Cross-case-study summary: three taxa, one panel

The three taxa and the cross-study question

The three shared case-study datasets span three corners of thermal physiology:

• brown shrimp Crangon crangon – a crustacean, lethal mortality, ungrouped;
• zebrafish Danio rerio – a fish, lethal mortality, across three life stages (young embryos, old embryos, larvae); this panel uses the zebrafish_lethal life-stage dataset — a separate experiment from the oxygen-gradient zebrafish_o2 data in vignette("case-study-zebrafish");
• vinegar fly Drosophila suzukii – an insect, lethal mortality, across the two sexes.

The cross-study question is descriptive: laid side by side, how do thermal limits compare across a crustacean, a fish (resolved by developmental stage), and an insect (resolved by sex)? CTmax (the critical temperature at a fixed reference exposure) places each taxon on the temperature axis; z (the change in temperature, in degrees Celsius, that multiplies tolerated exposure time tenfold) measures how sharply tolerated exposure responds to temperature: a smaller z is a steeper response (a small warming sharply cuts tolerated time), a larger z a more gradual one.

One caveat governs the whole panel. The reference exposures differ by study, because each follows the convention of its source assay:

Because a CTmax is defined at its reference exposure, the four CTmax values are not on a single common time scale: the fly number is the fitted relative midpoint temperature at 4 hours, whereas the shrimp and zebrafish fits use a 1-hour reference. The panel is therefore an illustrative cross-taxon synthesis, not a single common-scale comparison. z, by contrast, is a slope – degrees per tenfold change in time – and is comparable across taxa regardless of the reference exposure. Read the CTmax facet as four study-specific anchors and the z facet as a like-with-like comparison of duration sensitivity.

The three fits (version-stamped cache)

Each fit uses the configuration locked for its case study. The grouped fits (zebrafish, fly) estimate a separate CTmax and z per level with a shared shape (low, up, k); the ungrouped shrimp fit estimates one of each. All three use beta-binomial survival counts. The cache records the package/source version, R and TMB versions, input checksums, and the exact family, grouping, reference exposure, threshold, shape, and interval configuration. The build script is data-raw/build_case_study_summary_cache.R; the per-study vignettes and tests exercise the live paths and their data-adequacy warnings.

The vinegar-fly data ships per-individual (dsuzukii); we aggregate it to (temp, time, sex) counts in base R, exactly as the per-study vignette does.

summary_cache_path <- system.file(
  "extdata", "case_study_summary_cache.rds", package = "freqTLS"
)
if (!nzchar(summary_cache_path) || !file.exists(summary_cache_path)) {
  stop("The shipped cross-case-study cache is missing; reinstall freqTLS from a complete source tarball.")
}
summary_cache <- readRDS(summary_cache_path)
summary_cache$meta[c("schema_version", "generated_on", "freqTLS_version",
                     "freqTLS_source_commit", "R_version", "TMB_version")]
#> $schema_version
#> [1] 1
#> 
#> $generated_on
#> [1] "2026-07-12"
#> 
#> $freqTLS_version
#> [1] "0.1.0"
#> 
#> $freqTLS_source_commit
#> [1] "589e3af6c7c226c571ddcbf682f86a578f77ad9c"
#> 
#> $R_version
#> [1] "R version 4.6.0 (2026-04-24)"
#> 
#> $TMB_version
#> [1] "1.9.21"

Assembling the cross-study table

The panel is built from one combined data frame of (label, parameter, estimate, conf.low, conf.high), obtained by calling confint(..., method = "profile") on each fit. For the grouped fits we ask for the per-level parameter names (CTmax:young_embryos, z:M, and so on); for the ungrouped fits we ask for the bare CTmax and z. Every interval here is a profile-likelihood confidence interval, and – as the conf.status column confirms – every profile closes.

panel <- summary_cache$panel
# Six rows, two parameters: a tidy printout of the headline numbers.
panel_wide <- data.frame(
  Group           = panel$label[panel$parameter == "CTmax"],
  `CTmax estimate` = round(panel$estimate[panel$parameter == "CTmax"], 2),
  `CTmax 95% CI`  = sprintf("[%.2f, %.2f]",
                            panel$conf.low[panel$parameter == "CTmax"],
                            panel$conf.high[panel$parameter == "CTmax"]),
  `z estimate`     = round(panel$estimate[panel$parameter == "z"], 2),
  `z 95% CI`       = sprintf("[%.2f, %.2f]",
                            panel$conf.low[panel$parameter == "z"],
                            panel$conf.high[panel$parameter == "z"]),
  check.names = FALSE
)
knitr::kable(
  panel_wide,
  caption = "Six taxon/group rows: CTmax (at the study reference exposure) and z, each with its profile-likelihood 95% confidence interval."
)

Cross-taxon validation against bayesTLS and the two-stage estimator

The same headline numbers are shown beside the classical two-stage estimator and the bayesTLS posterior, read from the maintainer-built benchmark cache. The freqTLS and bayesTLS fits use the matched relative-midpoint, constant-shape configuration and the same per-study reference exposure. The classical estimator uses absolute LT50 and is therefore an approximate comparator for these lethal datasets, whose asymptotes lie near zero and one.

The panel: a Confidence Eye per taxon and group

Each row is drawn as an honest Confidence Eye – a pale, shallow horizontal lens whose width is exactly the 95% confidence interval and whose cosine taper is tallest at the estimate, with a hollow point marking the estimate itself. The two facets carry independent x-axes (CTmax in degrees Celsius; z in degrees Celsius per tenfold change in exposure time), because the two parameters live on different scales and, for CTmax, on different reference exposures.

The geometry follows the freqTLS Confidence-Eye contract: the shallow, wide lens reads as a confidence interval, not a probability density, and a profile that did not close would render as a hollow point with no lens. All 12 headline profiles here close (six groups for each of two parameters), so every row carries a lens.

# Build the honest Confidence-Eye geometry by hand so all three fits share one
# cross-taxon panel: a cosine-tapered pale lens (width = confidence interval)
# plus a hollow point, faceted by parameter with free x-axes. This reuses the
# package lens shape (see ?plot_confidence_eye) but spans every fit in one figure.
stopifnot(requireNamespace("ggplot2", quietly = TRUE))

# Fixed top-to-bottom row order (shrimp, zebrafish stages, fly sexes).
row_order <- c(
  "Shrimp",
  "Zebrafish: young embryos", "Zebrafish: old embryos", "Zebrafish: larvae",
  "D. suzukii: female", "D. suzukii: male"
)
panel$row <- match(panel$label, row_order)
panel$parameter <- factor(panel$parameter, levels = c("CTmax", "z"),
                          labels = c("CTmax (°C)", "z (°C / decade)"))

# One cosine-tapered lens polygon per row (tallest at the estimate, zero at each
# bound), built per facet so the free x-axes do not distort the taper.
lens_df <- do.call(rbind, lapply(seq_len(nrow(panel)), function(i) {
  x <- seq(panel$conf.low[i], panel$conf.high[i], length.out = 80)
  d_lo <- max(panel$estimate[i] - panel$conf.low[i], .Machine$double.eps)
  d_hi <- max(panel$conf.high[i] - panel$estimate[i], .Machine$double.eps)
  frac <- ifelse(x <= panel$estimate[i], (panel$estimate[i] - x) / d_lo,
                 (x - panel$estimate[i]) / d_hi)
  w <- 0.32 * cos((pi / 2) * pmin(pmax(frac, 0), 1))
  data.frame(id = i, parameter = panel$parameter[i], x = x,
             ymin = panel$row[i] - w, ymax = panel$row[i] + w)
}))

# Reference-exposure annotation for the CTmax facet only.
tref_lab <- data.frame(
  parameter = factor("CTmax (°C)",
                     levels = levels(panel$parameter)),
  row = panel$row[panel$parameter == "CTmax (°C)"],
  x   = panel$conf.high[panel$parameter == "CTmax (°C)"],
  lab = c("1 h", "1 h", "1 h", "1 h", "4 h", "4 h")
)

ggplot2::ggplot() +
  ggplot2::geom_ribbon(
    data = lens_df,
    ggplot2::aes(x = x, ymin = ymin, ymax = ymax, group = id),
    fill = "#1b7837", colour = NA, alpha = 0.30
  ) +
  ggplot2::geom_point(
    data = panel,
    ggplot2::aes(x = estimate, y = row),
    shape = 21, fill = "white", colour = "#1b7837", size = 3, stroke = 1
  ) +
  ggplot2::geom_text(
    data = tref_lab,
    ggplot2::aes(x = x, y = row, label = lab),
    hjust = -0.25, size = 2.7, colour = "grey35"
  ) +
  ggplot2::scale_y_continuous(
    breaks = seq_along(row_order), labels = row_order,
    trans = "reverse", expand = ggplot2::expansion(add = 0.7)
  ) +
  ggplot2::scale_x_continuous(
    expand = ggplot2::expansion(mult = c(0.05, 0.20))
  ) +
  ggplot2::facet_wrap(~ parameter, scales = "free_x") +
  ggplot2::labs(
    x = NULL, y = NULL,
    title = "Thermal limits across three taxa: CTmax and z",
    subtitle = "Confidence Eyes: pale lens = 95% confidence interval; hollow point = estimate.",
    caption = paste(
      "Profile-likelihood confidence intervals (freqTLS, no Stan).",
      "CTmax reference exposure differs by study (annotated): not a common time scale."
    )
  ) +
  ggplot2::theme_minimal() +
  ggplot2::theme(
    panel.grid.major.y = ggplot2::element_blank(),
    panel.grid.minor = ggplot2::element_blank(),
    plot.caption = ggplot2::element_text(hjust = 0)
  )

Within-taxon contrasts

The two grouped studies invite a within-taxon comparison: do the zebrafish life stages differ, and do the fly sexes differ? freqTLS requests a profile interval on the difference (dCTmax:A-B, dz:A-B) and uses its documented bootstrap fallback when a contrast profile does not close. Both are prior-free frequentist counterparts to the bayesTLS pMCMC bracket. A difference interval that excludes zero is a clear separation; one that spans zero is not.

contrast_tbl <- summary_cache$contrasts
knitr::kable(
  data.frame(
    Contrast   = contrast_tbl$parameter,
    Difference = round(contrast_tbl$estimate, 3),
    `95% CI`   = sprintf("[%.3f, %.3f]",
                         contrast_tbl$conf.low, contrast_tbl$conf.high),
    `Excludes 0` = ifelse(
      contrast_tbl$conf.low > 0 | contrast_tbl$conf.high < 0, "yes", "no"
    ),
    Method = contrast_tbl$method,
    check.names = FALSE
  ),
  caption = "Within-taxon contrasts: 95% confidence intervals from the requested profile or its documented bootstrap fallback."
)

For zebrafish, the one clear separation in CTmax is old embryos versus young embryos and larvae versus old embryos: old embryos sit about 1.46 degrees Celsius above young embryos, with a confidence interval well clear of zero, while larvae are essentially indistinguishable from young embryos in CTmax. None of the z contrasts excludes zero, so the three stages share a common duration sensitivity even where their critical temperatures differ. For D. suzukii, both the CTmax and the z sex contrasts span zero: there is no clear sex difference in either thermal limit, the same conclusion Ørsted et al. (2024) reached for these data.

What this shows

Two things stand out from the panel:

• Resolving a taxon by an internal axis can matter or not. Zebrafish life stage shifts CTmax by over a degree (old embryos are the most heat-tolerant stage), whereas D. suzukii sex shifts neither CTmax nor z detectably. The Confidence Eyes make this visible at a glance: the zebrafish lenses separate on the CTmax axis, while the two fly lenses overlap almost completely.

• Every estimate carries an honest, prior-free interval. All 12 headline profiles close (six groups for each of two parameters), so each row is a closed lens rather than a hollow point. Where a profile did not close (a weakly identified design or a boundary asymptote), the same display would show a hollow point with no lens – never a fabricated closed eye.

These are confidence intervals throughout: they summarise the likelihood’s support for each parameter and make no probability statement about the parameter itself.

See bayesTLS for the Bayesian multi-taxon ridge

This panel is the freqTLS (likelihood) counterpart to Manuscript Figure 5 of the bayesTLS supplement. Its full figure also includes the snow-gum PSII dataset, which is not redistributed here because its source is CC BY-NC 4.0. The Bayesian supplement draws the taxa as posterior ridge densities with median points and 95% credible bars. The two displays answer the same cross-study question from complementary inferential engines: the bayesTLS ridge is a posterior density shaped by priors and the data, whereas the freqTLS Confidence Eye is a prior-free likelihood interval that cannot be read as a posterior. For the Bayesian multi-taxon ridge plot, the within-taxon pMCMC contrasts, and the full posterior workflow, see bayesTLS (https://github.com/daniel1noble/bayesTLS); for the side-by-side posterior-versus-Confidence-Eye contrast on a single dataset, see vignette("comparing-to-bayesTLS").