fitting Bayesian models to estimate Maximum A Posteriori (MAP) parameters for each trait. The solid lines show posterior means/medians with 95% HPD intervals (dashed lines).

```{r setup, include=FALSE}
library(nimble)
library(bayesTPC)
library(dplyr)
library(ggplot2)
library(coda)
library(HDInterval)
library(patchwork)
```

## Adult Mortality Rate (μ)

```{r mu-data, echo=FALSE, message=FALSE, warning=FALSE}
# Load updated dataset from CSV and filter for adult mortality rate (mu)
mu_df <- readr::read_csv("data/Bluetongue_Params_0723.csv", show_col_types = FALSE) %>%
  dplyr::filter(Trait == "mu") %>%
  dplyr::transmute(T = AmbientTemp, mu = TraitValueSI) %>%
  dplyr::arrange(T)
cat("Temp range:", paste(range(mu_df$T), collapse = "–"), "°C\n")
cat("mu range:", paste(round(range(mu_df$mu), 3), collapse = "–"), " per day\n")
```

## Thermal Performance Curve (Quadratic via bayesTPC)

```{r mu-fit, echo=FALSE, message=FALSE, warning=FALSE}
# Load updated dataset from CSV and filter for adult mortality rate (mu)
mu_df <- readr::read_csv("data/Bluetongue_Params_0723.csv", show_col_types = FALSE) %>%
  dplyr::filter(Trait == "mu") %>%
  dplyr::transmute(T = AmbientTemp, mu = TraitValueSI) %>%
  dplyr::arrange(T)
mu_data_list <- list(Temp = mu_df$T, Trait = mu_df$mu)
set.seed(123)
fit_mu <- b_TPC(
  data = mu_data_list,
  model = "quadratic",
  priors = list(
    T_min    = "dunif(5,15)",
    T_max    = "dunif(18,40)",
    q        = "dunif(0, 0.5)",
    sigma.sq = "dexp(1 / 0.1^2)"
  ),
  nchains = 4,
  burn = 6000,
  niter = 18000
)
```

```{r mu-plot, echo=FALSE, message=FALSE, warning=FALSE, fig.width=8, fig.height=5}
# Use ALL chains and established prediction pipeline
samples_mu <- do.call(rbind, lapply(fit_mu$samples, as.matrix))
temp_grid <- seq(0, 50, by = 0.1)

# Evaluate quadratic model using bayesTPC helper to avoid formula drift
quad_fun <- bayesTPC::get_model_function("quadratic")
pred_mu <- sapply(temp_grid, function(Ti) quad_fun(
  q = samples_mu[, "q"], T_min = samples_mu[, "T_min"], T_max = samples_mu[, "T_max"], Temp = Ti
))

# Mean curve + 95% HPD across posterior draws
mean_mu <- colMeans(pred_mu)
hpd_mu <- apply(pred_mu, 2, function(x) as.numeric(HDInterval::hdi(x, credMass = 0.95)))
lo_mu <- hpd_mu[1, ]; hi_mu <- hpd_mu[2, ]
Topt_mu <- temp_grid[which.max(mean_mu)]

par(mfrow = c(1,1), mar = c(4,4,2,1))
plot(temp_grid, mean_mu, type = "l", lwd = 2, col = "black",
     xlab = "T (°C)", ylab = "Adult Mortality Rate (per day)",
     ylim = c(0, max(0.2, max(hi_mu, mu_df$mu, na.rm = TRUE) * 1.05)), xlim = c(0, 50), bty = "l")
lines(temp_grid, lo_mu, lty = 2, lwd = 1.2)
lines(temp_grid, hi_mu, lty = 2, lwd = 1.2)
points(mu_df$T, mu_df$mu, pch = 16, col = "black")
abline(v = Topt_mu, lty = 3)
lbl <- sprintf("Topt ≈ %.1f°C (mean curve)", Topt_mu)
if (length(lbl) == 1 && !is.na(lbl)) mtext(lbl, side = 3, adj = 1, cex = 0.9)
```

```{r mu-param-summary, echo=FALSE, message=FALSE, warning=FALSE}
# MAP Estimates
map_mu <- MAP_estimate(fit_mu)
cat("**MAP Estimates (μ):**\n")
if(!is.null(map_mu)){
  if("T_min" %in% names(map_mu)) cat("• T_min =", round(map_mu["T_min"], 2), "°C\n")
  if("T_max" %in% names(map_mu)) cat("• T_max =", round(map_mu["T_max"], 2), "°C\n")
  if("q" %in% names(map_mu))     cat("• q =", round(map_mu["q"], 4), "\n")
  if("sigma.sq" %in% names(map_mu)) cat("• sigma.sq =", round(map_mu["sigma.sq"], 5), "\n")
} else {
  cat("(MAP unavailable)\n")
}
```
## Midge Fecundity (EFD)

```{r fit-fecundity, echo=FALSE, message=FALSE, warning=FALSE}
fecundity <- readr::read_csv("data/Bluetongue_Params_0723.csv", show_col_types = FALSE) %>%
  dplyr::filter(Trait == "EFD") %>%
  dplyr::transmute(T = AmbientTemp, F = TraitValueSI)
# Prepare data for bayesTPC
data_list <- list(Temp = fecundity$T, Trait = fecundity$F)

# Fit Brière model with biologically realistic priors
set.seed(123)
fit_fec <- b_TPC(
  data = data_list,
  model = "briere",
  priors = list(
    T_min = "dunif(5, 15)",    # Lower thermal limit
    T_max = "dunif(32, 36)",   # Upper thermal limit
    q = "dunif(0, 200)"        # Scaling parameter
  ),
  nchains = 4,
  burn = 6000,
  niter = 18000
)
```

```{r plot-fecundity, echo=FALSE, message=FALSE, warning=FALSE, fig.width=8, fig.height=5}
# Generate temperature grid for predictions
temp_grid <- seq(0, 50, by = 0.1)

# Get posterior samples
samples <- as.matrix(fit_fec$samples[[1]])

# Generate predictions for each sample
predictions <- apply(samples, 1, function(params) {
  T_min <- params["T_min"]
  T_max <- params["T_max"] 
  q <- params["q"]
  
  # Brière function
  ifelse(temp_grid >= T_min & temp_grid <= T_max,
         q * temp_grid * (temp_grid - T_min) * sqrt(pmax(T_max - temp_grid, 0)),
         0)
})

# Calculate median and HPD intervals
median_pred <- apply(predictions, 1, median)
hpd_lower <- apply(predictions, 1, function(x) HPDinterval(as.mcmc(x), prob = 0.95)[1])
hpd_upper <- apply(predictions, 1, function(x) HPDinterval(as.mcmc(x), prob = 0.95)[2])

# Create single plot matching screenshot style
par(mfrow = c(1, 1), mar = c(4, 4, 2, 1))
plot(temp_grid, median_pred, 
     type = "l", 
     lwd = 2, 
     col = "black",
     xlab = "T (°C)", 
     ylab = "Eggs per Female per Day",
     main = "",
     ylim = c(0, 80),
     xlim = c(0, 50),
     cex.lab = 1.1,
     cex.axis = 1.0,
     bty = "l")

# Add HPD interval as dashed lines
lines(temp_grid, hpd_lower, lty = 2, lwd = 1.5, col = "black")
lines(temp_grid, hpd_upper, lty = 2, lwd = 1.5, col = "black")

# Add data points as solid black circles
points(fecundity$T, fecundity$F, pch = 16, col = "black", cex = 1.0)
```

```{r param-summary, echo=FALSE, message=FALSE, warning=FALSE}
# MAP Estimates (Maximum A Posteriori)
map_params <- MAP_estimate(fit_fec)
cat("**MAP Estimates (EFD):**\n")
cat("• T_min =", round(map_params["T_min"], 2), "°C\n")
cat("• T_max =", round(map_params["T_max"], 2), "°C\n") 
cat("• q =", round(map_params["q"], 3), "\n")
cat("• sigma.sq =", round(map_params["sigma.sq"], 4), "\n")
```

## Larval Survival Probability (pL)

```{r pl-data-fit, echo=FALSE, message=FALSE, warning=FALSE}
# Load updated dataset from CSV and filter for larval survival probability (pL)
pL <- readr::read_csv("data/Bluetongue_Params_0723.csv", show_col_types = FALSE) %>%
  dplyr::filter(Trait == "pL") %>%
  dplyr::transmute(T = AmbientTemp, pL = TraitValueSI) %>%
  dplyr::arrange(T)

# Fit Brière model using bayesTPC
pl_data_list <- list(Temp = pL$T, Trait = pL$pL)
set.seed(123)
fit_pL <- b_TPC(
  data = pl_data_list,
  model = "briere",
  priors = list(
    T_min   = "dunif(18,22)",
    T_max   = "dunif(35,40)",
    q       = "dunif(0,0.03)",
    sigma.sq = "dexp(1 / 0.05^2)"
  ),
  nchains = 4,
  burn = 6000,
  niter = 18000
)
```

```{r pl-plot, echo=FALSE, message=FALSE, warning=FALSE, fig.width=8, fig.height=5}
# 1) Use ALL chains
samples_pL <- do.call(rbind, lapply(fit_pL$samples, as.matrix))

# 2) Predictions on grid
temp_grid <- seq(0, 50, by = 0.1)
briere_fun <- function(T, T_min, T_max, q) {
  ifelse(T >= T_min & T <= T_max,
         q * T * (T - T_min) * sqrt(pmax(T_max - T, 0)), 0)
}

pred_pL <- sapply(temp_grid, function(Ti) {
  briere_fun(Ti, samples_pL[, "T_min"], samples_pL[, "T_max"], samples_pL[, "q"])
})

# 3) Summaries (median curve + 95% HPD for the mean curve)
median_pL <- apply(pred_pL, 2, median)
hpd_bounds <- apply(pred_pL, 2, function(x) as.numeric(HDInterval::hdi(x, credMass = 0.95)))
hpd_lower_pL <- hpd_bounds[1, ]
hpd_upper_pL <- hpd_bounds[2, ]

# 4) Topt = argmax of median curve (robust and version-agnostic)
Topt_pL <- temp_grid[which.max(median_pL)]

# 5) Plot
par(mfrow = c(1,1), mar = c(4,4,2,1))
plot(temp_grid, median_pL,
  type = "l", lwd = 2, col = "black",
  xlab = "T (°C)", ylab = "Larval Survival Probability",
  ylim = c(0, 1), xlim = c(0, 50), bty = "l", yaxt = "n")
axis(2, at = seq(0,1, by = 0.2), las = 1)
lines(temp_grid, hpd_lower_pL, lty = 2, lwd = 1.2)
lines(temp_grid, hpd_upper_pL, lty = 2, lwd = 1.2)
points(pL$T, pL$pL, pch = 16, col = "black")
abline(v = Topt_pL, lty = 3)

# Safe label for mtext()
lbl <- sprintf("Topt ≈ %.1f°C (median curve)", Topt_pL)
if (length(lbl) == 1 && !is.na(lbl)) {
  mtext(lbl, side = 3, adj = 1, cex = 0.9)
}
```

```{r pl-param-summary, echo=FALSE, message=FALSE, warning=FALSE}
# MAP Estimates (Maximum A Posteriori)
map_pL <- MAP_estimate(fit_pL)
cat("**MAP Estimates (pL):**\n")
cat("• T_min =", round(map_pL["T_min"], 2), "°C\n")
cat("• T_max =", round(map_pL["T_max"], 2), "°C\n")
cat("• q =", round(map_pL["q"], 4), "\n")
cat("• sigma.sq =", round(map_pL["sigma.sq"], 5), "\n")

```

## Biting Rate (a)

```{r a-data-fit, echo=FALSE, message=FALSE, warning=FALSE}
# Load updated dataset from CSV and filter for biting rate (a)
a_df <- readr::read_csv("data/Bluetongue_Params_0723.csv", show_col_types = FALSE) %>%
  dplyr::filter(Trait == "a") %>%
  dplyr::transmute(T = AmbientTemp, a = TraitValueSI) %>%
  dplyr::arrange(T)

# Fit Brière model using bayesTPC
a_data_list <- list(Temp = a_df$T, Trait = a_df$a)
set.seed(123)
fit_a <- b_TPC(
  data = a_data_list,
  model = "briere",
  priors = list(
    T_min    = "dunif(5,15)",
    T_max    = "dunif(32,36)",
    q        = "dunif(0,1)",
    sigma.sq = "dexp(1 / 0.1^2)"
  ),
  nchains = 4,
  burn = 6000,
  niter = 18000
)
```

```{r a-plot, echo=FALSE, message=FALSE, warning=FALSE, fig.width=8, fig.height=5}
# Use ALL chains and established prediction pipeline
samples_a <- do.call(rbind, lapply(fit_a$samples, as.matrix))
temp_grid <- seq(0, 50, by = 0.1)
briere_fun <- function(T, T_min, T_max, q) {
  ifelse(T >= T_min & T <= T_max, q * T * (T - T_min) * sqrt(pmax(T_max - T, 0)), 0)
}
pred_a <- sapply(temp_grid, function(Ti) briere_fun(Ti, samples_a[, "T_min"], samples_a[, "T_max"], samples_a[, "q"]))
median_a <- apply(pred_a, 2, median)
hpd_a <- apply(pred_a, 2, function(x) as.numeric(HDInterval::hdi(x, credMass = 0.95)))
lo_a <- hpd_a[1, ]; hi_a <- hpd_a[2, ]
Topt_a <- temp_grid[which.max(median_a)]

par(mfrow = c(1,1), mar = c(4,4,2,1))
plot(temp_grid, median_a, type = "l", lwd = 2, col = "black",
     xlab = "T (°C)", ylab = "Biting Rate (per day)",
     ylim = c(0, 1), xlim = c(0, 50), bty = "l", yaxt = "n")
axis(2, at = seq(0,1, by = 0.2), las = 1)
lines(temp_grid, lo_a, lty = 2, lwd = 1.2)
lines(temp_grid, hi_a, lty = 2, lwd = 1.2)
points(a_df$T, a_df$a, pch = 16, col = "black")
abline(v = Topt_a, lty = 3)
lbl <- sprintf("Topt ≈ %.1f°C (median curve)", Topt_a)
if (length(lbl) == 1 && !is.na(lbl)) mtext(lbl, side = 3, adj = 1, cex = 0.9)
```

```{r a-param-summary, echo=FALSE, message=FALSE, warning=FALSE}
# MAP Estimates (Maximum A Posteriori)
map_a <- MAP_estimate(fit_a)
cat("**MAP Estimates (a):**\n")
cat("• T_min =", round(map_a["T_min"], 2), "°C\n")
cat("• T_max =", round(map_a["T_max"], 2), "°C\n")
cat("• q =", round(map_a["q"], 4), "\n")
cat("• sigma.sq =", round(map_a["sigma.sq"], 5), "\n")
```

## Larva's Development Time (LDT)

```{r ldt-data-fit, echo=FALSE, message=FALSE, warning=FALSE}
# Use direct data from user (LDT values in days) - exactly as specified
ldt_df <- data.frame(
  T = c(20, 23, 27, 30, 35),
  LDT = c(34.5, 33.6, 26.5, 24.4, 16.7)
)

# Use a simple polynomial regression approach for better stability
# This captures the biological relationship: development time decreases with temperature
temp_centered <- ldt_df$T - mean(ldt_df$T)  # Center temperature for numerical stability
ldt_fit <- lm(LDT ~ temp_centered + I(temp_centered^2), data = data.frame(LDT = ldt_df$LDT, temp_centered))

# Extract coefficients for prediction
intercept <- coef(ldt_fit)[1]
linear_coef <- coef(ldt_fit)[2] 
quad_coef <- coef(ldt_fit)[3]
residual_sd <- summary(ldt_fit)$sigma

# Generate posterior-like samples for uncertainty quantification
set.seed(123)
n_samples <- 1000
intercept_samples <- rnorm(n_samples, intercept, summary(ldt_fit)$coefficients[1,2])
linear_samples <- rnorm(n_samples, linear_coef, summary(ldt_fit)$coefficients[2,2])
quad_samples <- rnorm(n_samples, quad_coef, summary(ldt_fit)$coefficients[3,2])
```

```{r ldt-plot, echo=FALSE, message=FALSE, warning=FALSE, fig.width=8, fig.height=5}
# Temperature grid for predictions (matching original: 0-50°C)
temp_grid <- seq(0, 50, by = 0.1)
temp_grid_centered <- temp_grid - mean(ldt_df$T)  # Center using same value

# Generate predictions using the quadratic model
pred_ldt <- sapply(1:n_samples, function(i) {
  intercept_samples[i] + linear_samples[i] * temp_grid_centered + quad_samples[i] * temp_grid_centered^2
})

# Calculate mean and 95% credible intervals
mean_ldt <- rowMeans(pred_ldt)
ci_ldt <- apply(pred_ldt, 1, function(x) quantile(x, c(0.025, 0.975)))
lo_ldt <- ci_ldt[1, ]
hi_ldt <- ci_ldt[2, ]

# Find optimal temperature (minimum development time)
Topt_ldt <- temp_grid[which.min(mean_ldt)]

# Plot matching the original paper's style
par(mfrow = c(1,1), mar = c(4,4,2,1))
plot(temp_grid, mean_ldt, type = "l", lwd = 2, col = "black",
     xlab = "T (°C)", ylab = "Larval Development Time (days)",
     xlim = c(0, 50), ylim = c(0, max(60, max(hi_ldt, ldt_df$LDT, na.rm = TRUE) * 1.05)), bty = "l")
lines(temp_grid, lo_ldt, lty = 2, lwd = 1.2)
lines(temp_grid, hi_ldt, lty = 2, lwd = 1.2)
points(ldt_df$T, ldt_df$LDT, pch = 16, col = "black", cex = 1.5)
abline(v = Topt_ldt, lty = 3)
lbl <- sprintf("Topt ≈ %.1f°C (min dev. time)", Topt_ldt)
if (length(lbl) == 1 && !is.na(lbl)) mtext(lbl, side = 3, adj = 1, cex = 0.9)
```

```{r ldt-param-summary, echo=FALSE, message=FALSE, warning=FALSE}
# Report the fitted quadratic model parameters (Maximum Likelihood Estimates)
cat("**Fitted Quadratic Model (LDT):**\n")
cat("• Intercept =", round(intercept, 2), "\n")
cat("• Linear coefficient =", round(linear_coef, 4), "\n") 
cat("• Quadratic coefficient =", round(quad_coef, 6), "\n")
cat("• Residual SD =", round(residual_sd, 2), "\n")
cat("• R-squared =", round(summary(ldt_fit)$r.squared, 3), "\n")

# Report the model equation
temp_center <- round(mean(ldt_df$T), 1)
cat("\n**Model:** LDT =", round(intercept, 2), "+", round(linear_coef, 4), "*(T -", temp_center, ") +", round(quad_coef, 6), "*(T -", temp_center, ")²\n")
cat("\n**Note:** Development time decreases with temperature (faster development at higher T).\n")
```

## Egg Development Time (EDT)

```{r edt-data-fit, echo=FALSE, message=FALSE, warning=FALSE}
# EDT data from user (Egg Development Time in days)
edt_df <- data.frame(
  T = c(20, 23, 27, 30, 35),
  EDT = c(63.6, 64.7, 61.4, 50.9, 57.1)
)

# Use a simple polynomial regression approach for stability
# Development time decreases with temperature (similar to LDT)
temp_centered_edt <- edt_df$T - mean(edt_df$T)  # Center temperature for numerical stability
edt_fit <- lm(EDT ~ temp_centered_edt + I(temp_centered_edt^2), data = data.frame(EDT = edt_df$EDT, temp_centered_edt))

# Extract coefficients for prediction
intercept_edt <- coef(edt_fit)[1]
linear_coef_edt <- coef(edt_fit)[2] 
quad_coef_edt <- coef(edt_fit)[3]
residual_sd_edt <- summary(edt_fit)$sigma

# Generate posterior-like samples for uncertainty quantification
set.seed(123)
n_samples_edt <- 1000
intercept_samples_edt <- rnorm(n_samples_edt, intercept_edt, summary(edt_fit)$coefficients[1,2])
linear_samples_edt <- rnorm(n_samples_edt, linear_coef_edt, summary(edt_fit)$coefficients[2,2])
quad_samples_edt <- rnorm(n_samples_edt, quad_coef_edt, summary(edt_fit)$coefficients[3,2])
```

```{r edt-plot, echo=FALSE, message=FALSE, warning=FALSE, fig.width=8, fig.height=5}
# Temperature grid for predictions
temp_grid <- seq(0, 50, by = 0.1)
temp_grid_centered_edt <- temp_grid - mean(edt_df$T)  # Center using same value

# Generate predictions using the quadratic model
pred_edt <- sapply(1:n_samples_edt, function(i) {
  intercept_samples_edt[i] + linear_samples_edt[i] * temp_grid_centered_edt + quad_samples_edt[i] * temp_grid_centered_edt^2
})

# Calculate mean and 95% credible intervals
mean_edt <- rowMeans(pred_edt)
ci_edt <- apply(pred_edt, 1, function(x) quantile(x, c(0.025, 0.975)))
lo_edt <- ci_edt[1, ]
hi_edt <- ci_edt[2, ]

# Find optimal temperature (minimum development time)
Topt_edt <- temp_grid[which.min(mean_edt)]

# Plot matching the original paper's style
par(mfrow = c(1,1), mar = c(4,4,2,1))
plot(temp_grid, mean_edt, type = "l", lwd = 2, col = "black",
     xlab = "T (°C)", ylab = "Egg Development Time (days)",
     xlim = c(0, 50), ylim = c(0, max(80, max(hi_edt, edt_df$EDT, na.rm = TRUE) * 1.05)), bty = "l")
lines(temp_grid, lo_edt, lty = 2, lwd = 1.2)
lines(temp_grid, hi_edt, lty = 2, lwd = 1.2)
points(edt_df$T, edt_df$EDT, pch = 16, col = "black", cex = 1.5)
abline(v = Topt_edt, lty = 3)
lbl <- sprintf("Topt ≈ %.1f°C (min dev. time)", Topt_edt)
if (length(lbl) == 1 && !is.na(lbl)) mtext(lbl, side = 3, adj = 1, cex = 0.9)
```

```{r edt-param-summary, echo=FALSE, message=FALSE, warning=FALSE}
# Report the fitted quadratic model parameters (Maximum Likelihood Estimates)
cat("**Fitted Quadratic Model (EDT):**\n")
cat("• Intercept =", round(intercept_edt, 2), "\n")
cat("• Linear coefficient =", round(linear_coef_edt, 4), "\n") 
cat("• Quadratic coefficient =", round(quad_coef_edt, 6), "\n")
cat("• Residual SD =", round(residual_sd_edt, 2), "\n")
cat("• R-squared =", round(summary(edt_fit)$r.squared, 3), "\n")

# Report the model equation
temp_center_edt <- round(mean(edt_df$T), 1)
cat("\n**Model:** EDT =", round(intercept_edt, 2), "+", round(linear_coef_edt, 4), "*(T -", temp_center_edt, ") +", round(quad_coef_edt, 6), "*(T -", temp_center_edt, ")²\n")
cat("\n**Note:** Egg development time varies with temperature following quadratic relationship.\n")
```

## Pupal Development Time (PuDT)

```{r pudt-data-fit, echo=FALSE, message=FALSE, warning=FALSE}
# PuDT data from user (Pupal Development Time in days)
pudt_df <- data.frame(
  T = c(20, 23, 27, 30, 35),
  PuDT = c(89.7, 65.5, 50.6, 39.1, 38.8)
)

# Use a simple polynomial regression approach for stability
# Development time decreases with temperature (similar to LDT and EDT)
temp_centered_pudt <- pudt_df$T - mean(pudt_df$T)  # Center temperature for numerical stability
pudt_fit <- lm(PuDT ~ temp_centered_pudt + I(temp_centered_pudt^2), data = data.frame(PuDT = pudt_df$PuDT, temp_centered_pudt))

# Extract coefficients for prediction
intercept_pudt <- coef(pudt_fit)[1]
linear_coef_pudt <- coef(pudt_fit)[2] 
quad_coef_pudt <- coef(pudt_fit)[3]
residual_sd_pudt <- summary(pudt_fit)$sigma

# Generate posterior-like samples for uncertainty quantification
set.seed(123)
n_samples_pudt <- 1000
intercept_samples_pudt <- rnorm(n_samples_pudt, intercept_pudt, summary(pudt_fit)$coefficients[1,2])
linear_samples_pudt <- rnorm(n_samples_pudt, linear_coef_pudt, summary(pudt_fit)$coefficients[2,2])
quad_samples_pudt <- rnorm(n_samples_pudt, quad_coef_pudt, summary(pudt_fit)$coefficients[3,2])
```

```{r pudt-plot, echo=FALSE, message=FALSE, warning=FALSE, fig.width=8, fig.height=5}
# Temperature grid for predictions
temp_grid <- seq(0, 50, by = 0.1)
temp_grid_centered_pudt <- temp_grid - mean(pudt_df$T)  # Center using same value

# Generate predictions using the quadratic model
pred_pudt <- sapply(1:n_samples_pudt, function(i) {
  intercept_samples_pudt[i] + linear_samples_pudt[i] * temp_grid_centered_pudt + quad_samples_pudt[i] * temp_grid_centered_pudt^2
})

# Calculate mean and 95% credible intervals
mean_pudt <- rowMeans(pred_pudt)
ci_pudt <- apply(pred_pudt, 1, function(x) quantile(x, c(0.025, 0.975)))
lo_pudt <- ci_pudt[1, ]
hi_pudt <- ci_pudt[2, ]

# Find optimal temperature (minimum development time)
Topt_pudt <- temp_grid[which.min(mean_pudt)]

# Plot matching the reference image scaling
par(mfrow = c(1,1), mar = c(4,4,2,1))
plot(temp_grid, mean_pudt, type = "l", lwd = 2, col = "black",
     xlab = "T (°C)", ylab = "Pupal Development Time",
     xlim = c(0, 50), ylim = c(0, 150), bty = "l")
lines(temp_grid, lo_pudt, lty = 2, lwd = 2, col = "black")
lines(temp_grid, hi_pudt, lty = 2, lwd = 2, col = "black")
points(pudt_df$T, pudt_df$PuDT, pch = 16, col = "black", cex = 1.5)
abline(v = Topt_pudt, lty = 3)
lbl <- sprintf("Topt ≈ %.1f°C (min dev. time)", Topt_pudt)
if (length(lbl) == 1 && !is.na(lbl)) mtext(lbl, side = 3, adj = 1, cex = 0.9)
```

```{r pudt-param-summary, echo=FALSE, message=FALSE, warning=FALSE}
# Report the fitted quadratic model parameters (Maximum Likelihood Estimates)
cat("**Fitted Quadratic Model (PuDT):**\n")
cat("• Intercept =", round(intercept_pudt, 2), "\n")
cat("• Linear coefficient =", round(linear_coef_pudt, 4), "\n") 
cat("• Quadratic coefficient =", round(quad_coef_pudt, 6), "\n")
cat("• Residual SD =", round(residual_sd_pudt, 2), "\n")
cat("• R-squared =", round(summary(pudt_fit)$r.squared, 3), "\n")

# Report the model equation
temp_center_pudt <- round(mean(pudt_df$T), 1)
cat("\n**Model:** PuDT =", round(intercept_pudt, 2), "+", round(linear_coef_pudt, 4), "*(T -", temp_center_pudt, ") +", round(quad_coef_pudt, 6), "*(T -", temp_center_pudt, ")²\n")
cat("\n**Note:** Pupal development time decreases with temperature (faster development at higher T).\n")
```